MASTER'S THESIS - DiVA Portal

MASTER'S THESIS

Meteors and Celestial DynamicsAssociation and Numerical analysis

Daniel Kastinen2016

Master of Science in Engineering TechnologySpace Engineering

Luleå University of TechnologyDepartment of Computer Science, Electrical and Space Engineering

Meteors and Celestial Dynamics

Association and Numerical analysis

D. Kastinen

August 9, 2016

D. Kastinen Meteors and Celestial Dynamics

Abstract

We have developed a skeleton version of a new toolbox for statistical small body dynamics in the Solar sys-

tem. The propagation parts of the software include perturbations from all major planets, radiation pressure

and the PoyntingRobertson effect. Currently, the software is constructed to generate clones of parent bodies

taking into account uncertainties in observational parameters and the parent body characteristics. To then

sample this distribution in a Monte Carlo fashion. These bodies then release test particles using sublimation

models. The parent bodies as well as the particle generation process are described by multivariate proba-

bility distributions. In our current usage, the distribution represented orbital elements, critical sublimation

radius, density, size and surface activity. The software designed to integrate the released particles over a

given time scale and examine close encounters with another body in the solar system. We have examined

close encounters with the Earth. We have also created module for calculating orbital similarity functions

and to find associations and classifications in data sets. This toolbox is entirely modular enabling the use of

every step individually.

Validation is performed by simulating known and observed meteor showers, we have simulated the 1933

and 1946 October Draconids as validation, and extended the simulations to the 2011 and 2012 October Dra-

conids. The simulation was performed by ejecting material from comet 21P/GiacobiniZinner during seven

perihelion passages between 1866 and 1972 and propagating the material forward in time. Each perihelion

passage was sampled with 50 orbital clones that produced meteoroid streams. In total 850 clones were prop-

agated. The clones were sampled from a multidimensional Gaussian distribution on the orbital elements

with width proportional to the given uncertainties. These orbital clones were then sampled from normal

distributions on the bulk density, surface activity factor, cometary mass and critical sublimation distance

from the Sun, with characteristic values from measurements of 21P/GiacobiniZinner. Each clone ejected

8,000 particles, each with an individual weight proportional to the mass loss (number of meteoroids) they

represented. This generated a total of 6.7 million test particles, out of which 43 thousand entered the Earth’s

Hill sphere during 1900-2020 and were considered encounters. Using the simulation we produced the unex-

pected and measured deviation of the meteor mass index from a power low in the 2012 October Draconids,

a feature not present in the 2011 October Draconids. We also predict a October Draconids outburst in 2018

with peak on the night between October 8 and 9 that should be larger than the 2011 and 2012 outbursts.

Lastly we present some analysis as a proof of concept for the future development of this toolbox.

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Preface

This master thesis is the culmination of a row of student projects and internships within the area of celestial

mechanics, meteor science, statistics, and numerical integrators.

I have in 2013 performed collaborative work on the implementation of a large aperture radar meteor database

called the Shigaraki Middle and Upper atmospheric Radar Meteor Head Echo Database (MURMHED). The

data will together with 20,000 events observed 2011-2012 be released in the form of an open database

containing trajectory and orbit information, hosted at the National Institute of Polar Research (NIPR),

Tokyo Japan. The construction of this database was supported by a grant from the Japan Society for the

Promotion of Science (JSPS) with P.I. Takuji Nakamura. Daniel Kastinen and Johan Kero developed the

database format.

In the F7005T Project in physics course I first reviewed the current field of association of meteoroids, both

D-criterions and phase-space metrics, and then introduced a new approach to the metrization of trajectories

using the physical trajectories themselves as a basis instead of phase spaces. The method uses Hausdorff

distance between the subspaces making up the trajectories. I also researched the theory of the meteoroid

complex, statistical analysis on clusters with different metrization functions. The constructed algorithms

and the results from the different metrics were compared to each other, the statistical distributions were

compared to real world data and an analysis to find new clusters in a head echo database is performed.

During summer/autumn 2014 I had a Swedish Institute of Space Physics (IRF), Kiruna, internship within

project 1217 (Johan Kero) titled Meteors and celestial dynamics - association and numerical analysis”. This

work was the continuation and completion of the work in the ”F7005T Project in physics course”.

I have participated in the 2015 JSPS Summer Program with an internship at the NIPR in Tokyo. Proper

methods of finding meteor showers in databases are an open area of research and the theoretical work sur-

rounding statistics and mathematics are far from complete. The purpose of this research is to develop proper

statistical methods of pattern recognition and machine learning in combination with advanced simulations of

orbital evolution and material ejection from celestial bodies. To execute this research a software platform is

being written to perform: simulations of dust material ejection from comets being propagated, generation of

Solar system initial condition from ephemerides, generation of parent bodies from debisaed population data,

propagation of test particles and close encounter determination using mercury6 software, orbital clone gen-

eration from observational errors using OrbFit software, several similarity functions such as D-criterions,

grouping methods such as kernel density estimation and hierarchical cluster analysis, association threshold

determination methods such as k-fold cross validation and association profile analysis, and analysis of large

scale meteor databases. During this summer almost all software described above has been developed with

the help of NIPR, Nihon University, Institute of Statistical Mathematics (ISM), and National Astronomical

Observatory of Japan (NAOJ). With the cooperation of several researchers in Japan, the software was taken

to such a state that the versatility of the developed platform can extend to many more research questions

outside the scope of the current work. Also, presentation of this work and surrounding questions has been

given at NIPR, NAOJ, and Nihon University. Three research trips were executed during the summer. The

first one was a visit Nihon University for discussion and presentations, the second on a visit to the Shigaraki

Radar facility for a tour and explanation of the system used to collect data to be analysed. And lastly to

Norikura Observatory for a meteor shower observation campaign using optical measurements.

During my research into meteor theory, association analysis, and celestial mechanics I ran into many speed

bumps due to unclear methodology and vital information being widely spread among many sources. This

resulted in a ”random walk” like experience while finding the required material and writing software. Many

of the mathematical concepts in literature where presented without rigour and central concept are sometimes

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overlooked by experimentalists. This is the reason so much of this work seems as basic knowledge. I have

decided to include all these basics in a simple and direct way keeping as much rigour as possible to create

a as a self contained overview of the required material to understand and reconstruct my research. Even

though i have applied the techniques and methods described on meteors and small body populations they

have a extremely wide range of use. The first half of this work consisting of theoretical background can be

used independently of my applications. Other applications will be mentioned when relevant and I will try

to give as much reference material on each subject as possible or as seen practical.

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Acknowledgements

I would like to thank everyone that has helped me with my education into the many different areas I have

studied. Both in courses and on my free time, as the preliminary studies needed far exceeded my expecta-

tions due to the broad field nature of the problem. And also my parents for their unwavering support of me

in my goal becoming a researcher.

I would also like to thank my supervisor Johan for his constant support and good advice. Without his

guiding and integration of me into the research community I would never have completed even a fraction of

the work described here.

Lastly would also like to thank my girlfriend for putting up with the many late nights of overtime needed to

be able to complete this work. Without her support and love I would not have been able to keep working at

the pace needed.

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Contents

Page

1 Introduction 1

I Classical celestial mechanics 3

2 Introduction to classical celestial mechanics 3

3 Hamiltonian mechanics 3

3.1 Classical mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3.2 From Lagrangian mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3.3 The two body problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.4 Keplarian elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.5 Hamiltonian form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.6 Poisson brackets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.7 Liouville’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.8 Phase space paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4 The problem of non gravitational perturbations 15

4.0.1 Radiation pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.0.2 Poynting-Robertson effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.1 Yarkovski effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.1.1 Yarkovsky O’Keefe Radzievskii Paddack effect . . . . . . . . . . . . . . . . . . . . . . 17

4.1.2 Relevance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5 Numerical integration 17

5.1 Symplectic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5.2 Hamiltonian splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5.2.1 Differential equation flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5.2.2 Order of a split . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5.3 Bulirsch-Stoer method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

5.3.1 Richardson extrapolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5.3.2 Modified midpoint method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5.3.3 Deufelhard serie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5.3.4 Neville’s Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

II Solar system population 24

6 Introduction to Solar system population 24

7 The Sun 24

8 The planets 24

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9 Small bodies 24

9.1 Comets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

10 Dust 26

10.1 Poynting-Robertson lifetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

10.2 Comets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

10.2.1 Dust ejection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

10.2.2 Critical sublimation radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

10.3 Asteroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

10.4 Main belt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

10.5 Interstellar dust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

10.6 Nanodust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

11 Databases 31

11.1 Pan-STARRS Synthetic Solar System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

11.2 JPL NAIF database . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

III Meteors 34

12 Introduction to meteors 34

13 Observations and mesurnemnts 34

13.1 Coordinate systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

13.2 Invariable plan and the ecliptic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

13.3 Meteors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

13.3.1 Visual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

13.3.2 Head echo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

13.4 Ablation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

14 Meteor and Meteoroid complex 38

14.1 Stream meteoroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

14.2 Sporadic meteoroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

15 Databases 40

15.1 MURMHED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

15.2 IAU meteor shower database . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

IV Associating and classifying meteoroids 43

16 Introduction to associating and classifying meteoroids 43

17 Similarity functions 43

17.1 Metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

18 Models of meteoroid complexes 45

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19 D-criterion 45

19.1 Southworth and Hawkins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

19.2 Drummond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

19.3 Valsecchi, Jopek, and Froeschle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

19.4 Jopek, Rudawska, and Bartczak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

20 Metric of phase-spaces 49

V Statistical methods 52

21 Introduction to statistical methods 52

22 Overview 52

23 Monte Carlo 53

24 Principal component analysis 54

25 Cluster analysis 55

25.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

25.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

25.3 Signal to noise ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

25.4 Previous usage of thresholds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

VI Software 63

26 Introduction to the software 63

27 Modular development 63

28 Dependancies 64

28.1 Programming language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

28.1.1 C++ 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

28.2 Boost library . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

28.3 SPICE library . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

28.4 Fortran 77 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

29 Monte Carlo Association Statistics module 66

29.1 Program flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

29.2 Execution time diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

29.3 Probability draw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

29.3.1 Dimensionality reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

29.4 Close encounter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

29.5 Parent body stream divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

29.6 Stream dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

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29.7 Data formatting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

29.7.1 Input files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

29.7.2 Output files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

29.8 Mass distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

29.8.1 Grun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

29.8.2 Uniform and logarithmic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

29.8.3 Custom file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

29.8.4 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

29.9 Probability formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

29.10Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

30 Parent Body Ejector module 101

30.1 Symplectic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

30.2 Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

30.3 Period syncing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

30.4 Ejection speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

30.5 Ejection direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

30.6 Particle mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

30.7 Ejection times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

30.8 Data formatting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

30.8.1 Input files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

30.8.2 Output files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

30.9 Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

31 Orbital Association Analysis module 113

31.1 Similarity functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

31.2 Similarity matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

31.3 Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

31.4 Parameter sweep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

31.4.1 Fixed step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

31.4.2 Adaptive step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

31.5 Error functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

31.6 Next update addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

31.7 Data formatting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

31.7.1 Input files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

31.7.2 Output files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

31.8 Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

32 Orbital Stability Estimation module 124

33 Simulation Merger module 124

34 Statistical Uncertenty Orbital Clones module 124

35 NASA Jet Propulsion Laboratory module 125

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36 Scripting of simulations 127

37 External software 128

37.1 Mercury6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

37.2 NASA Jet Propulsion Laboratory ephemeris software . . . . . . . . . . . . . . . . . . . . . . . 131

38 Data transformation logistics 132

38.1 State vectors and Kepler elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

38.1.1 From state vectors to Kepler elements . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

38.1.2 From Kepler elements to state vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

38.2 Geocentric coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

38.2.1 Ecliptic J2000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

38.2.2 Equatorial J2000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

38.3 Sun centred versions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

39 MATLAB visualization scripts 138

40 Future work 140

40.1 Purpose and aims . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

40.2 Future development plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

40.3 Significance of future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

40.4 Significance of community contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

VII Results 143

41 Introduction to Results 143

42 21P/GiacobiniZinner 143

42.1 Input state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

42.2 Summary of simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

42.3 Validation: 1933 and 1946 October Draconids . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

42.4 2011 October Draconids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

42.5 2012 October Draconids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

42.6 Mass transfer difference 2011-2012 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

42.7 Probable mass transfer cause . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

42.8 Year intensity overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

42.9 Cluster analysis calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

42.10Principal component Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

43 Pan-STARRS Short Period Comets 163

43.1 Input state 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

44 Concluding remarks 164

References 165

Page IX


Appendices 172

A Notation, abreviations, technical terms, and nomenclature 172

B List of figures 173

C List of tables 175

Page X


1 Introduction

Figure 1: The life of a meteoroid encountering the Earth. When caught by the Hill Sphere (the volume in

which Earth gravity is dominant), if the Earth gravitational pull directs the meteoroid towards the atmosphere

it will enter and begin to heat up. The meteoroid may disintegrate during its ablation phase, or in some cases

bounce of the atmosphere. The process of colliding with the atmosphere produces a meteor phenomena as the

collision turn small areas of atmosphere into plasma. If however the meteoroid survives atmospheric entry,

either due to the entry circumstances or its sheer size, a dark flight phase will begin. Then to finally impact

the surface and by doing so turning from meteoroid to meteorite.

The dynamics of small bodies in the Solar system form a theoretical basis to study, for example, meteoroid

streams (Soja et al., 2014) and their connected meteor showers (Vaubaillon et al., 2011) is a subject requiring

many different numerical tools. Whether it is the sporadic complex (Wiegert et al., 2009) of observed meteors

or the environment of newly formed meteoroid streams. Development of numerical simulations in this area

is needed to test and validate physical models, ranging from theories of dust ejection and ice sublimation of

comets (Ryabova, 2013), to the chaos mechanism of supplying the Solar system with new comets (Torbett,

1989). It is also highly relevant for technical models like the ESA Interplanetary Meteoroid Environment for

eXploration (IMEX) project (Soja et al., 2015).

Characterization of dust trails and meteoroid streams are important to assess the dust impact hazard to

Page 1


spacecraft. For example, the ESA communications satellite Olympus lost pointing control in 1993 likely due

to the impact of a Perseid meteoroid (Caswell et al., 1995). For comet and asteroid interaction missions,

such as the NASA space probe Deep Impact (AHearn et al., 2011), JAXAs Hayabusa 2, and future asteroid

deflection missions, accurately estimating debris dynamics will be vital.

In spite of the fact that the underlying orbital dynamics is well understood, it is still an open question

how much extraterrestrial material enters the Earth atmosphere. The most obvious missing feature in cur-

rent numerical models, which cover different aspects of Solar system small body dynamics, is the ability

to easily connect programs with different functionality. We believe this is the main reason why contempo-

rary techniques have not yet been combined. These include techniques such as Monte Carlo simulations of

meteoroid streams, orbital stability analysis, time tracing of orbital similarity functions to better analyse

measurements and neural network recognition of meteor showers. The lack of a comprehensive numerical

modelling approach appears to be the main bottleneck. Thus, the main focus of this work is dedicated to

developing a prototype version of a software toolbox which attempts to supplement the current research

area. We here present a skeleton version containing several of the above-mentioned features and we have

performed a feasibility check.

There are many other applications of numerical tools oriented towards statistical numerical integration

of bodies in the Solar system than the ones mentioned here. We shall however only focus on a few for com-

pression of the thesis, but keep concepts and implementation general enough so that the developed material

can have applications in other areas of research.

As this work is oriented towards people with a broad scientific background as well as the expert, every

section is provided with a small introduction reviewing what reader benefits the most from said section and

how it fits with the whole. For those being exposed to many new concepts while reading this first introduc-

tion, a quick illustration of the chain of events leading to the meteor phenomena are shown in figure 1. In

addition to this illustration it can be added that if the entry circumstances are right, even small particles

can survive atmospheric re-entry as shown by simulations in (Love and Brownlee, 1991). In appendix A we

have also listed a collection of abbreviations, technical terms, and commonly used symbols.

Page 2


Part I

Classical celestial mechanics

2 Introduction to classical celestial mechanics

This part is intended for those who are not comfortable with Hamiltonian mechanics and the mathematical

concepts relevant to numerical celestial mechanics, or new to celestial mechanics as a whole. Thus this part

covers the basics of Hamiltonian systems and briefly address perturbations to the Hamiltonian system in

form of electromagnetic forces, we do not cover proper Hamiltonian Perturbation theory as that is outside

the scope of this work.

The last section of this part is dedicated to numerical methods implemented in the software developed. If

standard numerical concepts within celestial mechanics is a new concept, this is chapter that should not be

skipped if the goal is to grasp the full context of the developed software.

3 Hamiltonian mechanics

3.1 Classical mechanics

Celestial mechanics has been of interest for as long as mankind had the ability to see the movements of the

planets and wonder what governed their motion and how one could predict their paths. The goal of classical

mechanics is to glance at the configuration of a system and determine its movement due to a set of forces,

whether it be backwards or forwards in time. The first real mathematical theory for this kind of problem

was developed by Isaac Newton and is called calculus, or the mathematics of change.

Tracing from this foundation of differential equations, integrals and functions of motion a first reformulation

of classical mechanics was made by a mathematician called Joseph-Louis Lagrange in 1788. This refor-

mulation is commonly called Lagrangian mechanics and is based upon the principle of stationary action.

This concept together with its use of generalized coordinates made it much more versatile than classical

mechanics.

3.2 From Lagrangian mechanics

Lagrangian mechanics are however not the preferred way to handle celestial mechanics since a set of mathe-

matical tools much more suited to the problem can be found in Hamiltonian mechanics. However, these two

theories of mechanics are very closely related and below a hint of their connection is supplied.

To begin grasping the underlying idea of Lagrangian and Hamiltonian mechanics we must cover action. The

concept of action is a way to talk about the path of a system through space-time. Lets consider a bijective

function that takes a path in space-time and transforms that path to a real number. Such a function also

needs to be a Lorentz invariant quantity, for it to be a covariant way to express unique paths. If we want to

form the basis of a realistic theory.

When considering the way classical mechanics is viewed, in deterministic motion, one can quickly realize

that for a real system the action has to be constant. The reason is because if the action was not constant,

the system could suddenly change to a different space time path. This is in other words simply a way of

stating that macroscopic objects does not suddenly teleport in space and time, of course there are deeper

meaning behind stationary action but this interpretation is sufficient for our purposes. This transformation

of a space-time path to a real number, or action S, is the space-time-integration of the Lagrangian density

L. Committing slight notational murder, we can express it as

Page 3


S =

∫all spacetime

Ld4x. (3.1)

This is, for example, what is used in Gauss’s law for gravity and in the Feynman path integral formulation

of quantum field theory.

Here it is important to make a distinction between the Lagrangian density and the Lagrangian. The La-

grangian density is integrated over all space-time to extract the action while the Lagrangian L is simply

integrated over time to give the action. The relationship between the Lagrangian L and the Lagrangian

density L is

L =

∫ ∫ ∫all space

Ldx1x2x3, (3.2)

if the spacial coordinates are (x1, x2, x3) and time x0. We know that this Lagrangian is dependant on the

generalized coordinates qi, their time derivatives qi, and time t. We can now write the action as

S =

∫ t1

t2

Ldt, (3.3)

and it is from this expression we will demand that the derivative of the action with respect to a phase-

space path disturbance evaluated at zero disturbance be equal to zero. Or in other words for those familiar

with variational calculus, we set the first variation, or the functional derivative, to zero and thus find the

extremal. We can then, through some manipulation and the cunning use of geometric algebra, as outlined

in (D’Orangeville and Lasenby, 2003), derive that the following equation,

d

dt

(∂L

∂qi

)− ∂L

∂qi= 0, (3.4)

must hold for every coordinate for this to be true. This is called the Euler-Lagrange equation and its solutions

yields the functions of motion for a system. However to derive the Hamiltonian formulation of classical

mechanics rather then the Lagrangian a Legendre transform will transform the Lagrangian, L(q, q, t) into a

Hamiltonian

H(q, p, t) =∑i

qipi − L(q, q, t), (3.5)

To understand what this transformation entails let us first realize that the Lagrangian is a function mapping

the configuration space of a system (its spacial coordinates) to the real numbers. The Hamiltonian however

is mapping a larger space, the phase space, to the real numbers. The phase space considers both the spacial

coordinates and their rate of change, their momentum. This may seem like a trivial difference since in

Lagrangian mechanics we use configuration space but we also considered the configuration rate of change in

the Lagrangian anyway. Even though this is the case, the fact that we are extending the space instead of

extracting the rate of change inside the function, comes with a few advantages. So, for a Hamiltonian system

the solution to the equations of motion will not only reside in the configuration space but in phase space,

effectively including the dynamics of the bodies momentum into the topology. How a Legendre transform

extends the configuration space involves creating a cotangent bundle of the space in question and is a more

mathematically rigours question not covered here.

Page 4


In the above expression we have denoted the momentum, or weighted time derivative, of the generalized

coordinate as pi, giving us the canonical coordinates (q,p) used to describe a system. Then by examining

the total differential of the Lagrangian, we can then some manipulation derive the Hamilton equations as

∂H

∂qi= −pi, (3.6)

∂H

∂pi= qi. (3.7)

As can be seen from this there will be as many equations as degrees of freedom in the described system.

These equations are equivalent to the Euler-Lagrange equations. So far the mentioned mathematics have

been general for all sets of mechanical systems, moving closer to celestial mechanics we can express the

Hamiltonian for a gravitational potential field in a N-body problem as

H =

N∑i=1

p2i2mi

−N−1∑i=1

N∑j=i+1

Gmimj

rij, (3.8)

where, if we use Cartesian coordinates,

rij =√

(qix − qjx)2 + (qiy − qjy)2 + (qiz − qjz)2, (3.9)

pi =√p2ix + p2iy + p2iz. (3.10)

Here we have changed the notation by instead of referring to the i’th coordinate we are referring to the

i’th particle with the associated three Cartesian coordinates, yielding a total of 6N variables in a system

of N bodies. Given that we reside in three dimensional space of course. Inserting the expression for the

Hamiltonian in equation 3.8 into equations 3.6 and 3.7 we can derive the following two equations of motion;

∂H

∂qnk=

∂

∂qnk

N∑i=1

p2i2mi

−N−1∑i=1

N∑j=i+1

Gmimj

rij

=

=∂

∂qnk

N−1∑i=1

N∑j=i+1

Gmimj

rij=

N∑i=1,i6=n

∂

∂qnkGmimn

rin=

=

N∑i=1,i6=n

Gmimn

(−1

2

)2(qnk − qik)

r3in=

N∑i=1,i6=n

Gmimnqnk − qikr3in

⇔

⇔ pnk = −N∑

i=1,i6=n

Gmimnqnk − qikr3in

, (3.11)

and

∂H

∂pnk=

∂

∂pnk

N∑i=1

p2i2mi

−N−1∑i=1

N∑j=i+1

Gmimj

rij

=

= − ∂

∂pnk

N∑i=1

p2i2mi

=pnkmn⇔

⇔ qnk =pnkmn

, (3.12)

Page 5


where n ∈ 1, 2, . . . , N and k ∈ x, y, z. These equations and methods are the basis for formulating a

numerical solution to a N-body problem. We can also combine equations 3.11 and 3.12 to write the equations

of motion for a general gravitational Hamiltonian on the form

miqik = −N∑

j=1,j 6=i

Gmjmiqik − qjkr3ji

⇔

⇔ qik = −GN∑

j=1,j 6=i

mjqik − qjkr3ji

, (3.13)

or in vector notation as

qi = −GN∑

j=1,j 6=i

mjqi − qj|qi − qj |3

. (3.14)

All of this can be found in any Celestial Mechanics or Hamiltonian mechanics text book, like (Danby, 1992).

3.3 The two body problem

When working with celestial mechanics one almost always comes across the term Keplerian elements. This

term refers to the variables describing a Keplarian orbit. If N in equation 3.14 is set to 2, there will be a

analytical solution to the differential equations yielding exactly what is called a Kepler or Keplerian orbit.

Even though Keplerian orbits and their propagation in many problems are sufficient one should not forget

that this is a very limited description of celestial mechanics. By the reduction of a more complex problem

down to just a two body problem, most of the interesting qualities are lost. The phrase Keplarian orbit has

several meanings, most often (Danby, 1992), as well as in this work, it means the set of all parametrization

of solutions to equations 3.11 and 3.12 when N = 2

p1k = −Gm2m1q1k − q2kr321

, (3.15)

p2k = −Gm1m2q2k − q1kr321

, (3.16)

q1k =p1km1

, (3.17)

q2k =p2km2

, (3.18)

which combined with qm = p, and, rij = rji gives

m1q1 = −Gm2m1q1 − q2

r312, (3.19)

m2q2 = −Gm2m1q2 − q1

r312. (3.20)

Performing a coordinate transformation into relative coordinates R = q1 − q2 and R = |R1 −R2| gives the

usually used form of the equations

Page 6


q1 − q2 = R = −Gm2R

R3−(−Gm1

−R

R3

)=

= −G(m1 +m2)

R3R. (3.21)

(3.22)

In a lot of literature the combined gravitational parameter µ = G(m1 + m2) is used, sometimes however

it is defined as µ = Gm1 without any explanation. This is always derived through the approximation

G(m1 +m2) ≈ Gm1 which is only applicable when the mass of the second body is a lot smaller relative to

the first.

Even tough not used explicitly in our work, for completeness we need to mention constants of motion. Since

the two body system covered here has analytical solutions, or is integrable, it will have a number of so called

constants of motion. In fact, as the system is integrable, it will have as many constants of motion as free

parameters. A constant of motion is simply a field on phase space that is constant along the flow of the

solution. Constants of motion are a useful concept because they can revel properties of the motion without

one solving the equations of motion. Some times even the trajectories themselves can be derived. This can

be the case since if the field is constant along a phase space path, one can also say that the motion of a body

follows intersections of multi dimensional ”level curves” of this field. In most cases the total energy of the

system is immediately known as a constant of motion, and we will later prove that for Hamiltonians we can

always re-write the system so that this is the case. But the energy need not always be constant. Although

conservation of energy is a fundamental law there are many systems where energy is not preserved due to

the physics behind the energy conversion not being considered. An example of this is radiation pressure, we

do not describe the photons or how the momentum from photons are absorbed by the body and re-emmited

in the actual Hamiltonian, instead we ad hoc add in the effect resulting in total energy no longer being

preserved. Such a system is called dissipative as energy can move into and out of the system.

3.4 Keplarian elements

In the previus section the two body solution was discussed and called a Keplarian orbit. Kepler however did

not derive his laws from a law a gravitation but from observation. But since the solution of the two body

problem coincide with Kepler’s laws of celestial motion:

(I) The orbit of a planet is an ellipse with the Sun at its pericenter,

(II) A line joining a planet and the Sun sweeps out equal sector areas during equal time intervals,

(III) The square of the orbital period of a planet is proportional to the cube of the semi-major axis of its

orbit,

often the term Keplerian elements and Keplarian orbit is used for things relating to a two body solution.

Despite their limitation to describing only orbits which are solutions to the two body problem, they can be a

valuable tool in examining more complex problems by looking at their deviation in time due to perturbations.

More on the explicit calculations of keplarian elements from state vectors can be read about in section 38.

The Keplerian (orbital) elements: are used to uniquely identify a specific orbit with a set of six

parameters, the choice of parameters often change but here we will most often adopt using the semi-

major axis a or perihelion distance q, the orbital inclination i, the numerical eccentricity e, the argument

of perihelion ω, the longitude of ascending node Ω, and if we also want to position the object on the

orbit, one of the anomalies, true anomaly ν, eccentric anomaly u or mean anomaly v. Illustration can

be found in figure 2 and figure 3.

Page 7


Figure 2: Digram illustrating and explaining various terms in relation to Orbits of Celestial bodies. (CC-BY-

SA-3.0-MIGRATED; Licensed under the GFDL by the author; Released under the GNU Free Documentation

License. http://commons.wikimedia.org/wiki/GNU_Free_Documentation_License)

Page 8

http://commons.wikimedia.org/wiki/GNU_Free_Documentation_License


Figure 3: Digram illustrating the three different anomalies where E is the eccentric anomaly, M the mean

anomaly, and ν the true anomaly. The central body is located at the point s and the orbiting body at point p.

It is also useful to introduce the so called longitude of perihelion as

ω = ω + Ω, (3.23)

and the mean longitude as

λ = M + ω + Ω, (3.24)

where M is the mean anomaly. These new variables are useful since Ω is not defined for zero inclination

orbits, and in the same way ω is not defined for zero eccentricity orbits. However, if we simply define these

angles as zero in these special cases and instead use the composite angles, we have a continues variable that

can still contribute information.

When working with the propagation of bodies along orbits a number of anomalies are used, as already

mentioned. These variables are different kinds of angles associated with the motion of a body along a elliptic

orbit.

To describe a specific point on a elliptic orbit we can use the true anomaly,

ν = cos−1e · r|e||r|

∀ r · v ≥ 0, (3.25)

ν = 2π − cos−1e · r|e||r|

∀ r · v < 0. (3.26)

This anomaly measures the angle from periapsis around the focus point. In the above expression e is the

so called eccentricity vector, r is the position relative the central body and the bodies v velocity. The

eccentricity vector, or the Laplace-Runge-Lenz vector, is a vector quantity associated with ellipses having

length equal to the eccentricity of the ellipse and orientation always pointing towards the pericenter of the

ellipse. The construction of this vector is purely a mathematical result of the geometry of oriented ellipses

Page 9


but due to its formulation it is a constant of motion in the two body problem and therefore useful when

calculating orbital elements from state vectors, or phase space coordinates.

Using the true anomaly one can calculate the so called eccentric anomaly,

E = cos−1(e+ cos ν

1 + e cos ν

)∀ r · v ≥ 0, (3.27)

E = 2π − cos−1(e+ cos ν

1 + e cos ν

)∀ r · v < 0. (3.28)

The eccentric anomaly is the angle representing a point on a auxiliary circle with radius equal to the semi-

major axis of the ellipse. This point is defined as where the line perpendicular to the semi major axis passing

through the orbiting body intersects the auxiliary circle.

Again using the eccentric anomaly the mean anomaly can be calculated,

M = E − e sinE.. (3.29)

This equation is commonly called the Kepler equation and this anomaly represents the angle required so

that the area swept on the auxiliary circle is equal to that swept on the ellipse. This is particularly useful

since Keplers second law states that the area swept by a line connecting a body and the ellipse pericenter

must be a constant of time, making propagation of this angle easy. Problems arise when this method of

propagation is put into action since equation 3.29 is a so called transcendental equation. Transcendental

equations are equations containing a transcendental function such that it cannot be solved for one factor in

terms of another using algebra (Danby, 1992).

More on the specific calculation of all these quantities will be covered in later sections, and algorithms for

solving the Kepler equation to constuct numerical propagators is not currently relevant to our scope.

3.5 Hamiltonian form

Now that we are getting more comfortable with the notion of how Hamiltonian mechanics work and what

they represent we need to generalize the concepts and properties of Hamiltonian systems. Starting with

identifying a Hamiltonian system. A system of ordinary differential equations on the form

dx

dt= F(x) (3.30)

is on Hamiltonian form if

x =

q1...

qNp1...

pN

, (3.31)

and there exists a function H(x, t) such that

Page 10


dqidt

=∂H

∂pi∀ i ∈ 1, . . . , N, (3.32)

dpidt

= −∂H∂qi∀ i ∈ 1, . . . , N. (3.33)

Then we can write the vector function F as

F =

∂H∂p1...∂H∂pN

− ∂H∂q1...

− ∂H∂qN

. (3.34)

This is a large class of differential equations, which is one reason why Hamiltonian formalism is so useful.

Considering Hamiltonian mechanics from the physicist point of view the Hamiltonian is not just a function

transform from phase space to real numbers or the Legendre transform of a Lagrangian, but it represents

the total energy of the system, in the same way the Lagrangian represents action by acting on configuration

space. Assuming the time tested formula that total energy is represented by kinetic and potential energy

H =|p|2

2m+ U(q), (3.35)

and that the force exerted on a particle due to a potential field is given by a second order differential equation

as

d2q

dt2= −∇U(q). (3.36)

All formulas of this type are of Hamiltonian form. This differential equation hold all dynamics caused by

classical gravity, however electromagnetic effects and general relativistic effects play a crucial roll in the

long term evolution of the Solar system and should not be dismissed easily. Analysing this particular set of

differential equations, it is useful to examine how the function H(x, t) evolves with time. By expanding the

full time derivative using the chain rule, denoting the scalar product of two vectors as 〈u,v〉,

dH

dt= 〈∇H, dx

dt〉+

∂H

∂t= 〈∇H,F(x)〉+

∂H

∂t(3.37)

where

∇ =

∂∂q1...∂∂qN∂∂p1...∂∂pN

. (3.38)

Page 11


We can then see that

〈∇H,F〉 = 0. (3.39)

If this is not obvious, it becomes clear by combining equations 3.32, 3.33 and 3.34. This shows that

dH

dt=∂H

∂t, (3.40)

thereby proving that all autonomous, or independent of time, Hamiltonians are constants of motion, and that

the systems total energy is preserved. This is a important result since a consequence of Hamiltonian formalism

is that all non-autonomous Hamiltonian can be turned into autonomous Hamiltonians by expanding the phase

space. Specifically one performs a transformation from the old phase space into a new one with a additional

position and conjugate momentum dimension by embedding the old time variable into the configuration

space and introducing a new time instead. If H = H(q,p, t) then introduce new phase space angle with

corresponding conjugate momentum τ = T giving a new Hamiltonian

H ′ = T +H(q,p, t) (3.41)

where it is easy to see that

dH ′

T= τ =

dτ

dt= 1⇔

⇔ τ = t. (3.42)

Since τ(t) = t is a direct substitution t 7→ τ makes the Hamiltonian autonomous, and therefore total energy

is a constant of motion yet again. If the simple addition of the conjugate momentum seems strange, the

operation can be justified by using equations 3.6 and 3.7. These are some basic facts about the result of

defining the Hamiltonian and deriving the first variations of the total energy integral, (Hand and Finch,

1998). We shall not cover the variational calculus proofs here, more information can be found in (Logan,

2013).

3.6 Poisson brackets

To later cover some of the specifics of Hamiltonian integration techniques we will quickly need to cover

Poisson brackets. Poisson brackets are a useful tool when working with phase space paths. Consider phase

space M , differential equation dxdt = F(x), x ∈M , and function f on M , we can define the Poisson brackets

as

f(x), g(x) =

N∑i=1

(∂f

∂qi

∂g

∂pi− ∂f

∂pi

∂g

∂qi

). (3.43)

It is straightforward to prove that the Poisson brackets are linear in their arguments

α1f1 + α2f2, g = α1f1, g+ α2f2, g, (3.44)

f, α1g1 + α2g2 = α1f, g1+ α2f, g2, (3.45)

Page 12


and that they are anti-commuting in their arguments

f, g = −g, f. (3.46)

We define the skew matrix, using the N-dimensional identity matrix IN , as

J =

(0 IN−IN 0

). (3.47)

Here, using the skew matrix defined in equation 3.47 we can redefine the Possion brackets as the skew scalar

product of the gradient of two scalar fields

f, g = 〈∇f, J∇g〉. (3.48)

This will be useful later when examining phase space paths.

3.7 Liouville’s theorem

We know from vector space analysis that the divergence of a vector field can also be seen as the normalized

evolution of a infinitesimal volume element along the vector field

1

∂V

d

dt∂V = div(F(x)) = 〈∇,F(x)〉. (3.49)

This can be interpreted as describing how a infinitesimal volume of initial conditions evolve along the flow

of the differential equation. It can then be shown for Hamiltonian systems the area element of position

and momentum must always stay constant. This implies that in a way volume in phase space is always

preserved when a Hamiltonian system evolves with time, a property also called Liouville’s theorem. From

this conclusion we can also imply that Hamiltonian systems does not have attractors. An attracting set for

a system is a closed subset of the phase space such that for ”many” initial values the system will evolve

towards the attracting set. This plays a very important role in the evolution of dynamical systems and more

on attractors can be read about in (Milnor, 1985).

From this theorem we mathematically conclude what we earlier conjectured that an N-degree of freedom

Hamiltonian is integrable if it has N independent constants of motion Φ1, . . . ,ΦN such that Φi,Φj =

0 ∀ i 6= j. There is no set way to find all constants of motion for a system although it is generally easier to

find these constants then to evaluate the integral solution of the system. A integrable Hamiltonian system

is a system where we can rewrite the differential equation on form 3.30 into a set of integral equations

∫ x(t)

x(0)

1

Fi(x)dxi =

∫ t

0

dτ ∀ i ∈ 1, . . . , 2N, (3.50)

where the left hand side of the equation can be evaluated, alas there is also here no set way to determine

if there are solutions to the integrals or not. This makes it hard to distinguish between systems that have

complicated integrals that one simply has not managed to solve and systems which are not integrable at all.

Page 13


3.8 Phase space paths

If f(x) is a function on M and x evolves according to the Hamiltonian flow, equations 3.32 and 3.33, a full

time derivative can be expressed by

df

dt= 〈∇f, dx

dt〉+

∂f

∂t, (3.51)

due to the chain rule. But we assume that ∂f∂t = 0 and from equation 3.48, 3.30, and 3.34 we can then

rewrite the derivative to

df

dt= f,H. (3.52)

This is a important result since we now know how to a field evolves along a phase space path due to the

Hamiltonian (Gel’fand and Dorfman, 1979). As most other literature is unclear on the derivation of the

infinite series describing motion that is later used for order determination in numerical methods, we shall

here do our own definition and derivation to clarify the process. Considering a coordinate as a function of

time we can write

dqidt

= qi, H, (3.53)

dpidt

= pi, H. (3.54)

Leaving one argument open in the Possion bracket of the Hamiltonian ·, H we can consider it as a differ-

ential operator acting on the function, propagating it by the equation dfdt = ·, Hf . This operator is often

called the Liouville operator. Using this, we can examine a differential equation on Hamiltonian form and

integrate its evolution. Assuming that the Liouville operator is not a function of time, which we know due

to the earlier proof on autonomous Hamiltonians, we can see that the equations are integrable. We now also

assume that we have a initial value at t = 0, then the resulting equation is a regular first order differential

equation with constant coefficients as

dx

dt= F(x)⇒ dxi

dt= Fi(x) = xi, H = ·, Hxi ⇔

⇔ xi = et·,Hxi(0). (3.55)

This relation can be used to describe the evolution of said phase space path. It may seem strange that one

can take the exponential of a differential operator but it will become clear by the use of Taylor expansion.

Consider the phase space path a time difference h from our initial time t0 as

xi(t0 + h) = ehddtxi(t) |t=t0 , (3.56)

through the use of the power series definition of exponential functions, or Taylor expansion, we can write

ehddt =

∞∑i=0

(h)i

i!

di

dti. (3.57)

Page 14


Thus the exponential of a differential operator is a infinite sum of differential operators. Here we also need

to consider what the different orders of time derivatives translate to in Liouville operators, let consider a

second order time derivative as

d2

dt2f =

d

dtf,H = f,H, H = ·, H2f. (3.58)

Applying this concept recursively in the Taylor series we get a expression

f(t) =

∞∑i=0

(t)i

i!·, Hif(0), (3.59)

that is usually called the Lie series of f under the flow of H (Fasso, 1990), this series is usually denoted as

StH for a flow according to Hamiltonian H for a time t. We can this way propagate any field or phase space

path a time h by the formula

f(x(t)) |t=t0+h = eh·,Hf(x(t)) |t=t0 (3.60)

where the differential operator eh·,H is acting on f(x). This is the central concept needed to describe the

main integration method for Hamiltonian integration software.

4 The problem of non gravitational perturbations

The big strength of Hamiltonians can pose a problem in numerical integrations of celestial mechanics. The

fact that Hamiltonian mechanics require total energy to be described and preserved excludes the concept of

physical effects not modelled properly. This means that if, for example, electromagnetic perturbations are to

be described one also has to model the photons that are absorbed, reflected, and emitted, the thermal energy

that is stored upon absorption, and then also keep describing these photons as they leave the body. If new

photons are produced even mass energy conservations has to be described, otherwise one will break the basic

notion of a constant energy in a system. There are however ways around this, most often one sees the phrase

’modified equations of motion’, this basically means that one has implanted a virtual potential field which

the objects can interact with to preserve the energy without actually describing the precise physics behind

the interaction, only describing the net effect. One can also model electromagnetic effects with something

called dissipative mapping techniques as in (Kehoe et al., 2003) which can be incorporated with some success

into already existing Hamiltonian integration schemes.

The important concept to grasp here is that after a proper Hamiltonian integration step has been performed,

we can apply whatever modifications due to other effects we like, as long as we do not do so inside the

Hamiltonian propagation step. This can be justified by the idea that each step takes a new initial condition

and propagates it, it does not matter what state we give the propagation, as long as we let the algorithm

run without interference, all desired properties shall be preserved. This can create a dissipative mapping,

with a symplectic step, which may seem like a contradiction at first.

4.0.1 Radiation pressure

We shall use the classical electromagnetic effect, the Sun’s radiation pressure, as a starting point. This

perturbation is due to photons being absorbed and remitted, either in a thermal or a reflective way. Since

momentum must be preserved during a process this means that simple radiation pressure will push a orbiting

Page 15


body outward proportional to the power of the incoming radiation, the pressure from absorbed photons can

be described by

Pa =W

ccos(α) (4.1)

where α is the angle between the objects normal vector and the incoming radiation of power W . The pressure

from reflected photons can instead be described by

Pr =2W

ccos2(α). (4.2)

This means that, together with a bodies albedo A and topology, the total radiative force acting on body can

be calculated. Most often when performing electromagnetic perturbation is celestial mechanics one derives

secular perturbation on the Kepler elements of a orbit. This is however a very rough approximation and

should be used with care.

4.0.2 Poynting-Robertson effect

The Poynting-Robertson effect, or radiation drag, is a secondary effect from that of radiation pressure due

to special relativity. The effect can be explained from the perspective of the body orbiting the Sun, the

Sun’s radiation appears to be coming from a slightly forward direction because of aberration of light. The

absorption of this radiation leads to a force with a component against the direction of movement. The

magnitude of the force is

FPR =v

c2W =

r2Ls4c2

√GMs

R5, (4.3)

where c is the speed of light, W is the power of the incoming radiation, r is the radius of the object being

radiated, G is the gravitational constant, Ms is the mass of the Sun, Ls is the Solar luminosity and finally R

is the orbital radius. As this force is dependant on the radius of the particle, a critical threshold can be found

between the radiation pressure and the radiation drag when the effect either pushes the particle away from

the Sun or causes it to spiral inwards. This is a important contribution to the appearance of the sporadic

sources in meteor science. One important result from early calculations of the effect on small particles is

the time of Solar impact for a circular orbit of initial semi major axis a. The calculations performed on a

spherical homogeneous sphere of density ρ and radius r that is absorbing and emitting all incoming Solar

radiation isotropically across the entire solid angle was found to have a time of Solar impact of

t = 7.0 · 106rρa2 y. (4.4)

This is small compared to the age of the Solar system ts ≈ 4.5 ·109 y. Thus, the sporadic complex of meteors

generated by dust particles entering the Earth atmosphere must be resupplied somehow. It is widely accepted

that the major supply mechanism is the sublimation of comets, more on this process will be covered in later

sections.

4.1 Yarkovski effect

The Yarkovsky effect is a force acting on a rotating body in space due to the anisotropic emission of thermal

photons. The part of this effect due to rotation around its own axis is called the diurnal effect. The diurnal

Page 16


effect however changes with orbital position around the source of radiation, and is called the seasonal effect.

From more detailed descriptions of this effect, one can conclude that the seasonal part always produces a net

decrease in semi-major axis while the diurnal effect both increase and decrease the semi-major axis. It will

also become evident that this effect vanishes for too large bodies due to the relation between its cross-section

and its moment of inertia. However, the effect vanishes for very small bodies as well due to the body’s

size being comparable to the penetration depth of the thermal wave. Of course since this is also dependant

on radiation flux the effect will be dependant on received radiation and surface conductivity. Under a few

assumptions the force can be calculated by the integral

FY =

∫S

− 2εσ

3mcT 4n⊥dS(u, v), (4.5)

where the surface of the object is S(u, v), ε is the surface thermal emissivity, σ is the Stefan-Boltzman

constant and n⊥ is the surface normal vector. More on this can be read about in (Bottke et al., 2006).

4.1.1 Yarkovsky O’Keefe Radzievskii Paddack effect

A secondary effect of the Yarkovsky effect is the YORP-effect. This is due to the fact that the reflection

and re-emission of photons from an objectss surface can also produce a thermal torque if the object has a

irregular shape. Over time, these torques will affect the spin rate and obliquities of small bodies in the Solar

System. The change of the internal spin and eccentricity vectors can be expressed by

dω

dt=

T · eC

, (4.6)

de

dt=

T− (T · e)e

Cω(4.7)

where ω is the rotation velocity of the object, C the objects moment of inertia, e is the eccentricity vector,

and T is the torque on the object due to thermal radiation, more on this can be read in (Bottke et al., 2006).

4.1.2 Relevance

From the above discussion we can conclude that only the Poynting-Robertson effect and radiation pressure

will be relevant to our integrations and thus we will neglect all other electromagnetic forces.

5 Numerical integration

5.1 Symplectic structure

Symplectic integrators are a scheme for numerically integrating a specific group of differential equations

connected to symplectic geometry that is a branch of differential geometry and differential topology. Since

this type integrator works on geometrical basis it is excellent for integrating physical problems such as

classical mechanics where the phase space of a system can take the form of a symplectic manifold. In our

case the set of differential equations are the Hamiltonian with the Poincare two-form dpi ∧ dqi, where qi is a

generalized coordinate and pi its momentum. These two elements together create our canonical coordinates,

phase space, and the symplectic integrator can be viewed as canonical transformations. This is directly

related to the concept in section 3.7 since symplectic integrator preserve the Poincare two-form and thus in

numerical integration total energy oscillate rather then diverge.

Page 17


Figure 4: Pendulum phase map with symplectic and non symplectic euler using the same stepsize.

Page 18


Figure 5: Relative error in total energy for different initial conditions for a pendulum.

A example of the difference between preserving symplectic structure and not doing so in a numerical inte-

gration can be seen in figure 4 and 5. In these pictures are shown what such a simple modification as to

make the method semi implicit can do for a numerical long term integration of a Hamiltonian system. It

should however be noted that symplectic integrators are only a special case of what is called geometric inte-

grators, there are several more geometric structures that can be preserved other then the Poincar two-form,

for example if one wants to preserve phase space volume or perhaps limit volume contraction to a constant

another approach to integration should be used. For more information see (McLachlan and Quispel, 2006).

For more information on the accuracy of symplectic integrators, see (McLachlan and Atela, 1992).

5.2 Hamiltonian splitting

The next step is to cover the Hamiltonian splitting technique. This technique attempts to separate and

isolate parts of the Hamiltonian, of a system with no analytical solution, into smaller parts who on their

own are integrable. Such a integrator is symplectic. For example in a usual N -body gravitational problem,

there is no solution unless N = 2. We can, loosely speaking, say that the problematic term is the sum of

gravitational interactions outside that of the Kepler orbit. Meaning that one can separate this problem into

several Kepler problems and the gravitational interaction between the smaller bodies. In other words, if we

have

H =

N∑i=1

p2i2mi

−N−1∑i=1

N∑j=i+1

Gmimj

rij, (5.1)

where the central mass possess index 1. We can then, as a example, split the Hamiltonian into (Wisdom

and Holman, 1991)

Page 19


H = Hkep +Hint (5.2)

Hkep =

N∑i=2

(p2i

2mi−Gmim0

ri1

)(5.3)

Hint =

N−1∑i=2

N∑j=i+1

Gmimj

rij(5.4)

where Hkep and Hint on their own can be solved but not together. This can be seen if we use equations 3.6

and 3.7 on the Hkep and Hint expressions separately. The importance of this kind of split will be evident in

coming chapters.

5.2.1 Differential equation flow

Let us consider the flow, Ψ, of a differential function x through its phase space M ,

x(t) = Ψtx(0), (5.5)

Ψt : M 7→M. (5.6)

This flow is a exact deterministic solution, but if the system has no analytical solutions, this flow is not

known. Thus we try to find maps, Φ, to approximate this flow. These maps are called integrators, let us for

example define the flow of an one step method on a Hamiltonian phase space x = (q, p), with step size h as

Φh : (qn, pn) 7→ (qn+1, pn+1) (5.7)

this function is called the numerical flow of the integration method.

5.2.2 Order of a split

We shall now outline a systematic way to derive an integration method using a Hamiltonian split and to find

its order. Even tough higher order integrators generally perform better, there is no guarantee that they will

do so. Also, there is usually a trade off point where the extra computational effort starts outweighing the

precession gain by increasing the order of integration.

The order is simply a measure of Taylor expansion similarity, in our case we will derive the propagation

scheme Φ and Taylor expand this flow. Φ is of order p if its Taylor expansion corresponds to the real flow Ψ

to term number p. To show this, let us take the exponential general solution to a phase space path evolution

as derived in equation 3.60,

f(q, p) |t=t0+h = eh·,Hf(q, p) |t=t0 (5.8)

and split the Hamiltonian into two parts H = HA +HB . We can write

f(q, p) |t=t0+h = eh·,HA+HBf(q, p) |t=t0 = eh(·,HA+·,HB)f(q, p) |t=t0 . (5.9)

since the Poisson brackets are linear in their arguments. We can not however assume that ·, HA and

·, HB commute, in other words ·, HA·, HB 6= ·, HB·, HA. This means that eh(·,HA+·,HB) 6=

Page 20


eh·,HAeh·,HB since this mathematical property of exponential functions require the commutative property

of the arguments. We can use this to derive the order of the splitting by expanding both expressions as

eh(·,HA+·,HB) =

∞∑i=0

hi(·, HA+ ·, HB)i

i!=

= 1 + h(·, HA+ ·, HB) +

∞∑i=2

hi(·, HA+ ·, HB)i

i!(5.10)

and

eh·,HAeh·,HB =

( ∞∑i=0

hi·, HAi

i!

)( ∞∑i=0

hi·, HBi

i!

)=

=

(1 + h·, HA+

∞∑i=2

hi·, HAi

i!

)(1 + h·, HB+

∞∑i=2

hi·, HBi

i!

). (5.11)

By subtracting the two expressions we can easily see that all first order terms vanish

eh·,HAeh·,HB − eh(·,HA+·,HB) =

= h·, HA

( ∞∑i=2

hi·, HBi

i!

)+

( ∞∑i=2

hi·, HAi

i!

)h·, HB+

+h·, HAh·, HB+

( ∞∑i=2

hi·, HAi

i!

)( ∞∑i=2

hi·, HBi

i!

)+

−∞∑i=2

hi(·, HA+ ·, HB)i

i!= O(h2) (5.12)

this means that

eh·,HA+HBf(q, p) |t=t0 = eh·,HAeh·,HBf(q, p) |t=t0 +O(h2). (5.13)

We can no state that our split into eh·,HAeh·,HB has an error equal to the difference between all terms

after the first order Taylor expansion term. This splitting is called a first order integrator, with error O(h2).

If we instead split

eh2 ·,HBeh·,HAe

h2 ·,HB (5.14)

we can see that the Taylor expansion of this expression compared with the Taylor expansion of the exact

solution compares to the second term, making this splitting of second order with error O(h3).

5.3 Bulirsch-Stoer method

The BulirschStoer method combines the three ideas of Richardson extrapolation, rational function extrapola-

tion in Richardson-type applications, and the modified midpoint method (Press et al., 1992). This technique

is not designed for differential equations containing non-smooth functions and it is not designed for differen-

tial equation with singular points inside the integration interval. Thus it is suited for integration Hamiltonian

systems with slight electromagnetic perturbations. The Bulirsch-Stoer method is widely recognized as one

of the best ways to calculate high accuracy solutions to ordinary differential equation with minimal compu-

tational time.

Page 21


5.3.1 Richardson extrapolation

Richardson extrapolation, or the deferred approach to the limit, is a method for convergence acceleration.

More explicitly, the idea behind Richardson extrapolation is that the value of convergence for a numerical

calculation in a limit is a function itself that is dependant on the number of terms in the calculation. If

this is the case we can fit a rational function, or a polynomial in our case, to a set of resulting values for

different number of terms and then evaluate the resulting function at the desired point. Since the goal is

to approximate a exact solution often one evaluates this function where it represents infinitely many terms.

By doing this one can quickly get a much better estimation of the convergence value at infinity rather than

calculating absurd amounts of terms. Obviously this will not work unless the sum of the terms is a relatively

smooth function. However, in celestial mechanics there is not a lot of non-smooth behaviour and as such

this will not be a problem.

As an example of Richardson extrapolation let us consider the series

A = limN 7→∞

N∑i=1

ai. (5.15)

let us consider a function

f(N) =

N∑i=1

ai, (5.16)

that will evaluate to limN 7→∞ f(N) = A. We cannot practically calculate A unless ai is some special

expression. If however we assume that f(N) is a rational function of N , even if we can never find f

we can fit a function to it by calculating a set of different series evaluations f(1), f(2), f(3), . . . with

corresponding input values 1, 2, 3, . . . and fit a function to this data f(N) ≈ f(N). If we then take the

now known approximate function and evaluate limN 7→∞ f(N) we have performed Richardson extrapolation

as limN 7→∞ f(N) ≈ A.

5.3.2 Modified midpoint method

The specific limit we are going to apply this to is the modified midpoint method with infinitely many steps,

or with a step size that tends to zero. The specific modification of the usual midpoint method is made

to remove oscillatory terms in the error of the method. This together with the asymptotic expansion of a

midpoint method in the square of the step size shows us that we can improve the method by the square of the

step with extrapolation instead of just by the step itself. The modified midpoint method can be expressed

as

hj =TN+1 − TN

nj(5.17)

uN = u(TN ) (5.18)

z0 = uN (5.19)

z1 = uN + hjf(z0, TN ) (5.20)

zm+1 = zm−1 + 2hf(zm, TN +mhj) ∀ m ∈ 1, . . . , nj − 1 (5.21)

unj

N+1 =1

2(znj + znj−1 + hnjf(znj , TN+1)) (5.22)

Page 22


5.3.3 Deufelhard serie

In order to fit a polynomial to Richardson’s deferred approach to the limit of infinitely many sub steps in a

modified midpoint method we need to choose data points for the fit. Several series have been proposed but

the one we deemed best for our purpose was the one Deufelhard (Kirpekar, 2003) suggested

nj = 2(j + 1) ∀ j ∈ N0. (5.23)

Since a numerical calculation needs a finite series, let us restrict j by a number k = 8, j < k, giving us a a

set of 8 n’s,

n = 2, 4, 6, . . . , 16. (5.24)

If at this point we deem that our fit is still not good enough, we instead must shorten the general step length

instead of the sub step length.

5.3.4 Neville’s Algorithm

Neville’s Algorithm is designed to build a polynomial that goes through a set of n points by combining lower

grade polynomials in sequence until a final expression is derived. For example, if given 5 points, the first

set of equations will be 5 constants through all 5 points, then these 5 constants will combine in a way that

we have 4 lines going through 2 of the points, then again forming 3 parabolas passing through 3 of the

points each, continuing until we have a grade 4 polynomial passing through all five points. The algorithm is

described by

Pj,j(h) = unj+1

N+1 ∀ j < k, (5.25)

Pj,i(h) =(hi − h)Pj,i−1 + (h− hj)Pj+1,i

hi − hj∀ 0 ≤ i < j < k. (5.26)

This is the the algorithm used to fit the Deufelhard series evaluation to a function which we can evaluate at

0 step length.

Page 23


Part II

Solar system population

6 Introduction to Solar system population

To create a useful simulation the input state must be carefully considered and designed. Unfortunately there

are many unknowns in the problem considered in our work, but we will here try to compile what is known,

to what accuracy, and to motivate the assumptions of the required parameters that are not known.

If the reader is knowledgeable on the constituents of the Solar system this part is not vital to progress

through the text, however the databases listed at the end may provide useful information.

7 The Sun

As we have not yet implemented nano-dust interactions with the Solar wind plasma we will not cover the

theory of Solar wind plasma to dust interactions at this time. One important factor to consider is the

electromagnetic radiation that emanates from the Sun causing the previously mentioned radiation pressure

and Poynting-Robertson drag perturbations. From mesurnments of the Solar radiation one can find a mean

total luminosity of L0 ≈ 3.846 · 1026 W integrated over the entire surface (Mamajek et al., 2015). As the

radiation expands at the speed of light in spheres the energy flux density will decrease with the distance

from the Sun R as

Ψ =L0

4πR2. (7.1)

We consider the Solar luminosity as a constant since cycle variations are small enough not to matter for the

current calculations (Willson and Hudson, 1991).

8 The planets

The prominent sources of gravitational perturbation influencing the dynamics of dust in the Solar system

are the eight planets. As such to create a appropriate initial state for the planets is crucial for a realistic

simulation. This can be done by using observational data gathered by the NASA Jet Propulsion Laboratory.

Using these observations one can also calculate the planetary masses and the Solar mass. Other then their

states and masses not much information about the planets are needed to perform our simulations.

9 Small bodies

Our goal is to simulate the dynamics of the small bodies in the Solar system in a statistical manner, and

thus it will be important to consider their populations and properties. We shall here only cover some basics

on the major small mass constituents of the Solar system.

9.1 Comets

A comet is a body considerably smaller then a dwarf planet that is made of rock and ice, typically a few

kilometres in diameter, orbiting the Sun. When such a body passes close to the Sun the Solar radiation

Page 24


begins heating the body until sublimation occurs and the comet outgases the previously icy material. This

produces a visible atmosphere or coma around the comet that some times produces a tail. In recent times

we have even begun to examine the small magnetic environment of the comet as it produces its own small

magnetosphere and plasma environment when interacting with the Solar wind and the Solar radiation. Also,

due to the jets and streams of outgassing, dust of various sizes and compositions are ejected making the

environment of a comet as it closes in on the Sun a very dynamic and unique object to examine.

The coma and tail are much larger then the actuall nuclei of the comet itself and can sometimes, if sufficiently

bright, be seen from the Earth without a telescope, such as the famous Hale-Bopp comet, formally designated

C/1995 O1. This comet could be seen from a clear sky in the recent perihelion passage of 1996 - 1997.

Short-period comets originate from the Kuiper belt and its scattered disc of material that lies beyond the

orbit of Neptune.

It has recently become widely acknowledged that long-period comets originate from the Oort cloud. The Oort

Cloud is a nearly spherical cloud of comets, dust, ice, and other derbies, which surrounds the Solar system.

Its width extends to heliocentric distances of more than 100 000 AU and is currently thought to inject about

12 comets with perihelion distance closer then 3 AU each year (Wiegert and Tremaine, 1999). Recently work

has been made on the dynamics of dust particles released from the Oort cloud comets (Nesvorny et al., 2011)

that is relevant to our purpose.

Figure 6: Histogram of the different distribution of orbital elements within the Pan-STARRS Solar system

synthetic model.

The Pan-STARRS synthetic Solar system model predicts the population of short period comets currently in

the Solar system to be distributed as shown in figure 6.

The main difference between an asteroid and an comet is the presence of an extended, gravitationally

unbound atmosphere surrounding the comet central nucleus. If however, a comet experiences so many

perihelion passages that it looses nearly all of its volatile ices it will come to resemble a small asteroid.

Page 25


10 Dust

Even tough dust in the Solar system may appear sparse it is a major part of the population. The inter-

planetary dust is distributed over the entire Solar system. This dust is also numerous enough to scatter

Solar light, called the zodiacal light, and produce measurable thermal emissions (Levasseur-Regourd, 1996).

This light becomes a predominant feature of the night sky in the 5 µm to 50 µm wavelength. The total

mass influx of dust onto the Earth has been estimated to be between 55 and 270 tones each day, in (Plane,

2012) a comprehensive comparison between 13 different mass influx models and measurements is compiled.

This mass influx is a huge number with a equally huge estimated range, and is still today a open question

in science. This mass influx has various effects on the Earth. For example the mass influx of meteoroids

produces a external input of atmospheric metals in the mesosphere and lower thermosphere as there have

been good correlation between amount of metallic ions in the lower thermosphere and meteor showers (Plane,

2003). Some of the effects of this input of extraterrestrial material into our atmosphere is sodium D-Line

airglow, persistent meteor trains, sporadic layers of metallic ions with impact on radio communications, and

noctilucent clouds.

It has also been shown that dust is a impact hazard to spacecraft. For example, the ESA communications

satellite Olympus lost pointing control in 1993 likely due to the impact of a Perseid meteoroid (Caswell et al.,

1995). During the same year due to the Perseids, NASA postponed its space shuttle launch to avoid the

meteor shower maximum, and diverted the pointing of the Hubble Space Telescope to avoid lens damage.

Thus it becomes evident that the presence of dust in the Solar system is no negligible matter. But to closer

examine the dynamics of dust and their effects one must begin with its origin.

10.1 Poynting-Robertson lifetimes

To find the origin of the dust observed to enter the Earth atmosphere and the general interplanetary dust

we must consider the previously mentioned Poynting-Robertson effect in section 4.0.2.

A quite logical first guess as to the origin of the interplanetary dust would be left over material from

the formation of the Solar system. We know that when the Sun ignited it generated a large wave of

radiation pressure that pushed all excess gas out of the Solar system creating a close to clean interplanetary

environment. This wave of radiation would however be too weak to push larger particles out of the Solar

system and thus they would possibly be left in the Solar system. This is where the effect of radiation drag

is of vital importance, as a direct consequence of this effect is that all particles either spirals inwards toward

the Sun due to the drag or outwards away from the Sun due to pressure. There is also the third cases where

the two forces are kept in equilibrium but due to the amount of gravitational perturbations in the Solar

system the particles will scatter and sooner or later collide with another object as they are forced to leave

such a equilibrium. As early as 1962 the time of Solar impact was calculated (Guess, 1962) for circular orbit

of initial semi major axis a, of a spherical homogeneous sphere of density ρ and radius r that is absorbing

and emitting all incoming Solar radiation isotropically across the entire solid angle. The resulting formula

t = 7.0 · 106rρa2 y. (10.1)

compared to the age of the Solar system ts ≈ 4.5682 · 109 y (Bouvier and Wadhwa, 2010) imply that all dust

present at the formation of the Solar system should today have collided with the Sun. But since we today

see a steady influx of mass into the atmosphere in the form of meteoroids and meteorites, we must assume

that the population of dust is refreshed by some mechanic. The best explanation to this seemingly steady

state meteor complex at the Earth is the continous ejection of dust from comets and asteroids.

But as many comets also have relatively short active lifetimes of ejection, the population of comets must also

have a mechanism for population renewal. There could be several such mechanisms but the most probable

Page 26


source for long-period and short-period comets are the so called ’kicking’ of comets out of the Oort Cloud

and the Kuiper Belt respectively. For example in (Levison et al., 2001) the origin of Halley-type comets are

examined by considering injection of comets from the Oort cloud.

10.2 Comets

Figure 7: Ejection speed as a function of the common logarithmic mass of the particle.

Since the most probable major source of dust in the Solar system seems to be comets, we must address their

relation to the creation of dust. The feature that has made comets visible to the naked eye and therefore

known to mankind for thousands of years is its dust and gas tail. This trail increases as the comet approaches

the Sun. The basic reason for this loss of mass was explained by a icy conglomerate model as early as 1950

in (Whipple, 1950). In this model the primary component of the nucleus is assumed to be ice with dust

particles embedded within it. Dust here is defined as any substance that is in a solid state at temperatures

of a few hundred kelvins. As the ice sublimes due to increasing Solar radiation, these grains are released.

Their subsequent motion away from the comet nucleus is due to interaction with the gas outflow, radiation

pressure, and gravity. This interaction is of course dependant on the size of the dust grains and this effect

forms the well known dust and gas trail. To model reasonable meteoroid streams or even their dissipation

into a sporadic complex one must model the ejection from the parent body in a sensible way since this is

the initial perturbation causing their trajectories to diverge from the comet itself.

10.2.1 Dust ejection

We previously mentioned the icy conglomerate model by Whipple as presented in (Whipple, 1950) and

(Whipple, 1951). We shall here quickly review this model and its evolution. Models like this considers the

absorption of sunlight by the cometary nuclei, which in consequence heats up to the point where sublimation

of the ices present on the surface of the nucleus occurs. When this ice generates gas a pressure difference is

generated and the gas shoots outwards. This is quite a violent process, and since the ice is polluted by dust

Page 27


particles they are ejected together with the gas. As such the ejection velocity of the particle is dependant

on the gas interaction with the dust, i.e. how much momentum the gas transfers to the particle. We will

not cover the entire derivation of the ejection physics but instead summarize some of the key points with

the models.

The Whipple model assumes that the sublimating gas uniformly expands from the sunlit hemisphere of the

nucleus, where the mean thermal velocity of the gas is ∝ r−τ and r is the heliocentric radius. Thus the mean

thermal velocity resolves to

vg =

(8kBTgπµ

)1/21

rτ. (10.2)

It is also assumed that the temperature of the gas is the maximal surface temperature Tg = 300r−1/2 K.

Assuming elastic collision between gas and dust particles Whipple then compensates with a drag coefficient

for the inelastic collisions that was chosen to kd = 26/9. Whipple also assumed a spherical nucleus of mass

M giving the mass loss due to absorbing sunlight with total luminosity L0 and subsequently sublimating as

M = πR2c

(L0

4πr2

)1

nH. (10.3)

Where the sublimation energy used was H = 1.88·106 J/kg, approximately the sublimation heat of ammonia,

and the coefficient n−1 ≈ 1. Here the nuclei radius Rc is a function of the mass Mc through the spherical

constant density assumption. Finally, it is assumed that the thermal velocity relation exponent is τ = 0.25.

Combining all these expressions with the differential equation for the acceleration of the dust, including the

gravitational attraction of the nucleus and assuming dust grains with shape parameter A and mass m, we

find two different equations for the escape velocity. The first one where the dust particles are slow compared

to the gas thermal velocity

V 2∞ =

kd2

MvgπRc

A

m− 2GMc

Rc, (10.4)

and the second where the dust is fast compared to the gas velocity,

V 2∞ =

kd2

M(vg − r)πRc

A

m− 2GMc

Rc. (10.5)

Where Whipple assumes that µ = 20 · 1.661 · 10−24 g allowing a reduction to the frequently used formula

v2 = 43Rc

(1

nsσr2.25− 8πG

3 · 43ρcRc

). (10.6)

Recently due to discrepancies between observation and the ejection speeds derived by Whipple, (Ma et al.,

2002) reinvestigated the problem starting from simple physical principles and derived new formulas for the

mass loss of the comet,

Mc =R2cL4H

(1

r2− 1

r2s

), (10.7)

Page 28


and the ejection speed of meteoroids,

v2 =WRcL8πHαsσ

(1

r2− 1

r2s

)− 2GMc

Rc, (10.8)

where W is the mean thermal velocity, Rc is the radius of the comet, L is the Solar luminosity, H is the

latent heat of vaporization, Mc is the mass of the comet, G is the gravitational constant, α is the fraction

of the comet that is active, s is the radius of the assumed spherical dust grain, σ it’s bulk density, and rs is

the critical heliocentric radius for sublimation to occur.

To get an idea of the ejection speed profile we have taken the values L = 4 · 1026 J/s, H = 2 · 106

J/kg, Mc = 1.55 · 1016 kg, Rc = 10 km, W for water to be 580 m/s at 273 K, σ = 3700 kg/m3, α = 0.15,

r = 1 AU, rs = 1.25 AU, and have chosen to vary the mass of the particle. Since we have also assumed

spherical particles with bulk density σ we are there thereby varying its radius s. The results of this can be

seen in figure 7, where a interesting cut off point where the square root becomes imaginary can be seen. This

represents the point where the gravitational attraction of the comet overcomes the gas ejection pressure force

and thus the particle never escapes, this has to be accounted for in numerical implementation of a ejection

simulation. This model by (Ma et al., 2002) has however been criticized since it appears to contain a few

mistakes in its derivation (Ryabova, 2013).

As we can see there is lot more to model then just fixing the ejection speeds of the dust grains. To perform

a realistic simulation of the formation of a meteoroid stream one must also consider many other questions.

Such as the fraction of the comet that is active, i.e. has areas of ices susceptible to Solar radiation absorption

facing the Sun. What the bulk density of the dust grains and the comet is. At which point the ices sublimes

at such a rate that gas outflow dominates over gravitation exerted by the nucleus. The electromagnetic close

contact bonds, or stickiness, of the particles. What the angular momentum of the comet around its own axis

is. The Solar cycle variations, the chemical composition of the comet, the amount of jets due to low possible

ejection areas but high absorption rates. Anomalous nuclei geometry, and so on.

At some point one has to make simplifications, fortunately the nature of Monte-Carlo simulations allow for

many of the simplification errors to be minimized since we can use averages and distributions instead of

specific assumptions. Also many of the singular effects are totally erased in the grand statistical scheme,

such as the ejection directions due to cometary nucleus angular momentum.

We shall lastly cover the modification of Whipple’s model by Hughes (Hughes, 2000) resulting in a im-

proved model. In Whipple the entire surface of the comet is active and undergoing sublimation but due

to observed jets and non active areas we know that this is not the case. In other words the term 2π in

Whipples formula was corrected to Ψπ, according to Hughes the typical value for Ψ is around 0.2 but can

be between 0.05 and 0.5. Since only a part of the nucleus is active and thus only a part is absorbing Solar

radiation. Thus the term πR2c is corrected to 2απR2

c . The typical value for α is assumed to be around 0.1.

In Whipple the drag coefficient was fixed to kd = 26/9, this is substituted by another parameter ζ. This

parameter has the value 2 if gas absorbed by the ejected particle is re emitted with the same velocity within

a short timespan. If the dust particle however has a higher temperature then that of the ice sublimation

then ζ > 2. If the particle absorbs gas molecules for a long timespan ζ < 2. Whipple assumed the radial

component out of the comet to be constant but many have pointed out that the gases should be accelerated

along the adiabatic pressure gradient as they expand into vacuum. Thus a coefficient θ ≈ 2 is introduced.

Lastly Hughes disputes Whipples assumption that the velocity of the sublimated gas changes as a function

of heliocentric distance and sets the exponent coefficient τ = 0.

This results in a formula where the ejection speed is given by

Page 29


V 2∞ =

2Mc

ΦπRc

(θvsrτ

)ζA

m− 2GMc

Rc, (10.9)

where the mass loss is given as

Mc = 2gπR2cS0

(r0r

)2 1

nH. (10.10)

For a good overview and more detailed description see (Ryabova, 2013) and references therein. Taking the

most common models, we can find that the ejection velocities deduced are within the range 10 to 103 m/s,

depending on the particle size, density, comet activity, perihelion passage, and so on. Thus, when examining

a broad range of mass and comet parameters using a model for the ejection can be vital (Vaubaillon et al.,

2005).

10.2.2 Critical sublimation radius

If the entire cometary nucleus is in thermal equilibrium with the Solar radiation, the critical radius at which

sublimation starts is located around 1.25 AU. However this strongly disagrees with observations, one reason

for this is that if only a fraction of the surface is active, i.e. parts of the comet is not radiating, sublimation

can start at much larger distances. Also ices such as CO2 and CO will sublime a lot earlier. Most comets

are observed to become active at around 3 AU (Delsemme, 1982) (Spinrad, 1987) and is a standard value

we have used in our general comet simulations. Sublimation can however occur at even larger distances and

when modelling specific comets, one would rather use observational values for this critical distance.

10.3 Asteroids

It has also been shown that asteroids can have significant dust production, for example 3200 Phaethon is

today considered the most likely candidate as the parent body for the Geminids meteor shower (Whipple,

1983). Dust production of near-Earth asteroids can be handled the same as cometary dust and as we have

not yet performed simulations on asteroids their more in depth theory will not be covered here.

10.4 Main belt

The main asteroid belt is considered a source of interplanetary dust as well due to its relatively high internal

collision rate between asteroids. These collisions break apart the colliding objects and the clouds of dust

scatter, being subject to perturbation they can eventally escape the main belt and become part of the

interplanetary dust complex. This may be a dominant source of meteoroids near Earth (Dermott et al.,

2002). This contribution will be considered in future simulations of the sporadic complex, but is irrelevant

in simulations of ejections from comets.

10.5 Interstellar dust

Interstellar dust has also been observed in the heliosphere (Grun et al., 1994). This population however is

too small compared to other sources to be taken into account in any kind of simulations. It would also be

very difficult to model sources for this population as the cosmic year, or the time it takes the solar system

to revolve around the galaxy center, is around 200 million Earth years. As such we cannot perform repeated

detection of the same galactic region but instead constantly pass trough new areas. However, in the future

such a model can very well be tested by the software developed here.

Page 30


10.6 Nanodust

Solid particles that are exposed to the Solar wind and the Solar UV radiation carry an electric surface charge.

They are therefore influenced by electromagnetic forces. Mann et al. (2007) showed that for dust particles

with sizes of several nanometres, the surface-charge-to-mass ratio is large enough for their dynamics in the

Solar wind to become similar to that of heavy ions. Such nanodust can reach a speed of the order of Solar wind

speed and move on orbits away from the Sun. An intermittent flux of nanodust near 1 AU, Earth orbit, was

indeed measured with the plasma wave instrument onboard the STEREO spacecraft (Meyer-Vernet et al.,

2009). While there are still many open questions concerning the detection of the nanodust impacts with

field measurements (Zaslavsky et al., 2012), with a similar type of field measurement dust impacts were also

measured onboard Cassini between 1 and 5 AU (Schippers et al., 2015). A possible evidence for nanodust has

also been found by Carpenter et al. (2007) through impact features on foil experiments at the International

Space Station (ISS).

The measurements with Stereo, Cassini and at ISS are in general agreement with trajectory calculations and

with the suggestion that nanodust forms in dust-dust collisions near the Sun where dust fluxes and collision

rates are highest (Mann et al., 2014). Czechowski and Mann (2010) studied the dynamics of nanodust in

the region inward from 1 AU through numerical simulations of dust trajectories to estimate the flux rates.

They found, in addition to those accelerated by Solar wind, a population of trapped grains within about

0.2 AU from the Sun. The presence of fast-flowing Solar wind, the magnetic field sector structure and the

heliospheric current sheet contribute to shaping the trajectories. Both the trapping zone and the acceleration

outward depend on parameters like dust surface charge, Solar wind speed and magnetic field at the location

of the particle, all of which can induce time-variability. If nanodust is generated by mutual collisions of

larger particles, also the stochastic nature of the sources near the Sun can generate a flux variation. Juhasz

and Horanyi (2013) have shown that even if the production is constant in time, the nanodust flux near the

ecliptic plane can be time-variable as a result of the interaction with the interplanetary magnetic field.

11 Databases

There are several sources of information available for the scientific community today. We will here cover the

databases used in our simulations.

11.1 Pan-STARRS Synthetic Solar System Model

Table 1: Data included in the Pan-STARRS Moving Object Processing System Synthetic Solar System Model.

File name Abbreviation Description Sample size

S0 NEOs Near-Earth Objects ∼270k

S1.01-10 MBOs Main Belt Objects ∼100k

S2 MBOs to NEOs - ∼50k

S.hilda - Hildas ∼1.8k

Sc SPCs Short-Period-Comets ∼28k

SH - Hyperbolic objects ∼8.3k

SL LPCs Long-period comets ∼9.4k

SR - Centaures ∼60k

SS - Scattered ∼11k

ST TNOs Trans-Neptunian Objects ∼48k

St4,5,6,7,8 - Trojan of Mars, Jupiter, Saturn, Uranus, Neptune -

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In this work we will frequently mention the Pan-STARRS Moving Object Processing System (MOPS) Syn-

thetic Solar System Model (S3M). This is a attempt at compiling a comprehensive flux-limited model of

the majority of the small-body populations in the Solar system. This database was first presented in (Grav

et al., 2011) and will be used in, for example, the Pan-STARRS 1 all-sky survey. The model consists of

synthetic populations of the objects listed in table 1. The data is compiled in ASCII files with spaces as

column separators and each row corresponds to a synthetic object, the current object parameters are in

order:

1. MOPS id

2. Semi-major axis (AU)

3. Eeccentricity

4. Inclination (degrees)

5. Longitude of ascending node (degrees)

6. Longitude of perihelion (degrees)

7. Mean anomaly at epoch (degrees)

8. Absolute magnitude

9. MOID

10. Opik collision probability with Earth

11. Epoch of orbits (yyyymmdd)

12. Expected year of discovery after start (years)

We have used this database as a sample for the initial distribution of orbital elements for SPCs to create

example simulations and shall in the future conduct several more simulation using this database.

11.2 JPL NAIF database

Table 2: JPL’s measurement database of Solar system major body gravitational parameters

Body AU3/days2 GMsun/GMbody km3/s2

MERCURY 0.491248045036476000D-10 6023682.155592 22031.780000

VENUS 0.724345233264412000D-09 408523.718658 324858.592000

EARTH 0.888769244512563400D-09 332946.048834 398600.435436

MARS 0.954954869555077000D-10 3098703.590291 42828.375214

JUPITER 0.282534584083387000D-06 1047.348625 126712764.800000

SATURN 0.845970607324503000D-07 3497.901768 37940585.200000

URANUS 0.129202482578296000D-07 22902.981613 5794548.600000

NEPTUNE 0.152435734788511000D-07 19412.259776 6836527.100580

PLUTO 0.217844105197418000D-11 135836683.768617 977.000000

SUN 0.295912208285591100D-03 1.000000 132712440041.939400

MOON 0.109318945074237400D-10 27068703.241203 4902.800066

The entire JPL database used can be found at http://naif.jpl.nasa.gov/pub/naif/generic_

kernels/ through the NAIF JPL NASA joint public file server. The use of these files, called Kernels,

Page 32

http://naif.jpl.nasa.gov/pub/naif/generic_kernels/

http://naif.jpl.nasa.gov/pub/naif/generic_kernels/


will be covered in section 28.3 describing the SPICE software. Available in this database is also the mass

parameter (GM) for the Sun, Moon, and Planets for three different unit systems that we have used in

simulations, listed in table 2.

Page 33


Part III

Meteors

12 Introduction to meteors

This part is dedicated to explaining the meteor phenomena and its different dynamics, to provide a under-

standing to those new to the area of meteor science about the measurement object and the various techniques

involved. The part requires no prior knowledge and is very general and broad, not diving into specific models

more than needed.

13 Observations and mesurnemnts

This section is dedicated to cover the the actual observation and measurements of the meteor phenomena

and consequently the meteoroid itself.

13.1 Coordinate systems

The equatorial coordinate system: is a celestial coordinate system that can be implemented in spherical

or rectangular coordinates, both have a geocentric origin and a fundamental plane consisting of the

projection of the Earth’s equator onto the celestial sphere. The primary direction is oriented towards

the vernal equinox, therefore the coordinate system does not rotate with the Earth. A right-handed

convention is followed. In most literature the variables given are the geocentric right ascension αG,

declination δG, and geocentric velocity vg. Illustration can be found in figure 8

The ecliptic coordinate system: is a celestial coordinate system that can be implemented in spherical

or rectangular coordinates. Both can have either geocentric or heliocentric origin and the fundamental

plane consists of the ecliptic, or the apparent path of the Sun on the celestial sphere. The primary

direction is oriented towards the vernal equinox, and a right-handed convention is followed. The

common variables in a geocentric system are longitude λ, latitude β and distance ∆, or in a heliocentric

system with longitude l, latitude b and distance r. Illustration can be found in figure 8.

13.2 Invariable plan and the ecliptic

The invariable plane in a planetary system is the plane normal to the total angular momentum vector of that

system and passing through the barycentre. This plane is sometimes refereed to as the Laplace’s invariable

plane and can be calculated given any set of state vectors of position and momentum (q,p) and body masses

m. Starting with calculating the center of mass as

X =

∑Ni=1mixi∑Ni=1mi

, (13.1)

and the total momentum as

P =

N∑i=1

pi, (13.2)

Page 34


Figure 8: Diagram of ecliptic coordinates, depicting the ecliptic, equator, ecliptic poles, longitude and latitude,

and the vernal equinox direction. (Tfr000 / Licensed under the Creative Commons Attribution-Share Alike

3.0 Unported license. http://creativecommons.org/licenses/by-sa/3.0/)

Page 35

http://creativecommons.org/licenses/by-sa/3.0/


the invariable plane is defined by the total angular momentum

H =

N∑i=1

(xi −X)× (pi −P). (13.3)

It would seem as this invariable plane, a fundamental quantity in physics, would be perfect as a reference

plane. This is however not so in most cases. Due to tradition we use the ecliptic, the orbital plane of the

Earth, system oriented at the J2000.0 equinox as a reference (Beutler, 2005). Here J2000.0 refers to the state

of the ecliptic at the fractional year 2000.0 in Julian calendar time. Since the total angular momentum of a

system is invariant, hence the name invariable plane, to transform from a ecliptic system to a invariable one

can be easily done with knowledge of the state vectors as we just need to find the Earth angular momentum

relative the total angular momentum.

If we set i to the angle between the ecliptic pole J2000.0 and the pole of the invariable plane. And set Ω

as the ecliptic longitude of the intersection of the invariable plane with the ecliptic. Then, with calculations

done on the DE405 JPL emphermis, which we will cover in detail later, we find these angles as i = 1.5787

and Ω = 107.5822 (Souami and Souchay, 2011).

This will however not yield the correct results if we do not also rotate the coordinate system, in the ecliptic

plane, so that the x axis is aligned towards the vernal equinox.

13.3 Meteors

Dust present in the Solar system may only be a small component of its total mass but this does not

reflect its importance to many areas of research. For example, dust dynamics is crucial in cosmological

models, in Solar system and planetary formation, modelling of accretion discs, in cometary and asteroidal

models and in knowing our own local space environment for the benefit of future space missions. Dust also

plays a role in the present Earth dynamics, for example when this dust encounters the Earth and interacts

with our atmosphere, producing what is called a meteor, a violent event of ablation takes place causing a

series of unusual phenomena to occur due to this extraterrestrial input and dispersion of material into our

atmosphere. It is therefore interesting to investigate how dust behaves dynamically in the Solar system due

to gravitational and electromagnetic perturbation over long time scales since this will be integral in modelling

our Solar system. It is through models that we can for example explain how our meteoroid complex today

is resupplied with material from comets and asteroids, and thereby find information on the small body

population of the Solar system. However due to the limitations on detecting dust in space due to its sparse

nature and small size one instead turns to Earth based observations. As such, to investigate the meteor flux,

the meteoroid orbits and their interactions with the upper atmosphere is important but at the same time

the method of investigation is limited, especially for precise measurements. High power large aperture radar

observation is a recent technique to provide useful information on these matters.

13.3.1 Visual

The first type of meteor observations were visual observation of the plasma trail generated when the mete-

oroid interacts with the atmosphere. By examining the trail one can quite accurately calculate a orbit by

compensating for Earth’s gravitational interaction and then converting the heading and speed of the object

into Keplerian elements. This however needs multiple simultaneous observations to be done.

The actual nature of trail and the head of the meteor is however much more complex than a simple ball

of plasma. For example, the meteor is not only dependant on the meteoroid but also the density of the

atmosphere, the translation speed due to air flow at the point of interaction, and the composition and

density of the meteoroid. All these factors and more affect the meteor and therefore the detection. From

Page 36


Figure 9: Concept image of a meteoroid head echo

visual data a spectral analysis can be performed to extract information on the chemical properties of the

meteoroid. This in combination with radar observation can provide additional information on the event.

13.3.2 Head echo

A meteor head echo is caused by radio waves scattered from the dense region of plasma surrounding and

co-moving with a meteoroid during atmospheric flight. The received signals Doppler shift and targets range

rate can therefore be used to accurately determine meteoroid velocity. This, as with visual observations, can

be calculated into Keplarian elements to reconstruct a orbit. A illustration of the detection of a meteor can

be seen in figure 9.

13.4 Ablation

Meteoroid ablation is the mass loss of meteoroids due to vaporization, fusion of molten material and frag-

mentation (Rogers et al., 2005). As previously explained, the meteor phenomena occur when dust encounters

the Earth and interacts with our atmosphere (Ceplecha et al., 1998). A meteor event detected by radar can

be split in two different parts; the head echo, which is a signal echo originating from the ball of plasma

co-moving with the meteoroid, and the trail echo, which is the echo from the trail of plasma left in the atmo-

sphere drifting with the local wind speed (Kero, 2008). Radar systems are usually optimized to detect either

head or trail echoes, but both kinds can be detected also with one and the same radar. If one configures the

radar for trail detection the amount of head echo’s detected will be lowered and vice versa. The trail echo is

traditionally split into two categories, the underdense and overdense trail echos. In an underdense echo each

free electron scatters the incident wave independently and the signal becomes the sum of the contributions

from all individual electrons. In a overdense echo the electron density is so high that the radio waves scatter

as from a cylindrical, metallic surface. Apart from radar observations of ablation processes, one can also

perform camera observation, most commonly visual observation, by recording the light emitted during the

ablation process and in the ’afterglow’ of the plasma trail. As mentioned, this can yield spectral information

Page 37


Figure 10: Contour plot of the logarithm of the total number of events per cell of the Hammer-Aitoff pro-

jection, entire MURMHED. The labels are the six sporadic sources: (SA) South apex,(NA) North apex,(ST)

South toroidal,(NT) North toroidal,(H) Helion and (AH) Anti-Helion.

of the meteoroid useful in determining composition, size and density. However the meteoroids need to be of

much larger size then the lower limit for radar detections since the ablation process have to produce sufficient

light to be recorded from ground observations, rather than just scatter the radio wave.

14 Meteor and Meteoroid complex

The meteoroid complex is a collection term for small bodies in the Solar System. Traditionally, the meteoroid

complex is split into two parts; stream meteoroids and the sporadic meteoroids. There is also the term meteor

complex, this is the collection term for all meteor events at the Earth. Thus, the meteor complex is a subset

of the meteoroid complex, of all meteoroids that encounter the Earth and produces meteors. Since detection

of meteoroids in space is difficult, one usually detects the meteor complex instead and use this to infer

properties about the meteoroid complex.

14.1 Stream meteoroids

Shower meteors or stream meteoroids are a concentration of radiants with characteristics similar enough

to proclaim that they originate from the same source. Practically all meteor showers are produced by the

meteoroid stream generated from a parent body like an asteroid or a comet. In the case of a comet, thermal

radiation from the Sun heats the surface, sublimating ice inducing jets of gas and dust being ejected into

space. A comet closing in on its perihelion passage will form two tails, one gas tail directed parallel with the

radiation pressure and one dust trail following in the wake of the comets orbit. We shall go further into this

process later. The particles in this trail will have seemingly random velocities away from the parent body

and their relative velocity to the parent body is small in comparison with their orbital speed thus making

Page 38


Figure 11: Hammer-Aitoff projection of all MURMHED meteor radiants with color encoded geocentric ve-

locities with showers Orionids and Geminids marked.

it possible to correlate them to the generating object. A dust trail from a comet or a asteroid produces a

meteor shower when the Earth orbit and a filament of dust crosses paths in space and the particles become a

part of our meteor complex. When considering this scenario we realize that, just like the exhaust fumes from

a jet plane, the meteoroid streams will dissipate over time but in the case of meteoroid streams the time for

dispersion ranges from hundreds to several thousand years if not more. More information about statistical

modelling of meteoroid streams by chi-squared density functions can be found in (Drummond, 2000). Also,

a more in depth description on the properties of comets and asteroids can be found in (Jenniskens, 2006).

In figure 11 a Hammer-Aitoff projection of the MURMHED database is shown with apex located at 270

and the Sun located at 0, this database will be covered in the coming section.

14.2 Sporadic meteoroids

The sporadic meteoroids is the most abundant source, yet it has for a long time been the least under-

stood part of the Earth’s meteoroid complex. A Hammer-Aitoff projection of the meteoroid radiants from

MURMHED is displayed in figure 10. Figure 10 is a Hammer-Aitoff projection logarithmic radiant density

plot which has been zenith distance corrected to represent the correct meteor flux according to (Zvolankova,

1983). As can be seen the sporadic complex seems to be split into several different sources. The sporadic

background must be understood for one to separate the stream meteoroids from the sporadic ones. The

origin of the sporadic background is quite simply dominated by stream meteoroids that has dissipated and

can no longer be traced back to a parent body. It seems today that our sporadic complex is in a fairly steady

state, meaning that by calculating the total influx of mass into the atmosphere due to meteoroids one can

also calculate the needed production of new particles to maintain the sporadic complex. A good summary

of the events is given in (Wiegert et al., 2009), to quote:

”Eventually, perturbations accumulate and disperse the meteoroid stream, the original close orbital relation-

ship between individual meteoroids becoming difficult to determine. At this point, the particles have become

Page 39


part of the second component of the meteoroid population, the ’sporadic meteoroids’, which form a more

diffuse but far from isotropic background flux of particles.”

15 Databases

15.1 MURMHED

The primary meteor database used in our work is from the Shigaraki Middle and Upper atmospheric Radar

Meteor Head Echo Database (MURMHED) (Kastinen et al., 2014). High power large aperture (HPLA)

radar observation is a recent technique to provide useful information on meteor influx and orbits, as well as

interactions with the atmosphere (Nakamura et al., 2014). Since 2009 orbital data of about 120,000 meteors

(Kero et al., 2012a) have been collected using a novel head echo analysis algorithm for the lower VHF band

(Kero et al., 2012b). The data was collected using the middle and upper atmosphere radar (MU radar)

of Kyoto University at Shigaraki (34.9N, 136.1S). The MU radar is a large atmospheric VHF radar with

46.5 MHz frequency, 1 MW output transmission power and 8330 m2 aperture array antenna. The database

currently holds 53 different parameters for each event and a number of associated time series consisting of

range, height, radar cross section, signal to noise ratio, radial velocity and meteoroid velocity. The database

parameters are listed below.

MJD

Year[UT]

Month[UT]

Day[UT]

Hour[UT]

Minute[UT]

Second[UT]

Duration[s]

RA[deg]

Dec[deg]

Az[deg]

Ze[deg]

Az uncorr[deg]

Ze uncorr[deg]

Ze correction[deg]

Obs initial vel[km/s]

Geocentric vel[km/s]

RCS[dBsm]

SNR[dB]

Start hgt[km]

End hgt[km]

Az of start point[deg]

Ze of start point[deg]

Az of end point[deg]

Ze of end point[deg]

Semimajor axis[AU]

Eccentricity

Perihelion dist[AU]

Lon of asc node[deg]

Inclination[deg]

Arg of periapsis[deg]

Period[yr]

Heliocentric vel[km/s]

Radiant ecl lon[deg]

Radiant ecl lat[deg]

Sol lon[deg]

RA error[deg]

Dec error[deg]

Az error[deg]

Ze error[deg]

Vel error[deg]

Semimajor axis error[AU]

Eccentricity error

Perihelion dist error[AU]

Inclination error[deg]

Arg of periapsis error[deg]

Period error[yr]

Heliocentric vel error

Ecl lon error[deg]

Ecl lat error[deg]

Hammer X

Hammer Y

In figures 12, 14, 13, 15, 16 and 17 a example entry of the times series for a echo is displayed.

Page 40


Figure 12: Meteoroid velocity time series of head echo

with MJD 55008.9589319839 from MURMHED

Figure 13: Radial velocity time series of head echo


Figure 14: Height time series of of head echo with

MJD 55008.9589319839 from MURMHED

Figure 15: Range-time intensity plot of head echo


Figure 16: Radar cross section time series of head

echo with MJD 55008.9589319839 from MURMHED

Figure 17: Signal to noise ratio time series of head

echo with MJD 55008.9589319839 from MURMHED

Page 41


As any measurement instrument the MU radar has a bias. One of these biases are due to the fact that the

antennas are not all seeing around the entire Earth. As they are stationed at a longitude and a latitude certain

parts of meteor outburst event may for example fall below the horizon. Not only this however, one cannot

assume that all is seen in the view limited by occupying a fixed point on the Earth as the antennas have a

gain pattern surrounding its azimuth. This gain pattern, both theoretical and experimental can be read more

about in (Kero et al., 2011). This can play an important roll in a statistical analysis. Another detection

specific variable is the mass thresholds detectable by the radar. Since different masses and compositions

create different targets for the radar echo, some configurations of masses cannot be detected by the MU

radar. The meteoroid complex is dynamic in its size and composition distribution so this will also matter

when applying theoretical models to a set of recorded data.

15.2 IAU meteor shower database

The IAU Meteor Data Center (MDC) is situated at the Astronomical Institute of the Slovak Academy of

Sciences and under the auspices of International Astronomical Union (IAU). The MDC is the international

organization that handles the designation of meteor showers and the collection of trajectory observations of

meteoroid orbits. They host a list of 697 showers, where 112 are well established and 37 are pro tempore (or

”for the time being”), the rest is not yet fully confirmed to be actual meteor showers. Since our software

intends to simulate meteor showers on the Earth by encounter with cometary dust trails such a database

will be very useful. The entire database can be found at http://www.astro.amu.edu.pl/˜jopek/

MDC2007/index.php.

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http://www.astro.amu.edu.pl/~jopek/MDC2007/index.php

http://www.astro.amu.edu.pl/~jopek/MDC2007/index.php


Part IV

Associating and classifying meteoroids

16 Introduction to associating and classifying meteoroids

This part is intended towards those interested in classification applications and the mathematical foundation

upon which it is formulated. Some of the result analysis relies on understanding of the D-criterions and

metrics given here. We also cover some historical progressions of the developed functions and methods.

At first glance the task of finding meteor showers seem trivial, but as ones dives into the subject, more

and more difficulties emerge from different areas of physics and mathematics. In this part we shall attempt

to create a review of the current area of research concerning association of meteoroids. We shall also later

contribute to it by developing new methods and ideas for addressing the classification meteoroids.

A lot of problems in astrophysics and celestial mechanics can be traced back to the comparison of ob-

jects, or collections of objects, and the resulting associations that are made. Often, this process involves

creating some sort of similarity or dissimilarity measure, like the commonly known D-criterion. Sometimes,

this measure can be an representation of distance, i.e. a metric. More often, however, it represents a subjec-

tive view of similarity. Using such a measure, a set of objects can be examined for patterns. Most common

types of association analysis for meteor measurements that we are aware of are the density function analysis,

the wavelet transform, and cluster analysis. Another common method is to manually look at the data. The

density function method will bin the space looking for regions of bins with high frequencies of meteors.

However, to avoid manual selection, it is often needed to use a critical threshold where one can say that

the density is large enough to imply the presence of a pattern. Sometimes this threshold can be derived by

long term continuous observations, while this may not always be the case. The same principle applies for

the wavelet analysis, with the difference that the specific pattern of the mother wavelet is searched for in

addition to an increase in meteor count. Lastly, the cluster analysis approach involves choosing a critical

distance, or threshold, at which a connection is made. If two objects are within this threshold they are

connected to each other. After a connection has been made, there are several ways to merge the clusters.

However, whatever method is used the clusters that are formed depend on the size of the critical threshold.

Thus, all these methods rely on knowledge of either the background, i.e. the sporadic meteor complex and

measurement uncertainties, or of the critical threshold needed for the shower to be selected.

In a straightforward and integrable system one can use models to exactly propose what restrictions to

apply to the data in order to associate elements. If, however, the system is either chaotic or the amount of

variables is so large that it appears impossible to predict a definite evolution of the system, statistics and

approximations have to be applied. When considering meteoroids or meteors, the reliability of an association

heavily depends on applied assumptions of, e.g. the correct value for the critical threshold (Kastinen et al.,

2014). This is one of the main reasons we have chosen to apply a statistical approach to the problem, in

order to start investigating the general statistics related to meteoroid streams.

17 Similarity functions

The concept of similarity is a highly subjective concept and similarities are viewed very differently depending

on context and observer. Similarity functions are used in a multitude of areas such as multimedia retrieval,

biometry, pattern recognition, manufacturing industry, and scientific research. This is the reason why it is

hard to corner a area of mathematics which can hold all similarity measures. With a similarity measure

Page 43


we are referring to a function mapping the Cartesian product of a set M with itself to the real numbers,

M ×M 7→ R. A similarity functions purpose is turning our concept of similarity between members of a

set into a quantitative measure, often represented by the real number line. This concept closely resembles

a area of mathematics called metric spaces, and often similarity functions can be proved to be metrics, but

this is also not always the case. A metric may not correctly model the intuitive notion of similarity, for

example single objects can be viewed as self-dissimilar while two distinct object can be viewed as identical.

The direction of comparison can affect the comparison, for example comparing a orange to a apple may not

be the same as comparing a apple to a orange. A similarity function does not even need to be transitive.

There are however also advantages to restricting ones functions to metrics as they are a very well studied in

mathematics with postulates that support common assumptions on similarity. From a more numerical point

of view metric indexing also allows efficient indexing and searching (Zezula et al., 2006).

17.1 Metric spaces

There are many areas within mathematics which use the concept of a metric. For example Topology is

the study of shapes and continuous deformations. This area focuses on the properties of a space that are

preserved under continuous deformations, like stretching and bending, but transformations, like cutting and

pasting or piercing, are not allowed. This field developed from examining geometry and set theory, as such

it is closely linked to the metric as this is a geometric measure of a distance.

When talking about metrics one should also mention the concept of a measure. A measure is a generalization

of the concepts of length, area, volume, and so on. In this thesis we will not go further into the mathematics

of these areas but only concern ourself with their application, for a more in depth explanation of measure

theory see (Bogachev, 2007).

A metric is a function that defines a distance between elements of a set, this function must follow cer-

tain rules. Denoting the set as E, x, y ∈ E, and define a function d : E × E 7→ R, then the pair (E, d) is

called a metric space iff

d(x, y) ≥ 0, the separation axiom (17.1)

d(x, y) = 0⇒ x = y, the coincidence axiom (17.2)

d(x, y) = d(y, x), the symmetric relation (17.3)

d(x, z) ≤ d(x, y) + d(y, z), the triangle inequality. (17.4)

The concept of a similarity does not need to follow these rules however, one can easily create a artificial

metric that is useful for ad hoc tasks, but at this point one should take care. If just creating a function

that maps our sense of similarity well in the cases that we thought of, this is fine. However, if we then

extend its use to cases that we didnt specifically test, consider, or develop it for its behaviour may be totally

unpredictable. A function that does not fulfil these metric requirements does not have all mathematical

properties that a metric should have and therefore may not behave as we expect it too.

As with all physics, one ends up in a pros versus cons situation. Should one use a metric as the base

of similarity of should one just model the similarity based on our own intuition? Are the more fundamental

properties of measuring distance between points superior to our own intuition? To answer these questions

one cannot do much else then experiment for the specific situation. It is very important to be aware of the

factors that can affect your system, be it physical or mathematical. Often too much thought is put into

forcing physical data into some kind of model and the mathematical description of ones conception may

suffer for it.

As a last note on metric spaces, it can be mentioned that from a single metric, a entire family of metrics can be

Page 44


found by applying metric preserving functions to the original. Some of these function are covered in (Dobos,

1998). One can also create products of metric spaces (Mi, di) so that M =∏ni∈1Mi (M,N(d1, . . . , dn)) is

a metric space where N(d1, . . . , dn) =√∑n

i=1 d2i and N is the euclidean norm defining N : M ×M 7→ R+

since di : Mi ×Mi 7→ R+. We shall not go deeper into these concepts since we have not yet implemented

numerical version of these concepts in the software. This is however planned for future work, to examine

different ways of applying metrics in this research field.

18 Models of meteoroid complexes

To be able to device a good similarity function, or validate a existing one, one needs not only observations

but also models of the objects to be examined. This is so that tests can be performed and compared between

simulations and observation. Also through synthetic data, calibrations of methods should be performed to

eliminate arbitrary ad hoc choices. For example the cluster analysis, that uses a critical threshold to associate

points, need a model to calibrate the critical threshold. If one does not use a model to justify this distance,

the findings are as good as eye-balling the data.

The best complete model we have found on the sporadic complex was developed and presented in (Wiegert

et al., 2009). Since then several articles suggesting improvements and different approaches has appeared,

like the recent (Nesvorny, 2013) and (Pokorny et al., 2014) which certainly could improve a model of the

sporadic background.

All these models have one thing in common; they generate a sporadic background by the simulation of

parent bodies producing dust to be exposed to long term perturbations before colliding with the Earth gen-

erating a subset of the meteor complex. Due to the nature of perturbation dynamics only certain types of

parent bodies, or dust producers, can contribute to certain parts of the sporadic complex. The perturbation

effects include those of the Sun, the planets, radiation forces and collisions. For example according to simu-

lations in (Wiegert et al., 2009) only minor contributions to the helion and anti-helion sporadic sources (see

figure 10) are made by Halley-family comets and the main contributor was determined to be the Jupiter-

family comets. There can however be high uncertainty in this kind of simulation if one does not consider

the affects of chaos and ensure that a very small change of the initial condition does not drastically change

the results. If the simulation is stable, it is perfect for examining sporadic statistics and therefore also for

finding unknown meteor showers. Alas, this type of simulation is difficult to setup as it contains a lot of free

parameters, such as dust production rates, that must be calibrated correctly.

19 D-criterion

Most of the previous attempts at creating a distance function for trajectories in celestial mechanics involve

creating a function for the many different phase spaces or configuration spaces of Keplerian orbits. One is

the so called D-criterion originally developed and presented in (Southworth and Hawkins, 1963) and then

extended on by (Porubcan, 1977), (Drummond, 1980), (Jopek, 1993), (Jopek and Froeschle, 1997), (Valsecchi

et al., 1999), and (Jopek et al., 2008). As this is a quite long list of evolution we shall only cover some of the

essentials. The D criterion is based on a sum of weighted differences between the orbits dependent variables,

we could however not find any literature that indicated that the D-criterion or its modifications satisfies

the metric space axioms, with one exception. We have also observed that some of them does not fulfil the

triangle inequality and as such the D-criterions should not be regarded as metrics.

Page 45


19.1 Southworth and Hawkins

The Southworth and Hawkins D-criterion, DSH (Southworth and Hawkins, 1963), was the first D-criterion

developed in 1963 and can be written in familiar elements as

D2SH = (eb − ea)2 + (qb − qa)2 + (2 sin

Iab2

)2 +

(ea + eb

22 sin

Πab

2

)2

, (19.1)

where

(2 sinIab2

)2 = (2 sinib − ia

2)2 + sin ia sin ib(2 sin

Ωb − Ωa2

)2 (19.2)

is the angle between the orbital planes and

Πab = ω2 − ω1 + 2 arcsin

(cos

ib − ia2

sinΩb − Ωa

2sec

Iab2

)(19.3)

is the difference between the longitudes of perihelion measured from the intersection of the orbits.

This function performs remarkably well for being the first of its kind. It is however desirable to gather as

much material for comparison and as it is still well used, it will be good to examine it further.

19.2 Drummond

Later in (Drummond, 1980) modified the function to create the DD criterion which was a more correctly

normalized version of DSH and can be written in familiar elements as

D2D =

(eb − eaeb + ea

)2

+

(qb − qaqb + qa

)2

+

(I

180

)2

+

(ea + eb

2

Θ

180

)2

, (19.4)

where

I = arccos (cos ia cos ib + sin ia sin ib cos (Ωb − Ωa)), (19.5)

and

Θ = arccos (sinβa sinβb + cosβa cosβb cos (λb − λa)). (19.6)

The ecliptic latitudes of the perihelion points can be expressed as

βa = arcsin (sin ia sinωa),

βb = arcsin (sin ib sinωb),

(19.7a)

(19.7b)

and the ecliptic longitudes of the perihelion points as

λa = Ωa + arctan (cos ia tanωa),

λb = Ωb + arctan (cos ib tanωb).

(19.8a)

(19.8b)

Page 46


The λ parameter has the additional condition of adding 180 if cosω < 0. This function is included in the

analysis due to its similarity to DSH since a similar statistical result will indicate that simple modifications

of distance functions are not desirable to obtain new statistical properties. More on the analysis on this

function can be read in (Drummond, 1981).

19.3 Valsecchi, Jopek, and Froeschle

In (Valsecchi et al., 1999) introduced yet another D-criterion, DN , but with another approach to the problem.

Instead of focusing on Keplerian elements they based their function on geocentric quantities and used the

geometric setup of Opik’s theory of close encounters:

D2N = (Ub − Ua)2 + w1(cos θb − cos θa)2 + (∆ξ)2, (19.9)

where |U| = U is unperturbed geocentric velocity derived through the Tisserad parameter which can be

used if one sets the mass of the Sun and the gravitational constant to unity and assume that the heliocentric

velocity of the Earth is also unity. The Tisserad is a reasonable variable to use for a orbital measure as it is

a constant of motion in the restricted 3 body problem. θ and φ are the two angles that give its geocentric

anti radiant and w1 is a suitable weight factor. The last component is somewhat more complicated and it

can be expressed as the minimum of two elements:

(∆ξ)2 = minw2(∆φI)2 + w3(∆λI)

2, w2(∆φII)2 + w3(∆λII)

2, (19.10)

where w2 and w3 weight the differences between the geocentric anti radiant angle, φ, and the Solar longitude,

λ,

∆φI = 2 sinφb − φa

2,

∆φII = 2 sin180 + φb − φa

2,

∆λI = 2 sinλb − λa

2,

∆λII = 2 sin180 + λb − λa

2.

(19.11a)

(19.11b)

(19.11c)

(19.11d)

The angles of U can be determined by:

θ = arccosUyU, (19.12)

φ = arctanUxU. (19.13)

Since U is based upon geocentric quantities it can easily be calculated from the declination, δG, and right

ascension, αG, and the geocentric velocity VG of the observed head echo:

UxUyUz

= Rz(λ)Ry(ε)VG29.7

− cos δG cosαG− cos δG sinαG

sin δG

, (19.14)

Page 47


where λ is the Solar longitude at the observation time and ε is the inclination of the ecliptic plane to the

celestial equator. The vector is described in a heliocentric ecliptic coordinate system making the transfor-

mations

Rz(λ) =

cosλ − sinλ 0

sinλ cosλ 0

0 0 1

, (19.15)

Ry(ε) =

cos ε 0 − sin ε

0 1 0

sin ε 0 cos ε

. (19.16)

This criterion is novel in its approach and it shows good qualities and will in the future be implemented

numerically, more about this criterion can be read about in (Valsecchi et al., 1999),(Jopek et al., 1999) and

(Jopek et al., 2003).

19.4 Jopek, Rudawska, and Bartczak

More recently in 2008 (Jopek et al., 2008) introduced the new criterion,

D2V = wc1(ci1 − cj1)2 + wc2(ci2 − cj2)2 + 1.5wc3(ci3 − cj3)2+

+ we1(ei1 − ej1)2 + we2(ei2 − ej2)2 + we3(ei3 − ej3)2 + 2wE(Ei − Ej)2, (19.17)

where wc1, wc2, wc3, we1, we2, we3 and wE are suitably chosen weight factors. In (Jopek et al., 2008) they

recommend using wck = (2σck)−2, wek = (2σek)−2, wE = (2σE)−2, k = 1, 2, 3 where σ denotes the expected

standard deviations of each variable from a stream. This distance function is defined in the domain of the

heliocentric orbital elements. Following the direction pointed out by Neslusan the full set of the vectorial

elements are utilized in the metric. A orbit is described by the triplet

Oi =

cieiEi

, (19.18)

where the Laplace-Runge-Lenz vector is

e =

e1e2e3

=r× c

µ− r

|r|, (19.19)

which magnitude is equal to the numerical eccentricity, |e| = e, and the area vector is

c =

c1c2c3

= r× r, (19.20)

and the energy is

Page 48


E =1

2r2 − µ

|r|, (19.21)

where µ is the standard gravitational constant. The elements used in the metric can be calculated from

r = (x, y, z)T and r which are the heliocentric vectors of the position and velocity of the meteoroid. More

can be read on this in (Neslusan, 2002).

This is the only D-criterion we have found that fulfils all the metric properties and has been constructed

independently by (Kholshevnikov, 2008) and can be made to coincide if the correct weights are chosen.

20 Metric of phase-spaces

Unlike the previously mentioned D-criterion that were developed for the specific task of associating orbits,

the functions in this chapter were developed to be proper metrics for the phase space of Keplerian orbits.

The question of topological and geometric structures on this space did not receive much attention until 1970

when the brilliant result of Moser (Moser, 1970) showed that the singularities in any system of elements for

elliptic orbits is due to the fact that the topology of the space of elliptic Keplerian orbits with fixed negative

energy is that of the product of 2-dimensional spheres.

In (Kholshevnikov and Vassiliev, 2004) published their work on the geometric structure of 5-dimensional

space of of elliptic Keplerian orbits were they generalized their previous developed Euclidean and Chebyshev

metrics with a Holder type metric. The spaces worked on are denoted E and E∗ and a orbit is denoted E.

Elements of E are oriented Keplerian ellipses with positive semi major axes and eccentricities, where the case

zero eccentricity produces a oriented circle without marked pericenter. Elements of E∗ are the same but the

case of a circle instead has its pericenter marked. These two functions respectively

ρ(p,E,E′) = min

(1

2π

∫sp(Q(u), Q′(u+ v))du

)1/p, (20.1)

for mapping E× E onto R+ and

ρ∗(p,E,E′) =

(1

2π

∫sp(Q(u), Q′(u))du

)1/p

, (20.2)

for mapping E∗ ×E∗ onto R+, where s(Q(u), Q′(u′)) is the R3 euclidean distance between the two points Q

and Q′ on the two ellipses having eccentric anomalies u and u′. In the case of the metric for E the distance

is heavily dependant on the lines of apsides even though for example two coplanar orbits with very low

eccentricity almost occupy the same set in R3. This is however not the case with the metric of E∗, which is

why we have chosen to only examine this function closer.

The normalized squared distance function W (u, u′), following the relation s(Q(u), Q′(u′)) =√

2aa′W (u, u′),

can be expressed as

W (u, u′) =W0 +W1 cosu+W2 sinu+W3 cosu′ +W4 sinu′+

+ 2(W5 cosu cosu′ +W6 cosu sinu′ +W7 sinu cosu′ +W8 sinu sinu′). (20.3)

Page 49


By using the eccentric and true anomaly, a point on a elliptic Keplarian orbit can be expressed as

r

a= P(cosu− e) + S sinu, (20.4)

giving the components of W (u, u′) as

4W0 = 2(α+ α′) + αe2 + α′e′2 − 4P ·P′ee′,

W1 = P ·P′e′ − αe,W2 = P′ · Se′,W3 = P ·P′e− α′e′,W4 = P · S′e,2W5 = −P ·P′,2W6 = −P · S′,2W7 = −P′ · S,2W8 = −S · S′.

(20.5a)

(20.5b)

(20.5c)

(20.5d)

(20.5e)

(20.5f)

(20.5g)

(20.5h)

(20.5i)

The P and S vectors can be expressed in familiar elements as

P =

cosω cos Ω− cos i sinω sin Ω

cosω sin Ω + cos i sinω cos Ω

sin i sinω

, (20.6)

S = ηQ = η

− sinω cos Ω− cos i cosω sin Ω

− sinω sin Ω + cos i cosω cos Ω

sin i cosω

, (20.7)

where η =√

1− e2, α = aa′ and α′ = a′

a .

To reduce computation time and to simplify the calculations the special case of p = 2 was choosen and

can, as pointed out in (Kholshevnikov and Vassiliev, 2004), be reduced to

ρ2(2, E,E′) = 2aa′(W0 −

√(W5 +W8)2 + (W6 −W7)2

). (20.8)

Later in 2008 K. V. Kholshevnikov (Kholshevnikov, 2008) further extended on this by presenting even more

metric spaces of Keplerian orbits and their subspaces. In (Kholshevnikov, 2008) the two spaces examined

were the space H(b), b ≥ 0 of curvilinear orbits and the space H of orbits, both curvilinear and rectilinear. The

space H does not however distinguish between ascending or descending orbits while the space of H(b), b ≥ 0

does not allow for retrograde orbits. This modeling will have to be done outside of the metric space function

and a post-distance calculation is trivial to perform. The distance function introduced into the space of

H(b), b ≥ 0 is

%2(ε1, ε2) =1

µ2L1|c1 − c2|2 + |e1 − e2|2, (20.9)

and the metric of H is

Page 50


%21(ε1, ε2) =1

µ2L1|c1 − c2|2 + |e1 − e2|2 +

L22

µ4(h1 − h2)2. (20.10)

where the c and e vectors are the same as in equations 19.20 and 19.19.

These are used to describe the orbit in the H(b), b ≥ 0 space, in the H space a orbit is described by these

vector together with the energy constant

h = E =r2

2− µ

|r|. (20.11)

However, since the spaces are supposed to describe Keplarian orbits the following restrictions must be met

in each space, limiting its dimensionality. In H(b), b ≥ 0

c · e = 0,

|c| > b,

(20.12a)

(20.12b)

and in H

c · e = 0,

2h|c|2 − µ2(e2 − 1) = 0.

(20.13a)

(20.13b)

Using these equations one can write the metric for both spaces in familiar terms as

%2(ε1, ε2) =1

L1(p1 + p2 − 2

√p1p2 cos ξ) + (e21 + e22 − 2e1e2 cos ζ) (20.14)

and

%21(ε1, ε2) = %2 +L22

4

(1

a1− 1

a2

)2

, (20.15)

where

cos ξ = cos i1 cos i2 + sin i1 sin i2 cos (Ω1 − Ω2), (20.16)

cos ζ =(cosω1 cosω2 + cos i1 cos i2 sinω1 sinω2) cos (Ω1 − Ω2)+

+ (cos i2 cosω1 sinω2 − cos i1 sinω1 cosω2) sin (Ω1 − Ω2)+

+ sin i1 sin i2 sinω1 sinω2. (20.17)

Although it has been mentioned in (Todorov, 2012) that the ”spaces of Keplerian curvilinear orbits, all orbits

and elliptic orbits with marked pericenter cannot carry a norm, compatible with their standard topology”,

this need not affect the usefulness of the derived metrics.

Page 51


Part V

Statistical methods

21 Introduction to statistical methods

This part is dedicated to explaining some basic concepts of statistics that are used in our work. If the reader

is unfamiliar with terms such as Monte Carlo methods, this part is highly recommended. We will also discuss

cluster analysis and its use in meteor science and how one should approach the method and the problem.

22 Overview

As our software in oriented towards being able to perform the integration of small body dynamics in a statis-

tical fashion by Monte Carlo sampling of a distribution, we must cover some basic statistical methods. Our

goal is also to build a platform on which we can easily generate synthetic data to test models and methods,

or calibrate analysis of observations. To this end we will also cover the concept of association and cluster

analysis. For example, this is used for fining head echoes originating from the same parent body. When

performing such a search the probability of false association is extremely important.

Statistics are merely functions performed on a entire set, for example the mean value, variance, and corre-

lation are all functions defined as

f : B(R) 7→ R. (22.1)

This directly makes all higher mathematical construction on the set unimportant to the characteristics of

statistics. For example, consider the travelling salesman problem. This problem asks the question: given a

set of cities, what is the shortest possible route that visits each city exactly once and returns to the origin

city? let us now assume that we are using different functions to determine the distance between the cities,

this can obviously affect what route is the most efficient. It does not however affect the fundamental set

of nodes and connections itself. So, for example, we can infer things about the map without ever choosing

a distance function by, for example, looking at the statistics of connections between nodes. Obviously this

is only useful when the amount of nodes and connections are far greater then any human can observe and

comprehend. We can also use statistics to map how these different functions affect the distances in general.

Taking the mean and variance of all distances and show how these change with the chaining functions will

give a fairly good picture of the dependence on the function. The important fact is that the statistical meth-

ods are just as valid regardless of the functions used, they are still just mapping a set of data, regardless

of how it was derived. One can however question the validity of the function from a physical point of view

thanks to statistics as well, such that it should actually represent distance travelled.

Using statistics as probability is a valid approach to many problems when one knows that a phenomena

is not subjected to large none-modelled perturbations. Then the actual distribution of event outcomes will

become the probability of that outcome, with a given level of reliability. For example, consider that we do

not know how a certain coin responds to a toss, it may not be a 50/50 % chance of heads or tails due to

irregularities in the coin or external factors. One way to make a prediction would be to start performing

tosses with said coin, after many tosses we may find that the distribution of results are instead 25/75 %. We

can then use this as our prediction model stating that ”there is a 25% chance of heads this toss”.

This kind of empirical method is not applicable if experiments cannot be performed, let us say that we only

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have a picture of the coin in question. Another approach would then be to first try to extract the physical

parameters from said picture, then create a model from classic mechanics, including air where the coin is

the be tossed, and numerically solving this model with different initial conditions. This method introduces a

multitude of challenges, firstly the model of the coin and toss must be correct, then choosing a sufficient set

of initial conditions is also needed. When all these have been fulfilled we can again perform statistics on this

generated data that is not measured but synthetic, and use this as a model of prediction. In the end this

method may yield a 28/72 % chance, which is close but not what experiments would have yielded. Often

this is due to the fact that the model of the event not exact. Most often it is nearly impossible to distinguish

such a error since the event being modelled occurs rarely, is very subtle with a low signal to noise ratio, or

has high uncertainties involved. In our example of the coin, assumptions on coin density and dimensions

can be wrong, wind speed in the room where the toss is performed is highly unpredictable and may affect

outcome, and so on.

What this above example illustrates directly translates to our work. We have measurements of parent

bodies and from this we try to derive a distribution as a result of repeated simulations. From the resulting

synthetic data we then want to draw conclusions about outcome. And these conclusion need not always be

predictions but also aspects of the outcome that we cannot observe, essentially generating a missing piece

of the puzzle ourself. As one of our case studies covered in section 42, where we find the mass distribu-

tion transfer between a comet and the Earth, something that cannot be measured accurately by current

instruments.

23 Monte Carlo

The Monte Carlo method in statistical analysis and mathematics as a whole is more oriented towards

experimental mathematicians. This term may be somewhat unfamiliar to most but a clarification of this

may be useful for putting this work into the correct context. The core difference between a theoretical

mathematician and a experimental mathematician is that the theoretical deduces conclusion from postulates

while the experimental one infers conclusions from observations and calculations. The Monte Carlo method

is a perfect example of this difference.

Monte Carlo methods are now an essential part of statistical analysis in contemporary science (Robert and

Casella, 2013), and thus only a brief review is given here to better explain the methodology of the software.

One of the features of Monte Carlo methods is that they rely on sampling and on concepts of probability.

To illustrate this let us consider the calculation of π using uniform distributions. This may seem very

roundabout, but it is a good example. Consider a two dimensional uniform distribution

U(x, y) =1

A∀ x ∈ [−r, r], y ∈ [−r, r], (23.1)

where the A = (2r)2 = 4r2 is the area within which the distribution is located. A circle centred in the origin

and with radius r will have area Ac = πr2. The ratio of these two areas are

AcA

=π

4. (23.2)

Since the probability distribution is uniform, we also know that this ratio is equal to the ratio between the

probabilities of a random point ending up inside the circle and a random point ending up in the square, or

in other words

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Figure 18: Example illustrating the concept of a Monte Carlo simulation calculating the value of π simply

from the use of uniform distributions.

Pcircle

Psquare=

∫ r0

∫ 2π

0U(x(ρ, θ), y(ρ, θ))ρdρdθ∫ r−r∫ r−r U(x, y)dxdy

=

∫ r0

∫ 2π

01Aρdρdθ∫ r

−r∫ r−r

1Adxdy

=1Aπr

2

1A4r2

=π

4. (23.3)

Thus, as we draw more and more points N from this uniform distribution and label the amount of points

that fulfil√x2 + y2 ≤ r as Nc we know that as we approach infinity the ratio of these two number will

approach the ratio of the probabilities and thus the areas as

limN 7→∞

NcN

=π

4, N ∈ N. (23.4)

We can now use this to approximate π by

π ≈ 4NcN. (23.5)

A example implementation of this is shown in figure 18.

Our application of Monte Carlo is in the sense that our initial distribution is known but it will affect the

outgoing distribution in a unknown fashion due to the complexity of the in-between numerical propagation.

More on how to interpret the data output of the software will be given in the next part.

24 Principal component analysis

Principal component analysis, or PCA, is a algorithm based on orthogonal transformations to convert a

set of data points with possibly correlated variables to a set of points of linearly uncorrelated variables

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called principal components. The number of principal components are always equal to or less than the

dimensionality of the input data, where the less then part apply if all the data has zero variance in a certain

direction.

This analysis procedure is constructed in such a way that the first principal component has the largest

possible variance, and each succeeding principal component has the largest possible variance while being

orthogonal to the preceding components. Normalizing the vectors we will find a new set of orthonormal

uncorrelated basis vectors. This can be recognized as the Gram-Schmidt process of finding a orthonormal

set of basis vectors.

The principal components are orthogonal since they are eigenvectors of the covariance matrix. It is however

very apparent that such a analysis is very sensitive to variable scaling. Thus one should, unless all direction

have the same scaling, use the principal component analysis on the z-score

z =ξ − µσ

, (24.1)

of the data instead, where µ is the mean value and σ is the standard deviation. By doing this the principal

components are instead solely based on correlations and not dependant on relative scaling.

The strength of applying such a method is that one can from data with high dimensionally; reduce the

number of dimensions in the data and create lower dimensional data-sets that only include the directions

with variance, or patterns. Also, reducing data in this way provides a way to produce graphical represen-

tations of high dimensional data sets wile loosing minimal information since most of the variations are still

included in the representations.

As a example consider a large data set of different apples where there are 20 measurements taken on each

apple. All these apples are from the same breed of apple tree but we want to find in-species trends and

variance. To find and visualize this data to know what direction the research should take we can apply the

PCA algorithm on the z-score of our data set and examine the eigenvalues and directions. We here use the

z-score since most of the variables should be in different measurement regimes, let us say that some of the

variables are size measurements while other are chemical compositions. Upon examining the eigenvalues we

find that only 2 out of 20 dimensions actually have high variance. This indicates that inter-individual vari-

ances inside a species only occur on a 2-dimensional manifold, and that we have 18 dimensions of redundant

data processing. This would also allow us to accurately visualize this variation since we are not projecting

away structure. When we then examine these two directions of most variance, or the plane of most variance,

we perhaps find that the basis for this plane can reveal other trends in the data.

In this example we can see that the PCA is not directly a tool for generating scientific results, but rather to

indicate in what direction one should focus the research efforts. It is also a great tool for examining if one

has enough information in ones data to find patterns. If the plane of most variance only shows a big ball

of distributed data without any visible structure, there is probably not enough information in the data to

deduce any sort of patterns.

25 Cluster analysis

25.1 Introduction

Cluster analysis or clustering is a method for grouping a set in such a way that objects in the same group are

more similar to each other then they are to those in any other group. This measure of similarity is subjective

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Figure 19: Example of clustering within 6 nodes in a two dimensional plane where the similarity between

nodes are their geometric distance. Clustering is performed using a arbitrary critical distance.

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and is the reason we covered similarity function in section 17. These similarity function tells the clustering

how similar objects are and enables it to perform the grouping.

Often one uses cluster analysis in exploratory data mining and statistical data analysis, for example machine

learning and pattern recognition. Cluster analysis is not a single algorithm or a single method but rather

the idea behind it, clustering can be done in many ways and we shall here cover only a few.

One of the first uses was in anthropology as early as 1932 and biometrics (Edwards and Cavalli-Sforza, 1965),

as such extensive work has been done in the subject and plenty of material is available, like (Romesburg,

2004).

The idea behind clustering is that one cycles through every node in a set of nodes and asks the question: is

this node similar enough to any other node? If the answer is yes, then those two nodes become associated and

form a cluster. Depending on the rule for which one merges clusters with, different configurations emerge.

An example of clustering as shown in figure 19. In this simple example a single link adds nodes to a cluster

and the geometric two dimensional distance is the nodes similarity.

25.2 Definition

The purpose of this section is to take the analysis procedure and put it in a abstract context to more clearly

extract the important mathematical and statistical choices that are made along the way. Also this will make

it easier to propagate choices into effects on the final results. It is highly recommended that readers of this

section is well versed in mathematical notation and basic set theory (Jech, 2013).

let us consider a space M, a subset of this space X ⊂ M, and a function mapping f : M × M 7→ R+

which we shall call the similarity function.

We can introduce a grouping algorithm to assign each element in the subset X to a group, or essentially

breaking it into several smaller subsets. These subsets shall be denoted Xi ⊂ X. There are many grouping

methods to choose from, but in all cases there is the need for a additional input of data to create a specific

grouping configuration. In the case of cluster analysis and density analysis which we shall call the additional

parameter the critical threshold, Dc.

CLUSTERING ALGORITHM

INPUT : (X,Dc)

OUTPUT : Xii∈I< Algorithm statement... >

...

return Xii∈I

Where in the above algorithm I is the indexing set containing all the labels for Xi. For example, if 3 clusters

X1, X2, X3 where formed then the indexing set will be I = 1, 2, 3. Just considering clustering as our

grouping method, there are still several ways to create clusters, so the method will have to be tailored to

application. A large group of clustering methods are called hierarchical clustering methods, these depend

on a merging algorithm. The starting point of the clustering is that each element in X is in its own cluster,

to do this easily since we know that our set has finite cardinality we can introduce the indexing set as

I ⊂ N : #I = #X so that

Xi := ai : ai ∈ X, i ∈ I.

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At this point #Xi = 1, and we can see that⋃iXi = X will always be true. To perform a merge between

two clusters we check a linkage criteria between two clusters: if this criteria is fulfilled, they are merged into

a new and larger one

∀ (Xi, Xj) : i 6= j, i ∈ I, j ∈ Iif CRITERA

Xi := Xi ∪Xj

I := I \ j

This process is repeated until no new merges happen. Using a single linkage method the linkage criteria is

min (f(a, b) : a ∈ Xi, b ∈ Xj) < Dc. In words, this criteria is: If any pair of two nodes from two different

groups have a connection, the groups merge. If we use the complete linkage criteria f(a, b) < Dc ∀ a ∈Xi, b ∈ Xj the clustering dynamics will most often look completely different. This linkage criteria can be

put into words as: If all pairs of two nodes from two different groups have connections, the groups merge.

We are now ready to construct a complete algorithm, using single linkage as a example we can write

CLUSTERING ALGORITHM

INPUT : (X,Dc)

OUTPUT : Xii∈II ⊂ N : #I = #X

Xi := ai : ai ∈ X, i ∈ Ido

I0 := I

∀ (Xi, Xj) : i 6= j, i ∈ I, j ∈ Iif min (f(a, b) : a ∈ Xi, b ∈ Xj) < Dc

Xi := Xi ∪Xj

I := I \ jwhile I0 6= I

return Xii∈I

This algorithm is very inefficient in computation but instead easy to follow in reasoning.

We can now introduce a classification we shall call the association of the clustering. To do this we will be

classifying objects into two categories, nodes with a connection and nodes without. The amount of associ-

ation after a clustering is performed is the ratio between the nodes with connections and all nodes. Thus,

if all nodes have a connection, the entire cluster is associated. This measure is designed to show how many

members of the set is considered similar to another member.

Since we cannot know at what value the variable Dc produces the desired clustering we will have to de-

termine this somehow. By sweeping from Dc = 0 to Dcmax we can get the complete clustering profile of the

input set using the a certain linkage criteria. We define Dcmax as the point where the nodes and connections

form a complete graph or where no more cluster merges occur, depending on the analysis type. We will only

need to sweep until the entire set X is clustered into a single set #I = 1 to extract all the information on

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the clustering since no other merges can happens after this. One advantage of using a metric function as

similarity function is that this will guarantee a finite value of Dcmax if the subset X has a finite diameter.

Introducing a set of real data into this concept, our goal is to run a clustering algorithm on our real world

measurement data set X, with a specific Dc, and achieve a certain configuration of clustering that revels

meteor showers. We do not know what the clustering should look like in this particular set so we cannot

simply sweep Dc and pick the right one. We do however know what physical laws and dynamics the system

that generated our measurements are governed by. And the governing dynamics of the data can give hints

about what critical threshold to pick.

This reasoning means that we can create a artificial data set governed by the same dynamics, examine the

clustering in this synthetic data, and from the results of clustering calibrate our critical threshold applied

on real data.

Now it is important to think about the problems involved with this approach; The success of using the

same Dc that created the correct clustering in a simulation on real world data is totally dependant on the

accuracy of the simulation and the accuracy of the measurement. If we where to be able to simulate the

exact scenario to a very precise degree or measure the objects properties perfectly, we would probably not

need clustering methods since there are no ambiguities. Thus we must assume that our knowledge about

the real world scenario is limited to at lest a range of possibilities. The best way to concentrate a range of

possibilities into a single calibration is a Monte Carlo type simulation. We will thus run through our entire

range of possibilities and every time find a new Dc that produced the desired clustering. The average value

of these multiple clustering sweeps will give a good representation of what value to use in the real world data

analysis.

25.3 Signal to noise ratio

A signal to noise ratio may seem like a strange concept to introduce in cluster analysis but the translation

from regular signal analysis to clustering is straight forward. Consider a clustering configuration, where

we in one of the clusters consisting of 100 nodes have 90 nodes that are very similar. But the similarity

function is not perfect, and the linkage criteria is not really sufficient either, so 10 of the nodes have been

miss-classified. They should actually belong to a different cluster. We can then say that this cluster has a

signal to noise ratio of SNR = 90/10 = 9, or in SNRdB = 10 log10(90/10) = 9.54 dB.

In our application, a signal to noise ratio in cluster analysis can be artificially estimated by simulating

a meteor shower, and by simultaneously simulating the sporadic complex. Performing the clustering on both

data sets but with the knowledge of what nodes are actually a part of showers and what nodes are not, we

can consider two limits:

1. Dc shower is the limit at which meteor showers start clustering

2. Dc sporadic is the limit at which the sporadic complex start clustering

Somewhere between these two limits we can find a good clustering threshold for meteor showers in real world

data. If the value of Dc shower is ”close” to the value of Dc sporadic the clustering method chosen is unreliable.

If Dc sporadic is lower than Dc shower the method is not able to perform the desired clustering at all. Only

if Dc sporadic is larger and sufficiently separated from Dc shower is the method a valid algorithm for finding

association in the real world data set.

We have performed this type of analysis on the same example as in figure 19. In figure 20 we see the

result of a background to signal count ratio of 1 to 3. The signal is distributed in a Gaussian fashion and the

background is uniform. The top graph shows geometric representation of clusters, nodes, and connections

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Figure 20: Example of cluster analysis with background data on a two dimensional plane where the similarity

between nodes are their geometric distance. Top graph shows geometric representation of clusters, nodes,

and connections where red nodes are signal data and blue nodes background data. Bottom graph shows the

sweep of critical threshold and the respective association. Background to signal count ratio of 1 to 3.

Figure 21: Background to signal count ratio of 1 to 1.

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Figure 22: Background to signal count ratio of 5 to 1.

where red nodes are signal data and blue nodes background data. Bottom graph shows the sweep of critical

threshold and the respective association. In figure 21 we see the same analysis but with a ratio of 1 to 1.

And finally in figure 22 we see a background to signal node count ratio of 5 to 1. We can from figure 20

realize that even tough we have good association in our signal and high signal to background count, the

cluster is resolved and seems to contain most of the desired data but we still have clusters that form in the

background data. This indicates that one must disregard clusters of certain sizes depending on the data

configurations. This may also be a indication that single linkage clustering may not be suitable since it does

not represent density the same way average linkage does. Thus the samples statistical properties, such as

density, variation, size, and so on, will affect the cluster analysis itself and is important to consider before

application.

25.4 Previous usage of thresholds

Many of the papers published searching for meteor showers using cluster analysis area does not mention

exactly how the critical threshold is derived, if the uncertainty of that choice has been investigated, or what

the sample statistical properties are. This can easily lead to misleading results. For example the thresholds

that (Southworth and Hawkins, 1963) derived

DSHc = 0.2

(360

N

)1/4

, (25.1)

and (Lindblad, 1971) derived

D′SHc = 0.8N−1/4, (25.2)

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where N is the sample size. This formula is very hard to evaluate and where derived in a empirical manner

without consideration of their bias. Now that computer power has increased to such a degree that we can run

proper simulations to generate synthetic data such formulas should not be used any more. In (Welch, 2001)

and (Jopek and Froeschle, 1997) a similar statistical analysis to the one developed here is used. However

due to lack of information about the generation of the sporadic background in these papers it is difficult to

compare the results, also in (Welch, 2001) no reliability level is calculated.

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Part VI

Software

26 Introduction to the software

This section is intended to explain the bulk of the work performed and should not be skipped as it is vital

to interpret the results correctly. Also this part functions like a instruction manual for the use of the software.

Even though the plethora of applications many of the computational software’s used by scientists are out-

dated by modern standards. One such example is the mercury6 software (Chambers, 1999) commonly

used today for integrations of the Solar system. This code has been the base of follow up version, Swift,

and later Swifter, which recently was ported to C/CUDE as cuSwift to do GPU calculations. This is

the most up to date dedicated Newtonian gravitation Hamiltonian integrator software to our knowledge.

We are thus disregarding short term integrators not utilizing specialized Hamiltonian techniques such as

the ESA NEOprop2, which contains most relevant non-gravitational effects in addition to the gravitational

perturbations, but relines on regular integration techniques even though symplectic in nature. Many of these

older programs lack the versatility possible with modern programming languages and few are managed as

easily assessable open source software. Advances have been made in the subject of symplectic integrators

but few have actually been implemented in publicly available software.

The perhaps most missed feature in current models is the ability to easily connect programs with differ-

ent functionalities. We believe this is the main reason why many contemporary techniques such as Monte

Carlo simulations of meteoroid streams, orbital stability analysis, time tracing of orbital similarity functions

to better analyse measurements, neural network recognition of meteor showers, and so on, have not yet been

combined. And this was also the bottleneck to perform the research presented in this thesis. Thus we have

developed such a software attempting to supplement the current research area.

The recent progress in measurements, like the Rosetta mission, means that a multitude of data is avail-

able to use with this software, for example to improve and test models of meteoroid ejections, composition,

and propagation through the Solar system. Also, with improved meteor observations, like with the future

radar system EISCAT˙3D (McCrea et al., 2015), it will be important to perform statistical simulations to

evaluate the meteor data correctly. This software gives an opportunity to use measurements gathered from

a variety of ESA missions in previously unrelated procedures.

27 Modular development

One of the most challenging tasks when creating new software is broad functionality without loosing usabil-

ity. To maximize the functionality of all the developed software we have chosen to proceed in a modular

fashion, where every significant self contained part of the analysis and simulation is made into a independent

module. One module acts as a master program that has the ability to call several independent modules.

The modules can run independently of the master program. The user can therefore link modules in different

orders, choose to distribute data between modules in different ways, and tailor case studies in a powerful

fashion. The parent bodies can be hand-picked for case studies, or generated from probability distributions

like e.g. the Pan-STARRS Synthetic Solar System Model. In the example simulation of the random stream

environment the software was configured so that the the distributions represent orbital elements, critical

sublimation radius, density, size, and surface activity. The software then integrated ejected particles over

a chosen time and examined close encounters with the Earth. The real power of the software lies in its

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statistical approach. The ability to choose distributions and then do repeated simulations implies that one

can assess the most probable outcomes based on the uncertainty of the input parameters rather than relying

on assumptions. The software enables a long-term statistical approach, which is needed to investigate the

sporadic meteoroid complex and its origin.

Below a flowchart of the modular nature of the developed software is shown, where the MCAS program

is the master program. Currently the Celestial Mechanics Simulator module and the Orbital Stability Es-

timation module have not yet been implemented in code, also the Statistical Uncertainty Orbital Clones is

currently functional at a bare minimum level. This is due to the fact that the current version is a proof of

concept for the full scale project and thus non vital functions and modules where not implemented.

MCASMonte Carlo Association Statistics

CMS/mercury6Celestial Mechanics Simulator

OAAOrbital Association Analysis

SUOCStatistical Uncertainty Orbital Clones

PBEParent Body Ejector

JPLJPL ephemeris calculator

OSEOrbital Stability Estimation

28 Dependancies

28.1 Programming language

When starting development of this software several languages was considered and some preliminary testing

was done using MATLAB integrators. While very high level languages like Python did not provide the com-

plexity required to performed efficient scientific calculations and data handling very low level and restricted

languages like Fortran are too ridged to provide flexibility during development and future modification. Thus

we have chosen the trade off point of C++, where the object oriented programming language can provide

enough ”behind the scenes” functionality to make the code easily modifiable and dynamic while allowing

complex programming functions closer to the machine hardware to speed up calculations. Thus complex

algorithms, like integrators, will be written in C and wrapped in C++. Also C++ is a language with plethora

of libraries already available thus lowering the development work needed.

28.1.1 C++ 11

Some of the functions or syntax made available in the C++ 11 has been used to provide easy modification

of the source code as many of the functions implemented in this C++ dialect reduces the coding experience

needed by the user. However the majority of the code is written according to C++ 98 standard and only

small modification is required to compile using this dialect instead.

28.2 Boost library

To provide easy development of file handling and system interaction we have used several of the Boost

libraries. Boost provides free peer-reviewed portable C++ source libraries with strong usability. The specific

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libraries and function used will not be covered here, for this we refer to the future release of the code. However

to re-compile the developed software the boost libraries must be installed with on the compiling system. For

example if a Ubuntu type Linux distribution is used one should install the boost dependencies with:

Listing 1: Example command

1 sudo apt-get install libboost-all-dev

More on boost can be found in (Schaling, 2011).

28.3 SPICE library

SPICE is an information system designed to assist scientists in planning and interpreting observations from

instruments and to assist engineers in modelling, planning and executing space activities. This tool is

developed by The Navigation and Ancillary Information Facility (NAIF), acting under the directions of

NASA’s Planetary Science Division.

SPICE is designed to use data files refereed to as ”kernels” containing observational data and complementary

information such as leap second data to compute the desired parameters referring to a object in the Solar

system. The kernels also come with more information about that specific file provided in a ”readme”

document included in the folder. The different types of kernels available are:

• DSK - Digital Shape Kernel:

• FK - Frames Kernel

• LSK - Leapseconds Kernel

• PCK - Planetary Constants Kernel

• SPK - Spacecraft and Planet Kernel (ephemeris kernel)

In out case we have used the Toolkit version for C to develop a small module that can compute the geometric

positions of the planets and other objects in the desired coordinate system. This toolkit is needed to compile

the JPL module and can be found on https://naif.jpl.nasa.gov/naif/toolkit.html, where

installation guides and extensive documentation is available.

The system currently includes the following aberration corrections to correct for a one way light time and

stellar aberration error.

• ”NONE”: Apply no correction, gives the geometric state.

• ”LT”: Apply correction for one way light time using Newtonian formulation.

• ”LT+S”: Apply correction for one way light time and stellar aberration using Newtonian formulation.

• ”CN”: Apply correction using converged Newtonian light time.

• ”CN+S”: Apply correction using converged Newtonian light time and stellar aberration.

In current applications however we are only interested in the geometric state of the planets and in MCAS this

has been fixed to ”NONE”.

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https://naif.jpl.nasa.gov/naif/toolkit.html


28.4 Fortran 77

Since we have substituted the CMS module for mercury6 until development of this module is mature enough

for early deployment, a Fortran 77 compiler is required to run the entire system. For example is a Ubuntu

type Linux distribution is used one should install the Fortran 77 dependencies with:

Listing 2: Example command

1 sudo apt-get install fort77

29 Monte Carlo Association Statistics module

Monte Carlo Association Statistics or MCAS is developed to manage a simulation of bodies that produce small

particles in our Solar system in a Monte Carlo fashion. The parent bodies are generated from probability

distributions of some variable or set of variables. In its current stage the available variables to distribute are

orbital elements, critical sublimation radius, parent body mass, parent body size, particle mass distribution,

and surface activity. The software is then designed to integrate these particles over some time scale and

examine close encounters with a major body (planet) in the Solar system. We have concentrated on close

encounters with the Earth.

As mentioned in the software introduction, a Monte Carlo type simulation is preferable to solve many of the

contemporary science questions mentioned. As we have covered the basic basic principles of such a method in

section 23 we shall now more closely specify our intentions and applications of this principle in the developed

software.

The following chapters will specify the implementation specifics of the MCAS module.

29.1 Program flow

Below is a diagram illustrating the flow of the software. The purpose of the diagram is to quickly get a

overview of the practical implementation and capabilities of the program.

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Start

SUOC

Parent body distribution

Pick random Parent body Will body produce particles?

PBE

Propagate and eject particles

No

Yes

mercury6

Propagate ejected particles and parent body

Particle mass distributionJPL

Simulation initial state

Save data on all close

encounters inside Hill Sphere

and their parents

OAA

Association analysis

on generated showers

OSE

Orbital stabillity check

of parent bodies

Load time series of close encounters

Examine divergence from parent bodyMonte Carlo done?

Done

No

Yes

29.2 Execution time diagnostics

In almost every type of numerical calculation or simulation software there is a bottleneck for the execution

time. In Hamiltonian propagation and cluster analysis these bottlenecks all stem from the fact that inter-

particle calculations scale as O(N2) and thus grows beyond normal computing requirements quite quickly.

There are however several other pitfalls for the inexperienced programmer, such as faulty balancing of RAM

saving versus hard drive write and read speeds. Thus to find these bottlenecks and further optimize a newly

developed software execution timers are a excellent way to improve either the software itself or the simulation

settings by perhaps lowering resolution in certain bottleneck areas where possible.

The current implementation of MCAS has been divided into 9 time sections within which the execution time

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of that section is saved to a file. The tenth section is the execution of the entire software, the file containing

this data is specified in the output section. This is useful both for development and also for post-simulation

analysis, the sections functions can be found in the below list as:

1. Parent body generation

2. PBE time

3. mercury6 time

4. Snapshot time

5. close6 time

6. element6 time

7. Stream dissipation time

8. Parent body divergence and file handling

9. Cluster analysis

10. Total time

Figure 23: Execution time for different segments of MCAS, stream dissipation calculation enabled.

As an example of this kind of time execution analysis we have run 2 different simulations, each with the same

exact integration parameters and settings, except we have turned on Stream dissipation analysis in one of

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Figure 24: Execution time for different segments of MCAS, stream dissipation calculation disabled.

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the simulations. The output time data can be found in figure 23 for the simulation with stream dissipation

analysis and in figure 24 for the simulation without. The simulation was performed with a maximum of

1000 test particles per ejection, one perihelion ejection passage with the parent body picked from the Pan-

STARRS cometary database. The particles where integrated for 220 years in mercury6 using a step length

of 16 days and the association analysis was performed with 4 similarity function conducting a parameter

sweep of single linkage clustering. These graphs indicate the problem of calculating stream dissipation in the

current fashion. A solution to this problem is proposed by creating a custom output solution in CMS that

creates files allowing the calculation to be more efficient by grouping data in files of time intervals rather

than object time series, and additionally by printing larger time steps even though the propagation was

performed with smaller steps. This would also require a function estimating the amount of RAM that can

be used by the stream dissipation calculation to optimize the amount of data loaded from the CMS output

at once.

29.3 Probability draw

To be able to numerically simulate a process dependant on a stochastic variable, or events based on proba-

bility, one needs to generate random values with the correct frequencies. In C++ the only random number

generator available by default generates a uniform distribution between two numbers. To be able to draw

events based on general probability density function we must specify a algorithm to transform uniform dis-

tributions to a desired distribution. We shall hereby refer to the action of finding a random value based on

a general probability distribution function as a draw from that distribution.

The most basic version of such a function is to draw from the one dimensional histogram. This function,

given a set of bins and a set probabilities or frequencies for each bin, one can define a algorithm as:

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DRAW FROM 1D DISTRIBUTION

INPUT : Histogram bins: X

Histogram frequencies: F

OUTPUT : Random draw from histogram

Save size of histogram vector: s = SIZE(F )

Generate uniform random number between 0 and 1: r = RAND

Normalize the histogram: F′ =F

SUM(F)

Start cumulative adding at 0: H = 0

FOR i FROM 1 TO s

Check if we found the selected bin: IF H <= r AND r < (H + F′[i])

return X[i]

ELSE

Otherwise prepare for the next check: H = H + F′[i]

END

END

This algorithm will converge to produce exactly the given histogram as one continues to draw. This algorithm

is the fastest of the ones implemented in the software but it does not provide a satisfactory result given a

more carefully considered input distribution. As the vast majority of the simulation time in the current

application of the software is spent on time integration of the Solar system the speed of this algorithm is not

important. If however a application is considered where many generations of random initial conditions must

be done compared to the amount of time integrations, one can consider this simplified algorithm. The major

fault of this algorithm is the discrete return values that are the center of the bin. To rectify this we make

the return value a continous distribution by returning, instead of the bin center, a value randomly chosen

inside the bin by a uniform distribution:

DRAW FROM 1D DISTRIBUTION

...

FOR i FROM 1 TO s


Draw the ’in bin’ value: r2 = RAND

return1

2(X[i− 1] + X[i]) +

r22

(X[i+ 1]−X[i− 1])

...

To show the effect of this improvement in a practical application we have taken some input data for the

SUOC module, namely the comet population of the Pan-STARRS Synthetic Solar System Model. In figure

25 we can see the orbital shape parameters, semi major axis and eccentricity, as histograms and as a density

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Figure 25: Comparison between the described first method of drawing a random value from a distribution

versus the real multi dimensional distribution

Figure 26: Comparison between the described improved first method of drawing a random value from a

distribution versus the real multi dimensional distribution

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map where the upper row show the actual data and the bottom row show graphs of the first histogram

generation algorithm. In figure 26 the improved continuous version of the histogram generation algorithm

is shown in the same kind of graphs. We can see the significant improvement in the density map since this

shows the smoothing effect of the continuous rather then the discrete return values, however the histograms

seems mostly unaffected. This effect is something that one should be very aware of since a histogram may

fail to represent important trends in data.

Using a one dimensional histogram assumes that variables are uncorrelated, and this simplification can be

seen in the figures as the shapes of the two dimensional distributions in the top and bottom row of figure

26 are not the same. To better illustrate the effect we have created a artificial distribution suitable for this,

as shown in figure 27. To correctly represent the data we have to assume that the parameters need not be

independent, so instead each object is treated as an N dimensional point. A psudo code for the continuous

version algorithm for such a multidimensional function is given below:

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DRAW FROM ND DISTRIBUTION

INPUT : Histogram bins: XHistogram frequencies: F

OUTPUT : Random draw from histogram

Save size of histogram vector: s = SIZE(F)

Save dimension of histogram volumes: N = DIM(X )

Calculate bin sizes and save: Xs = BIN SIZE(X )

Generate uniform random number between 0 and 1: r = RAND(1)

Normalize the histogram: F′ =F

SUM(F)

Start cumulative adding at 0: H = 0

FOR i FROM 1 TO s


Draw the ’in volume’ value: r = RAND(N)− 0.5

return r ·Xs[i] + X [i]

ELSE

Otherwise prepare for the next check: H = H + F′[i]

END

END

Here X is a list of all N dimensional points in the distribution representing the center of histogram hyper

volumes, each housing a frequency listed in F. Here we can see that the non trivial programming does not

lie in the drawing function but the creation of the histogram bin list in a sufficient fashion as to give a good

frequency to resolution ratio. As a last note on this example, r·Xs[i] denotes the element wise multiplication,

not matrix algebra or scalar product. The result of implementing this on the artificial data and the orbit

shape data can be seen in figures 28 and 29.

The reason behind the extensive discussion of random point generation is that the method one generates

the initial distribution is crucial to understanding the end distribution. In our use of the software many

of the variables are picked randomly to, over time, eliminate the possibility of assumption errors. One

should however be aware of the fact that the more stochastic space one has to explore the less specific the

results become and the more computation time is needed. Currently the variables generated initially from

probability distributions are:

• Parent body semi major axis

• Parent body eccentricity

• Parent body inclination

• Parent body argument of perihelion

• Parent body longitude of ascending node

• Parent body epoch of orbital elements

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Figure 27: Comparison between the described improved first method of drawing a random value from a

distribution versus the artificially generated multi dimensional distribution

Figure 28: Comparison between the described multi dimensional method of drawing a random value from a

distribution versus the Pan-STARRS multi dimensional distribution

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Figure 29: Comparison between the described multi dimensional method of drawing a random value from a

distribution versus the artificially generated multi dimensional distribution

• Parent body radius

• Parent body ejected material bulk density

• Parent body surface sublimation activity factor

• Parent body critical heliocentric sublimation radius

• Parent body mass

Then at later stages the following variables are also generated in such a fashion

• Ejected particle mass

• Ejected particle direction

One can thus, by setting some of these distributions to Dirac-delta functions, examine the statistical outcome

of any input distribution or any combinations of the above.

29.3.1 Dimensionality reduction

In section 24 we covered the principal component analysis and briefly covered dimensionality reduction.

Such a feature is not yet implemented but is planned in future versions. In light of the problem with

multidimensional distribution we know that their sampling size increases exponentially with the amount of

dimensions they are spread across. This is why such a dimensionality reduction function would be useful,

since we cannot assume that the datasets available are of appropriate size to sample the entire space, or that

we have enough computing power to cover the entire space.

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To illustrate this we shall consider the Pan-STARRS set of comets previously discussed. Using 5 orbital

parameters and ignoring the orbit epoch as a parameter we are left with a 5 dimensional space. Let us also

consider a set of resolutions of ∆a = 0.5 AU, ∆e = 0.05, ∆i = 5,∆ω = 10, ∆Ω = 10 to be sufficient

for representing the data. This set contains 27 501 artificial comets, let us label this N . We can assume

the parameters to range from a ∈ [2 AU, 10 AU],e ∈ [0, 1), i ∈ [0, 180],ω ∈ [0, 360], Ω ∈ [0, 360].

We now know the size of the sample and the size of the space it is distributed in, let us now explore the

possible resolutions we can find acceptable. For the purpose of the example let us say that we cannot give

any statistical significance to a average histogram frequency of less then 100. Let us also assume that the

data for now is uniformly distributed in this space. In the case of just simple histograms for each variable,

i.e. if all parameters are independent, we would get a set of resolutions of ∆a = 0.03 AU, ∆e = 0.0036,

and so on. This is very well below what we defined as acceptable. If however we instead partition the space

into hypercubes and calculate the resolutions with 100 samples in each cube we find the resolutions to be

∆a = 2.6 AU,∆e = 0.32,∆i = 58,∆ω = 117, ∆Ω = 117. Thus, it is impossible to reach any reasonable

resolution with a sample size of only 27 501 if all the 5 variables are dependant, for this we would need a

sample size of around 1 500 000 000. However, let us say that only the eccentricity and inclination where

found to be dependant on each other while the rest where totally independent. Then we can perform the

same calculation and find a set of multi dimensional resolutions of ∆a = 0.48 AU,∆e = 0.06, while the rest

are the one dimensional resolutions. Thus, depending on the sample size, dimensionality reduction can be

essential.

This is only a problem however if one wants to draw a number of orbits randomly that exceeds the distribution

size. If the number of draws is smaller then the sample size one can simply integrate the sample points

themselves instead of creating a continuous distribution function.

29.4 Close encounter

When examining interactions between particles and planets, and in our case the meteor phenomena, there

is a significant difference between a close encounter and a actual collision. This is due to the fact that even

if the meteoroid enters the Earth’s field of superior gravitational influence, its Hill Sphere, the particle may

never encounter the atmosphere and undergo ablation. The Hill sphere of a body can be approximated as

rH ≈ a(1− e)( m

3M

), (29.1)

where rH is the Hill sphere radius, a and e is the familiar orbital shape parameters, and m is the smaller

body mass while M is the larger body mass. In our case, m would be the mass of the Earth and M the

mass of the Sun. The parameters a and e would be the orbital parameters of the Earth.

Even tough there is a option in the configuration of MCAS to change what is regarded as a meteor with

regards to the close encounters closest distance we have in our demo simulation chosen this to be 1 Hill

radii. The reason behind this simplification is that the probability of encounter for a single particle greatly

decreases when instead picking only particles that passes through the atmosphere, requiring magnitudes

of more particles and thus a computation time of O(N2) increase as we add more particles. This is not

applicable when performing our kind of statistics based simulations. Thus we have chosen to regard all

encounter within 1 Hill radii as meteors.

This simplification is not as bad as it seems due the same reason that the approximation was implemented.

Since there is not computational power to represent every ejected dust grain from the comet during subli-

mation each test particle integrated represents millions and millions of dust particles. All these particles will

have small perturbations in their initial conditions and thus we can instead of regarding each test particle as

one single meteoroid, we can view them as small swarms of meteoroids. When viewed this way it becomes

reasonable that even if the particles only comes within one Hill radii of Earth, some part of this ”swarm”

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will still produce meteors.

29.5 Parent body stream divergence

One interesting aspect of a meteoroid stream is its age. This is however something that is hard to determine

from observations since the only aspect detectable is the meteor showers orbital configuration. Thus it may

be interesting to, in simulations, calculate how far from the parent body the orbits have deviated before

encounter. Such a calculation may give a good indication of what epoch trail the detected meteor shower

was a part of. By this we mean that if the parent body is known, and we have just detected a meteor shower

with a certain orbital elements distribution, one can make a fast first assumption on what trail caused the

shower by comparing the orbits of the current parent body and the detections. As such, to calculate such a

parent body divergence time series we are only interested in meteoroids that actually encountered the Earth

and in the simulation we shall only perform this analysis on those. Since there are many orbital similarity

function available we have implemented this divergence calculation using the 4 already implemented function

in the OAA module. The calculation logistics are straight forward, we load the time series for the parent

body, and then in succession we load the time series for each encountered test particle and calculate the time

time series of the similarity function. Below we demonstrate some pseudo code to clarify the method for one

similarity function:

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PARENT BODY METEOROID DIVERGANCE

INPUT : Particles time series: Oi(t0 + n∆t) ∀ n ∈ [0, N ], i ∈ [1,M ]

Time series for parent body: OPB(t0 + n∆t) ∀ n ∈ [0, N ]

Similarity function: F

OUTPUT : Divergance time series: Di(t0 + n∆t) ∀ n ∈ [0, N ], i ∈ [1,M ]

FOR i FROM 1 TOM

FOR k FROM 0 TO N

Calculate the orbital similarity: Di(t0 + k∆t) = F (Oi(t0 + k∆t),OPB(t0 + k∆t))

END

END

Figure 30: A highly perturbed meteoroid stream encountering Earth over 400 years.

In the above example O(t) are the orbital elements needed for the similarity function at time t, whether it

be geocentric quantities, positions and velocity vectors, or Kepler elements. As a example we have chosen

two different iterations of the Pan-STARRS simulation. The first is one of a stream that experiences a

relatively small amount of perturbation, as can be seen of the shower orbital plot in figure 30. Using the

orbital similarity functions one can quantify the difference in the perturbation effect on the orbit compared

to the parent body by using the algorithm above, the resulting data for the Earth encountered particles can

be seen in figure 32. The second one illustrates a stream in a region with a high amount of perturbations,

or low e-folding time, causing the orbits to spread over a large area qickly as can be seen in figure 31. The

resulting criterion divergence time series can be found in figure 33.

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Figure 31: A slightly perturbed meteoroid stream encountering Earth over 400 years.

Figure 32: The DSH criterion measure of the stream divergence from its parent body in a highly perturbed

meteoroid stream encountering Earth over 400 years.

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Figure 33: The DSH criterion measure of the stream divergence from its parent body in a slightly perturbed

meteoroid stream encountering Earth over 400 years.

29.6 Stream dissipation

To quantify stream dissipation is not a easy question as the morphology of a stream may evolve in may

different ways depending on the gravitational environment in the general area of the parent body. If the area

is dominated by gravitational resonances chaotic effects becomes predominant and the stream will quickly

scatter over the Solar system. Previously most stream morphology examination have visually shown cross

section of the stream, or snapshots of its evolution at different stages. Or, in its early life, examined its cross

section particle density distribution. All these methods however become troublesome when summarizing over

many different streams, examining many iterations of the same stream simultaneously, or doing a statistical

analysis and thus we cannot utilize the regular methods.

To resolve this we will use the previously mentioned orbital similarity functions. This is advantageous since

we are no longer relying on regular 3-dimensional distances in space but rather the similarity of the orbits in

the stream, and how this similarity evolves with time. This also removes the need to find some sort of ”cross

section” of the stream, as the orbit configuration is not dependant on where in the orbit the particle is. It

only considered how the particle has been perturbed, i.e. the particles that ”lag behind” the comet does not

give a large distance if their orbits still coincides with the other particles. One way to do this would be to

calculate the distance matrix

Dij(tn) = D(Oi(tn),Oj(tn)), (29.2)

in each time step. Here we have denoted the similarity function as D and a orbits describing parameters at

time tn as O(tn). However, the calculation of this matrix has the problem of having a scaling of O(N2). Due

to the fashion that mercury6 outputs time series and the inherent O(N2) scaling of the calculation RAM

and hard drive usage will be a problem when using this to estimate stream dissipation. To resolve this we

will only use the distance matrix as a intermediate calculation and the actual output will be the mean value

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and standard deviation of this matrix. As we would only want to load a few times series at a time and we

cannot load all particle data for one time we will calculate the mean

µN =1

N

N∑i=1

xi, (29.3)

and standard deviation

σ2N =

1

N

N∑i=1

x2i −

(1

N

N∑i=1

xi

)2

, (29.4)

with a sample increment formula as given by

µN+1 = µN + ∆µ, (29.5)

∆µ =xN+1 − µNN + 1

, (29.6)

and

σ2N+1 = σ2

N + ∆σ, (29.7)

∆σ =x2N+1 − σ2

N +Nµ2N

N + 1− µ2

N+1. (29.8)

Or in direct formulas as

µN+1 = µN +A =

= µN

(1− 1

N + 1

)+xN+1

N + 1, (29.9)

σ2N+1 = σ2

N +A =

=

(1

N

N∑i=1

x2i

)(1− 1

N + 1

)+x2N+1

N + 1− µ2

N+1. (29.10)

Of course we also need to account for the sample bias, so the output variable will be

s2N =N

N − 1σ2N ⇔ s =

√N

N − 1σN . (29.11)

This will be used in such a way that each unique pair of time series are loaded, the pairs similarity function

time series are calculated, and added by the above formulas to the output data. A set of preliminary results

using different similarity function can be found in the results section.

29.7 Data formatting

Essential in any data analysis of results is to have complete knowledge of the data formatting of the output

files. Thus this entire section is dedicated to the specification of all the output and input files of this software.

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29.7.1 Input files

Table 3: family<x>.data, <x> represents the index number of the parent body distribution

Column separator : ’Space’ 0x20

Input File description: Parent body characteristics

Row Description Data type Units Notes

1 Comet radius Double m -

2 Bulk density Double kg/m3 -

3 Comet activity fraction Double - Fraction of comet subliming

4 Critical distance for sublimation Double AU Heliocentric

5 Comet mass Double AU Optional row

Column Description Data type Units Notes

1 Distribution type Char - U: Uniform, N: Normal distribution

2 Distribution value 1 Double - U: minimum value, N: mean

3 Distribution value 2 Double - U: maximum value, N: standard deviation

Table 4: mass.data


Input File description: Major body masses (Sun and all planets)


1 Mass Double Several choices, see configuration file -

Table 5: mass_dist.txt


Input File description: Custom mass distribution input file


1 Bin center value Double grams -

2 Bin probability Double - Does not need to be normalized

29.7.2 Output files

Table 14: error_profile_<f>.txt, <f> represents the name of the used criterion.


Output File description: Error function evolution during cluster analysis

Note: -


Odd Critical value used Double - -

Even Error function value Double - -

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Table 6: execute_time.txt


Output File description: Execution time of sections in software simulation. May have unequal number of

columns for each row.


All Section iteration Double seconds -


1 Parent body generation section Double seconds -

2 PBE section Double seconds -

3 mercury6 section Double seconds -

4 Snapshot section Double seconds -

5 close6 section Double seconds -

6 element6 section Double seconds -

7 Stream dissipation section Double seconds -

8 PB divergence section Double seconds -

9 Cluster analysis section Double seconds -

10 Entire software Double seconds Only one column

Table 7: snapshot_pos.txt


Output File description: Takes a snapshot of all test particles positions at specified time


1 Simulation id Integer - Also MC run ID

3 X position Double AU Ecliptic heliocentric J2000.0 equinox

4 Y position Double AU Ecliptic heliocentric J2000.0 equinox

5 Z position Double AU Ecliptic heliocentric J2000.0 equinox

Table 8: snapshot_vel.txt


Output File description: Takes a snapshot of all test particles velocities at specified time



3 X velocity Double AU/day Ecliptic heliocentric J2000.0 equinox

4 Y velocity Double AU/day Ecliptic heliocentric J2000.0 equinox

5 Z velocity Double AU/day Ecliptic heliocentric J2000.0 equinox

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Table 9: snapshot_earth.txt


Output File description: Takes a snapshot of 10 or less if too close to integration start Earth positions at

specified time



2 Relative time from simulation start Double years -

3 X position Double AU Ecliptic heliocentric J2000.0 equinox

4 y position Double AU Ecliptic heliocentric J2000.0 equinox

5 Z position Double AU Ecliptic heliocentric J2000.0 equinox

6 X velocity Double AU/day Ecliptic heliocentric J2000.0 equinox

7 y velocity Double AU/day Ecliptic heliocentric J2000.0 equinox

8 Z velocity Double AU/day Ecliptic heliocentric J2000.0 equinox

Table 10: time_data.txt


Output File description: Simulations Earth time data


1 Parent body id Integer - Also MC run ID

2 Parent body sync from simulation start Double Days -

3 PBE start time Double JD -

4 PBE stop time Double JD -

5 mercury6 start time Double JD -

6 mercury6 stop time Double JD -

7 PB perihelion passage 1 Double JD -...

6+n PB perihelion passage n Double JD -

Table 11: PB_kep_dist.txt


Output File description: Histogram


1 Semi-Major axis Double AU Bin centre

2 Frequency Integer - -

3 Eccentricity Double - Bin centre


5 Inclination Double Degrees Bin centre


7 Argument of perihelion Double Degrees Bin centre


9 Longitude of ascending node Double Degrees Bin centre


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Table 12: function_divergance_<f>.txt, <f> represents the name of the used criterion.


Output File description: Time series of criterion between parent body and a encountered particle, each

simulation ID marks its section of the file


1st of section Time Double Days Repeated for each simulation ID

Rest of section Criterion value Double - -


1 Parent body ID Integer - Designates data section

Rest Data - - -

Table 13: stream_dissipation_data_<f>.txt, <f> represents the name of the used criterion.


Output File description: Time serie of criterion between parent body and a encountered particle, each

simulation ID marks its section of the file


1st of section Time Double Days Repeated for each simulation ID

2nd of section Mean criterion Double - On stream internal distances

3rd of section Criterion standard deviation Double - On stream internal distances


1 Parent body ID Integer - Designates data section

Rest Data - - -

Table 15: association_profile_<f>.txt, <f> represents the name of the used criterion.


Output File description: Fraction of sample associated during cluster analysis

Note: Number of rows = 2 × showers generated


Odd Critical value used Double - -

Even Association amount Double - -

Table 16: particle_weight_stat.txt


Output File description: Number of meteoroids each test particle represents for each parent body

Note: Only shower generating parent bodies


1 Number of meteoroids weight Double - -

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Table 17: particle_ejection_v_stat.txt


Output File description: Ejection velocity relative parent body

Note: -


Odd Heliocentric distance Double AU -

Even Ejection speed Double m/s -


1 Parent body ID Integer - -

Rest Data - - -

Table 18: PB_kep_timeS_data.txt


Output File description: Time series data for all shower producing parent bodies


1 Parent body ID Integer - Also shower ID

2 Time Double Days -

3 Semi-Major axis Double AU -

4 Eccentricity Double - -

5 Inclination Double Degrees -

6 Argument of Perihelion Double Degrees

7 Longitude of Ascending Node Double Degrees -

8 True anomaly Double Degrees -

9 Mass Double kg -

Table 19: earth_encounter_data.txt


Output File description: Earth orbital elements for each encounter



2 Time Double Year -

3 Closest distance Double AU -




7 Longitude of Perihelion Double Radians NOT argument of perihelion

8 Longitude of Ascending Node Double Radians -

9 Mean anomaly Double - Mean longitude if Eccentricity < 1.e-8

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Table 20: met_encounter_data.txt


Output File description: Meteoroid orbital elements for each encounter (at encounter)




3 Closest distance Double AU -




7 Longitude of Perihelion Double Radians NOT argument of perihelion

8 Longitude of Ascending Node Double Radians -

9 Mean anomaly Double Radians Mean longitude if Eccentricity < 1.e-8

10 Mass Double kg -

Table 21: PB_kep_data.txt


Output File description: All generated parent body Kepler elements


1 Parent body ID Integer - -




5 Argument of Perihelion Double Degrees -


9 True anomaly Double Radians -

Table 22: PB_M_stat.txt


Output File description: Shower generating parent body mass change


All Parent body mass Double kg New entry every change in mass, different row length

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Table 23: met_before_encounter_data.txt


Output File description: Meteoroid orbital elements at integration step before each encounter









8 Mean anomaly Double Degrees -

9 True anomaly Double Degrees -

10 Mass from mercury Double Solar masses Set to 0

Table 24: ejector_char_data.txt


Output File description: Parent body characteristics



1 Comet radius Double m -


3 Comet activity fraction Double - Part of comet subliming

4 Critical distance for sublimation Double m Heliocentric

Table 25: ejector_kep_data.txt


Output File description: Parent body characteristics








6 True anomaly Double Degrees At initial state

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Table 26: ejector_n_data.txt


Output File description: Parent body count data

Note: Only subliming parent bodies


1 Parent body ID Integer - Number of generated parent bodies

2 Total particles created Integer - -

2 Particles created this propagation Integer - -

2 Total Earth encountered particles Integer - -

2 Earth encountered particles this propagation Integer - -

29.8 Mass distributions

The PBE software takes a custom input of ejected particle mass distribution. During ejection each particle

is assigned a random mass according to the supplied distribution Ψ(m). The MCAS software automatically

configures the PBE mass distribution input according to 4 options:

• The Grun interplanetary flux model

• Uniform distribution

• Common logarithmic uniform distribution

• Custom file

29.8.1 Grun

The perhaps most commonly used meteoroid flux model at Earth is the one developed in (Grun et al., 1985).

It is based on function fitting of measurement data and gives the total average meteoroid flux in terms of

the integral flux. The flux function is given by

F (m) = 3.15576 · 107 (F1(m) + F2(m) + F3(m)) , (29.12)

F1(m) =(2.2 · 103m0.306 + 15

)−4.38, (29.13)

F1(m) = 1.3 · 10−9(m+ 1011m2 + 1027m4

)−0.36, (29.14)

F1(m) = 1.3 · 10−16(m+ 106m2

)−0.85, (29.15)

(29.16)

where the function F1(m) refers to particles with mass over 10−9 g, function F2(m) refers to the mass range

between 10−9 g and 10−14, and lastly the function F3(m) refers to the smallest particles with mass under

10−14 g. Usually when this function is used one modulates this flux with a Earth shielding and Gravitational

focussing term which we shall neglect due to the fact that the simulations concern particle distributions at

the edge of the Earth Hill sphere.

To use this as a viable probability distribution the software converts the particle flux into a histogram

type data function and uses this as input to PBE,

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ΨGRUN(m) =F (m)

αGRUN, (29.17)

αGRUN =

∫ mmin

mmax

F (m)dm. (29.18)

29.8.2 Uniform and logarithmic

Given two boundary values mmin and mmax the resulting probability distribution functions will be

ΨLIN(m) =1

mmax −mmin, (29.19)

for a uniform representation and

ΨLOG(m) =1

m(log10(mmax)− log10(mmin)), (29.20)

for a common logarithmic uniform representation. The common logarithmic representation is useful when

doing specific mass propagation analysis over large ranges of masses spanning several magnitudes.

29.8.3 Custom file

The file input functions on a item probability basis, each column is a value with a respective probability

to acquire this item when a readout of the distribution is done. As can be seen in table 5 the input file is

constructed like a histogram where the bin center is given with its frequency or probability. Thus generating

bin centres that are not equally spaced will distort the probability function as the probabilities are spread

over the entire bin.

29.8.4 Comments

Due to the fact that particle propagation calculation has a time complexity of O(N2) where N is the number

of particles it is very hard to get good statistical coverage using a model such as the Grun model. This is

due to the fact that one type of mass is orders of magnitude more abundant than the rest, thus making

the probability of the smaller particles being ejected so small that their resulting statistics are often useless.

This is the reason why the logarithmic model was introduced, so that mass transfer over large mass ranges

can be examined with statistical coverage.

More recently, a technical model called the Interplanetary Meteoroid Environment for eXploration (IMEX)

is currently under development on a European Space Agency (ESA) contract for the planning of space

missions (Soja et al., 2015). IMEX is designed to produce a static database of all released particles, taking

into account an important, but limited, fraction of the Solar system comet population. The model currently

uses a static set of 428 short period comets integrated for 200-400 years with a Runge-Kutta integrator. The

ejection was done using a static sublimation radius from 3 AU ejecting a mass distribution of 8 masses 100

microns to 1 cm. The model was verified using comet trails comparison with IR images and performing a

case study of Leonids in 2001 with Dust from 55P/Tempel Tuttle ejected between 1690 and 1998.

29.9 Probability formulation

We shall now address the actual probability formulation of the simulation. The basic concept is that the

dynamics are too vast and difficult to visualize by simply examining the data and the governing equations.

Page 91


Also due to uncertainties in measurements in a region where chaos is present, one must instead let the

simulation work out how the input distributions change as time passes. This concept can be summarized

easily in the below flowchart.

Initial distribution ξ Simulation MCAS Result distribution χ

Compare and deduce

Here the initial distribution is sampled in a Monte Carlo fashion and thus generates the resulting distribution.

By comparing the distributions before and after the effects of time we can, if used correctly, deduce the answer

to our questions.

Let us consider a input probability distribution of orbital elements of a parent body, ξOrb ∈ RN . We will

propagate that parent body and release particles until the sublimation process is completed. The sublimation

is however dependant on the characteristics of the body, which themselves can be distributions, ξPB ∈ RM .

Lastly we have the distributions of ejection direction, ξDir ∈ R3, and of particle mass ξM ∈ R1.

This can be seen as the complete set of input distributions. After ejections are performed the set of initial

conditions for the ejected particles are input to the long term propagator generating our output distribu-

tions. Any later calculations done on these parameters, such as cluster analysis or coordinate changes, only

transforms the output distributions further. Of course one can set some of the parameter distributions to

Dirac-delta function thus eliminating it as a variable but it is still a input distribution even though its

constant nature.

Even though not essential for our understanding of the results we need to consider the fact that the distri-

bution of ejected particles in phase space is directly dependant on the distributions of the parent body. This

can be seen as a function dependant on stochastic variables such as ηParticles itself is a distribution

ηParticles(ξOrb, ξPB, ξDir, ξM) ∈ R6. (29.21)

The next step is, as mentioned, to compare and deduce. To explain how such a deduction can be done by

comparison of distributions we shall consider a few general examples. In the following examples P (statement)

denotes the probability of statement, E(distribution) denotes the expected value of a distribution, or the

mean of a sample, and # means ”the number of”. In the following examples we will refer to a demo simulation

of the software done with the Pan-STARRS cometary database, the simulation was performed using:

• 500 Generated meteor showers

• Ejected in the year 1800

• Propagated until 2200

• Meteoroid passing within 0.01 AU considered a Meteor

• Standard physical characteristics for comets

• Each sublimating comet ejects a maximum of 8000 particles

This simulation is just a demo simulation and was performed while the software was till under construction

and is therefore not a complete case study of nay kind, instead it serves as a illustrative example. Let us

begin with the simple question: What parent bodies produce meteor showers?. The answer to this question

is trivial as we need only select all parent bodies in the output data that ejected a particle which encounters

Earth. Using our example simulation we produce the distributions in figure 34. This plot suffers from

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Figure 34: The histograms describing the orbital parameters of generated parent bodies. Blue histogram bars

represent the initial distribution while red shows the resulting distribution that generated meteor showers on

Earth.

the previously discussed problem of simple histograms not showing the true multidimensional distributions,

however in this example the visualization is sufficient.

Even simpler questions can give definitive answers such as: What is P (A comet produces a meter shower)

given our initial distribution? This can be found approximatively by our simulation as,

P (A comet produces a meter shower) ≈ # Generated showers

# Generated comets. (29.22)

In the future the ≈ shall be omitted as we define the numerical probability as the statement dependant on

the simulation results. This straight forward reasoning can be extended to several other question. Given our

demo simulation, we can find that the results are:

• 500 Generated meteor showers

• 995 Generated meteoroid streams

• 3370 Generated comets

• 7801548 Generated particles

• 29598 Earth encountered particles

We can translate this into numerical probability as

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P (A comet produces a meteoroid stream) =# Generated streams

# Generated comets= (29.23)

= 29.5%,

P (A comet produces a meter shower) =# Generated showers

# Generated comets= (29.24)

= 14.8%,

P (A cometary meteoroid stream encounters earth) =# Generated showers

# Generated streams= (29.25)

= 50.2%,

P (A particle of cometary origin encounters earth) =# Earth encountered particles

# Generated particles= (29.26)

= 0.38%.

The probability given in equation 29.26 is a useful measure for future simulation runs. Using that fraction

determine that to achieve a resolution on Earth encountered particles of 100 000 data points, we will need

to generate around 100000/0.0038 = 26315789 particles.

Here we should also address the apparently high amount of streams that seem to encounter the Earth and

what is considered a ”shower”. In equation 29.25 we can see that 50.2% of all generated meteoroid streams

in this example using the Pan-STARRS distribution produced a meteor ”shower” on Earth. This result will

need to be considered with four things in mind:

1. Only 3 370 points where drawn from a initial sample of 27 501 comets.

2. There are only short period comets in the initial sample.

3. A ”stream” is here defined as all particles from a single parent body, no matter the dispersion of said

stream

4. A meteor ”shower” is defined as a close encounter, regardless of the shower intensity

As we can see here, when calculating such probabilities one should be careful to draw conclusions before

the definition bias has been addressed. We can say nothing about the intensity of the stream encounters

contributing to the 50.2% as it could as well be a single meteoroid hitting the Earth at one time and it would

count as if that stream produced a meteor shower in the simulation software. Also, the stream could be in a

region with such low e-folding time, i.e. with high dispersion rate due to chaos, that the stream would after

just 100 years be spread over almost the entire Solar system and should not be considered a ”stream” any

more but rather sporadic background. All these considerations are useful when defining better and more

precise measures of data analysis. Let us consider the two last points concerning the stream and shower

definitions and implement them in a improved cometary stream encounter probability calculation as

P (A cometary meteoroid stream produces meteor shower) =

=# Generated showers with > 10 test particles and mean DSH < 0.4/century comet divergance

# Generated streams=

= 20.2%. (29.27)

In equation 29.27 we can see that if we constrained the streams so that they could not dissipate large amounts

over a century and if we required at lest 10 test particles out of 8 000 to encounter the Earth we reduced the

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Figure 35: The distribution of stream to parent body mean divergence among the Pan-STARRS distribution

of comets.

Page 95


probability to 20.2%. We have however not yet explained the concept of mean comet divergence measured in

DSH . Here we have calculated the gradient per century of the encountered particles parent body divergence,

previously shown in figure 32 and 33, and taken the mean of each curve. This gives a mean divergence

per century for each individual particle in a shower. Then we have taken the mean of the particles mean

gradient in a single shower and we call that the mean criterion divergence per century of the shower. This

is a relativity crude method mostly developed for stream dynamic exploratory reasons. The histogram of all

these means are shown in figure 35.

There are many more easily calculated results using the generated output data, depending on the application

of the software. One general formula for deriving a probability based on the simulation is

P (Event X occuring) =# Simulations containing event X

# Simulations. (29.28)

This is the reasoning we used to find the expressions for equations 29.23 and 29.23 as each simulation

generated one comet.

Figure 36: Probability of encountering a random cometary meteoroid stream ejected in the year 1800 in a

given year.

Such a concept is indeed very basic but can be extended to a more sophisticated examination by using the

same equation as in 29.28 but distributed over a parameter, like for example time. To illustrate this, we

can use the same model simulation as before but instead check what the probability is that a stream ejected

from a random comet encounters Earth as a function of time, as can be seen in figure 36. In this figure

equation 29.24 was used but on every individual year instead, counting the number of comets that generated

a shower that year of all the generated ones. Obviously this is not proper probability distribution function

over time but rather a collection of individual probabilities. This is due to the fact that the same shower can

be counted for more then one year as it can be a reoccurring shower, if one where to remove the shower when

it encounters once, the diagram would represent a proper probability distribution function that integrates

to 1.

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Figure 37: The mean time after ejection to encounter for each meteoroid stream.

Figure 38: The variance from the mean of time after ejection to encounter within a each stream.

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As the setup of the software is not a specialized simulation of a particular parent body but instead represents

a wide population the results does not predict shower occurance times and to extinguish different showers

in the data is not useful. However, the fact that the probability quickly rises to a steady probability of

encounter indicates how cometary streams spread in the Solar system. It would seem since around half of

every stream, as calculated with 29.25, within 400 years dissipates and encounter Earth one can assume a

very different rate of dissipation among different cometary streams. Of course this graph is also affected by

the number of particles generated since the entire initial distribution was not sampled with high frequency,

only 995 points for a wide high dimensional distribution, but it is enough as a demonstration tool. To find

out more about how and why this graphs seems to flat out, we can compare the ejection time with the

encounter time. To do this in a meaningful manner one should not mean over every encountered particles

arrival time since that would mean mixing particles of different origin. Instead we will mean within one

generated shower produced by only one comet, and examine the distribution of means, , illustrated in figure

37, and standard deviations, illustrated in figure 38, of the internal arrival time.

From this we can see that the initial distribution of comets can be split into two families focusing on the

recurrence of the meteor showers. In figure 38 we can see a clear peak around 0 standard deviation in

encounter time internally for showers. We can also observe a much wider distribution centred on 80-100

years. This indicates that there is one group of meteor showers that occur only once, thus having around 0

in time deviation internally, and the other population is of reoccurring meteor showers, either reoccurring

over many years or returning few times with large times between shower returns.

This result also illustrates the concept of a sporadic complex very well as the streams spread very differently

but still over half of all streams encounter the Earth at one point or another, as can be seen from equation

29.25. In the final part, such methods as illustrated here shall be used to arrive at real conclusion regarding

a case study. They will be applied on other types of simulations, such as orbital clones sampling the

measurement uncertainty of a single comet and deriving the most probable encounter time for the Earth

observed meteor shower.

To complete this probabilistic picture we need to further also solidify the concept of a mean simulation.

Since the Monte-Carlo fashion of sampling here does not draw the entire sample at once but rather draws

only one point and runs the numerical methods on that single point, we call this a single simulation. When

mentioning a mean simulation we thereby refer to the summarization of all simulations and mean forming

over the number of Monte Carlo iterations. This will provide a type of mean simulation. This concept is

more suited for distribution representing clones of the same object or a very small population, for example

trying to derive a ”mean cometary meteor shower” is not useful since most observed meteor showers are very

different in nature, also the initial distribution of comets is wide and thus a mean will not provide useful

data. However, what can be interesting to examine it not the specifics of the shower itself but rather the

mean dissipation of the meteoroid streams that encounter the Earth, and the mean ”density distribution”

of the generated meteor shower. This will give a good idea on what kind of deviations from the sporadic

background to look for when searching for new meteor showers in observational data.

Also, one should always check the convergence of such a mean simulation. And the convergence should be

specified to the application. For example, if simulating a meteor shower in the year 2000 and 2001 at the

same time. The convergence of the mean configuration distribution for the 2000 meteor shower could occur

well before the convergence for the 2001 meteor shower distribution since particles may not propagate in the

same numbers to this year.

To conclude, the manner in which one should analyse the data and what comparisons should be done depends

entirely on the type of simulation that is performed. However the idea described here are a few of the general

probabilistic and statistical methods that apply to all output data, and can be applied on most of the output

parameter space.

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29.10 Configuration

To streamline this work we have omitted listing all code implementation choices, however to provide a more

complete picture of the constructed software we have below included a sample configuration file. The below

file can be used to run the software module as the comments are omitted by the phrasing function.

Listing 3: MCAS Configuration example

1 #%%%%%%%%%%%%%%%% CONFIGURATION FILE %%%%%%%%%%%%%%%%

2 # Lines starting with # are ignored

3 #----------------------------------------------------

4 #Calculation settings

5 #

6 #Integrators:

7 # 0: CMS

8 # 1: mercury6

9 #

10 #Solarsystem IC generator:

11 # 0: JPL (invariable plane, no equanox orientation)

12 # 1: JPL SPICE-c (J200.0)

13 #

14 #Family:

15 # 0 = NEOs (~270k)

16 # 1= MBOs (~100k)

17 # 2= MBOs --> NEOs (~50k)

18 # 3 = Hildas (~1.8k)

19 # 4 = Comets (~28k)

20 # 5 = Hyperbolic (~8.3k)

21 # 6 = LPCs (~9.4k)

22 # 7 = Centaures (~60k)

23 # 8 = Scattered (~11k)

24 # 9 = TNOs (~48k)

25 # 10 = BOTKKE NEO’s

26 # 11 = Statistical uncententy orbital clone generator (SUOC)

27 #

28 #Particle mass distribution:

29 # 0 = GRUN particle model (only min mass)

30 # 1 = Uniform representation, can also be single size if min = max

31 # 2 = 10 logarithmic uniform representation (equal probabillity for each exponent)

32 # 3 = Custom file (mass_dist.txt), first line masses, second line their respective

probabillites

33 #

34 #Particle mass distribtuion resolution:

35 # The resolution of the resulting histogram used in generating the random particles

36 #

37 #Encounter end date (Julian days):

38 # If not zero, then ignore integration time and use this date instead allowing

variable integration time

39 #

40 #Distribution type:

41 # 0: Pick random bodies from the supplied set

42 # 1: Create histogram over orbital elements and pick random combinations

43 # 2: Create multidimensional map over distribution of supplied set and draw a set of

coordinates from this

44 # 3: Reduced multidimensional map using principal component analysis

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45 #

46 #

47 #Planet mass input format:

48 # 0: kg

49 # 1: AU**3/DAY**2

50 #

51 #

52 #X types:

53 # 0: Showers

54 # 1: Clones

55 #----------------------------------------------------

56 Integrator =1

57 Solarsystem IC generator =1

58 Start date (Julian days) =2424861.2071

59 Integration time for encounter (years) =0

60 Encounter end date (Julian days) =2458849.5

61 Family =11

62 Distribution type =0

63 Calculate clusters =1

64 Calculate stream dissipations =0

65 Particle mass distribution =2

66 Particle mass distribtuion resolution =100

67 Lower mass bound (g) =1e-6

68 Upper mass bound (g) =1e2

69 Memory allocation (Mb) =1e+03

70 X type =1

71 Number of X to generate =50

72 Close enoucnter treshold( hill radii) =1.2

73 Planet mass input format =1

74 Snapshot date =0

75 #----------------------------------------------------

76 #OUTPUT settings

77 #

78 #Logfile:

79 # 0 = No file

80 # 1 = Write file

81 #----------------------------------------------------

82 Logfile output =1

83 #----------------------------------------------------

84 # mercury6 settings

85 #----------------------------------------------------

86 #Mercury integrator:

87 # 0 = Hybrid

88 # 1 = BS

89 #

90 #----------------------------------------------------

91 Mercury integrator =0

92 Time setp (days) =1

93 Include PR effect =1

94 Ejection distance (AU) =100

95 Hybrid changeover (hill radii) =5

96 Max particles in mercury.inc =10000

97 #----------------------------------------------------

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98 # SUOC settings

99 #----------------------------------------------------

100 # Sigma range:

101 # Normal distribution sigma range

102 #

103 # Distribution resolution:

104 # Normal distribution numerical resolution i.e.

105 # the number of bins.

106 #

107 #----------------------------------------------------

108 Number of clones =5000

109 Sigma range =5

110 Distribution resolution =5000

111 %%%%%%%%%%%%%%%%%%%% END OF FILE %%%%%%%%%%%%%%%%%%%%

30 Parent Body Ejector module

The Parent Body Ejector module is designed to simplify the creation of long term initial condition generation.

Often authors will eject particles only at perihelion and in only two directions, as in (Vaubaillon et al., 2011),

thus creating a easily scripted generation of initial conditions for propagation. Due to the cumbersome nature

of generating more complex initial condition continous ejection in different direction at different speeds are

not considered, practically omitting much of the in-stream structure while only looking at the extremes.

This module will make generation of such initial conditions easy where several ejection models are already

available, parent body parameters are simply input and the output is a simulation state once ejection of

particles is complete.

30.1 Symplectic structure

Symplectic integrators are a scheme for numerically integrating a specific group of differential equations con-

nected to symplectic geometry that is a branch of differential geometry and differential topology. Since this

type integrator works on geometrical basis it is excellent for integrating physical problems such as classical

mechanics where the phase space of a system can take the form of a symplectic manifold. In our case the set

of differential equations are the Hamiltonian with the two-form dpi∧dqi, where qi is a generalized coordinate

and pi its momentum. These two elements together create our canonical coordinates, phase space, and the

symplectic integrator can be viewed as canonical transformations.

A example of the difference between preserving symplectic structure and not doing so in a numerical inte-

gration can be seen in figure 4 and 5. In these pictures are shown what such a simple modification as to

make the method semi implicit can do for a numerical long term integration of a Hamiltonian system.

30.2 Propagation

As only relatively short time spans should be integrated using this module we have so far implemented only

two different integration methods, the Bulirsch-Stoer method described in section 5.3, and a symplectic 8

part Hamiltonian split method. The split concept and order determination was described in section 5.2.2.

The particular symplectic split implemented originates from a newly discovered family of splits (Blanes et al.,

2013), where we have picked the 8 order H = Ha + Hb splitting, where Ha is the kinetic energy and Hb is

the potential. This split can be written as

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Ha1a Hb1

b Ha2a Hb2

b Ha3a Hb3

b Ha4a Hb4

b Ha5a Hb4

b Ha4a Hb3

b Ha3a Hb2

b Ha2a Hb1

b Ha1a (30.1)

with numerical values of the coefficients as (Blanes et al., 2013)

a1 = 0.03809449742241219545697532230863756534060

a2 = 0.1452987161169137492940200726606637497442

a3 = 0.2076276957255412507162056113249882065158

a4 = 0.4359097036515261592231548624010651844006

a5 = −0.6538612258327867093807117373907094120024

b1 = 0.09585888083707521061077150377145884776921

b2 = 0.2044461531429987806805077839164344779763

b3 = 0.2170703479789911017143385924306336714532

b4 = −0.01737538195906509300561788011852699719871

(30.2)

This means that we shall, in each step, apply the separate Hamiltonians Ha and Hb each with the total

time step modified by the above coefficients. In other words, we apply the effect of kinetic energy by a time

a1Tstep, then we apply the effect of the gravitational pull of all bodies by a time b1Tstep, and so on.

Both of these methods have electromagnetic corrections implemented following the form from (Burns et al.,

2014),

mpdv

dt=

(SAQpr

c

)((1− v • S

c)S− v

c

), (30.3)

where mp is the particle rest mass, A the geometrical cross-section of the particle, Qpr the radiation efficiency

factor, and S is the energy flux density in the radiation beam. In our implementation we have set the Qprcoefficient to 1 as that is a reasonable assumption and used standard mean luminosity coefficients for the

Sun as given by (Mamajek et al., 2015).

30.3 Period syncing

When designing a propagator one cannot assume all inputs to be initially time synchronized, thus we shall

consider two possibilities; that the initial state of the Solar system is given at a different time than that of

the parent body, and that the initial state of the parent body is inside the sublimation sphere. To account for

the first possibility we will simply employ a method that propagates the Solar system forwards or backwards

to match the time of the parent body.

To resolve the second problem is somewhat more difficult due to the fact that we cannot know the parent

body propagated backwards in time will have a stable orbit. As such, one should be able to chose between

two different algorithms. One that propagates backwards until periapsis for short term stable orbits, and

one that simply propagates backwards until the parent body is outside the sublimation radius, with some

termination condition if the propagation goes on for too long for general trajectories. There is also the

possibility that the orbit never leaves the sublimation sphere, if that is the case the algorithm should stop

at a apsis point.

The first kind can be summarized by:

FIND APHELION

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INPUT : Simulation state at t1

OUTPUT : Simulation state at t0 < t1

Find approximate orbital period of PB: TPB = 2π

√a3PB

µSun

Current time simulation time is t

Begin backwards propagation: WHILE t− t1 < TPB

Propagate the state one step : t 7→ t−∆t

Check if the orbital motion has turned : IF (Previous |rPB • vPB|)(Current |rPB • vPB|) < 0

Break while loop

END IF

END WHILE

return Simulation state at: t0

And the second can be written as

FIND APHELION

INPUT : Simulation state at t1

Termination condition Γ

OUTPUT : Simulation state at t0 < t1

Current time simulation time is t

Begin backwards propagation: WHILE Γ

Propagate the state one step : t 7→ t−∆t

Check if outside sublimation sphere : IF |xPB − xSun| > Rsublim

Break while loop

END IF

END WHILE

return Simulation state at: t0

In the above algorithm the termination condition Γ will vary depending on the application. It can be a max

time Γ = t− t1 < Tmax, or it can be some orbital element condition Γ = aPB > amax, thus this will have to

be changed from application to application. This dynamic change has not yet been implemented so currently

Γ = t− t1 < 0.75TPB.

30.4 Ejection speed

To proceed further with the ejection of particles, we need to calculate the ejection speed of the particles

from the parent body. In section 10.2.1 we have defined several models and more thoroughly described their

derivation, thus here we shall only give the equations implemented in the software:

• (I) Model from (Whipple, 1951)

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• (II) Model from (Hughes, 2000)

• (III) Model from (Ma et al., 2002)

• (IV) Linear heliocentric model

The formula for (I) is given by,

V 2∞ =

kd2

McvgπRc

A

m− 2GMc

Rc, (30.4)

where the comet mass loss in the case of a spherical body is

Mc = πR2cS0

(r0r

)2 1

nH. (30.5)

The used coefficient values are, kd = 26/9 for the particle gas drag, sublimation heat of H = 1.88 · 106 J/kg,

and vg is derived from the mean thermal velocity of gas

vg =

(8kBTgπµ

)1/21

rτ(30.6)

where we assume µ = 20 · 1.661 · 10−24 g, Tg = 300r−1/2 K, and τ = 0.25. The values of Mc and Rc are

inputs of the parent body characteristics

For the ejection model in (II), winch is a modification of (I), the ejection speed is given by

V 2∞ =

2Mc

ΦπRc

(θvsrτ

)ξA

m− 2GMc

Rc, (30.7)

where the mass loss is given as

Mc = 2gπR2cS0

(r0r

)2 1

nH, (30.8)

and the thermal velocity of the gas is

vg =

(8kBTgπµ

)1/2

. (30.9)

Using this model we have chosen static coefficients for the surface absorption factor g = 0.1, the free drag

coefficient ξ = 2.0, the adiabatic acceleration θ = 2, τ = 0. The coefficients n and H are as in model (I).

The coefficient Ψ, or the surface activity factor, is a input of the parent body characteristics and A and m

are given by the input bulk density and mass distributions.

The ejection speed formula used in model (III) is given by

v2 =WRcL8πHαsσ

(1

r2− 1

r2s

)− 2GMc

Rc, (30.10)

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and the cometary mass loss is given by

Mc =R2cL4H

(1

r2− 1

r2s

). (30.11)

Here the mean thermal velocity is set to W = 580 m/s, the Solar luminosity L is chosen as previously, the

latent heat of vaporization H = 2 · 106 J/kg. In this formula the input fraction of the comet that is active is

α. s is the radius of the assumed spherical dust grain, σ it’s bulk density, and rs is the critical heliocentric

radius for sublimation to occur, all of which are dominated by input data. In all the above formulas G is

the gravitational constant. All these parameters and the models are discussed in (Ryabova, 2013).

Lastly the formula for (IV) is simply a increase in ejection velocity which is linear in the heliocentric distance,

starting from 0 m/s at the input rs and peak at a input parameter at perihelion.

In the future we plan to implement full control over the model coefficients by creating configurations files

for each model and also to implement more models. The ability to chose these coefficients as distributions

is also planned for future versions. This can be very useful since one can examine how subtle differences

propagate through time, such as the value of the sublimation. A variable that is dependant on what ices are

present on the comet.

30.5 Ejection direction

Figure 39: Two spherical distributions, the left illustrating a spatial angle uniform distribution, the right

illustrating a true spherically uniform distribution.

The question of how ejection points on the sphere representing the parent body is picked is a relevant one due

to several factors. The main reason is that there is a sensitivity in the resulting deviation from the parent

orbit to the direction of the initial perturbations. For example in cometary ejections due to sublimation

classical dust stream simulations (Vaubaillon et al., 2011) are commonly set-up with only two perturbation

directions, both tangent to the orbit and in the orbital plane (Sato, 2003).

However there are many advantages to use a random uniform distribution on the sphere. One such is

Page 105


the ”filling” of the dust stream such that the entire range of possible perturbations are sampled, and not

only the extreme perturbation cases. It may be argued that a uniform distribution is unrealistic due to

jets and non active areas of cometary nuclei. The individual shape and activity of the individual simulated

comet can however be ignored if repeated simulations are performed on a wide population. The activity

configuration can also be ignored due to the spin of the comet. There are many mechanisms that can alter

the spin state of a nucleus, most are stochastic in nature or have large time scales. However the out gassing

torque is the primary mechanism of spin-state change (Samarasinha et al., 2004). Since out gassing is the

main effect for producing dust streams we know that this effect is present in simulations of cometary dust

streams.

One cannot simply create a uniform distribution by picking the polar coordinates θ and φ from uniform

distributions since the spherical area element dΩ = sin(φ)dθdφ is itself dependant on φ. Instead one can

use the rules of quaternion transformation to transform uniform distributions U(a, b) to a uniform US(r)

spherical distribution (Marsaglia et al., 1972). This can be done by using the four dimensional stochastic

vector

x =

ξ0ξ1ξ2ξ3

, (30.12)

ξi ∈ U(−1, 1) ∀ i ∈ [0, 3], (30.13)

and restricting this distribution to only

x • x < 1. (30.14)

Then the resulting distribution that is left can be transformed to the surface of the sphere by

x =2(ξ1ξ3 + ξ0ξ2)

x • x, (30.15)

y =2(ξ2ξ3 − ξ0ξ1)

x • x, (30.16)

z =ξ20 + ξ23 − ξ21 − ξ22

x • x, (30.17)

Y =

xyz

∈ US(1) (30.18)

The resulting distribution can be seen in figure 39 together with the non uniform distribution defined by

uniformly distributing the polar angles.

30.6 Particle mass

In section 29.8 we covered the types of input mass, let us now address how these distributions are used. First

we need to find to define a mean particle mass, to be used in the mass timing ejection calculation. Given a

vector with masses m and a vector of the same size with the corresponding probabilities Pm for each mass

to be ejected. If the probability vector is normalized we can write the mean mass as a simple scalar product,

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µm = m •Pm. (30.19)

As we draw masses from Pm the mean of the masses will converge to µm with sufficient high draw counts.

This indicates that with a sufficient amount of particles per ejection, N , this can estimate the amount of

mass ejected as Nµm.

30.7 Ejection times

Now that we have defined at what speed the particles are to be ejected, and in what directions, finally

we have to address at what times the ejections are to be. This is less straight forward then expected as

depending on the amount of test particles one cannot eject at each discrete time step of the simulation, since

one sometimes want very small accurate time steps while ejecting few particles. Thus we have chosen to

implement a type of cumulative ejection, where we have as user input a maximum amount of ejected test

particles and a maximum amount of ejection instances. This provides relatively large ejection configuration

control.

However, since the mass loss of the comet increases as it approaches the Sun it will eject more meteoroids

with higher velocities. Thus one would in most application want more precision in ejections around the area

of highest mass loss and largest ejection velocities.

To fulfil these requirements we will first propagate the synchronized system once without ejection particles.

During this propagation we only calculate the mass loss of the comet in each step to estimate the total

mass loss of the comet. Since few body propagation for short time frames such as one or two orbits is very

fast with modern computers this can be done with little computational cost. Using this total mass loss of

the comet we shall, given the input ejection instances, choose a mass per ejection limit. When the second

propagation starts we calculate the mass loss in each step, and add it to a cumulative value. Once this value

exceeded the mass per ejection limit, a set of particles are ejected. The amount of particles in each ejection

depend on the maximum number of particles and the maximum number of ejection instances. Why we refer

to this as a ”maximum” number is due to the fact that the first integration was an estimation, and some

ejections may fail. Also since the particles are randomly generated from mass distributions it may occur

that small variations in mass ejections may occur if too few test particles are chosen. Lastly, a possible

variation in ejection particles and instances can occur if the particles generated are unable to escape the

comets gravitational field. Since the mass distribution input allows for any mass there can be, depending on

the ejection model, some masses that are not accelerated enough by the outflow of the gas to reach escape

velocity from the comet. If this happens, the simulation will find that the particle could not escape and

draw a new random mass for this particle. If this happens too often, the simulation will be slowed down too

much and it will skip that specific ejection moment. But it will not completely omit the mass loss that the

comet has experienced, it will instead propagate one step, increase the cumulative mass loss, and attempt

to eject particles again with the new increased mass loss. Thus we shall not lose mass loss representation

if the particle mass range cannot be expressed computationally efficiently at a point but instead eject more

particles once we have statistically enough ejection of that particle mass to represent the ejection given the

number of allowed particles. Below are psudo code algorithms for the discussed procedures.

ESTIMATE MASS LOSS

Page 107


INPUT : Initial state:(qi,pi)|t=t0 ∀ i ∈ [1, N ]

Integration step: ∆t

Max particles: Pmax

Max ejections: Emax

Parent body data: G

OUTPUT : Mass loss per ejection: Me

Particles per ejection: Pe

Initialize total mass loss: ∆M = 0

Initialize propagation: t = t0

Estimate mass loss: WHILE t < t1

Propagate one step: t 7→ t+ ∆t

Calculate mass loss: M = SUBLIM(qPB,pPB,G)

∆M = ∆M + M

END WHILE

Estimate mass loss per ejection: Me =∆M

Emax

Particles per ejection: Pe =Pmax

Emax

CUMULATIVE EJECTION

INPUT : Initial state:(qi,pi)|t=t0 ∀ i ∈ [1, N ]

Integration step: ∆t

Mass loss per ejection: Me

Particles per ejection: Pe

Parent body data: G

OUTPUT : System state after ejection: (qi,pi)|t=t1 ∀ i ∈ [1, N +NP ]

Page 108


Initialize cumulative mass loss: ∆M = 0

Initialize total mass ejected: Mt = 0

Initialize propagation: t = t0

Estimate mass loss: WHILE t < t1

Propagate one step: t 7→ t+ ∆t

Calculate mass loss: M = SUBLIM(qPB,pPB,G)

∆M = ∆M + M

IF ∆M > Me

Eject particles: EJECT(Pe)

Check if signifficant ejection possible: IF Enough ejections sucseed

Add ejected particles to simulation state: ADD PARTICLES

Add ejected mass to total: Mt = Mt + ∆M

Reset cumulative mass loss: ∆M = 0

END IF

END IF

END WHILE

Lastly, we can estimate the particle weight Wp as

Wp =Mt

NPµm, (30.20)

where NP is the number of generated particles, Mt is the total ejected mass in particles, and µm is the mean

particle weight.


30.8.1 Input files

Table 27: mass.data,


Input File description: Simulation state mass vector


1 Object mass Double kg -

Table 28: pos.data,


Input File description: Simulation state position vector, one row for each object


1 x-component Double m -

2 y-component Double m -

3 z-component Double m -

Page 109


Table 29: vel.data,


Input File description: Simulation state velocity vector, one row for each object


1 x-component Double m/s -

2 y-component Double m/s -

3 z-component Double m/s -

Table 30: mass_dist.data


Input File description: Ejection mass distribution


1 Bin center value Double grams -

2 Bin probability Double - Does not need to be normalized

Table 31: body_data.data


Input File description: Parent body data


1 Position - - -


1 x-component Double m -

2 y-component Double m -

3 z-component Double m -


2 Velocity - - -


1 x-component Double m/s -

2 y-component Double m/s -

2 z-component Double m/s -


3 Properties - - -


3 Nucleus radius Double m -


3 Nucleus activity factor Double 1 -

3 Critical sublimation radius Double m heliocentric

3 Nucleus mass Double kg -

3 Perihelion passage difference Double days Relative system initial state

Page 110


30.8.2 Output files

Table 32: particle_q_data.data,


Output File description: Simulation state position vectors, one row for each object


1 x-component Double m If equatorial J2000.0 equinox: Vernal equinox component

2 y-component Double m Perpendicular to equinox in plane

2 z-component Double m Perpendicular to reference plane

Table 33: particle_v_data.data,


Output File description: Simulation state velocity vectors, one row for each object


1 x-component Double m/s If equatorial J2000.0 equinox: Vernal equinox component

2 y-component Double m/s Perpendicular to equinox in plane

2 z-component Double m/s Perpendicular to reference plane

Table 34: particle_m_data.data,


Output File description: Simulation state mass vector, one column for each object


1 Object mass Double kg -

Table 35: particle_t_data.data,


Output File description: Simulation ejection time vector, one column for each object


1 Time of ejection Double s Relative simulation start

Table 36: particle_abs_v_data.data,


Output File description: Simulation ejection velocity vector, one column for each particle


1 Heliocentric distance Double AU -

2 Ejection velocity relative parent body Double m/s -

Page 111


Table 37: particle_kep_data.data,


Output File description: Kepler data after ejection, one row for each object




3 Inclination Double rad -

4 Argument of Perihelion Double rad -

5 Longitude of Ascending Node Double rad -

6 True anomaly Double rad -

7 Mean anomaly Double rad -

8 Argument of Latitude Double rad -

9 True Longitude Double rad -

10 Longitude of Perihelion Double rad -

11 Semilatus Rectum Double m -

Table 38: body_m_data.data,


Output File description: Simulation state parent body mass, one row for each instance


1 Time Double days Relative simulation start

2 Mass Double kg Change varies with sublimation model

Table 39: sim_data.data,


Output File description: Simulation summary data


1 Particle representation weight Double 1 Same for all particles from this simulation

1 Total simulations time Double s -


1 Parent body end mass Double kg -

1 Time synchronization required Double s From input start to PBE start

1 Orbits simulated for ejection Double 1 -

30.9 Configuration




Listing 4: PBE Configuration example

1 #%%%%%%%%%%%%%%%% CONFIGURATION FILE %%%%%%%%%%%%%%%%%

2 #-------------------------------

3 #Simulation settings

4 #-------------------------------

Page 112


5 #Ejection model:

6 # 0 = Ma et al. 2002

7 # 1 = Hughes et al. 2000

8 # 2 = Whipple 1951

9 # 3 = from 0 to input at perihelion

10 # 4 = Only at perihelion

11 #

12 #-------------------------------

13 Ejection model =1

14 Time step (days) =16

15 Number of orbits =1

16 Max Particles =8e+03

17 Maximum number of ejection instances =20

18 Print precision =20

19 Output type (helio 0 /bary 1) =0

20 Max velocity for model 3 & 4 [m/s] =30

21 %%%%%%%%%%%%%%%%%%%% END OF FILE %%%%%%%%%%%%%%%%%%%%

31 Orbital Association Analysis module

The Orbital Association Analysis module is designed to implement a multitude of methods to associate orbits

and analyse the results. One common approach we have implemented is the use of orbital similarity function

to create associations and cluster analysis to examine the made associations. In the future this will become

a platform for testing novel association and analysis methods. Such novel methods could be topological

similarity functions rather than function based on orbital element differences, or machine learning methods

rather then static density analysis or wavelet coefficient thresholds.

In this section we will cover the current implementation state of the OAA module and define what methods

are implemented and in what fashion.

31.1 Similarity functions

We have chosen to define similarity functions such that 0 indicate complete similarity and the maximum

value of the function indicates complete dissimilarity. In the current version we have implemented 4 similarity

functions:

• Standard D-criterion: DSH

• Normalized D-criterion: DD

• Holder p=2 Metric: ρ2

• Euclidean composite Metric: %1

The actual implementation of the functions are described below. The variables q, a, e, i, ω,Ω refer to the

standard orbital elements unless otherwise stated.

The Southworth and Hawkins D-criterion, DSH (Southworth and Hawkins, 1963), is defined as;

D2SH = (eb − ea)2 + (qb − qa)2 + (2 sin

Iab2

)2 +

(ea + eb

22 sin

Πab

2

)2

, (31.1)

Page 113


where

(2 sinIab2

)2 = (2 sinib − ia

2)2 + sin ia sin ib(2 sin

Ωb − Ωa2

)2 (31.2)

is the angle between the orbital planes and

Πab = ω2 − ω1 + 2 arcsin

(cos

ib − ia2

sinΩb − Ωa

2sec

Iab2

)(31.3)

is the difference between the longitudes of perihelion measured from the intersection of the orbits. The

Drummond D-criterion, DD (Drummond, 1980), a more correctly normalized version of DSH ;

D2D =

(eb − eaeb + ea

)2

+

(qb − qaqb + qa

)2

+

(I

180

)2

+

(ea + eb

2

Θ

180

)2

, (31.4)

where

I = arccos (cos ia cos ib + sin ia sin ib cos (Ωb − Ωa)), (31.5)

and

Θ = arccos (sinβa sinβb + cosβa cosβb cos (λb − λa)). (31.6)

The ecliptic latitudes of the perihelion points can be expressed as

βa = arcsin (sin ia sinωa),

βb = arcsin (sin ib sinωb),

(31.7a)

(31.7b)

and the ecliptic longitudes of the perihelion points as

λa = Ωa + arctan (cos ia tanωa),

λb = Ωb + arctan (cos ib tanωb).

(31.8a)

(31.8b)

The λ parameter has the additional condition of adding 180 if cosω < 0. Since the only difference between

DSH and DD is a normalization procedure, it is interesting to examine the different association patters due

to such a change of the similarity function.

Then we also implemented a Holder type metric acting on the space of oriented Keplerian ellipses with

positive semi major axes and eccentricities E, where the case zero eccentricity produces a oriented circle

without marked pericenter. This metric was constructed (Kholshevnikov and Vassiliev, 2004) and is described

by

ρ(p,E,E′) = min

(1

2π

∫sp(Q(u), Q′(u+ v))du

)1/p, (31.9)

for mapping E × E onto R+. In the special case of p = 2 that we have chosen to implement; the metric

integration reduces to

Page 114


ρ2(2, E,E′) = 2aa′(W0 −

√(W5 +W8)2 + (W6 −W7)2

), (31.10)

where the W coefficients are found by

4W0 = 2(α+ α′) + αe2 + α′e′2 − 4P ·P′ee′,

W1 = P ·P′e′ − αe,W2 = P′ · Se′,W3 = P ·P′e− α′e′,W4 = P · S′e,2W5 = −P ·P′,2W6 = −P · S′,2W7 = −P′ · S,2W8 = −S · S′.

(31.11a)

(31.11b)

(31.11c)

(31.11d)

(31.11e)

(31.11f)

(31.11g)

(31.11h)

(31.11i)

The P and S vectors for each orbit is found by

P =

cosω cos Ω− cos i sinω sin Ω

cosω sin Ω + cos i sinω cos Ω

sin i sinω

, (31.12)

S = ηQ = η

− sinω cos Ω− cos i cosω sin Ω

− sinω sin Ω + cos i cosω cos Ω

sin i cosω

, (31.13)

where η =√

1− e2, α = aa′ and α′ = a′

a and the prime denotes the second input orbit. This is a interesting

metric to implement since it function on a different principle, the integrated total orbital distance, than the

D-criterions.

The fourth and last function implemented so far is another type of metric on the space H of orbits, both

curvilinear and rectilinear. The space H does not however distinguish between ascending or descending

orbits but instead acts on fundamental orbital quantities. This function was presented in (Kholshevnikov,

2008) and its definition is

%21(ε1, ε2) =1

µ2L1|c1 − c2|2 + |e1 − e2|2 +

L22

µ4(h1 − h2)2, (31.14)

where the c and e vectors are the area vector, or the specific angular momentum vector, and the Laplace-

Runge-Lenz vector respectively. This metric can be reduced to the software implemented version of

%21(ε1, ε2) =1

L1(p1 + p2 − 2

√p1p2 cos ξ) + (e21 + e22 − 2e1e2 cos ζ) +

L22

4

(1

a1− 1

a2

)2

. (31.15)

Where L1 and L2 are scaling constants and ξ and ζ are calculated by

cos ξ = cos i1 cos i2 + sin i1 sin i2 cos (Ω1 − Ω2), (31.16)

Page 115


cos ζ =(cosω1 cosω2 + cos i1 cos i2 sinω1 sinω2) cos (Ω1 − Ω2)+

+ (cos i2 cosω1 sinω2 − cos i1 sinω1 cosω2) sin (Ω1 − Ω2)+

+ sin i1 sin i2 sinω1 sinω2. (31.17)

By comparing these four function we can perform a number of comparisons. For example the difference

between two metric approaches, the integral orbital distance versus orbital element distances, and the effect

of normalization in similarity functions.

31.2 Similarity matrix

Given a general similarity function f(xi, xj) acting on two objects xi and xj from a set of N objects

xi : 1 ≤ i ≤ N we can define the similarity matrix of that set as

Dij = f(xi, xj) ∀ 1 ≤ i ≤ N 1 ≤ j ≤ N. (31.18)

When f is a metric, the diagonal will always be zero due to the metric properties but with similarity functions

this need not be the case as objects can be non self similar, i.e. have a non zero distance to itself. This has

a calculation complexity of O(N2) but the amount of calculations can be reduced from N2 to (N − 1)N/2 if

the function is commutative, if the self similarity value is not of interest the amount of calculations can be

reduced to (N − 2)(N − 1)/2, which is still O(N2) but more effective.

31.3 Clustering

Currently only one cluster merging function is implemented, the single linkage clustering rule. This rule can

be written as

min (f(a, b) : a ∈ Xi, b ∈ Xj) < Dc, (31.19)

where the two sets that are being checked for merging is Xi and Xj , the critical clustering threshold is Dc

and the similarity function is f . If this statement is resolved as true, the two clusters being checked are

merged to a single one.

To implement this in a C++ algorithm we have constructed a recursive node walk function. Given a set of N

nodes xi : 1 ≤ i ≤ N, we can consider a function that, given a node; finds all other nodes within a critical

distance Dc of the input node. If this function then calls itself on all found nodes, we easily see that given a

node, this function will always explore the entire cluster of the given node and noting else. Thus calling this

recursive function once for each node, and having the function skip all nodes already visited once by it, will

search the entire node network. The clusters are found by having this function record every node it visits

with each calling, since one calling will completely explore one cluster and disable all nodes in that cluster

so that the function skips them, the algorithm will only visit each node once and reduce computing time.

To speed up calculations the function finds nodes that are withinDc by a boolean look-up matrix. This matrix

is the boolean value of the inequality posed on the similarity matrixGij = Dij < Dc = f(xi, xj) < Dc ∀ i 6= j.

Thus, if the input node is xk the function will call itself with node xj as input if Gkj is true.

31.4 Parameter sweep

In the parameter sweep we call the clustering function while increasing the critical threshold Dc from zero,

where in a set of N nodes there will be N clusters, to the point of termination. Currently there are 2 types

of termination points:

Page 116


1. Association maximum

2. Error function minimum

For the first termination criterion, as previously, we have here defined the amount of association in a set as

the amount of node that has a connection. For example, let us say we have 10 nodes, and 5 clusters, each

with 2 nodes in the cluster. This way all nodes are in some way associated with another node and we have

defined the association to be at its maximum, even though the clustering process is not complete, i.e. when

all nodes form one big cluster. If the first termination point is chosen, the parameter sweep will terminate

once every node has at least one connection. In the case of single linkage clustering there is also another

state to consider, one where the set of nodes and their connections form a complete graph. A complete graph

is when every node in a N set has N − 1 connections to every other node.

As the level of association to be examined can vary between different research and the time to perform the

parameter sweep is dependant on the size of the sample and the similarity function, there may be the desire

to implement ones own definition of a correct clustering. Thus we have also implemented the use of custom

error function.

31.4.1 Fixed step

As the parameter sweep searches the value space of the critical threshold it will have to step through this

space in some manner. Depending on the structure of the examined association algorithm and the similarity

function using a fixed step may be advantageous to adaptive step methods and vice versa, thus both are

included in the implementation.

Currently one can set a fixed step that the algorithm steps with until the termination criteria has been

fulfilled. There is currently a hard coded upper threshold on the number of steps that the sweep can perform

without any change in association state, once this limit has been passed the sweep self terminates. In future

implementations this will not be a hard coded value but instead a input option.

This safety measure is in place since we have found that certain data configurations using some function can

create vast differences for some nodes. For example in the case of the orbital similarity functions, using a

function where one of the dependant variables are the semi major axis instead of the perihelion distance, on

a set of orbits all with small semi major axis seems fine. However, in some cases, if a gravitational slingshot

effect occurs during the simulation sending one body out of the Solar system at very high speeds, if the Solar

system escape velocity is not exceeded and the orbit does not turn hyperbolic, the semi major axis of the

object can be any value between 0 and ∞. A fixed step would in this case essentially step forever trying to

associate this one last point if the termination criterion requires all nodes to have at least one connection.

This threshold is currently set to 1000 steps without any state change.

31.4.2 Adaptive step

The adaptive step algorithm function on the notion of state change of the evaluation function. Currently, the

evaluation function is the same as the termination criterion function, i.e. the error function or association

amount. If the state of the evaluation function does not change, i.e. no new important association are made,

the step size in increased according to some rule. The complete algorithm for step increase can be found

below.

ADAPTIVE STEP INCREASE

Page 117


1.Calculate association state using current Dc

2.Calculate evaluation function using current association state

3.WHILE Next step is not OK

3.1. Step Dc by current step toDc2

3.2. Calculate association state using Dc2

3.3. Calculate evaluation function using next association state

3.4. IF Next evaluation step differs from current

3.4.1. Next step is OK

ELSE

3.4.2. Increase current step size by rule

END IF

END WHILE

The above pseudo code may seem more ineffective than a normal fixed step but considering that the step

size increment rule can be dependant on the step size, we can find many scenarios when this is more effective

than a fixed step. For example a multiplicative function that doubles the step size each time the next step

check is performed and fails, would quickly traverse huge gaps in the parameter space.

However we immediately find a drawback; the more the step size is increased the less resolution we have,

thus we need a counter to this effect with a adaptive step decrease algorithm, described below.

ADAPTIVE STEP DECREASE

1.Calculate association state using current Dc

2.Calculate evaluation function using current association state

3.WHILE Next step is not OK

3.1. Step Dc by current step toDc2

3.2. Calculate association state using Dc2

3.3. Calculate evaluation function using next association state

3.4. IF Next step differ too much from current

3.4.1. Decrease current step size by rule

ELSE

3.4.2. Next step is OK

END IF

END WHILE

Here, the definition of the association states differing too much is very ambiguous and can be chosen very

differently depending on the features that one want to examine. We have chosen this to be defined as when

more than one cluster merging occurs. This means that if for example, after a large value of no state changes,

we in one step suddenly associate our entire sample, then the step size will continue to decrease until we can

resolve every single cluster merge.

Thus we have combined both of the above algorithms to create our adaptive step. This step algorithm is

only faster in certain situations but it has the advantage of using less saved data as only steps that changes

Page 118


the state evaluation function are saved. However due to possible computational problems the adaptive step

algorithm uses a minimum and maximum step size as well to limit run-away behaviour. If the step is changed

to this maximum or minimum value, a new step is forced instead of a step change. The reason for this can

be illustrated easily if one considers a case where one cluster in the middle of two other is about to merge

the three clusters into a single one. If the merging distance between the middle cluster and the side ones are

infinitesimally close to each other the adaptive step decrease will change the step size forever to resolve the

two different merges. Thus a minimum step size limit has to be set.

31.5 Error functions

The use of a error function is commonly used in areas such as pattern recognition and machine learning.

Its purpose is to represent the amount of error in the performed algorithm. These will be tailored to the

research to be performed and currently only two error functions are implemented. These function are to take

the output of the association as input and quantify the performance of the algorithm where a low value is

the best. The functions in the case of cluster analysis take the clusters as input and are currently defined as

Ferr =# number of nodes

# of nodes in largest cluster, (31.20)

Ferr =# of clusters

# number of nodes. (31.21)

As can be seen both of these function are at a minimum when all nodes are in a single cluster, however

they emphasise very different properties when examining their behaviour during a parameter sweep. The

function in equation 31.20 always has a minimum of 1, but instead has a maximum that is the number of

nodes. This function also decrees in error rapidly if there is a single large cluster and thus emphasises the

formation of large connected structures over many small ones. However the function in equation 31.21 has

a fixed maximum error of 1, and a minimum error of 1 divided by the number of nodes. This function does

not care about the relative node distribution among the clusters, only how many clusters there are. For

example, let us consider the case of 50 nodes in two clusters. If the node distribution in this example was

so that one cluster contained 49 of the nodes and the other cluster only contained the last node, function

31.20 would only be one step from its minimum, and so would the function in 31.21. However, if the node

distribution instead was so that each of the two clusters had 25 nodes, then 31.21 would have the exactly

same error value, but now 31.20 would only be half way between its maximum and its minimum. This

example illustrates how important it is to consider the emphasis of ones association interpretation when

considering if the algorithm did well or not.

One could for example imagine a error function that counts the number of total connections in the set and

compares this to the number of possible connections. This would instead emphasize interconnectivity of

nodes and thus the error function would decrease if a new connection between two nodes already in the same

cluster is made, even if no new clusters are merged.

The error function is a tool to examine what algorithm configurations produce desirable results, it is also a

tool to examine how the change of the input configuration changes the algorithm performance. For example

a very steep error function rapidly falling in value after a certain value may indicate some finer structure of

the data or of the similarity function that was not considered before.

31.6 Next update addition

In the next step of implementation we have planned to implement 3 new similarity functions:

• Valsecchi geocentric D-criterion DN

Page 119


• Hausdorff metric

• Custom time dependant similarity function

We also plan to implement four new cluster merge functions,

• Complete linkage: f(a, b) < Dc ∀ a ∈ Xi, b ∈ Xj

• Average linkage:(

1#Xi

1#Xj

∑a∈Xi

∑b∈Xj

f(a, b))< Dc

• Centroid linkage, if ci is centroid of Xi: f(ci, cj) < Dc.

• Ward’s linkage method

Many of these cluster merge function are more similar to density analysis since they emphasize the in-cluster

variation much more then single linkage. The advantage to clustering methods versus density analysis is

that as phase space dimensions increase the density field calculation increases exponentially while the cluster

analysis does not and is thus a much faster algorithm in essence.

Some preliminary plans have also been prepared for implementation of Big data versions of similarity matrices

and merging functions. These big data function will use compressed data files saved to the hard drive as

temporary storage of calculation data that is not to be used immediately to reduce RAM usage.


31.7.1 Input files

Table 40: OAA_input.txt,


Input File description: Node data, each row corresponds to one node

Currently only Kepler element node data input





4 Longitude of Perihelion Double Degrees NOT argument of perihelion


31.7.2 Output files

Table 41: clusters.txt,


Output File description: Cluster state where each row is one cluster

This file has variable row sizes as it lists all indexes for cluster members on one row


n Cluster member n node ID Integer - -

Page 120


Table 42: error_profile_<f>.txt, <f> represents the name of the used similarity function.


Output File description: Error function evolution during cluster analysis


1 Critical value used Double - -

2 Error function value Double - -

Table 43: association_profile_<f>.txt, <f> represents the name of the used similarity function.


Output File description: Fraction of sample associated during cluster analysis


1 Critical value used Double - -

2 Association amount Double - -

31.8 Configuration




Listing 5: OAA Configuration example

1 #%%%%%%%%%%%%%%%% CONFIGURATION FILE %%%%%%%%%%%%%%%%

2 # Lines starting with # are ignored

3 #----------------------------------------------------

4 #Main Calculation settings

5 #

6 # INPUT FORMAT (0 index): MUMRHED

7 # VAR: COL:

8 # a 26

9 # e 27

10 # i 30

11 # omega 31

12 # Omega 29

13 # ra 9

14 # dec 10

15 # v_g 17

16 # lambda 36

17 #

18 # INPUT FORMAT (0 index): MCAS

19 # VAR: COL:

20 # a 3

21 # e 4

22 # i 5

23 # omega 6

24 # Omega 7

25 #Analysis type:

26 # 0 = Entire database: Analysis is performed on entire input file at once.

27 # 1 = BIG DATA entire database: Special matrix partition functions are used to allow

computing extreamly large databases.

Page 121


28 #

29 #Input format:

30 # 0 = MURMHED

31 # 1 = MCAS

32 #

33 #Cluster analysis:

34 # 0 = Do not use cluster analysis

35 # 1 = Use cluster analysis

36 #

37 #Cluster analysis type:

38 # 0 = Single linkage cluster analysis

39 # 1 = Mean linkage cluster analysis

40 # 2 = ...

41 #

42 #Parameter sweep:

43 # 0 = No parameter sweep, will use the static settings in the association type

configurations

44 # 1 = Will sweep the critial parameters to find a sucsess function maxima, using

settings from association type

45 # configurations.

46 #

47 #Parameter sweep adaptive step:

48 # 0 = Disable adaptive step

49 # 1 = Enable adaptive step

50 #

51 #Parameter sweep termination:

52 # 0 = Error function is minimum

53 # 1 = All nodes are clustered

54 # 2 = Only one cluster exists

55 #

56 #Error function:

57 # 0 = ERR_FUNC is # number of nodes / # of nodes in largest cluster

58 # 1 = ERR_FUNC is # of clusters / # number of nodes

59 #

60 #----------------------------------------------------

61 Analysis type =0

62 Input format =0

63 Relative object row =1

64 Cluster analysis =1

65 Cluster analysis type =0

66 Parameter sweep =1

67 Parameter sweep adaptive step =0

68 Adaptive step enable limit (fraction) =0.2

69 Adaptive step multiplyer =1.7

70 Parameter sweep termination =0

71 Error function =1

72 Memory allocation (Mb) =1e+03

73 #

74 #----------------------------------------------------

75 #OUTPUT settings

76 #

77 #Logfile:

78 # 0 = No file

Page 122


79 # 1 = Write file

80 #----------------------------------------------------

81 Logfile output =1

82 #

83 #----------------------------------------------------

84 # D-criterion options

85 #----------------------------------------------------

86 #

87 #Criterion selection:

88 # 0 = Disable criterion

89 # 1 = Enable criterion

90 # D_SH = Southwork & Hawkins /cite

91 # D_D = Drummond /cite

92 # D_J = Jopek /cite

93 # D_N = ...

94 #

95 #Static value: If no parameter sweep is to be performed use

96 # this single critical value for the analysis.

97 #

98 #

99 #----------------------------------------------------

100 D_SH criterion =1

101 D_SH static value =0.1

102 D_SH step =0.0001

103 D_D criterion =1

104 D_D static value =0.2

105 D_D step =0.001

106 #

107 #----------------------------------------------------

108 #Metric and psudo-metric options

109 #----------------------------------------------------

110 #

111 #Metric selection:

112 # 0 = Disable metric

113 # 1 = Enable metric

114 # rho2 = Kholshevnikov cite

115 # varrho2 = again cite

116 #

117 #Static value: If no parameter sweep is to be performed use

118 # this single critical value for the analysis.

119 #

120 #

121 #----------------------------------------------------

122 rho2 metric =1

123 rho2 static value =0.2

124 rho2 step =0.0001

125 varrho1 metric =1

126 varrho1 static value =0.2

127 varrho1 step =0.0001

128 %%%%%%%%%%%%%%%%%%%% END OF FILE %%%%%%%%%%%%%%%%%%%%

Page 123


32 Orbital Stability Estimation module

The orbital stability estimation module will function on integration of the variational equation to estimate the

Lyapunov exponent of a orbit, this module is however not yet ready for early deployment or documentation.

One of the very useful applications of such a function is to, before Monte Carlo sampling of the initial distri-

butions, to examine the phase space dissipation, or e-folding time, of a certain initial condition to regulate

the sampling so that sufficient resolution can be used and that the simulation can omit certain propagations.

Changing the uncertainties sampling and the integration time would be based on the dissipation of the end

state distribution due to initial conditions with low or high e-folding times compared to the integration time.

Some research into the interplay between Monte-Carlo methods and chaotic system has been done. Such

as (Leitao et al., 2014) where the effect of different complexities of chaotic phase spaces on the efficiency of

importance sampling in Monte Carlo simulations is examined.

33 Simulation Merger module

The simulation merger module currently only performs index-preserving merges of simulation data. This

lets us, for example, take two parallel simulations that each produce 10 meteor showers and merge them

together. In each of the simulation the index of each shower will range from 1 to 10, and the generated

parent bodies will range from 1 to whatever number of clones was necessary to produce 10 showers. When

performing data analysis it may be desirable to combine simulation data, if for example one is examining the

joint effect of two different bodies on Earth at the same time or the effect of one parent body at one certain

time for two different ejection epochs. Thus, the data merger software will instead of duplicating simulation

indexes, create one long unique index list, where in the merged simulation the showers would range from 1

to 20 in index. It does this with all index dependant output files, as described in the data output section. A

example syntax is shown below:

Listing 6: SM module execution syntax

1 ./SM simulation_folder_1/ simulation_folder_2/ merged_simulation_folder/

This allows for quick set up of multi computer and multi core simulations.

34 Statistical Uncertenty Orbital Clones module

One of the fundamental principles of our software is to use distributions rather then single assumptions, and

as such it is very relevant how the initial distributions are created. Thus we have dedicated an entire module

to the task of, from different sources of data such as samples or observations or pure uncertainties, creating

a appropriate distribution to transfer to the main module. Currently the module only allows for input of

Kepler orbital elements distributed as a multivariate normal distribution with diagonal covariance matrix.

Given a space x ∈ RN we can define the multivariate normal distributions density function as

Σij = Cov(xi, xj), (34.1)

f(x) =1√

(2π)NDet(Σ)exp

(−1

2(x− µ)TΣ−1(x− µ)

)(34.2)

where Σ is the symmetric positive definite covariance matrix and µ is the multi dimensional expected value.

The input file currently only supports a diagonal covariance matrix

Page 124


Σij = δijσi, (34.3)

in orbital elements, input format can be found in 44.

Some preliminary considerations on Markov Chain Monte Carlo (MCMC) methods (Oszkiewicz et al., 2009)

and measurement uncertainty (Weisman et al., 2014) have been conducted and we are planning on adding

such functionality in a soon coming update.

Table 44: orb_in.data, in the case of Kepler element input configuration


Input File description: Data required to generate orbital clones


1 Mean µ Double - -

2 Standard deviation σ Double - -







6 True Anomaly Double Degrees Optional

35 NASA Jet Propulsion Laboratory module

Currently the JPL module only encompasses the most basic of SPICE functionalities; it loads a set of input

kernels and returns the J2000.0 Sun centric ecliptic state vectors of the desired object given the corrections

stated. Currently, we have included the following SPICE kernels:

• Leap second file: naif0011.tls

• JPL planetary and lunar ephemeris DE430: de430.bsp

• JPL planetary and lunar ephemeris DE431 part 1: de431 part-1.bsp

• JPL planetary and lunar ephemeris DE431 part 2: de431 part-2.bsp

Allowing for the software to extract state vectors between 1549 December 31 00:00:00.000 2650 January 25

00:00:00.000 for:

• MERCURY BARYCENTER relative SOLAR SYSTEM BARYCENTER

• VENUS BARYCENTER relative SOLAR SYSTEM BARYCENTER

• EARTH BARYCENTER relative SOLAR SYSTEM BARYCENTER

• MARS BARYCENTER relative SOLAR SYSTEM BARYCENTER

• JUPITER BARYCENTER relative SOLAR SYSTEM BARYCENTER

• SATURN BARYCENTER relative SOLAR SYSTEM BARYCENTER

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• URANUS BARYCENTER relative SOLAR SYSTEM BARYCENTER

• NEPTUNE BARYCENTER relative SOLAR SYSTEM BARYCENTER

• PLUTO BARYCENTER relative SOLAR SYSTEM BARYCENTER

• SUN relative SOLAR SYSTEM BARYCENTER

• MERCURY relative MERCURY BARYCENTER

• VENUS relative VENUS BARYCENTER

• MOON relative EARTH BARYCENTER

• EARTH relative EARTH BARYCENTER

From these bodies all internal relative distances can be found. These values can be found with added

aberration corrections, covered in section 28. The time for the desired state vectors is given in seconds past

J2000 TDB and currently, all positions returned are relative to the Sun. Now that we have covered the

functionally of the program we shall address the calling arguments. The first argument is the target body,

the second specifies if the returned data is the barycentre of the body or the geometric center. The third

argument specifies the time. The fourth argument the target folder for the output files containing the state

vectors. The rest of the arguments specifies all kernels to be loaded. The software will create the output file

if not present, and if a file is present the state vectors are appended to the end of the file. The file names

for the position vector is pos.data and for the velocity vector vel.data.

A example call of the module to extract the state vectors for the Neptune barycentre relative to the Sun at

1900 November 26 12:00:00 UT without any aberration corrections is given below:

Listing 7: JPL module execution example

1 ./JPL_SPICE NEPTUNE BARYCENTER -3.127248e+09 OUTPUT/ NONE /de430.bsp

/de431_part-1.bsp /de431_part-2.bsp /naif0011.tls

As can be seen the time is given as -3.127248e+09, this value can be found by the Julian date of the J2000

TDB time of 2451545.0 and calculating the number of seconds past this date

tJ2000s = 86400(JD− 2451545.0). (35.1)

A Julian date, or JD, is calculate by a Julian day number, or JDN, and time of day by

JD = JDN +hour− 12

24+

minute

1440+

second

86400. (35.2)

And lastly, to convert between a common day Gregorian calendar date to a Julian day number we need to

calculate three coefficients

a =

⌊14−month

12

⌋, (35.3)

y = year + 4800− a, (35.4)

m = month + 12a− 3, (35.5)

where bxc denotes the floor function. Using these coefficients we can find the JDN as

Page 126


JDN = day +

⌊153m+ 2

5

⌋+ 365y +

⌊y4

⌋−⌊ y

100

⌋+⌊ y

400

⌋− 32045. (35.6)

As practical calculations example we can calculate the JD and then the tJ2000s of 1900 November 26 12:00:00

in 10 steps:

1. a =⌊14−11

12

⌋= 0

2. y = 1900 + 4800− a = 6700

3. m = 11 + 12a− 3 = 8

4.⌊153m+2

5

⌋= 245

5.⌊y4

⌋= 1675

6.⌊y

100

⌋= 67

7.⌊y

400

⌋= 16

8. JDN = 26 + 245 + 365 · 6700 + 1675− 67 + 16− 32045 = 2415350

9. JD = 2415350 + 12−1224 + 0

1440 + 086400 = 2415350.0

10. tJ2000s = 86400(2415350.0− 2451545.0) = −3127248000

Currently the coordinate system the state vectores are to be returned in is fixed as the ecliptic J2000.0

reference. This is a inertial frame, in other words it is, non-rotating with respect to stars, it has a non-

accelerating origin, and it typicly has zero velocity. The inertial frames suported by spice at the moment is

J2000 (also called ICRF or EME2000) and ECLIPJ2000. The J2000 frame is based on the Earths equator

and equinox while ECLIPJ2000 is based on the Earth’s orbital plane and equinox.

36 Scripting of simulations

It can sometimes be very useful to script runs of multiple simulations running on different cores and config-

urations. Thus below we have included a small bash script example that was used to handle the simulation

logistics of the runs performed in this work. Here we had prepared a set of different configuration files and

orbital element files for each simulation. We then ran the simulations with a bash script that took two input

numbers. The first number represented the simulation core number and the second the simulation run, this

avoids manual input of configuration files in large simulation batches.

Listing 8: Example bash script for use with MCAS

1 #!/bin/bash

2

3 echo ’Setting up simulation core’ $1 ’with simulation id’ $2;

4 rm -v MCAS_v2_sim$1/SETTINGS/MCAS_config.cfg;

5 rm -v MCAS_v2_sim$1/SUOC/orb_in.data;

6 cp -v SIMSETTINGS/sim$2/MCAS_config.cfg MCAS_v2_sim$1/SETTINGS/MCAS_config.cfg;

7 cp -v SIMSETTINGS/sim$2/orb_in.data MCAS_v2_sim$1/SUOC/orb_in.data;

8 echo ’Simulation starting’;

9 cd MCAS_v2_sim$1/;

10 ./MCAS;

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37 External software

37.1 Mercury6

Until development of the CMS module is complete the mercury6 software package (Chambers, 1999) will

be used to perform the long term N-body integrations. mercury6 is written in Fortran 77 and currently

includes the following N-body algorithms:

• a second-order mixed-variable symplectic algorithm incorporating simple symplectic correctors

• a general Bulirsch-Stoer

• conservative Bulirsch-Stoer

• Everhart’s RA15

• Hybrid symplectic/Bulirsch-Stoer integrator

The standard release version of mercury6, the 6.2 version, includes only the effects of Newtonian gravi-

tational forces between point masses and not electromagnetic perturbation. However the code contains a

empty function able to include the effects of other forces. By modifying this subroutine in the source code,

for example the non-gravitational forces described in (Marsden et al., 1973) for comets can be implemented

in mercury6. The most relevant non-gravitational force for the purposes of our current work is Radiation

pressure and the Poynting-Robertson effect, previously covered in section 4.0.2. There has been some con-

fusion over the implementation of electromagnetic effects previusly, however as per the re-consideration in

(Burns et al., 2014) where the process was discussed in a straight forward manner, we have used

mpdv

dt=

(SAQpr

c

)((1− v • S

c)S− v

c

). (37.1)

In the above equation mp is the particle mass, v is the particle velocity, S is the unit vector in the direction

of the incoming radiation beam, S is the Solar radiation flux density at the position of the particle, Qpr is

the radiation pressure efficiency factor, A is the cross-section of the particle, and c is the speed of light. If

in heliocentric coordinates the radiation source vector can be written as

S =x

|x|(37.2)

and the Solar radiation flux density can be written as

S =L0

4π|x|2, (37.3)

where L0 is the Solar total luminosity. We have also assumed spherical grains thus the parameter A will be

A = πr2, (37.4)

where r is the particle radius. Given the particle mass we can find the radius by the bulk density as

Page 128


mp = ρV = ρ4πr3

3⇔ r =

(3mp

4πρ

) 13

. (37.5)

Inserting these into equation 37.1 we find

dv

dt=

1

mp

L0

4π|x|2AQprc

((1− v • x

c|x|)

x

|x|− v

c

)=

=

(3mp

4πρ

) 23 L0Qpr

4|x|2cmp

((1− v • x

c|x|)

x

|x|− v

c

). (37.6)

We can assume Qpr ≈ 1 (Burns et al., 1979) and L0 ≈ 3.846 · 1026 W as mentioned in section 7.

However, the mercury6 code does not allow for additional body parameters to be passed to the user force

function. This creates the problem that we cannot have particles with different bulk density and size without

recompiling the entire software. And since every comet in the current application of the software will have

varied properties these properties will have to be input parameters. Thus we have modified the mercury6

code to allow for these parameters to be passed to the program through the so called small.in file. The

complete modification of the mercury6 code will be made available when testing is complete. Also, units

internally in mercury6 is in AU for length, days for time, and Solar masses for mass, and thus all the SI

input units will have to be converted inside the function. The complete implementation can be found below.

Listing 9: MFO USER.FOR subroutine

1 c Applies an arbitrary force, defined by the user.

2 c

3 c If using with the symplectic algorithm MAL_MVS, the force should be

4 c small compared with the force from the central object.

5 c If using with the conservative Bulirsch-Stoer algorithm MAL_BS2, the

6 c force should not be a function of the velocities.

7 c

8 c N.B. All coordinates and velocities must be with respect to central body

9 c ===

10 c------------------------------------------------------------------------------

11 c

12 subroutine mfo_user (time,jcen,nbod,nbig,m,x,v,a,params,

13 % output_flag,global_params)

14 c

15 implicit none

16 include ’mercury.inc’

17 c

18 c Input/Output

19 integer nbod, nbig, output_flag

20 real*8 time,jcen(3),m(nbod),x(3,nbod),v(3,nbod),a(3,nbod)

21 real*8 params(3,nbod),global_params(4)

22 c

23 c Local

24 integer j

25 real*8 xt,yt,zt,R2,R

26 real*8 vxt,vyt,vzt

27 real*8 L0,Q_pr,c0,m_p,rho

28 real*8 coef1,coef2,VX_dot

29 c

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30 c------------------------------------------------------------------------------

31 c

32 c default accelerations are zero

33 do j = 1, nbig

34 a(1,j) = 0.d0

35 a(2,j) = 0.d0

36 a(3,j) = 0.d0

37 end do

38

39 c Solar LUM

40 L0=5.5739511D-12

41 c PI is already defined

42 c Speed of light

43 c0=173.144632327D0

44 c Radiation presure efficiency factor

45 Q_pr=1.0D0

46 do j = nbig+1, nbod

47 if (params(1,j).NE.0.AND.params(2,j).NE.0) then

48 xt=x(1,j)

49 yt=x(2,j)

50 zt=x(3,j)

51 vxt=v(1,j)

52 vyt=v(2,j)

53 vzt=v(3,j)

54

55 rho=params(2,j)*1683.60312587D0

56 m_p=params(1,j)/1.98855D30

57

58 R2 = xt*xt + yt*yt + zt*zt

59 R = R2**0.5D0

60

61 coef1=((m_p*0.75D0/(PI*rho))**(2.D0/3.D0))

62 coef2=L0*Q_pr/(4*R2*c0*m_p)

63 VX_dot = (vxt*xt + vyt*yt + vzt*zt)/R

64

65 a(1,j) = coef1*coef2*((1 - VX_dot/c0)*(xt/R) - vxt/c0)

66 a(2,j) = coef1*coef2*((1 - VX_dot/c0)*(yt/R) - vyt/c0)

67 a(3,j) = coef1*coef2*((1 - VX_dot/c0)*(zt/R) - vzt/c0)

68 else

69 a(1,j) = 0.d0

70 a(2,j) = 0.d0

71 a(3,j) = 0.d0

72 end if

73 end do

74 c

75 c------------------------------------------------------------------------------

76 c

77 return

78 end

79 c

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37.2 NASA Jet Propulsion Laboratory ephemeris software

As a early development alternative to SPICE the current version of MCAS supports use of a modified version

of the JPL ephemeris software (Hardy, 2004). Even though this is to be phased out in later development

stages we shall here go over the modifications and its use. This software library is designed to read and write

NASA’s Jet Propulsion Laboratory planetary ephemeris files, convert ephemeris file types, and calculate

positions and velocities from ephemeris. The software itself was originally written in Fortran but later

ported to C. We have used the C version that was made available for free under the Lesser Gnu Public

License. The library contains the following functions,

• append: Joins binary ephemeris files.

• convert: Creates binary ephemeris files.

• extract: Extracts segment from binary ephemeris files.

• print header: Extracts header data from binary ephemeris files.

• read record: Reads one record of binary ephemeris file data.

• scan records: Lists start/stop times of ephemeris file records.

• rdeph: Calculate positions and velocities of a planet at the supplied time.

We have only modified rdeph by making the program also write the positions and velocities to a file instead

of just printing the data to the screen. We shall now detail the step by step requirements to to set up a

ephemeris file for use in our software.

• (I): Download the appropriate set of ephemeris files from (JPL, 2016) in ASCII format, we have used

version 430 from the year 1550 to 2550 in this example.

• (II): Change the defined EPHEMERIS variable value found in file ephem˙types.h to the version

downloaded, in this cased 430. Then also change the elif if clause to the same version so that error

does not occur.

• (III): Compile the convert and the append programs.

• (IV): Convert all the ASCII files to binary, example command for one of the files: ./convert header.430˙572

ascp1550.430 bin1550.430

• (V): Create the combined binary base to append to using the chronologically first file, example com-

mand: cp bin1550.430 bin1550to2550.430

• (VI): Append ALL binary files in chronological order to the combined file using the append program,

example command for FIRST append: ./append bin1550to2550.430 bin1650.430

• (VI): Place the completed bin1550to2550.430 file in the JPL/ folder in the MCAS structure and rename

it to ephBin.000

the modified JPL software can now generate initial conditions for the Solar system within the time frame

available in the binary file. The file distributed with the complete MCAS software is the one described in the

example using the 430 ephemeris (Folkner et al., 2014).

Page 131


38 Data transformation logistics

As many authors present their results in different coordinate basis it is preferable to specify how exactly the

transformation from default output to other representations is performed. Here we shall define a few basic

transformations.

38.1 State vectors and Kepler elements

The concept of a set of state vectors comes from the formulation of a Hamiltonian system. As mentioned

before a set of state vectors for a object describes its state in phase space, we will denote state vectors as

a pair of three dimensional vectors representing position and velocity (r,v) together with the gravitational

parameter of the object and the central mass as covered in section 3.3. Kepler elements are defined as

described in section 3.4. We will denote the orthonormal coordinate base as ex, ey, ez and the projection

of a vector onto this base as u • ex = ux.

38.1.1 From state vectors to Kepler elements

Given the body state (r,v) and its gravitational parameter µ = G(m + M) we find the Kepler elements

systemically by first introducing

r = |x|, (38.1)

v = |v|, (38.2)

vr =(r

r

)• v. (38.3)

The semi major axis can then be calculated for elliptic orbits as

a = − µ2ε, (38.4)

and for hyperbolic as

a =µ

2ε, (38.5)

where

ε =v2

2− µ

r. (38.6)

We then calculate the specific angular momentum as

h = r× v (38.7)

h = |h|. (38.8)

As this quantity is perpendicular to the orbital plane we can calculate the inclination of the orbit as

i = cos−1(hzh

). (38.9)

Page 132


Here we can see that the inclination runs from 0 to π where an inclination over π/2 represents a retrograde

orbit. We then calculate the vector pointing towards the ascending node

n = ez × h, (38.10)

n = |n|, (38.11)

and use it to calculate the longitude of the ascending node as

Ω = cos−1(nxn

)∀ ny ≥ 0, (38.12)

Ω = 2π − cos−1(nxn

)∀ ny < 0. (38.13)

We notice that if h only has a ez component the vector pointing towards the ascending node will be the zero

vector. This is due to the fact that if a orbit lies in the reference plane, it will never ascend or descend into

that plane as it is always in the plane. Thus we define the longitude of the ascending node to be zero in this

case

Ω = 0 ∀ i = 0. (38.14)

This is where one uses the longitude of perihelion rather than the longitude of the ascending node

ω = ω + Ω. (38.15)

Continuing on to the eccentricity of the orbit. This can be calculated by the use of the eccentricity vector

which can be expressed by

e =1

µ

((v2 − µ

r)r− rvrv

), (38.16)

e = |e|. (38.17)

The argument of periapsis can now be calculated by

ω = cos−1(n • e

ne

)∀ ez ≥ 0 (38.18)

ω = 2π − cos−1(n • e

ne

)∀ ez < 0. (38.19)

Here we also notice a degenerate case where the orbit will not have a argument of periapsis in a perfect

circular orbit as all points are equally close the central focus point. Thus also this is defined as zero when

the eccentricity is zero

ω = 0 ∀ e = 0. (38.20)

A consequence of this is that all circular orbits with inclination seems to have a periapsis in the ascending

node, and one should be aware of this. Lastly we need to calculate the position on the orbit to complete the

transformation,

Page 133


ν = cos−1(e • r

er

)∀ vr ≥ 0 (38.21)

ν = 2π − cos−1(e • r

er

)∀ vr < 0. (38.22)

38.1.2 From Kepler elements to state vectors

Starting with a set of 6 Kepler elements we first define the co-planar orbit equations as

o =

r cos(ν)

r sin(ν)

0

= a1− e2

1 + e cos(π + ν)

cos(ν)

sin(ν)

0

(38.23)

o =

√µ

a(1− e)

− sin(ν)

cos(ν)− e0

. (38.24)

The position dependant radius is found by

r = a1− e2

1 + e cos(π + ν), (38.25)

as we have defined the apoapsis to point towards the reference direction ex. We can then find the orbit in

the reference frame in which the orbital elements are given by the rotations

r = Rz(−Ω)Rx(−i)Rz(−ω)o, (38.26)

v = Rz(−Ω)Rx(−i)Rz(−ω)o, (38.27)

where the rotation matrecies are defined as

Rz(θ) =

cos(θ) − sin(θ) 0

sin(θ) cos(θ) 0

0 0 1

, Rx(θ) =

1 0 0

0 cos(θ) − sin(θ)

0 sin(θ) cos(θ)

. (38.28)

38.2 Geocentric coordinates

38.2.1 Ecliptic J2000

The concept of a ecliptic geocentric coordinate system when examining meteor radiants is to have a general

reference system for the direction from which the meteor phenomena is originating. Thus the point describing

a single meteor will not represent the meteoroid position, or its destination, but rather the direction from

which it enters the Earth gravitational field before gravitational focusing applies. The reference is the vernal

equinox, i.e the x-axis of the system is aligned in the equinox direction, and the reference plane is the ecliptic,

i.e the y-axis lies in the ecliptic and the z axis is almost parallel to the Earth orbital momentum vector.

Let us consider a meteoroid with orbital parameters an, en, in, ωn,Ωn, νn where ν is the true anomaly

at the point right before gravitational focusing has become predominant. Let us also consider the Earth

with orbital parametersaE , eE , iE , ωE ,ΩE , νE where νE is the Earth true anomaly at the same time as the

meteoroid has true anomaly νn.

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To find the ecliptic radiant we need both the velocity vectors of the meteoroid and Earth, thus we begin

with transforming the orbital elements to state vectors, as described in section 38.1. The end result will be

a meteoroid state (xn,vn) , and a Earth state (xE ,vE).

The inclination is defined such that it is the inclination to the reference plane at the ascending node and

thus a orbit with inclination below 90 is prograde and a orbit with i > 90 is retrograde. In this definition

the co-planar, apsis aligned orbit has focus in origin and apoapsis pointing in the direction of positive x and

counter clockwise movement as prograde motion. One common mistake is to define the system so that the

periapsis and not the apoapsis is pointing towards the equinox before spatial rotations has been performed.

The first matrix rotates the orbit in the orbital plane around the orbital plane normal, the second matrix

then inclines the orbit. If we incline along the x axis the ascending node will point towards vernal equinox

when the longitude of the ascending node is 0. Another common mistake to incline along the y axis,

resulting in a orbit that will initially have a longitude of the ascending node relative equinox of −90 and

a increased angle to the argument of periapsis of 90. Lastly we will rotate the ascending node away from

the reference direction and thus find our J2000.0 ecliptic equinox oriented orbit given the 5 orbital elements

and the orbital position.

Now that we have the velocity vectors calculated for both the Earth and the meteoroid at the point of

encounter we will find the geocentric velocity vector for the meteoroid as

v′n = vn − vE . (38.29)

This represents the direction the meteoroid is travelling relative the Earth, we are interested in pointing to

the apparent direction the meteor is coming from, and thus we shall define

u = −v′n, (38.30)

u = |u|. (38.31)

We can proceed to calculate the relevant geocentric ecliptic coordinates as

λ = tan−1(uxuy

), (38.32)

β = 180 − cos−1(uzu

), (38.33)

where λ is the ecliptic longitude, β is the ecliptic latitude, and u is the geocentric velocity.

38.2.2 Equatorial J2000

To transform coordinates from the ecliptic coordinate system to the equatorial we need to know the obliquity

of the ecliptic. This is the term used to describe the inclination of Earth’s equator with respect to the ecliptic.

Its mean value is currently around 23.4, however it is also decreasing by around 0.013 ever hundred years

due to planetary perturbations. To transform between the ecliptic and the equatorial coordinate systems is

easy if the coordinate system used is aligned towards the vernal equinox. The vernal equinox is the direction

of intersection between the ecliptic plane and the equatorial, and thus, one only needs to rotate around the

vernal equinox direction to swhich between the too coordinate systems. However, since the obliquity of the

ecliptic changes due to planetary perturbation, we need to account for this change. This can be done by a

Taylor series expansion and numerical integrations as done by (Laskar, 1986). There the mean obliquity of

the ecliptic was calculated to

Page 135


ε′′0(T ) =− 4680.93T − 1.55T 2 + 1999.25T 3 − 51.38T 4 − 249.67T 5+

− 39.05T 6 + 7.12T 7 + 27.87T 8 − 5.79T 9 − 2.45T 10, (38.34)

in arc seconds, where T is defined as hundreds of Julian centuries from J2000.0, or

T =1

100

JD − 2451545.0

36525=JD − 2451545.0

3652500. (38.35)

This formula has a probable accuracy of around 0.02 arc seconds, or 5.56 ·10−6 degrees, after 1000 years and

accurate within a few arc seconds up to 10 000 years. To convert from arc seconds to degrees we simply take

ε0(T ) =ε′′0(T )

3600(38.36)

This is however the mean obliquity of the ecliptic, to find the instantaneous obliquity we must include the

nutation of the equator ∆ε,

ε(T ) = ε0(T ) + ∆ε(T ) (38.37)

The nutation of the obliquity has been defined in the International Astronomical Unions (IAU’s) 2000B

theory section, which is based on (McCarthy and Luzum, 2003). Here the nutation in longitude is defined

as

∆Ψ(T ) =

78∑i=1

(Ai + 100A′iT ) sin(θ) +A′′i cos(θ), (38.38)

and the nutation in obliquity is defined as

∆ε(T ) =78∑i=1

(Bi + 100B′iT ) cos(θ) +B′′i sin(θ), (38.39)

where

θ =

5∑i=1

NiFi. (38.40)

Here the coefficients Ni are integers and Fi the five fundamental arguments

1. Mean anomaly of the Moon

2. Mean anomaly of the Sun

3. Mean longitude of the Moon

4. Mean elongation of the Moon from the Sun

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5. Mean longitude of the ascending node of the lunar orbit

The coefficient lists A and B can be found at the IAU website. Once the nutation and the mean obliquities

has been calculated one can proceed to find the total nutation matrix defined as

N (T ) = Rx(−ε)Rz(−∆Ψ)Rx(ε0). (38.41)

Thus to find a true equinox of date position x from a mean equinox of date position x0 one simply applys

the nutation matrix

x = Nx0.

Figure 40: Calculation of the variations in the obliquity of the ecliptic for two different time scales, ± 10

years from J2000.0 and ± 10 000 years from J2000.0.

These variations are however on the order of a few tenths of arc seconds and can be disregarded due to

observational uncertainties, at lest in meteor studies. In figure 40 we can see the variations of the obliquities

of the ecliptic for two different time frames, using the above methods.

To calculate the coordinates in a equatorial we simply rotate the geocentric velocity in equation 38.29 from

the ecliptic as

Rx(ε)v′n = V′n, (38.42)

and again define the direction of origin as

U = −V′n, (38.43)

U = |U|. (38.44)

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We can proceed to calculate the relevant geocentric equatorial coordinates as

α = tan−1(UxUy

), (38.45)

δ = 180 − cos−1(UzU

), (38.46)

where α is the right ascension, δ is the declination, and U is the geocentric velocity.

38.3 Sun centred versions

One can define Sun centred version of these coordinate systems by accounting for the longitude of the Sun,

defined as the angular distance along the Earth’s orbit measured from the intersection of the ecliptic and the

celestial equator. In other words relative vernal equinox. The Sun is furthermore defined to move from south

to north. A Solar longitude coordinate gives the position of the Earth on its orbit and can be calculated

from the date, but not vice versa unless given the year, due to time corrections. In such a coordinate system

the Sun, and the Earth apex will be stationary making it a convenient representation when visualizing the

sporadic meteoroid complex apparent sources.

39 MATLAB visualization scripts

Figure 41: The graphical user interface used to call all the data analysis scripts developed.

One must always complement the computational power of the developed software with sensible visualization

and data summation tools since the output data is generally too large for manual evaluation. Thus a extensive

library of plotting and data summarization scripts with a accompanying GUI was developed in MATLAB. A

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example of the GUI can be seen in figure 41. This group of scripts is the one used to generate all graphs

included in the results part of this work.

Currently we have developed scripts to visualize

• Statistics on mass ejection

• Statistics on ejection speeds

• Statistics on parent bodies

• Statistics on meteoroids

• Statistics on test particles

• Total simulation statistics

• Similarity function divergence from parent bodies for shower causing particles

• Orbit configurations

• Meteor association parameter sweeps

• Ejected mass distribution

• Encountered mass distribution

• Snapshot of simulation state

• Specific data filters

• Mass transfer functions

• Trail transfer functions

• Principal component analysis

• Calculation of geocentric coordinates

• Stream dissipation analysis

• Execution time diagnostics

• Statistics on Earth at encounter times

• Mass encounter difference between 2 years

• Check of electromagnetic coefficient for perturbation

• Mass distribution as a function of trail origin

• Yearly clustering configurations

• Orbit criterion evolution in trail

• Control of statistical convergence of Monte Carlo sampling

• Time dependant initial distribution dissipation plots

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40 Future work

40.1 Purpose and aims

At the moment we are planning a continuation project that would address three major questions within

Solar system small body dynamics:

(I) What is the mass propagation of material from parent bodies to Earth?

(II) What are the dominant effects on the fluxes of Solar system nanometer-sized dust?

(III) What is the influence of chaos on meteoroids streams and when does it become predominant?

In spite of the fact that the underlying orbital dynamics is well understood, it is still an open question how

much extraterrestrial material enters the Earth atmosphere. The most obvious missing feature in current

numerical models, which cover different aspects of Solar system small body dynamics, is the ability to

easily connect programs with different functionality. We believe this is the main reason why contemporary

techniques have not yet been combined. These include techniques such as Monte Carlo simulations of

meteoroid streams, orbital stability analysis, time tracing of orbital similarity functions to better analyse

measurements and neural network recognition of meteor showers. The lack of a comprehensive numerical

modelling approach appears to be the main bottleneck. Thus, the main focus of the future project is

dedicated to developing a software toolbox which attempts to supplement the current research area. The

work presented here works as a proof of concept providing a solid basis for full development.

40.2 Future development plan

The major workload in this proposed project would be software development and performing the simulations

needed to address the science questions. Reviewed numerical software all seems to have been designed with

a specific task in mind, thus requiring extensive programming skills and development time for its re usage.

To avoid such a development process, every module described below will be designed with re usability and

ease-of-modification in mind.

Below, the proposed work is divided into four Work Packages (WP), the first three aimed towards one science

question each.

WP I contains development of the Celestial Mechanics Simulator (CMS), Statistical Uncertainty Orbital

Clones (SUOC), and Parent Body Ejector (PBE) modules needed to address mass propagation of material

from parent bodies.

• I.1: Implement Hamiltonian integration methods in CMS with electromagnetic effects, close encoun-

ters, and time zone integration, replacing mercury6. Modify MCAS to utilize all the functionality of

CMS in the statistical calculation.

• I.2: Using statistical software libraries like STAN, implement in SUOC Bayesian statistical inference

and other methods needed to procure all input probability distributions. Modify MCAS to utilize the

entity of SUOC’s functionality.

• I.3: Review Rosetta results and implement additional features in PBE: e.g. non-uniform spherical

ejection, jet ejection, size-dependent particle density.

• I.4: Perform case studies addressing mass propagation from parent bodies.

WP II contains development of modules needed to address the flux of nanodust.

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• II.1: Modify MCAS so that various nanodust distributions can be used as input instead of ejection of

dust from comets with PBE. Add Solar wind conditions and interaction corrections to CMS.

• II.2: Perform simulations to investigate which effects dominate the fluxes of nanodust.

• II.3: Investigate the time variable flux at Earth and other parts of the Solar system.

WP III includes a new module called Orbital Stability Estimation (OSE) needed to address chaos and orbital

stability.

• III.1: Implement numerical integration of the variational equations needed to estimate chaos effects

and information dissipation rates in OSE. Modify MCAS and CMS to utilize this integration module.

• III.2: Derive a method for characterizing the rate of information loss of a certain initial orbit using

the Lyapunov Characteristic Exponents.

• III.3: Perform simulations to characterize chaotic and stable orbit families.

WP IV contains an additional module called Orbital Association Analysis (OAA) in order to develop a

systematic method of meteor shower detection in observational data for validation purposes of all three

science questions.

• IV.1: Add several clustering methods, density approximations, wavelet transforms and pattern recog-

nition methods to OAA.

• IV.2: Perform case studies to characterize strengths and weaknesses of the different methods.

• IV.3: Construct systematic methods for calibrating meteor searches in databases.

• IV.4: Using calibration results from the IV.3 results, construct a long-term sporadic model, or use

an existing one, and derive portable definitions for SNR.

The only module in continuous development for the duration of the project is the MCAS module. This allows

for disjointed completion of the different WP-s. Also, possible programming issues and software updates

within one WP will not adversely affect the other ones.

40.3 Significance of future work

Each module will be developed with the general needs of the science community in mind. All software

will be published as open source and designed for reuse. This will enable collaborative case studies with a

variety of research groups. Here we only cover some of the possible future uses in addition to our planned

investigations.

The mass influx onto Earth as a function of mass, time and place is one of the outstanding questions

in meteor science. We would statistically investigate the mass transfer from parent bodies to Earth with

numerical modelling and compare with measurements and published results. Our investigation would shed

light on how to combine the observations, leading to an improved estimation. The reusable approach of

the modules will allow for additional studies such as the mass transfer efficiency to Earth from yet-to-be

discovered parent bodies. Also, accurately estimating debris dynamics will be vital for specific events such

as comet and asteroid interaction missions, e.g. the NASA space probe Deep Impact, JAXAs Hayabusa 2,

and future asteroid deflection missions, and this will be possible to address.

We will extend the basic model of dust trajectories used by (Czechowski and Mann, 2010) by including time

variations in the nanodust surface charge and the effect of Solar wind fluctuations on the motion of charged

dust grains in order to estimate fluxes. The importance of these effects on fluxes of nanodust have not been

addressed in previous investigations (Mann et al., 2014).

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Our work on chaos and orbital stability will provide a method to estimate the range of time that simulations

and calculates are valid, or in other words, the rate of information loss in certain orbital regions. Such

estimations are useful for determining propagation limits in parent body searches and better estimations of

the reliability of results. In this context we will also investigate the effects that simplifying assumptions has

on model outcomes, and assess improvements. One such example is the frequent assumption of parent bodies

ejecting particles only at a single point on their orbit at perihelion (Vaubaillon et al., 2011). Determining

which factors are important to consider will enable significantly improved meteor shower predictions.

The detection of meteor showers in data can be seen as the separation of the influx rates between new

showers versus the sporadic complex (old showers). An SNR definition, or significance estimation, is a basic

requirement for machine learning, which has had wide success in many areas and should be implemented

also in the systematic search for meteor showers. The next generation atmosphere and geospace research

radar EISCAT˙3D will be able to measure up to hundreds of thousands of meteoroid orbits per day with

unprecedented accuracy (Pellinen-Wannberg et al., 2016a). Such vast amounts of orbital data will benefit

from the development of advanced orbit association analysis methods.

40.4 Significance of community contributions

As the software modules near a state where they can be released as open source to the scientific community,

the importance of community contributions to the development will gradually increase. As the developer

of the software has specific research goals to fulfil with the software, many dynamic aspect will be lost

or overlooked and it is in this regard that the community can contribute immensely. By adding features

allowing other types of research be performed or simply informing that the next update should introduce a

new feature or algorithm modification will quickly increase the software quality in terms of usefulness to the

scientific community.

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Part VII

Results

41 Introduction to Results

We have chosen to perform two initial simulations to both showcase the capabilities of the described software

and also to contribute to the scientific community.

The first case study is a multiple trail simulation of 21P/GiacobiniZinner, a comet which was recognized as

the parent body for the October Draconids meteor shower (Davidson, 1915) as early as 1915. This comet

has been extensively examined throughout the years after its discovery on December 20, 1900 by Michel

Giacobini. The second observation in 1913 by Zinner resulted in its current naming. The reason behind its

extensive observations and fast meteor shower connection is its relatively short orbital period of six years

and that its perihelion distance is very close to that of the Earth. Thus making the produced dust stream

a ideal candidate for being responsible for a strong meteor shower on Earth. And as it happens, in the

years 1933 and 1946 a set of spectacular meteor showers came to pass with hourly rates of thousands of

meteors. Recently simulations of the meteor shower has been performed by (Vaubaillon et al., 2011) and

(Ye et al., 2013b) to explain the recent meteor shower outbursts of 2011 and 2012 respectively. Both of

these meteor showers have also been observed by the MU radar in (Kero et al., 2012c) and (Fujiwara et al.,

2016). However, when comparing the two observations it was discovered that there was a significant mass

difference detected between the two years. This conclusion was drawn by using simultaneous radar and visual

observations. As visual observation are only generated by larger grains such simultaneous observations can

give some insight into the mass distribution.

To explain the actual observed deviation from standard models we must explain the mass index for a meteor

shower. The mass index can be calculated using either the underdense echos or the overdens echos covered

in section 13. Both these calculations rely on assuming that the cumulative number of echos is proportional

to a simple power law. In the case of underdense echoes the cumulative number of echoes N with amplitude

greater than A is proportional to A1−s (McIntosh, 1968). For overdense echoes however the cumulative

number N is instead proportional to the duration T rather than the amplitude with a relation T 0.75(1−s).

In these formulas the aforementioned mass index is the number s, generally assumed to be between 1 and 2.

During the 2012 October Draconids visual observers yielded a much lower rate of around 200 ZHR compared

to radar observations, indicating that the magnitude dependant cumulative number is not represented by

the power law that was just explained. Thus we have performed simulations to determine the actual mass

transfer of GiacobiniZinner to the Earth for these meteor showers in a attempt to further our understanding

of this difference.

The second simulation is a demo simulation designed to showcase the capabilities of the developed software

and also as a platform for developing the data analysis procedures needed. The simulations where performed

with the previously, in section 11.1, covered Pan-STARRS synthetic Solar system model population of short

period comets as a initial distribution.

42 21P/GiacobiniZinner

42.1 Input state

To generate the data needed a set of 18 simulations was set up parallel to each other. One simulation

considered one perihelion passage of 21P/GiacobiniZinner each, as per the list:

1. 1866

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2. 1873

3. 1880

4. 1887

5. 1894

6. 1900 Nov. 28

7. 1907 May 19

8. 1913 Nov. 2

9. 1920 May 18

10. 1926 Dec. 11

11. 1933 July 15

12. 1940 Feb. 17

13. 1946 Sept.18

14. 1953 Apr. 16

15. 1959 Oct. 26

16. 1966 Mar. 28

17. 1972 Aug. 4

The simulations after and including the passage of 1900 used orbital elements from JPL small body database,

these orbital elements are calculated from observations. There where however two passages not observed, the

years 1920 and 1953, and thus these passages must be numerically integrated from the closest observation.

The same is true for the passages prior to 1900, thus we have calculated them ourself using the built in time

synchronization function in the Parent Body Ejector module previously described in section 30. Due to small

variations in the orbital elements during 1900, when propagated backwards through the large perturbation

by Jupiter, the variation on the passage dates blow up from hours to days and in extreme cases months,

which is why no month and date is provided for passages prior to 1900 in the list.

Each simulation was set to produce 50 clones of GiacobiniZinner and then propagate the resulting meteoroid

stream until 2020. The ejection model used was the one described as (II) in 10.2.1 implemented from

(Hughes, 2000).

We considered all passages of particles within the Earth hill sphere as close encounter and regarded them as

meteors with a number weight calculated from the sublimation simulation.

The Cluster analysis was configured to performed a non-adaptive parameter sweep of a single linkage cluster

algorithm using all currently implemented similarity functions.

The time step mercury6 was set to 1 day and the step of PBE was set to 16 days. PBE utilized a Bulirsch-

Stoer algorithm with multiple sub steps according to the previously described series, and mercury6 was

configured to use the hybrid symplectic integrator, both including electromagnetic forces.

Finally, the mass distribution for ejection of particles was set to a logarithmic uniform distribution between

10−9 kg to 10−1 kg.

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Figure 42: Histogram of the orbital element distribution for orbital clones of 21P/GiacobiniZinner.

Figure 43: Histogram of the orbital element distribution for encountered test particles ejected from the clones

of 21P/GiacobiniZinner.

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42.2 Summary of simulations

The resulting simulation yielded:

• 850 clones of 21P

• All 850 clones underwent sublimation

• 821 of the 850 meteoroid trails encountered Earth before 2020

• On average 52 test particles from each stream encountered Earth

• Using the mass loss weight, this corresponds to on average 7.69 · 109 meteors per trail.

• In total 6 714 499 test particles where propagated.

• Out of them 43 637 encountered the Earth.

In figure 42 we can see the orbital distribution of the 850 generated 21P clones, and in figure 43 we can see

the collective orbital distribution of all test particle encounters.

42.3 Validation: 1933 and 1946 October Draconids

Figure 44: Histogram of Probability of encounter and mean shower intensity for the 1933 October Draconids.

A common method to validate software is to make back-predictions on already established data, both nu-

merically and observationally. The meteor shower of 1946 October Draconids has been extensively simulated

and there are also observations available, making it a very good candidate for software validation. As done

in (Vaubaillon et al., 2011) the simulations of the October Draconids should show a meteor shower on Earth

during 1933 October 9 and 1946 October 10, both at ≈ 197 Solar longitude.

Our results for the encounter probability distributed along Solar longitude for the year 1933 can be found

in figure 44, the mean flux trail contribution distribution can be found in figure 45, and the trail encounter

probability can be found in figure 46. In figures 47, 48, and 49 the respective plots for the 1946 meteor

showers can be seen.

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Figure 45: Density maps for the mean trail meteoroid flux distribution for the 1933 October Draconids.

Figure 46: Density maps for the trail distribution probability for the 1933 October Draconids.

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Here the discussed probability and mean simulation formulations in section 29 come into use when examining

meteoroid trail encounter events. In figures 44 and 47 the left side of the plots refer to the probability of the

encounter occurring, i.e. how many of the simulations that contains meteor events at that Solar longitude

bin compared to all performed simulations. The right side however refers to the amount of encountered

test particles weighted by the sublimation calculation weight of how many real meteoroid that test particle

approximately represents. In the pure histogram versions, the difference is negligible between probability

and mean flux. However when looking at the trail distribution in figures 48 and 49 we see a clear difference

between probability of meteors occurring and the their mean numbers. The most prominent feature is that

there is high probability of the Earth encountering meteoroids from the trails from 1926 to 1940 in the year

1946 but even if such a encounter occurs, the majority of the flux present will be from the 1933 trail alone.

These simulated data corresponds well with previous simulation and we regard the validation as accomplished.

42.4 2011 October Draconids

One of the science questions these simulations where to address where the difference between the 2011

and 2012 October Draconids. In figures 50, 51, and 52 the same types of encounter probability and trail

distribution plots can be seen as for the 1933 and 1946 October Draconids. One interesting feature we see

here is that there is a high flux coming from the trail formed during the year 1972, however the probability

of this flux occurring is relatively low compared to other trails. This highlights one of the key points of the

future work proposed here. As there are no uncertainties given for the orbital elements of the 1972 epoch

on the JPL Small body database that was used for the input configuration, we chose an arbitrary Gaussian

distribution around the given elements. That the probability is low, indicates that in many of the simulations

the flux does not occur at all since this measure does not reflect the amount of particles as explained earlier.

As such one would assume that a part of the input Gaussian distribution is untrue, because if the flux is

observed then the parts of the distribution that did not generate it cannot contain the true parent body

and vice versa. Also, this flux is around 4 to 8 hours after the main October Draconids peak and should

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Figure 53: Density maps for the mean meteoroid flux mass distribution for the 2011 October Draconids.

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Figure 54: Density maps for the mass distribution probability for the 2011 October Draconids.

be distinguishable from the other trails in observational data. Unfortunately the radiant for the dedicated

meteor radar Canadian Meteor Orbit Radar (CMOR) that did detailed observations (Ye et al., 2013a)

descended below the horizons and could not observe at the time where this intensity increase would have

appeared. In (Kero et al., 2012c) there is however one trailing observation with high uncertainty indicating

a second peak, and also in the International Meteor Organization (IMO) visual data quicklook website

http://www.imo.net/live/draconids2011/, we can see a trailing single observation at 195.382

Solar longitude with high ZHR but also large uncertainties. This may be indication that this event actually

occurred but no real conclusions can be drawn from the current data.

This kind of event is a good example of why the probability distribution approach is a useful analysis type,

if we would have had the completed SUOC module and was able to do a MCMC raining (Oszkiewicz et al.,

2009) of the ephemeris the probability distribution may have altered the uncertainty of this extra intensity

peak drastically.

Moving on to the next part of the analysis, we will here introduce a new type of density map, the mass

transfer map, as illustrated in figures 53 and 54. Here we have used the same analysis type as with the

perihelion passage distribution but instead with particle masses versus time,or Solar longitude. In figure 53

we see the mean mass flux configuration and in figure 54 we see the probability that a certain mass type will

propagate to the Earth, not considering the amount of that mass that is propagated. Since the input mass

distribution of all iterations are a logarithmic uniform function a equal amount of particles in each mass bin

is ejected. This means, that we, by looking at the total amount of particles that propagates to Earth we

can find the mass transfer efficiency of a certain parent body. However, if we instead of looking at the total

amount of mass transferred to Earth, instead look at the deviation from a uniform distribution observed at

Earth, we can infer information about how well different mass propagates from different times and parent

bodies to specific years, on Earth. This can be seen as a decoding chart, where if we observe one certain

distribution at Earth and de-bias, we can actually make estimations on the ejected mass distribution at the

comet itself.

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http://www.imo.net/live/draconids2011/


42.5 2012 October Draconids



To complete the comparison between the 2011 and 2012 events we only need to examine the 2012 October

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Figure 57: Density maps for the mean meteoroid flux mass distribution for the 2012 October Draconids.

Figure 58: Density maps for the mass distribution probability for the 2012 October Draconids.

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Draconids as well. Their respective plots can be found in figures 55, 56, 57, and 58. These simulations also

correspond well to measurements (Fujiwara et al., 2016).

42.6 Mass transfer difference 2011-2012

Figure 59: The normalized difference in mass distribution between the October Draconids of 2011 and 2012.

As the difference between the overall number of meteors each outburst is less interesting we instead focus

on the relation between the masses internally. To accomplish this, we first select all the simulated Earth

encounters in one of the years, create a mass distribution plot for this year and normalize it. Then we take

the difference between the normalized mass distributions from both years, resulting in figure 59. As we can

see here there is a clear difference between the two years in mass propagation efficiency from 21P to the

Earth. In 2011 the visual magnitude meteoroids in the mass rage between 10−6 kg to 10−4 kg is up to 30-40%

more numerous than in 2012, when compared to the other rates for 2011. We now also have to consider the

fact that our input distribution was a logarithmically uniform one, which is not what one expects to find

ejected from a comet in real life. As such, even if here, the 0.1 to 0.01 kg meteoroids are relatively more

numerous in 2012 than in 2011, these are so rarely ejected that they probably would still not probably have

been observed even if they should have occurred in greater numbers.

42.7 Probable mass transfer cause

Having reproduced the observed mass relation difference in a simulation we are able to probe the cause of

this difference. One way to do this is to create a subset of simulations, in this case the passages of:

1. 1900 Nov. 28

2. 1907 May 19

3. 1913 Nov. 2

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Figure 60: The normalized difference in mass distribution between the October Draconids of 2011 and 2012

using only a subset of perihelion passages.

Figure 61: The normalized difference in mass distribution between the October Draconids of 2011 and 2012

using only a subset of perihelion passages and propagation without electromagnetic perturbations.

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4. 1920 May 18

5. 1959 Oct. 26

6. 1966 Mar. 28

7. 1972 Aug. 4

We select and use only this data from the above simulation and performed the same mass transfer difference

analysis, the result showing in figure 60. We have selected these perihelion passages as they are the most

probable, from our simulations, to contribute mass to the 2011 and 2012 October Draconids.

Then we re-run the set of simulation with exactly the same settings, but with one difference: we turn

of electromagnetic perturbations. We only do this in the long term propagation however, not in the ejection

propagation.

The resulting graph can be seen in figure 61. Thus, due to this lack of electromagnetic perturbation, we

can see that the mass distribution situation has changed. The range of visual meteors are now much more

similar while two differences of equal size has emerged in the radar meteor range. Since exact meteoroid

mass determination from observations of meteors is difficult, this kind of di-pole type diffence of the rates

in nearby meteoroid mass bins may be undetectable in radar data. Even so, we have just shown that such

a mass difference is not what occurs when a more complex model is used. This di-pole difference is most

likely due to the fact that a continous ejection model with radiation pressure and sublimation momentum

transfer was used rather than set initial velocity. This effectively creates a initial filament structure due to

particle mass. Since the gravitational perturbation is ∝ R−2, if there is a filament structure in the correct

direction when, for example, Jupiter makes a close passage of the stream, this filament feature is enhanced

by the square. The reason the ejection model was not simplified further was that we had not predicted in

the software development that we needed to reduce the models complexity and thus there was not time to

set up such a simulation, it is however planned for the future.

42.8 Year intensity overview

As we have selected 4 specific years to plot and examine amongst our data, a overall overview of the mean

yearly flux distribution is a good point to start from when choosing years to performed specific studies of,

as done here. Such a plot can be found in figure 62 together with markers showing the showers that are

examined here. From this plot we can see that we have covered some of the major peaks in the activity

of the October Draconids in our analysis, and also that in the year 2018, we predict a major outburst of

the October Draconids. From examining the Solar longitudes of the 2018 October Draconids simulation we

predict the shower to peak in intensity at the night between October 8 and October 9, corresponding to

195.4.

42.9 Cluster analysis calibration

When searching for October Draconids in RADAR data one must use a calibrated set-up, as discussed in

previous chapters. We shall here try to give some first indications on how such a calibration process can start

by using a cluster analysis parameter sweep on the meteor showers generated in our simulation. To do this

we will introduce two new plots, the association function plot the error function plot. We have performed

the clustering with 4 similarity function, as described in section 31. Since the clustering the meteors are very

dependant on how long they have been perturbed in space, it would be misleading to perform a association

sweep that disregards time. As such we have produced a year by year version of the analysis procedure for

this case where a specific parent body is being repeatedly simulated. Thus we take the total configuration

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Figure 62: Bar plot of the weighted amount of encountered particles per year for all simulated streams of

21P/GiacobiniZinner.

Figure 63: Cluster analysis parameter sweep of critical threshold for cluster merging. Plot of the association

as a function of critical threshold for all the simulated versions of 2011 October Draconids.

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Figure 64: Cluster analysis parameter sweep of critical threshold for cluster merging. Plot of the error as a

function of critical threshold for all the simulated versions of 2011 October Draconids.

Figure 65: Cluster analysis parameter sweep of critical threshold for cluster merging. Plot of the association

and error as a function of critical threshold for the MURMHED observations of 2011 October Draconids.

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of a meteor storm in one specific year resulting from all perihelion passages, and performed the parameter

sweep on this data. In figure 63 we can see the plot of the association functions for the year 2011. In this

plot the black lines represent the association function of all the different variations of the meteor shower

simulated. The circles represent when the maximum amount of association is reached and the green line is

the fraction of shower simulations that has reached maximum. The similar plot for the error function can be

seen in figure 64 where we have used the error function in equation 31.21. Here the the blue lines represent

the error function as the critical threshold is varied and the circles when each simulated shower reaches its

minimum error. The red line is the fraction of showers that has reached minimum.

To compare we have performed the cluster analysis on the 13 selected meteors observed during 2011 that

was assumed to be a part of the October Draconids (Kero et al., 2012c). Their orbital element data was

calculated from the head echo observation and the resulting curves are shown in figure 65. We must however

consider several factors here: that the observations are biased as where our simulation is not, how the

selection was made in (Kero et al., 2012c), and how clustering is affected by sample size. The last point

was discussed in length in section 25, and we can see how a larger particle sample can affect the association

curve, however in this case it should not matter very much as the density of the sample has not changed

dramatically. The bias correction calculation and measurement uncertainty correction is outside the scope

of this work at the moment and shall be ignored, such calculations is proposed in future research. We then

have the question of selection left, as observation originating from the expected October Draconids radiant

where quite sparse the observation selection was made by simply taking all head echoes inside a box centred

on the expected radiant. This may very well, either miss some of the shower signal or include sporadic

meteors in the selection. Comparing figures 63 and 64 with 65 we can see that the association quickly

rises to a acceptable level within the observations in the same size range as with the simulated association

levels. However the critical threshold almost doubles before the clustering process has minimized error and

maximized association in the observational data compared with the simulations. This may indicate that a

few of the selected meteors from the measurement have very high uncertainties. As when we more closely

examine the events listed in (Kero et al., 2012c) we see that two events have high errors in the orbital

elements. In fact, they have so high uncertainties that they cover almost the entire predicted shower volume

in parameter space. These two events are also, upon closer examination, the reason for the doubling before

association maximum. Thus one should aim to both include this measurement uncertainty in the analysis

in a better way, and to better fit the simulated data to the likeness of a real observation by including bias

calculations as well. In the future, when the next generation of measurements become available (Pellinen-

Wannberg et al., 2016b), this kind of analysis can pre-calibrate meteor radars to detect specific meteor

showers and performed the selection automatically with a before hand calculated high reliability level, based

on also sporadic simulation.

42.10 Principal component Analysis

Lastly we have, as a proof of concept, performed two sets of principal component analysis, as described

in section 24. Firstly on all the encountered test particles, regardless of encounter time where the base

space was the Kepler elements. The resulting plane of most standardized variance can be seen in figure 66,

with the principal component representation shown in figure 67. In the plane of most variance we can see

some structure and patterns, finding such structure using PCA is a first indication that pattern recognition

methods can work on meteor data. We have in the figure also color coded the perihelion passage that the

particle was ejected from, i.e. the trail origin. We can see a certain pattern in the data where a drift seem

to occur as later passages produces new particles but this pattern is not distinguished enough to implement

a direct application on observational data. It may however be more interesting to only examine a single

event or observation campaign, thus we instead restrict our data to only 2011 and calculate the principal

components and the new basis coordinates. We can see the result in figure 68 and the principal components

in figure 69. As there is some structure difference regarding the 1972 trail encounter in this coordinate basis

Page 160


Figure 66: Plane of most standardized variance as extracted with principal component analysis, performed

on the set of all encountered particles ejected from 21P.

Figure 67: Principal component representation in the Kepler orbital element basis.

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Figure 68: Plane of most standardized variance as extracted with principal component analysis, performed

on the set of encountered particles during the 2011 October Draconids.

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Figure 69: Principal component representation in the Kepler orbital element basis for the 2011 October

Draconids analysis.

it would be interesting to examine the PCA of a Kernel Density Estimation (KDE) of the observed meteors.

One could in such a KDE use measurement uncertainties and instead extract the regions of most cumulative

probability of meteors and overlay with the simulation data. This can give a indication of trail origin of the

measurements. Unfortunately the measurement uncertainties are too large and cover several classifications of

trail origin simultaneously, this will however be possible with the next generation of measurement instruments

(Pellinen-Wannberg et al., 2016b). There was not enough time to implement the KDE functionality in the

MATLAB analysis scripts and as such we cannot showcase the results of such a analysis. For the future

we plan to implement Kernel Density Estimation techniques and many other machine learning methods to

complement this kind of analysis (Murphy, 2012).

43 Pan-STARRS Short Period Comets

43.1 Input state 1

As many of the analysis procedures and algorithms explained in part VI needed visual examples and a general

case we designed a simple and easy to explain simulation of the Pan-STARRS short period comets synthetic

model. The sampling was sampled using only one simulation, set to generate 500 streams that encountered

the Earth within 400 years of ejection. To simplify simulation setup and data analysis we set all perihelion

passages to occur at the same date during the year 1800 and integrated to the year 2200. Close encounters

with a closest distance under one Earth Hill radii where considered meteor producing test particles. The

time step mercury6 was set to 8 day and the step of PBE was set to 16 days. PBE utilized a Bulirsch-Stoer

algorithm with multiple substeps according to the previously described series, and mercury6 was configured

to use the hybrid symplectic integrator, both including electromagnetic forces. The ejection model used was

the one described as (III) in 10.2.1 implemented from (Ma et al., 2002), from with also meteoroid weights

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where calculated. Finally, the mass distribution for ejection of particles was set to a logarithmic uniform

distribution between 10−9 kg to 10−1 kg.

As most of the results from this simulation where covered as explanations and examples we will not evaluate

further on the results from this simulation at this time. This is due to the fact that any further analysis

of showers and stream would be unreliable as the synthetic model comes with a set perihelion passage date

that we have disturbed by setting all passages to the year 1800, thus a more correct simulation needs to be

setup to extract the real resulting distributions.

44 Concluding remarks

We can conclude that the software developed is a promising simulation tool as all the above meteor shower

simulations and the derived conclusions where performed on two personal computers over a 1 week period,

using minimal setup preparations and moderate data analysis work. This can allow for precise simulation

of almost any situation resulting in a meteoroid stream or large set of small particles and estimation of the

results. When the proposed additions in section 40 is completed the software toolbox will be highly versatile

and we intend to continuously, through future development, apply it to many other case studies to further

the standard of numerical simulations in meteor science.

Page 164


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Appendices

A Notation, abreviations, technical terms, and nomenclature

Table 45: Technical terms and nomenclature

Term Description

Meteor The phenomena in the atmosphere of a dust particle

ablating

Ablation Removal of material an object by vaporization, chip-

ping, or other erosive processes

Meteoroid Dust particle in the Solar system

Meteorite Meteoroid that has survived ablation and lands on

the surface of the Earth

Sublimation The phase transition from solid to gas

Table 46: Common notation

Symbol Description

a Semi-major axis

e Eccentricity

i Inclination

ω Argument of pericentre/perihelion

Ω Longitude of the Ascending Node

q Perihelion distance

Table 47: Abbreviations

Abreviation Description

ESA European Space Agency

NASA National Aeronautics and Space Administration

PCA Principal Component Analysis

KDE Kernel Density Estimation

IAU International Astronomical Union

IMO International Meteor Organization

MU radar Middle and Upper atmospheric radar

MURMHED MU Radar Meteor Head Echo Database

JPL Jet Propulsion Laboratory

NAIF Navigation and Ancillary Information Facility

Pan-STARRS Panoramic Survey Telescope and Rapid Response System

MC Monte-Carlo

MC Markov-Chain

MCMC Markov-Chain Monte-Carlo

PDF Probability Density Function

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B List of figures

List of Figures

1 The life of a meteoroid encountering the Earth. When caught by the Hill Sphere (the volume in

which Earth gravity is dominant), if the Earth gravitational pull directs the meteoroid towards

the atmosphere it will enter and begin to heat up. The meteoroid may disintegrate during

its ablation phase, or in some cases bounce of the atmosphere. The process of colliding with

the atmosphere produces a meteor phenomena as the collision turn small areas of atmosphere

into plasma. If however the meteoroid survives atmospheric entry, either due to the entry

circumstances or its sheer size, a dark flight phase will begin. Then to finally impact the

surface and by doing so turning from meteoroid to meteorite. . . . . . . . . . . . . . . . . . . 1

2 Digram illustrating and explaining various terms in relation to Orbits of Celestial bodies. (CC-

BY-SA-3.0-MIGRATED; Licensed under the GFDL by the author; Released under the GNU

Free Documentation License. http://commons.wikimedia.org/wiki/GNU_Free_Documentation_

License) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Digram illustrating the three different anomalies where E is the eccentric anomaly, M the

mean anomaly, and ν the true anomaly. The central body is located at the point s and the

orbiting body at point p. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4 Pendulum phase map with symplectic and non symplectic euler using the same stepsize. . . . 18

5 Relative error in total energy for different initial conditions for a pendulum. . . . . . . . . . . 19

6 Histogram of the different distribution of orbital elements within the Pan-STARRS Solar sys-

tem synthetic model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

7 Ejection speed as a function of the common logarithmic mass of the particle. . . . . . . . . . . 27

8 Diagram of ecliptic coordinates, depicting the ecliptic, equator, ecliptic poles, longitude and

latitude, and the vernal equinox direction. (Tfr000 / Licensed under the Creative Com-

mons Attribution-Share Alike 3.0 Unported license. http://creativecommons.org/

licenses/by-sa/3.0/) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

9 Concept image of a meteoroid head echo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

10 Contour plot of the logarithm of the total number of events per cell of the Hammer-Aitoff

projection, entire MURMHED. The labels are the six sporadic sources: (SA) South apex,(NA)

North apex,(ST) South toroidal,(NT) North toroidal,(H) Helion and (AH) Anti-Helion. . . . 38

11 Hammer-Aitoff projection of all MURMHED meteor radiants with color encoded geocentric

velocities with showers Orionids and Geminids marked. . . . . . . . . . . . . . . . . . . . . . 39

12 Meteoroid velocity time series of head echo with MJD 55008.9589319839 from MURMHED . 41

13 Radial velocity time series of head echo with MJD 55008.9589319839 from MURMHED . . . 41

14 Height time series of of head echo with MJD 55008.9589319839 from MURMHED . . . . . . 41

15 Range-time intensity plot of head echo with MJD 55008.9589319839 from MURMHED . . . . 41

16 Radar cross section time series of head echo with MJD 55008.9589319839 from MURMHED 41

17 Signal to noise ratio time series of head echo with MJD 55008.9589319839 from MURMHED 41

18 Example illustrating the concept of a Monte Carlo simulation calculating the value of π simply

from the use of uniform distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

19 Example of clustering within 6 nodes in a two dimensional plane where the similarity between

nodes are their geometric distance. Clustering is performed using a arbitrary critical distance. 56

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20 Example of cluster analysis with background data on a two dimensional plane where the sim-

ilarity between nodes are their geometric distance. Top graph shows geometric representation

of clusters, nodes, and connections where red nodes are signal data and blue nodes background

data. Bottom graph shows the sweep of critical threshold and the respective association. Back-

ground to signal count ratio of 1 to 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

21 Background to signal count ratio of 1 to 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

22 Background to signal count ratio of 5 to 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

23 Execution time for different segments of MCAS, stream dissipation calculation enabled. . . . . 68

24 Execution time for different segments of MCAS, stream dissipation calculation disabled. . . . 69

25 Comparison between the described first method of drawing a random value from a distribution

versus the real multi dimensional distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

26 Comparison between the described improved first method of drawing a random value from a

distribution versus the real multi dimensional distribution . . . . . . . . . . . . . . . . . . . . 72

27 Comparison between the described improved first method of drawing a random value from a

distribution versus the artificially generated multi dimensional distribution . . . . . . . . . . . 75

28 Comparison between the described multi dimensional method of drawing a random value from

a distribution versus the Pan-STARRS multi dimensional distribution . . . . . . . . . . . . . 75

29 Comparison between the described multi dimensional method of drawing a random value from

a distribution versus the artificially generated multi dimensional distribution . . . . . . . . . . 76

30 A highly perturbed meteoroid stream encountering Earth over 400 years. . . . . . . . . . . . . 79

31 A slightly perturbed meteoroid stream encountering Earth over 400 years. . . . . . . . . . . . 80

32 The DSH criterion measure of the stream divergence from its parent body in a highly perturbed

meteoroid stream encountering Earth over 400 years. . . . . . . . . . . . . . . . . . . . . . . . 80

33 The DSH criterion measure of the stream divergence from its parent body in a slightly perturbed

meteoroid stream encountering Earth over 400 years. . . . . . . . . . . . . . . . . . . . . . . . 81

34 The histograms describing the orbital parameters of generated parent bodies. Blue histogram

bars represent the initial distribution while red shows the resulting distribution that generated

meteor showers on Earth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

35 The distribution of stream to parent body mean divergence among the Pan-STARRS distribu-

tion of comets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

36 Probability of encountering a random cometary meteoroid stream ejected in the year 1800 in

a given year. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

37 The mean time after ejection to encounter for each meteoroid stream. . . . . . . . . . . . . . 97

38 The variance from the mean of time after ejection to encounter within a each stream. . . . . . 97

39 Two spherical distributions, the left illustrating a spatial angle uniform distribution, the right

illustrating a true spherically uniform distribution. . . . . . . . . . . . . . . . . . . . . . . . . 105

40 Calculation of the variations in the obliquity of the ecliptic for two different time scales, ± 10

years from J2000.0 and ± 10 000 years from J2000.0. . . . . . . . . . . . . . . . . . . . . . . 137

41 The graphical user interface used to call all the data analysis scripts developed. . . . . . . . . 138

42 Histogram of the orbital element distribution for orbital clones of 21P/GiacobiniZinner. . . . 145

43 Histogram of the orbital element distribution for encountered test particles ejected from the

clones of 21P/GiacobiniZinner. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

44 Histogram of Probability of encounter and mean shower intensity for the 1933 October Draconids.146

45 Density maps for the mean trail meteoroid flux distribution for the 1933 October Draconids. . 147

46 Density maps for the trail distribution probability for the 1933 October Draconids. . . . . . . 147

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53 Density maps for the mean meteoroid flux mass distribution for the 2011 October Draconids. 151

54 Density maps for the mass distribution probability for the 2011 October Draconids. . . . . . . 152



57 Density maps for the mean meteoroid flux mass distribution for the 2012 October Draconids. 154

58 Density maps for the mass distribution probability for the 2012 October Draconids. . . . . . . 154

59 The normalized difference in mass distribution between the October Draconids of 2011 and 2012.155

60 The normalized difference in mass distribution between the October Draconids of 2011 and

2012 using only a subset of perihelion passages. . . . . . . . . . . . . . . . . . . . . . . . . . . 156

61 The normalized difference in mass distribution between the October Draconids of 2011 and

2012 using only a subset of perihelion passages and propagation without electromagnetic per-

turbations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

62 Bar plot of the weighted amount of encountered particles per year for all simulated streams of

21P/GiacobiniZinner. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

63 Cluster analysis parameter sweep of critical threshold for cluster merging. Plot of the associa-

tion as a function of critical threshold for all the simulated versions of 2011 October Draconids.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

64 Cluster analysis parameter sweep of critical threshold for cluster merging. Plot of the error as

a function of critical threshold for all the simulated versions of 2011 October Draconids. . . 159

65 Cluster analysis parameter sweep of critical threshold for cluster merging. Plot of the associ-

ation and error as a function of critical threshold for the MURMHED observations of 2011

October Draconids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

66 Plane of most standardized variance as extracted with principal component analysis, performed

on the set of all encountered particles ejected from 21P. . . . . . . . . . . . . . . . . . . . . . 161

67 Principal component representation in the Kepler orbital element basis. . . . . . . . . . . . . . 161

68 Plane of most standardized variance as extracted with principal component analysis, performed

on the set of encountered particles during the 2011 October Draconids. . . . . . . . . . . . . . 162

69 Principal component representation in the Kepler orbital element basis for the 2011 October

Draconids analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

C List of tables

List of Tables

1 Data included in the Pan-STARRS Moving Object Processing System Synthetic Solar System

Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2 JPL’s measurement database of Solar system major body gravitational parameters . . . . . . 32

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3 family<x>.data, <x> represents the index number of the parent body distribution Column

separator : ’Space’ 0x20 Input File description: Parent body characteristics . . . . . . . . . . 83

4 mass.data Column separator : ’Space’ 0x20 Input File description: Major body masses (Sun

and all planets) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5 mass_dist.txt Column separator : ’Space’ 0x20 Input File description: Custom mass dis-

tribution input file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

14 error_profile_<f>.txt, <f> represents the name of the used criterion. Column sepa-

rator : ’Space’ 0x20 Output File description: Error function evolution during cluster analysis

Note: - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6 execute_time.txt Column separator : ’Space’ 0x20 Output File description: Execution

time of sections in software simulation. May have unequal number of columns for each row. . 84

7 snapshot_pos.txt Column separator : ’Space’ 0x20 Output File description: Takes a snap-

shot of all test particles positions at specified time . . . . . . . . . . . . . . . . . . . . . . . . 84

8 snapshot_vel.txt Column separator : ’Space’ 0x20 Output File description: Takes a snap-

shot of all test particles velocities at specified time . . . . . . . . . . . . . . . . . . . . . . . . 84

9 snapshot_earth.txt Column separator : ’Space’ 0x20 Output File description: Takes a

snapshot of 10 or less if too close to integration start Earth positions at specified time . . . . 85

10 time_data.txt Column separator : ’Space’ 0x20 Output File description: Simulations Earth

time data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

11 PB_kep_dist.txt Column separator : ’Space’ 0x20 Output File description: Histogram . . 85

12 function_divergance_<f>.txt, <f> represents the name of the used criterion. Column

separator : ’Space’ 0x20 Output File description: Time series of criterion between parent body

and a encountered particle, each simulation ID marks its section of the file . . . . . . . . . . . 86

13 stream_dissipation_data_<f>.txt, <f> represents the name of the used criterion.

Column separator : ’Space’ 0x20 Output File description: Time serie of criterion between

parent body and a encountered particle, each simulation ID marks its section of the file . . . 86

15 association_profile_<f>.txt, <f> represents the name of the used criterion. Column

separator : ’Space’ 0x20 Output File description: Fraction of sample associated during cluster

analysis Note: Number of rows = 2 × showers generated . . . . . . . . . . . . . . . . . . . . 86

16 particle_weight_stat.txt Column separator : ’Space’ 0x20 Output File description:

Number of meteoroids each test particle represents for each parent body Note: Only shower

generating parent bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

17 particle_ejection_v_stat.txt Column separator : ’Space’ 0x20 Output File descrip-

tion: Ejection velocity relative parent body Note: - . . . . . . . . . . . . . . . . . . . . . . . . 87

18 PB_kep_timeS_data.txt Column separator : ’Space’ 0x20 Output File description: Time

series data for all shower producing parent bodies . . . . . . . . . . . . . . . . . . . . . . . . . 87

19 earth_encounter_data.txt Column separator : ’Space’ 0x20 Output File description:

Earth orbital elements for each encounter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

20 met_encounter_data.txt Column separator : ’Space’ 0x20 Output File description: Me-

teoroid orbital elements for each encounter (at encounter) . . . . . . . . . . . . . . . . . . . . 88

21 PB_kep_data.txt Column separator : ’Space’ 0x20 Output File description: All generated

parent body Kepler elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

22 PB_M_stat.txt Column separator : ’Space’ 0x20 Output File description: Shower generating

parent body mass change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

23 met_before_encounter_data.txt Column separator : ’Space’ 0x20 Output File descrip-

tion: Meteoroid orbital elements at integration step before each encounter . . . . . . . . . . . 89

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24 ejector_char_data.txt Column separator : ’Space’ 0x20 Output File description: Parent

body characteristics Note: Only shower generating parent bodies . . . . . . . . . . . . . . . . 89

25 ejector_kep_data.txt Column separator : ’Space’ 0x20 Output File description: Parent

body characteristics Note: Only shower generating parent bodies . . . . . . . . . . . . . . . . 89

26 ejector_n_data.txt Column separator : ’Space’ 0x20 Output File description: Parent

body count data Note: Only subliming parent bodies . . . . . . . . . . . . . . . . . . . . . . . 90

27 mass.data, Column separator : ’Space’ 0x20 Input File description: Simulation state mass

vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

28 pos.data, Column separator : ’Space’ 0x20 Input File description: Simulation state position

vector, one row for each object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

29 vel.data, Column separator : ’Space’ 0x20 Input File description: Simulation state velocity

vector, one row for each object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

30 mass_dist.data Column separator : ’Space’ 0x20 Input File description: Ejection mass

distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

31 body_data.data Column separator : ’Space’ 0x20 Input File description: Parent body data 110

32 particle_q_data.data, Column separator : ’Space’ 0x20 Output File description: Simu-

lation state position vectors, one row for each object . . . . . . . . . . . . . . . . . . . . . . . 111

33 particle_v_data.data, Column separator : ’Space’ 0x20 Output File description: Simu-

lation state velocity vectors, one row for each object . . . . . . . . . . . . . . . . . . . . . . . 111

34 particle_m_data.data, Column separator : ’Space’ 0x20 Output File description: Simu-

lation state mass vector, one column for each object . . . . . . . . . . . . . . . . . . . . . . . 111

35 particle_t_data.data, Column separator : ’Space’ 0x20 Output File description: Simu-

lation ejection time vector, one column for each object . . . . . . . . . . . . . . . . . . . . . . 111

36 particle_abs_v_data.data, Column separator : ’Space’ 0x20 Output File description:

Simulation ejection velocity vector, one column for each particle . . . . . . . . . . . . . . . . 111

37 particle_kep_data.data, Column separator : ’Space’ 0x20 Output File description: Ke-

pler data after ejection, one row for each object . . . . . . . . . . . . . . . . . . . . . . . . . . 112

38 body_m_data.data, Column separator : ’Space’ 0x20 Output File description: Simulation

state parent body mass, one row for each instance . . . . . . . . . . . . . . . . . . . . . . . . 112

39 sim_data.data, Column separator : ’Space’ 0x20 Output File description: Simulation sum-

mary data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

40 OAA_input.txt, Column separator : ’Space’ 0x20 Input File description: Node data, each

row corresponds to one node Currently only Kepler element node data input . . . . . . . . . 120

41 clusters.txt, Column separator : ’Space’ 0x20 Output File description: Cluster state

where each row is one cluster This file has variable row sizes as it lists all indexes for cluster

members on one row . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

42 error_profile_<f>.txt, <f> represents the name of the used similarity function. Col-

umn separator : ’Space’ 0x20 Output File description: Error function evolution during cluster

analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

43 association_profile_<f>.txt, <f> represents the name of the used similarity function.

Column separator : ’Space’ 0x20 Output File description: Fraction of sample associated during

cluster analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

44 orb_in.data, in the case of Kepler element input configuration Column separator : ’Space’

0x20 Input File description: Data required to generate orbital clones . . . . . . . . . . . . . . 125

45 Technical terms and nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

46 Common notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

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47 Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

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