The Determination of Orbits from Spacecraft Imagingastrosun2.astro.cornell.edu/~mwe/MW_Evans_Nov2008_thesis.pdf · either post-docs or postgrads at Queen Mary, if I have forgotten

The Determination of Orbitsfrom Spacecraft Imaging

A thesis submitted to the University of London

for the degree of Doctor of Philosophy

Michael Wyn Evans

Queen Mary, University of London

November 2000

1

Abstract

Orbits are derived for the saturnian satellites Atlas, Prometheus and Pandora

from images taken by the Voyager 1 and Voyager 2 spacecraft.

The process of geometrical correction of Voyager images, whereby distortions

introduced by the vidicon imaging system are removed, is discussed. The IDL

routines written to perform this process are described.

The mechanics of image navigation, where the precise pointing direction of the

camera at the time a image was taken, are explained. Circular features in the outer

saturnian ring system and Saturn’s limb are used as fiducials for navigating the

Atlas, Prometheus and Pandora images..

Orbits are determined by fitting a uniformly precessing, inclined ellipse to the

observations. A differential correction process is used to minimise the rms residuals

between the calculated and observed locations of the object of interest.

The orbital elements of Atlas, Prometheus and Pandora at the Voyager 1 and

Voyager 2 epochs are derived. Possible explanations for Prometheus’ observed mean

longitude lag in 1995 are discussed. The effect of Pandora’s close proximity to the

3:2 co-rotation eccentricity resonance with Mimas are investigated.

The relationship between the accuracy of an orbit determination and such factors

as number of observations, range of spacecraft at observation time and epoch range

of the observations is investigated. This is then used in the planning of observations

of the small saturnian satellites for the Cassini mission.

2

Acknowledgments

First and foremost my thanks go to Prof. Carl Murray, my supervisor. He saw

something in a candidate with a somewhat checkered academic history. The idea

for the subject of this thesis was all his, although I think his original vision involved

less orbit determination and more on rings. Even when the work was progressing

slowly he remained supportive and provided encouragement. Carl appears to be on

first name terms with everybody working on rings and their associated satellites,

and through him I too have met many of them. There is a saying that “its not what

you know its who you know”, Carl knows both the “what” and the “who” and has

endeavored to pass on both to me. He has attempted to improve my spelling and

grammatical style, reading drafts of this thesis must have been painful for him.

My special thanks to Dr. Terry Arter, my personal postscript consultant. The

better diagrams would not exist in the same form without him acting as a “human

interface”. He also bravely submitted himself to torture by thesis, the current

version makes much more sense due to his puzzlement.

This work would not have been possible without the help of several people.

Dr. Carolyn Porco gave permission for me to use the MINAS software package,

without which the task of geometrical correction and image navigation would have

been much more difficult. Her comments during visits to Queen Mary highlighting

areas where I had previously seen no problems.

Dr. Mark Showalter aided in locating images of Prometheus, providing details

of searches he had performed for that satellite. Mark made available images that

are not currently on the PDS Voyager data CD-ROMs. I spent an enjoyable week

at NASA Ames with Mark as my host, an experience I thank him for.

Though he knows it not, Dr. Mitch Gordon has provided invaluable aid in my

work. Mitch was also Carl Murray’s student and his thesis involved geometrically

correcting and navigating Voyager images from the Saturn encounter in the search

for additional satellites. His thesis showed me the nuts and bolts of performing

many necessary tasks. The C-kernels I used for initial pointing directions for my

images were complied by Mitch from raw data.

My thanks to Prof. Iwan Williams for the opportunity I had to observe the

3

Leonids from La Palma in 1998, tens of fireballs a minute putting your average

meteor shower to shame. I spent a week on top of a volcano in November 1999

observing saturnian satellites with Iwan. It is due to him that I am classed as an

‘experienced’ observer.

Discussions with Dr. Dick French about my work and his on Prometheus and

Pandora have proved useful. Dr. Phil Nicholson has provided data on the orbit

of the F ring and passed on unpublished orbits for Prometheus and Pandora from

Dr. Bob Jacobson. I thank Dr. Don Taylor for his Technical Note which finally

enabled me to fit precessing ellipses to observations.

I was only able to study for a Ph. D. with a studentship from the Particle Physics

and Astronomy Research Council. My thanks to them and a British government

who still provide some funding for astronomy research.

Then there are the people who kept me sane (in no particular order) Kevin,

Tolis, Angela, Helena, Phil, Tom, Agueda, John, Matt, Paul, Chad and Nick. All

either post-docs or postgrads at Queen Mary, if I have forgotten anyone its simply

an oversight.

For all my computing needs there was always Richard Frewin and STARLINK.

There are good systems managers and there are helpful systems managers, Richard

is both.

And finally my mother. It must be worrying when you have a child (in his late

twenties) who still has not decided what he wants to be when he grows up. Not

only did she not criticise me for lack of direction but supported me both morally

and financially when I had no right to expect it.

4

I know that I am mortal by nature, and ephemeral; but when I trace at my

pleasure the windings to and fro of the heavenly bodies I no longer touch earth with

my feet: I stand in the presence of Zeus himself and take my fill of ambrosia, food

of the gods.

Ptolemy, epigram to Syntaxis

circa 145 A.D.

5

For my mother, and her patience with a son

in his thirties who is still at school.

6

Contents

1 Introduction 16

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.2 The initial research goal . . . . . . . . . . . . . . . . . . . . . . . . 18

1.3 Possible origins of Prometheus’ lag . . . . . . . . . . . . . . . . . . 19

1.3.1 A co-orbital companion . . . . . . . . . . . . . . . . . . . . . 19

1.3.2 Periodic encounters with the F ring . . . . . . . . . . . . . . 20

1.3.3 Cometary impact . . . . . . . . . . . . . . . . . . . . . . . . 21

1.3.4 Gravitational interaction with the F ring . . . . . . . . . . . 21

1.3.5 Other mechanisms . . . . . . . . . . . . . . . . . . . . . . . 21

1.4 Pandora develops a lag . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.5 The accuracy of determined orbits . . . . . . . . . . . . . . . . . . . 22

1.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2 Orbits and Orbit Determination 24

2.1 The geometry of elliptical orbits . . . . . . . . . . . . . . . . . . . . 24

2.1.1 The position of an object in an unperturbed Keplerian orbit 29

2.1.2 Orbital and reference frames . . . . . . . . . . . . . . . . . . 31

2.2 Osculating and geometric elements . . . . . . . . . . . . . . . . . . 34

2.3 Orbit determination . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.4 Preliminary orbit determination . . . . . . . . . . . . . . . . . . . . 36

2.4.1 Laplace’s method . . . . . . . . . . . . . . . . . . . . . . . . 36

2.4.2 Gauss’ method . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.4.3 The f and g functions . . . . . . . . . . . . . . . . . . . . . . 40

7

2.4.4 The suitability of Laplace’s and Gauss’ methods for planetary

satellites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.4.5 Determining the orbits of planetary satellites from spacecraft 42

2.5 Improving the orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.5.1 Integration of the full equations of motion for an entire system 43

2.5.2 The precessing ellipse model . . . . . . . . . . . . . . . . . . 44

2.5.3 Generation of an ephemeris . . . . . . . . . . . . . . . . . . 47

2.6 The differential correction of elements . . . . . . . . . . . . . . . . . 47

2.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.7.1 Integration of the full equations of motion . . . . . . . . . . 50

2.7.2 The precessing ellipse model . . . . . . . . . . . . . . . . . . 51

2.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3 Voyager Images 53

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.2 Voyager spacecraft and instruments . . . . . . . . . . . . . . . . . . 53

3.3 Voyager images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.4 The geometrical correction of Voyager images . . . . . . . . . . . . 56

3.4.1 Overview of the geometrical correction process . . . . . . . . 58

3.4.2 Reseau marks . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.4.3 Location of reseau marks . . . . . . . . . . . . . . . . . . . . 60

3.4.4 The detection of reseaus in a raw image . . . . . . . . . . . 62

3.4.5 The generation of ‘pseudo-reseau’ marks . . . . . . . . . . . 62

3.4.6 Mapping image-space locations into object-space . . . . . . . 62

3.4.7 The nearest neighbour pixel mapping technique . . . . . . . 68

3.4.8 Pixel interpolation method . . . . . . . . . . . . . . . . . . . 68

3.4.9 Comparison of the two methods . . . . . . . . . . . . . . . . 68

3.5 VICAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.6 MINAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.7 Routines for reseau location and geometrical correction . . . . . . . 71

3.7.1 resloc.pro: locating reseau marks . . . . . . . . . . . . . . 72

3.7.2 geoma.pro: Performing the Geometrical Correction . . . . 76

8

3.8 Comparison of VICAR and MINAS results . . . . . . . . . . . . . . 78

4 Image Navigation 80

4.1 Introduction to image navigation . . . . . . . . . . . . . . . . . . . 80

4.2 Methods of image navigation . . . . . . . . . . . . . . . . . . . . . . 81

4.3 Software for image navigation, MINAS . . . . . . . . . . . . . . . . 82

4.4 Navigating an image . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.5 Calculating the pointing vector to an object in an image . . . . . . 86

5 Atlas 88

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.2 Search methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.3 Geometrical correction . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.4 Image navigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.5 Generating a set of observations . . . . . . . . . . . . . . . . . . . . 93

5.6 Orbit determination . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.7 Identified images . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.8 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.9 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.10 The distribution of longitudes at observation mid-times . . . . . . . 100

5.11 The orbit of Atlas in JPL Ephemerides . . . . . . . . . . . . . . . . 103

5.12 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6 Prometheus 108

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6.2 Search methodology and orbit determination . . . . . . . . . . . . . 111


6.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119


6.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

6.7 The orbit of Prometheus in JPL Ephemerides . . . . . . . . . . . . 129

6.8 Comparison of the derived orbits with explanations for the origins of

Prometheus’ lag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

9

6.8.1 A co-orbital companion . . . . . . . . . . . . . . . . . . . . . 131

6.8.2 Periodic encounters with the F ring . . . . . . . . . . . . . . 132

6.8.3 Cometary impact . . . . . . . . . . . . . . . . . . . . . . . . 132

6.8.4 Gravitational interaction with the F ring . . . . . . . . . . . 133

6.8.5 Other mechanisms . . . . . . . . . . . . . . . . . . . . . . . 133

6.8.6 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

6.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

7 Pandora 135

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

7.2 Search methodology and orbit determination . . . . . . . . . . . . . 135


7.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142


7.6 Discussion of the orbit fits . . . . . . . . . . . . . . . . . . . . . . . 148

7.7 The orbit of Pandora in JPL Ephemerides . . . . . . . . . . . . . . 149

7.8 The 3:2 near-resonance with Mimas . . . . . . . . . . . . . . . . . . 151

7.8.1 The planetary disturbing function . . . . . . . . . . . . . . . 152

7.8.2 Lagrange’s planetary equations . . . . . . . . . . . . . . . . 154

7.8.3 The effects of the Pandora-Mimas 3:2 CER from theory . . . 155

7.8.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

8 Planning Future Observations 162

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

8.2 The error in e from geometry . . . . . . . . . . . . . . . . . . . . . 164

8.2.1 Determining F(SSL) . . . . . . . . . . . . . . . . . . . . . . 167

8.2.2 Other orbital parameters . . . . . . . . . . . . . . . . . . . . 167

8.2.3 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

8.3 Expected accuracy of realistic observations . . . . . . . . . . . . . . 168

8.4 Planning Cassini observations of the small saturnian satellites. . . . 169

8.4.1 Introducing synthetic ‘observational errors’ . . . . . . . . . . 170

8.4.2 Fitting an orbit . . . . . . . . . . . . . . . . . . . . . . . . . 170

10

8.4.3 Examples of calculated orbit determination accuracy for Cassini171

8.4.4 The effects of varying the number of observations and as-

sumed observational error . . . . . . . . . . . . . . . . . . . 174

9 Summary and Discussion 177

9.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

9.2 Accuracy of determinations . . . . . . . . . . . . . . . . . . . . . . 178

9.3 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

11

List of Figures

2.1 The geometry of an ellipse . . . . . . . . . . . . . . . . . . . . . . . 25

2.2 Definition of the eccentric anomaly E. . . . . . . . . . . . . . . . . 26

2.3 The geometry of an orbit in 3-dimensions. . . . . . . . . . . . . . . 28

2.4 Orbit and reference frame coordinates . . . . . . . . . . . . . . . . . 31

2.5 The orbital frame for a body. . . . . . . . . . . . . . . . . . . . . . 32

2.6 The J2000 reference frame. . . . . . . . . . . . . . . . . . . . . . . . 33

2.7 Laplace’s method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.8 Reference frame for precessing elliptical orbit. . . . . . . . . . . . . 46

2.9 Reference frame for orbital elements of a satellite. . . . . . . . . . . 50

3.1 Raw Voyager image orientation. . . . . . . . . . . . . . . . . . . . . 56

3.2 Raw space mapped onto image space . . . . . . . . . . . . . . . . . 57

3.3 A raw Voyager image showing reseau marks . . . . . . . . . . . . . 59

3.4 Object-space reseau mark locations . . . . . . . . . . . . . . . . . . 61

3.5 Object-space pseudo-reseau mark locations . . . . . . . . . . . . . . 63

3.6 Object-space triangular grid . . . . . . . . . . . . . . . . . . . . . . 64

3.7 Image-space triangular grid . . . . . . . . . . . . . . . . . . . . . . 65

3.8 Pixel interpolation 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.9 Pixel interpolation 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.10 9 × 9 pixel region around the location of a reseau mark. . . . . . . . 73

3.11 Index numbers for a 4 × 4 pixel image. . . . . . . . . . . . . . . . . 77

4.1 Image displayed by Minas, pre-navigation . . . . . . . . . . . . . . . 84

4.2 Image displayed by Minas, post-navigation . . . . . . . . . . . . . . 85

4.3 Instrument frame for Voyager cameras . . . . . . . . . . . . . . . . 86

12

5.1 Atlas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.2 Voyager 1 positions of Atlas . . . . . . . . . . . . . . . . . . . . . . 101

5.3 Voyager 2 positions of Atlas . . . . . . . . . . . . . . . . . . . . . . 102

6.1 Prometheus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.2 Wide angle Prometheus . . . . . . . . . . . . . . . . . . . . . . . . 110

6.3 Voyager 1 positions of Prometheus . . . . . . . . . . . . . . . . . . 125

6.4 Voyager 2 positions of Prometheus . . . . . . . . . . . . . . . . . . 126

7.1 Pandora . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

7.2 Voyager 1 positions of Pandora . . . . . . . . . . . . . . . . . . . . 146

7.3 Voyager 2 positions of Pandora . . . . . . . . . . . . . . . . . . . . 147

7.4 The position vectors r and r′ of m and m′ . . . . . . . . . . . . . . 153

7.5 Time variation of the resonant argument of Pandora . . . . . . . . . 157

7.6 Time variation of the mean motion of Pandora . . . . . . . . . . . . 157

7.7 Time variation of ∆Λ for Pandora. Epoch: 2444839.6682 JED . . . 158

8.1 Geometry of observations 1 . . . . . . . . . . . . . . . . . . . . . . 164

8.2 Geometry of observations 2 . . . . . . . . . . . . . . . . . . . . . . 165

8.3 Relationship between F(SSL) and SSL . . . . . . . . . . . . . . . . 168

8.4√N against ∆ values from Table 8.5 . . . . . . . . . . . . . . . . . 176

13

List of Tables

1.1 1995 ring plane crossing mean longitudes . . . . . . . . . . . . . . . 17

1.2 Expected errors in mean longitudes in 1995 . . . . . . . . . . . . . . 18

5.1 Constants used in the image navigation and orbit determination pro-

cess. Where f and σ are the focal length and scale factor of the

camera as used in Eqns. 4.3-4.5 . . . . . . . . . . . . . . . . . . . . 92

5.2 Voyager 1 Images of Atlas . . . . . . . . . . . . . . . . . . . . . . . 96

5.3 Voyager 2 Images of Atlas. . . . . . . . . . . . . . . . . . . . . . . . 97

5.4 The orbital elements of Atlas . . . . . . . . . . . . . . . . . . . . . 98

5.5 Voyager 1 orbital elements of Atlas . . . . . . . . . . . . . . . . . . 100

5.6 Orbital elements of Atlas from vg2 sat.bsp . . . . . . . . . . . . . . 105

5.7 Orbital elements of Atlas from sat081.4.bsp . . . . . . . . . . . . . . 106

5.8 The adopted orbital elements of Atlas: epoch 2444839.6682 . . . . . 107

6.1 Voyager 1 Images. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6.2 Voyager 2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

6.3 Voyager 1 Results: epoch 2444839.6682 JED. . . . . . . . . . . . . . 120


6.5 Voyager 1 Results using Synnott et al. ’s 18 images . . . . . . . . . 121

6.6 Voyager 2 Results using Synnott et al.’s 27 images . . . . . . . . . . 121

6.7 Voyager 1 Results: epoch 2444513.5 JED . . . . . . . . . . . . . . . 122

6.8 Voyager 1 Results using Synnott et al.’s images at Voyager 1 epoch 122

6.9 Combined Voyager 1 & 2 Results: epoch 2444839.6682 JED . . . . 123

6.10 Combined Synnott et al.’s (1983) Voyager 1 & 2 Results . . . . . . 123

6.11 Mean longitudes for Prometheus from various authors . . . . . . . . 128

14

15

6.12 Orbital elements for Prometheus from vg2 sat.bsp . . . . . . . . . . 130

6.13 Orbital elements for Prometheus from sat081.4.bsp . . . . . . . . . 131

7.1 Voyager 1 Images of Pandora . . . . . . . . . . . . . . . . . . . . . 138

7.2 Voyager 2 Images of Pandora . . . . . . . . . . . . . . . . . . . . . 140



7.5 Voyager 1 Results using 28 of Synnott et al. ’s 32 images . . . . . . 144

7.6 Voyager 2 Results using 37 of Synnott et al. ’s 39 images . . . . . . 144

7.7 Combined Voyager 1 & 2 Results: epoch 2444839.6682 JED. . . . . 145

7.8 Combined 65 of Synnott et al. ’s(1983) Voyager 1 & 2 Results . . . 145

7.9 Orbital elements for Pandora from vg2 sat.bsp . . . . . . . . . . . . 150

7.10 Orbital elements for Pandora from sat081.4.bsp . . . . . . . . . . . 151

7.11 The orbital elements of Pandora including the 3:2 resonance with

Mimas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

7.12 The orbital elements of Pandora including the 3:2 resonance with

Mimas, best fit phase . . . . . . . . . . . . . . . . . . . . . . . . . . 160

7.13 The orbital elements of Pandora of French et al. (2000) . . . . . . . 161

8.1 90 test observations of Prometheus . . . . . . . . . . . . . . . . . . 171

8.2 90 test observations of Pandora . . . . . . . . . . . . . . . . . . . . 172

8.3 90 test observations of Epimetheus . . . . . . . . . . . . . . . . . . 173

8.4 Observations of Prometheus, constant number of images . . . . . . 174

8.5 Observations of Prometheus, constant σ . . . . . . . . . . . . . . . 175

Chapter 1

Introduction

1.1 Background

The smaller satellites in the saturnian system are generally not detectable using

ground-based observations. They were discovered using images from the Voyager

1 spacecraft. They should however be visible during ring plane crossings. Close to

the time when the Earth crosses the saturnian ring plane the brightness of the rings

drops and satellites that are normally too faint to be observed can be detected.

The only Sun crossing since the Voyager encounters occurred on November 19

1995, with Earth crossings on May 22 1995, August 10 1995 and February 11 1996.

During the May event all observed satellites were at their expected positions except

for Prometheus (e.g. Nicholson et al. 1996). An object subsequently identified as

Prometheus was detected although it was 19.7 from the position expected (Bosh

and Rivkin 1995, Bosh and Rivkin 1996). Prometheus orbits between the narrow

F ring and the outer edge of the A ring.

Further observations made during the August and November events again sho-

wed that Prometheus was lagging behind its predicted location, by 18.74 and

18.82 respectively (Nicholson et al. 1996). An error in the Voyager ephemeris was

deemed unlikely. Nicholson et al. (1996) showed that for the August and November

events the fitted orbits for Mimas, Janus, Epimetheus and Pandora all agreed with

their previously determined elements. Furthermore their mean longitudes were all

within 1 of their expected values, consistent with the propagation of small errors

16

CHAPTER 1. INTRODUCTION 17

in the mean motions in the 15 years since the Voyager encounters. Table 1.1 shows

our reconstruction of the expected positions of Mimas, Prometheus and Pandora

as compared to their observed locations during the Earth crossing of the saturnian

Table 1.1: Comparison of predicted and observed mean longitudes at epoch, λ0,for satellites during the saturnian ring plane crossing of August 1995. The epochis 2449940.0 JED. The observed mean longitudes, λ0,obs are from Nicholson et al.

(1996), the calculated mean longitudes, λ0,calc, for Prometheus and Pandora arefrom Jacobson (private communication) while that for Mimas is from Harper andTaylor (1993).

Satellite Predicted Observed ∆λ

λ0,calc λ0,obs (λ0,calc − λ0,obs)

Mimas 176.07 177.11 ± 0.05 −1.04

Prometheus 358.32 339.23 ± 0.10 −19.09

Pandora 95.59 95.68 ± 0.18 −0.09

ring plane on August 10th 1995. Janus and Epimetheus have not been included in

Table 1.1 due to the difficulty in calculating their expected longitudes accurately

in 1995 because of their co-orbital configuration. Nicholson et al. (1996) state that

∆λJanus = −0.33 and ∆λEpimetheus = 0.76.

The error associated with the calculated mean longitude of a satellite at time t

can be estimated from Eqn. 2.12 (Section 2.1),

λ = n (t− t0) + λ0

where t0 is the epoch, λ0 the mean longitude at epoch and n the mean motion

determined at epoch. The relationship between the standard error in λ and the

standard errors in n and λ0 is given by

σλ =√

(σnt)2 + (σλ0

)2 (1.1)

Eqn. 1.1 is based on the calculation of errors from Squires (1976). Table 1.2 is

constructed using the mean motions and mean longitudes at the Voyager 2 epoch

of Prometheus and Pandora. Eqns. 2.12 and 1.1 are used to calculate the expected

mean longitudes during the August 10th 1995 ring plane crossing to illustrate the

maximum expected size of the deviations between the two epochs.


Table 1.2: Illustration of expected errors in mean longitudes at 2449940.0 JEDbased on Voyager values of mean motion and mean longitude at epoch. The valuesfor λ0 and n are from Jacobson (private communication) the error in n is fromSynnott et al. (1983) and the error in λ0 is an estimate based on Smith et al. (1981)and Smith et al. (1982)

Satellite λ0 n λ

Prometheus 188.54 ± 0.20 (587.2892 ± 0.0005)/day 358.32 ± 2.56

Pandora 82.15 ± 0.20 (572.7889 ± 0.0005)/day 95.59 ± 2.56

Table 1.2 clearly shows that the errors in the mean longitudes for Prometheus

and Pandora in 1995, based on Voyager epoch elements, could be as high as ∼2.6. The maximum expected errors in the mean longitudes of the other satellites

should be of approximately the same size. Table 1.1 along with ∆λJanus = −0.33

and ∆λEpimetheus = 0.76 (Nicholson et al. 1996) shows that the observed satellites

were all within the error bars of their expected mean longitudes during the ring

plane crossing event in August 1995, except for Prometheus. The lag between

Prometheus’ predicted and observed position was ∼ 19, a difference of ∼ 8σ.

Analysis of Hubble Space Telescope (HST) observations during the oppositions

of 1996, 1997 and 1998 showed that this lag was increasing by approximately

0.6 deg /year (French et al. 1998). Additional observations obtained in 1994, 1999

and 2000 when combined with the ring plane crossing data from 1995 and the 1996,

1997 and 1998 HST observations indicate that Prometheus’ lag is increasing by

0.57year−1 (French et al. 2000).

With the observed positions of Pandora, Janus, Epimetheus and Mimas all

agreeing with their previously determined elements (being well within the expected

errors (Table 1.2)) the ∼ 19 lag in Prometheus’ mean longitude from the expected

value based on the Voyager orbit was real and required explanation.

1.2 The initial research goal

There are two possible explanations for the observed change in the orbit of Prome-

theus:


1. The initial orbit determinations of Synnott et al. (1983) and Jacobson (private

communication) for the Voyager epoch are incorrect.

2. The Voyager epoch elements are correct and Prometheus’ orbit had changed

between Voyager and the ring plane crossing in 1995.

The initial goal of this research is to address the problem presented by the anomalous

motion of Prometheus. The task of determining which of the two explanations

are correct is approached by re-examining the Voyager data set and obtaining

the elements of Prometheus’ orbit at the Voyager 1 and Voyager 2 epochs both

separately and for a combined Voyager result. It is hoped that the inclusion of

additional images, not used by Synnott et al. (1983), will lead to a more accurate

determination of Prometheus’ orbital elements. If the derived orbits are comparable

with those of Synnott et al. (1983) and Jacobson (private communication) then the

simplest explanation would be that Prometheus’ orbit had indeed changed. If the

Voyager epoch orbit is indeed accurate then the mechanism responsible for changing

the orbit of Prometheus could have operated in the interval between the Voyager

1 and Voyager 2 encounters. Such a change in orbital elements would make the

use of combined data ill-advised. In this case any significant difference between

the elements at the two epochs would place important constraints on any model

explaining the origin of the lag.

1.3 Possible origins of Prometheus’ lag

What mechanisms could account for the observed difference in the orbit of Prome-

theus between the Voyager epoch and HST observations from 1994 to 2000?

Any model for explaining the origin of Prometheus’ lag must reproduce both

the observed mean longitudes and mean motions at all epochs simultaneously. Just

fitting the mean longitudes or the mean motions is insufficient.

1.3.1 A co-orbital companion

Prometheus could be maintaining a small co-orbital companion satellite in a horse-

shoe orbit (Murray and Giuliatti-Winter 1996, Nicholson et al. 1996). In such a


configuration each satellite has two distinct values for its mean motion, n. For half

the time the companion satellite orbits exterior to Prometheus’ orbit with

ncompanion = n0 − ∆ncompanion (1.2)

and the other half of the time interior to the orbit of Prometheus with

ncompanion = n0 + ∆ncompanion (1.3)

The periodic changes in ncompanion would occur near the times of closest approach

between Prometheus and the companion. Conservation of orbital angular momen-

tum leads to Prometheus experiencing similar changes in mean motion. When the

companion is exterior to Prometheus’ orbit

nPrometheus = n0 + ∆nPrometheus (1.4)

and when it is interior

nPrometheus = n0 − ∆nPrometheus (1.5)

The values of ∆nPrometheus, ∆ncompanion and n0 are constants. The value of nPrometheus

could remain constant for years before changing to a different value, remaining

constant at this new value for years before switching back once again to the original

value. With the whole cycle repeating on a regular basis. If Prometheus’ mean

motion switched to a lower value after Voyager and back again just before 1995 the

∼ 19 lag could be accounted for. For a full discussion of co-orbital satellites see

Dermott and Murray (1981a, 1981b).

1.3.2 Periodic encounters with the F ring

Murray and Giuliatti-Winter (1996) pointed out that Prometheus should experience

encounters with the F ring every 19 years. The apocentre of Prometheus’ orbit being

exterior to the pericentre of the F ring’s orbit with differential precession bringing

the two into alignment every ∼ 19 years.

A collision with an object, or objects, in the F ring between 1981 and 1995 could

quite easily lead to a reduction in Prometheus’ mean motion thus accounting for

the observed lag.


1.3.3 Cometary impact

The impact of a comet, or asteroid, between 1981 and 1995 could have resulted in a

reduction in Prometheus’ mean motion, accounting for the observed lag (Nicholson

et al. 1996). The end result is identical to that from the F ring encounter scenario

(subsection 1.3.2).

1.3.4 Gravitational interaction with the F ring

Prometheus’ could be undergoing gravitational interactions with F ring material,

either clumps or embedded moonlets, producing a ‘pseudo-random’ walk in its orbit

(Showalter 1999).

1.3.5 Other mechanisms

Gross errors in the Voyager ephemeris, back reaction from density wave torques

and secular effects are all considered by Nicholson et al. (1996).

1.4 Pandora develops a lag

While Pandora occupied its expected mean longitude during the ring plane crossings

of 1995 (Nicholson et al. 1996), data published after the start of this research project

indicated that its motion too was anomalous (French et al. 1998, French al. 2000).

Observations using the Hubble Space Telescope in the period 1994-2000, along with

the 1995 ring plane crossing data, indicated that Pandora was lagging behind its

Voyager predicted position by −1.27year−1 (French et al. 2000). There was also

an additional oscillatory component to the lag with a period of ∼ 600 days and

amplitude ±0.78, consistent with the perturbation due to Pandora’s proximity

to the 3:2 co-rotation eccentricity resonance (CER) with Mimas (see for example

Murray and Dermott 1999).

In light of this new information it was decided to extend the investigation of

the orbit of Prometheus to include Pandora. An orbit changing from its calculated

orbital elements implies the action of some ‘unseen’ mass which hasn’t been allowed

for in the orbit model used. If Prometheus is being perturbed by an ‘invisible’ mass


in the F ring region, the mass could have observable effects on Pandora as well.

It seems logical to further extend the goals to include the orbit of the other small

satellite in the F ring region, Atlas. There have been no observations of Atlas since

Voyager 2 in 1981, although Bosh and Rivkin (1996) did identify an object as Atlas

which was unsupported by other authors.

French et al. (2000) have stated that the observed lag in the orbit of Pandora

is entirely consistent with the effects of the 3:2 co-rotation eccentricity resonance

with Mimas. The perturbations due to this resonance are investigated in detail in

Section 7.8.

1.5 The accuracy of determined orbits

When planning a series of observations of satellites it is essential to know how

many images will be needed to achieve a desired result. For the Cassini spacecraft

with many instrument packages and even more investigators competing for limited

resources (time and memory space) observations have to be prioritised.

Is the scientific case for making a particular set of observations strong enough?

Does another instrument team want to make observations at the same time but

pointing in a different direction? Is there enough memory space available on the

spacecraft’s solid state recorders?

The case for a particular set of observations is strengthened if it can be shown

exactly how many individual observations are required to achieve a desired result.

In the case of orbits, this is how many observations are necessary to achieve a

specific accuracy for determined orbital elements.

We investigate the dependency of the accuracy of a determined orbit with a

number of factors. We specifically use proposed Cassini observations of small sat-

urnian satellites to investigate this dependency.

1.6 Summary

We will search the Voyager dataset for images of Atlas, Prometheus and Pandora.

Orbits will be determined for each of these satellites using the Voyager 1 and Voy-


ager 2 images both separately and in combination. Any differences in the derived

orbital elements at the two Voyager epochs will be discussed.

In light of the elements derived, possible explanations for the observed lags in

the orbits of Prometheus and Pandora will be evaluated. The determinations of

the orbits of Atlas, Prometheus and Pandora from Voyager images are discussed

separately in Chapters 5, 6 and 7.

The precessing ellipse model of Taylor (1998) will be used to investigate the rela-

tionship between the accuracy of determined orbital elements for saturnian satellites

and various parameters. These parameters include semi-major axis, observational

error, distance from Saturn, number of images and sub-spacecraft latitude. The

theoretical accuracy of determined elements using observation sets proposed for the

Cassini mission will be discussed.

Chapter 2

Orbits and Orbit Determination

2.1 The geometry of elliptical orbits

To a good approximation, all orbiting objects move in elliptical paths around their

centres of motion (Newton 1687). Objects that are not in bound orbits have

parabolic or hyperbolic trajectories e.g. some comets. We are only interested in

bound and thus approximately elliptical orbits. Fig 2.1 illustrates the geometry

of an ellipse in a plane. The term longitude is used in celestial mechanics for an

angle which has been measured with respect to a reference line which is fixed in

inertial space. In Fig 2.1 the angle θ is called the true longitude and is measured

with respect to a fixed reference direction. The polar equation for an ellipse is

r =a (1 − e2)

1 + e cos (θ −)(2.1)

where r is the distance from the ‘filled’ focus to the point on the ellipse with true

longitude θ. The pericentre and apocentre are respectively the points on the ellipse

that are closest to and furthest from the ‘filled’ focus. The angle f , called the true

anomaly, is simply θ −, i.e. the angle as measured from the radius vector to the

pericentre. Eqn. 2.1 can also be written as

r =a (1 − e2)

1 + e cos (f)(2.2)

In one orbital period, T , the true longitude increases from θ to θ+2π (or θ− 2π for

retrograde motion). The angular velocity, f , of the radius vector from the ‘filled’

24

CHAPTER 2. ORBITS AND ORBIT DETERMINATION 25

apocentrepericentre

‘empty’ focus ‘filled’ focus

position of object

reference direction

ae f

x

yr

a

b

ϖ

θ = 0

θ = ƒ + ϖ

Figure 2.1: The geometry of an ellipse of semi-major axis a, semi-minor axis b,eccentricity e and longitude of pericentre .

focus to the object is not constant (Kepler’s second law). However, we can define a

‘mean’ angular velocity, called the mean motion, n, as

n =2π

T(2.3)

. A more general form of which is

n =

√

µ

a3(2.4)

which is a form of Kepler’s third law of planetary motion, where µ = G×Mass . G

is the Universal gravitational constant and Mass is the mass of the object at the

centre of motion i.e. the body being orbited.

Some way is needed to uniquely locate the position of an object on the ellipse

at a specific time. The only variable in Eqn. 2.1 that is a function of time is θ.

Unfortunately since the angular velocity, θ, is not constant θ does not vary linearly

with time. What is needed is an angle that is not only 2π periodic but also varies


linearly with time. The angle which we use is the mean anomaly,M , which is defined

in terms of the mean motion, n, and the time when the object passes through the

pericentre, τ , by

M = n (t− τ) (2.5)

The mean anomaly, M , has no simple geometrical representation but it can be

related to an angle that does. This angle is the eccentric anomaly, E. Consider

E f

ellipse

circumscribed circle

centre of circle focus of ellipse

a

r

x

y

object

Figure 2.2: Definition of the eccentric anomaly E.

an ellipse with semi-major axis a and eccentricity e. Place a circumscribed circle

of radius a concentric with the ellipse. A line is extended from the major axis

of the ellipse through the point occupied by the orbiting object, perpendicular to

the major axis of the ellipse, intersecting the circle. The eccentric anomaly, E, is

defined as the angle between the major axis of the ellipse and the radius vector

from the centre of the circle to the intersection point on the circle’s circumference.

The geometry is illustrated in Fig 2.2. Converting the polar coordinates (r, θ) into

cartesian coordinates (x, y) is accomplished using

x = a (cosE − e) (2.6)


y = a√

1 − e2 sinE (2.7)

The mean and eccentric anomalies are related through Eqn. 2.8, Kepler’s equation.

M = E − e sinE (2.8)

At this stage a final angle is introduced, the mean longitude, λ, which is defined by

λ = M + (2.9)

Since λ depends only on M and the constant it is a linear function of time which

has no geometrical interpretation. Combining Eqns. 2.5 and 2.9 gives

λ = n (t− τ) + (2.10)

if at time t = t0, λ = λ0 then using Eqn. 2.10

λ0 = n (t0 − τ) + (2.11)

subtracting Eqn. 2.11 from Eqn. 2.10 gives

λ = n (t− t0) + λ0 (2.12)

In celestial mechanics a fixed time, called an epoch, is used when defining orbits and

reference frames. The value of an orbital variable, usually called an orbital element,

at the chosen epoch will be denoted by the subscript 0. So the mean longitude at

epoch is λ0, the mean anomaly at epoch M0 and so on.

In a system consisting of only two bodies, which always orbit each other in the

same plane, there is no need to extend a 2-dimensional representation of an elliptical

orbit as illustrated in Fig 2.1 to a 3-dimensional one. When a system contains three

or more bodies, the orbits are usually not confined to a single plane and a fixed

reference plane therefore needs to be defined. The introduction of a reference plane

and the move from a 2- to a 3-dimensional representation of elliptical orbits requires

the introduction of an additional 3 orbital variables. The angle of inclination of the

orbital plane to the reference plane is the inclination of the orbit, i. The line of

intersection of the orbital plane and the reference plane is the line of nodes. The

point in both planes where the orbit crosses from below the reference plane to above


it is the ascending node. The angle between the reference direction, which lies in

the reference plane, and the radius vector to the ascending node is the longitude of

ascending node, Ω. The angle between the radius vector to the ascending node and

the radius vector to the pericentre is the argument of pericentre, ω. The geometry

of an elliptical orbit in 3-dimensions is illustrated in Fig 2.3.

planereference

orbit plane

referencedirection

pericentre

ascending node

object

Ωω

f

i

direction oforbitalmotion

Figure 2.3: The geometry of an orbit in 3-dimensions.

The inclination always lies in the range 0 ≤ i ≤ 180. The motion is prograde if

i < 90 and retrograde if 90 ≤ i ≤ 180. The longitude of pericentre, , is defined

as

= Ω + ω. (2.13)

When i = 0, is the angle between the reference direction and the radius vector

to the pericentre.

It is usually stated in the literature that the unperturbed elliptical orbit of a

body is defined by six orbital elements (e.g. Danby 1992 and Roy 1988). For these

six orbital elements we will use a, λ, e, , i and Ω. Knowledge of these six orbital

elements at a particular epoch does indeed uniquely define the position of the body

at that epoch but it does not allow the position of the body to be calculated at

any time. A seventh orbital parameter is required in order to determine the body’s


position at any time. Examination of Eqn. 2.12 clearly shows that the calculation of

the mean longitude, λ, at any time given the mean longitude at epoch, λ0, requires

the mean motion, n. We will use n as the seventh orbital parameter. The mean

motion as the seventh orbital parameter is usually overlooked. Kepler’s third law of

planetary motion is often used to calculate the orbital period, T , and mean motion,

n, directly from the semi-major axis, a using Eqns. 2.3 and 2.4. Use of Kepler’s third

law in this way requires that the exact orbital period and semi-major axis already

be known for another body orbiting the same central object as the body of interest.

This criterion is not always satisfied e.g. when Dactyl was discovered orbiting the

asteroid Ida by the Galileo spacecraft in 1993 there were no other satellites, let

alone ones with accurately determined a and T . Knowing the mass of the central

body would also allow Kepler’s third law to be utilised, Eqn. 2.4, but again this

is not always known a priori. Therefore in the most general case an unperturbed

elliptical orbit has six orbital elements and a seventh orbital parameter.

2.1.1 The position of an object in an unperturbed Keplerian orbit

Given the 6 orbital elements (or the six orbital elements and the seventh orbital

parameter in the most general case) of an object at epoch its location at any time

in the past or future can be calculated. Assuming that the mass of the object being

orbited is known, the 6 orbital elements at epoch are a0, λ0, e0, 0, i0 and Ω0. The

epoch is t0 and the time at which the location of the orbiting object is required is t.

The mean motion, n, is calculated using Kepler’s third law, Eqns. 2.3 or 2.4. Since

we are considering an unperturbed Keplerian ellipse the only one of the orbital

elements which is time dependent is the mean longitude, λ. All the other elements

retain their epoch values.

Using Eqn. 2.12 the value of λ at time t is calculated,

λ = n (t− t0) + λ0

then Eqn. 2.9 is used to determined the mean anomaly, M ,

M = λ+0


Kepler’s equation, Eqn. 2.8, is then solved for the eccentric anomaly, E,

E − e0 sinE = M

Kepler’s equation is transcendental in E and in general it is impossible to express

E as a simple function of M . Kepler’s equation can be solved numerically, e.g. the

Newton-Raphson method, or by using a series solution. The solution of Kepler’s

equation is well covered in the literature (e.g. see Murray and Dermott 1999 or

Danby 1992) and we shall not detail it here.

With E known the x and y coordinates in the cartesian system illustrated in

Fig. 2.2 are calculated using Eqns. 2.6 and 2.7.

x = a0 (cosE − e0)

y = a0

√

1 − e20 sinE

The (x,y) coordinates of the object must then be transformed into a 3 dimensional,

inertial Cartesian reference frame with axes X, Y and Z which is illustrated in

Fig. 2.4. The chosen reference direction lies along the X axis of the reference frame.

Orbital plane coordinates are transformed into reference plane coordinates using

(Murray and Dermott 1999)

X

Y

Z

= r

cos Ω cos(ω + f) − sin Ω sin(ω + f) cos i

sin Ω cos(ω + f) − cos Ω sin(ω + f) cos i

sin(ω + f) sin i

(2.14)

where we have assumed that the object of interest lies in the orbital plane and

r =√x2 + y2.

The pointing direction to the object from an observer at reference frame coor-

dinates Xobserver, Yobserver, Zobserver is P where

Px

Py

Pz

=

X

Y

Z

−

Xobserver

Yobserver

Zobserver

(2.15)

and the unit pointing vector from the observer to the object is P. We are ignoring

the effects of the finite speed of light, c, and planetary aberration at this stage.


X

Y

Z

x

y

z

pericentre of orbit

orbit plane

referenceplane

i

Ω

ω

Ascending node

reference direction

x

X reference plane

orbit plane axes

axes

directionof motion

f

object

Figure 2.4: Illustration of relationship between orbit plane coordinates x, y, z andreference frame coordinates X, Y, Z.

Following the described procedure allows the position of an object in an unper-

turbed Keplerian orbit to be calculated at any time. The pointing vector to the

object from an observer can then be calculated providing the coordinates of the ob-

server is known in reference frame coordinates. If the observer is also in a Keplerian

orbit its position in reference frame coordinates must also be determined.

2.1.2 Orbital and reference frames

An object’s orbital frame has its origin at the centre of the object’s motion. In the

Solar System it is usually the Sun, heliocentric, for planets, asteroids and comets.

For planetary satellites it is the parent planet, planetocentric. In certain cases using

the centre of mass, the barycentre, of a system of objects as the origin provides a

better representation of the orbit. For Atlas, Prometheus and Pandora a planeto-


centric origin provides a much better fit to the data.

The object’s velocity vector, along with the centre of its motion, defines the

orbital plane. The x- and y-axes are in the orbital plane with the x-axis point-

ing towards some pre-chosen reference direction, we use the radius vector to the

pericentre (after Murray and Dermott 1999). The z-axes is perpendicular to the

orbital plane, parallel to the angular momentum vector of the object. The y-axis is

orthogonal to both x and z as defined by the vector cross-product y = z × x. The

orbital frame is shown in Fig. 2.5 As long as , Ω and i remain constant the orbital

x

y

z

orbital plane

pericentredirection of motion

centre of motion

Figure 2.5: The orbital frame for a body.

frame will remain fixed in inertial space. When a system contains more than two


bodies, as do all real world systems, mutual interactions between the bodies cause

the orbital elements to change, rotating the orbit and its orbital frame through in-

ertial space. This change in the orbital frame due to the effects of the perturbations

of additional objects is called precession.

The z coordinate of a body in its own orbital frame is always z = 0 but other

bodies may have z 6= 0 in that particular frame. The orbital frame of one body

usually will not coincide with that of another, although the orbital planes of bodies

within a system are often nearly co-planar.

Reference planes are chosen such that the X-axis is parallel to a fixed vector,

the reference direction. The Z-axis is parallel to another fixed vector which is

orthogonal to the reference direction with the Y-axis defined by the vector cross-

product Y = Z×X. The J2000 reference frame has the X-axis parallel to the vector

from the centre of the Earth pointing towards the intersection of the ascending

node of the mean Earth equator and the ecliptic at J2000 ( 2000 JAN 01 12:00:00.0

TDB). The Z-axis is parallel to the spin angular momentum vector of the Earth

at J2000 (i.e. parallel to the North polar axis) as shown in Fig. 2.6. The origin of

x

yz

ecliptic

Earth equator plane

North pole

Figure 2.6: The J2000 reference frame.


the J2000 frame can be translated through space without changing the orientation

of the cartesian axes. For example, observations in the J2000 frame are centered

on the observer. The position and velocity of an object at a specified time with

respect to an origin is called a state, or state vector. States in the J2000 frame need

to have the origin specified, usually chosen to be the Sun, a planet or the system

barycentre. The J2000 frame is the current standard frame for observations, star

positions and states.

Previous standard frames include the B1950.0 frame, which is similar to the

J2000 frame but uses the mean Earth equator and North polar axis at 1949 DEC

31 22.09:07.2 TDB. All the Voyager data has observations and states in the B1950

reference frame. Transforming from one reference frame to another involves a simple

rotation.

2.2 Osculating and geometric elements

A body moving in a Keplerian orbit, or Keplerian ellipse, obeys Kepler’s three laws

of planetary motion exactly.

1. Objects move in ellipses with the centre of motion at one focus.

2. A radius vector from the centre of motion to the orbiting object sweeps out

equal areas in equal times.

3. The square of an object’s orbital period is proportional to the cube of its

semi-major axis.

For a Keplerian orbit, the conversion of orbital elements to states and vice versa

is straightforward (e.g. see Taylor 1998 or Danby 1992). The osculating elements of

an orbit are those calculated from an instantaneous state assuming perfect Keplerian

motion. For a system consisting of only two point masses the osculating elements

describe the exact trajectory followed by one mass with respect to the other. This

was demonstrated by Newton (1687).

When there are three or more masses in the system, the masses do not follow

perfect Keplerian curves due to the perturbations of the other objects as proved by

Poincare (1892-1899). In the n-body case (n > 2) the osculating elements do not


describe the exact trajectory followed by the objects. The exact trajectory followed

by an object is described by its geometric elements. The geometric elements take

into account the perturbations of other masses in the system or the effects of an

oblate mass (Greenberg 1981). The major differences are usually in n and a while

and Ω are no longer constant but precess.

For the planets the osculating elements describe the actual trajectories very

well. For planetary satellites the effects of oblate primaries make the osculating

elements an inadequate description of the trajectories. Throughout this work all

orbital elements will be the geometric elements unless specifically stated otherwise.

2.3 Orbit determination

The problem of determining the orbit of an object divides naturally into two parts.

1. The determination of a preliminary, approximate orbit from a minimal number

of observations.

2. Improving the preliminary orbit by using as many observations as possible.

A very common problem in celestial mechanics is finding the orbit of a newly discov-

ered asteroid or comet using a very small number of observations. The preliminary

orbit is used to predict the object’s position in the future so that additional obser-

vations can be made thus allowing the orbit to be improved.

The two main methods used for preliminary orbit determination were developed

from the techniques of Laplace and Gauss. These approaches, called Laplace’s

method and Gauss’ method after their originators, are extensively described in the

literature (e.g. Escobal 1965, Roy 1988, Marsden 1985 and Danby 1992) and so

will only be summarised here. In both Laplace’s and Gauss’ method of initial orbit

determination the position, velocity and acceleration (R, R and R) of the observer

is known at all times.


2.4 Preliminary orbit determination

2.4.1 Laplace’s method

Consider an observer and an object with position vectors R and r with respect to

the centre of motion of the object. The position vector of the object with respect

to the observer is p as in Fig. 2.7. The relationship between the vectors R, r and

R

r

Observer

Object

Central body e.g. the Sun

p

Figure 2.7: Laplace’s method

p is

p = r −R (2.16)

or

pp = r − R (2.17)

where the position of the observer, R, and the unit pointing vector to the object,

p are known. The two unknowns are the radial vector to the object, r, and the

observer-object separation, p. Squaring Eqn. 2.17 leads to

p2 = r2 − 2r · R +R2 (2.18)


differentiating Eqn. 2.17 with respect to time gives

p ˙p + pp = r − R (2.19)

and differentiating again gives

p¨p + 2p ˙p + pp = r − R (2.20)

A minimum of three closely space observations are required, i.e. three values of

p. From three observations the values of ˙p and ¨p can be calculated numerically

using Lagrangian interpolation (Escobal 1965), where t2 is taken to be 0.

p(t) =t(t− t3)

t1(t1 − t3)p1 +

(t− t1)(t− t3)

t1t3p2 +

t(t− t1)

t3(t3 − t1)p3 (2.21)

with

˙p =(2t− t3)

t1(t1 − t3)p1 +

2t− t3 − t1t1t3

p2 +2t− t1

t3(t3 − t1)p3 (2.22)

and

¨p =2

t1(t1 − t3)p1 +

2

t1t3p2 +

2

t3(t3 − t1)p3 (2.23)

where pi and ti (i = 1, 2, 3) are the unit pointing vectors and times of the three

observations and p(t) is the pointing vector at time t. Eqn. 2.20 is multiplied by

(p× ˙p) and utilising the fact that

(p × ˙p) · p = 0 (2.24)

(p × ˙p) · ˙p = 0 (2.25)

becomes

p¨p · (p× ˙p) = (p × ˙p) · r − (p× ˙p) · R (2.26)

Using Newtonian gravitation

r = −GMr3

r (2.27)

so

(p× ˙p) · r = −GMr3

(p× ˙p) · r (2.28)

which using Eqn. 2.17 becomes

(p× ˙p) · r = −GMr3

(p × ˙p) · (pp + R) (2.29)


or

(p× ˙p) · r = −GMr3

(p(p × ˙p) · p + (p× ˙p) · R) (2.30)

which since (p× ˙p) · ˙p = 0 is

(p× ˙p) · r = −GMr3

(p× ˙p) · R (2.31)

Substituting Eqn. 2.31 back in Eqn. 2.26 gives

p¨p · (p × ˙p) = −GMr3

(p× ˙p) · R − (p × ˙p) · R (2.32)

Eqns. 2.32 and 2.18 are a coupled set of equations with two unknowns, r and p,

which are usually solved iteratively. Once p is known r immediately follows from

Eqn. 2.17 and r from Eqn. 2.19.

2.4.2 Gauss’ method

Again considering the geometry of object, observer and central body as illustrated

in Fig. 2.7. Let the radius vector of the object at three times be r1, r2 and r3. Since

the plane of the object’s orbit remains constant (assuming no perturbations) there

are scalars c1 and c3 such that

r2 = c1r1 + c3r3 (2.33)

The constants c1 and c3 can be calculated using Gauss’ sector-triangle ratios (the

following is from Danby 1992). Let [ri, rj] represent the area of the triangle formed

by ri and rj. Let the area swept out by the radius vector in moving between ri and

rj be (ri, rj). The constants c1 and c3 are

c1 =|r2 × r3||r1 × r3|

(2.34)

=[r2, r3]

[r1, r3](2.35)

c3 =|r1 × r2||r1 × r3|

(2.36)

=[r1, r2]

[r1, r3](2.37)

the sector-triangle ratios y1, y2 and y3 are defined such that

y1 =(r2, r3)

[r2, r3](2.38)


y2 =(r1, r3)

[r1, r3](2.39)

y3 =(r1, r2)

[r1, r2](2.40)

Eqn. 2.36 can be written

c1 =[r2, r3]

[r1, r3]

(r2, r3)

(r2, r3)

(r1, r3)

(r1, r3)(2.41)

which using Eqns. 2.39 and 2.40 becomes

c1 =(r2, r3)

(r1, r3)

y2

y1

(2.42)

Using Kepler’s second law of planetary motion, a radius vector sweeps out equal

areas in equal times i.e. area = kt where k is a constant. Eqn. 2.42 becomes

c1 =k(t3 − t2)

k(t3 − t1)

y2

y1

or

c1 =(t3 − t2)

(t3 − t1)

y2

y1(2.43)

and in a similar way

c3 =(t2 − t1)

(t3 − t1)

y2

y3(2.44)

Using Eqn. 2.17 in Eqn. 2.33 we obtain

c1p1p1 − p2p2 + c3p3p3 = c1R1 −R2 + c3R3 (2.45)

Eqn. 2.45 is a coupled set of three equations with three unknowns and can be solved

iteratively. Initial values of c1 and c3 are calculated assuming that the sector-triangle

ratios are exactly equal to one so

c1 =t3 − t2t3 − t1

c3 =t2 − t1t3 − t1

and p1, p2 and p3 solved for. The corresponding values of ri are obtained from

Eqn. 2.17. The values of c1 and c3 at the next iteration are calculated using

Eqns. 2.43 and 2.44 with y1, y2 and y3 determined using ri from the previous

iteration. The determination of the sector triangle ratios, yi, is covered in great

detail in Danby (1992). Once the solution has converged r1, r2 and r3 are known.


2.4.3 The f and g functions

Laplace’s and Gauss’ method only calculate the position of the object, not its

velocity. The velocity is obtained from the position using the f and g functions.

For an object in a Keplerian orbit the radius vector at time t can be expressed in

terms of the radius and velocity vectors at time t0

r(t) = fr0 + gv0 (2.46)

Given the f and g functions and the radius vectors at two times, the velocity at one

of those times can be calculated. The f and g functions can be calculated using

the radius vectors, the sector-triangle ratios and µ as part of Gauss’ method as

described by Danby (1992). The less accurate truncated f and g series have to be

used to obtain the velocities using Laplace’s method

f = 1 − 1

2

µ

r30

(t− t20) + . . . (2.47)

g = (t− t0) −1

6

µ

r30

(t− t0)3 + . . . (2.48)

The additional terms in the two series include the velocities so have to be discarded.

When both the position and velocity of the object are known its osculating orbital

elements can be calculated (section 2.2).

2.4.4 The suitability of Laplace’s and Gauss’ methods for planetary satellites

Laplace’s method may be used for any number of observations, with the minimum

number being three. In its original form it could only be used for three observations,

Herget (1948) suggested that the inclusion of a fourth observation would improve the

accuracy of the orbit determination. Lagrangian interpolation allows any number

(≥ 3) of observations to be utilised (Escobal 1965). The greater the number of

observations used the more accurate the determination of the orbit. Laplace’s

method requires that the observations be made very close together, each observation

occurring a short time after the previous one. This is not a problem for observations

of asteroids or planets which have orbital periods of years. Observations separated

by days are typically ∼ 0.01 of an orbital period apart, close enough for Laplace’s

method to be valid. When a satellite with an orbital period of ∼ 0.5 days is being


observed from a spacecraft it is rarely possible to take a series of observations say

every 12 minutes for an entire orbital period. Successive observations of a particular

satellite may be separated by several orbital periods. In this case Laplace’s method

is not valid.

The use of the truncated f and g series to calculate the velocities introduces

additional errors into the derived orbital elements. For asteroids, comets and planets

the effects of an oblate primary (section 2.2) are not a consideration. For planetary

satellites, particularly those of the giant planets, the effects of the oblate planet on

the orbit of a satellite are significant.

Gauss’ method may only be used for exactly three observations. If a large data

set is available different combinations of three observations would have to be used

in turn and a mean set of elements determined. It has the advantage that the f

and g functions are used to determine the velocity not the truncated f and g series.

Successive observations do not have to be closely spaced, they can be separated by

several orbital periods. Again the effects of an oblate primary are not accounted

for.

Only states and osculating elements can be obtained from Laplace’s or Gauss’

method. As discussed in section 2.2 the actual geometric shape of a satellite’s orbit

around an oblate primary is not accurately described by osculating elements.

There is no guarantee that Laplace’s or Gauss’ method will work for a particular

set of observations. Sometimes the solutions just will not converge. We programmed

Gauss’ method to determine orbits from three observations. In tests the determined

orbits of objects were fairly accurate unless one or more of the observations was

made from a position further away from the primary than the object. In this case

the solution failed to converge. We were unable to determine whether this is an

intrinsic flaw in the method or a fault with the algorithm used. Since spacecraft

observations of satellites are generally made from positions exterior to the satellite’s

orbit this difficulty is a great limitation.

Laplace’s and Gauss’ method are very useful where there is a minimal set of ob-

servations available. Where there are large data sets they are incapable of achieving

accuracies commensurate with the number of observations. These methods are use-


ful for making an initial orbit determination. When there are more than a minimal

number of observations other techniques model the actual dynamics with greater

accuracy.

2.4.5 Determining the orbits of planetary satellites from spacecraft

For most planetary satellites assuming a circular, prograde, equatorial orbit pro-

vides sufficiently accurate starting conditions for improving an orbit. To date all

planetary probes have approached a planet in a trajectory inclined to the planet’s

equatorial plane. In this situation one observation is sufficient to determine a cir-

cular, prograde, equatorial orbit.

A difficulty with many spacecraft (e.g. the Voyager probes) is that they do not

orbit a planet but simply fly by. This limits the time over which observations can be

made. The Voyager observations of Prometheus only span ∼ 30 days. Fortunately

this will not be a difficulty for Cassini with a four year tour planned.

2.5 Improving the orbit

Once an approximate orbit is known for an object the only way to improve the

orbit, so that it more accurately represents the actual dynamics of the system

under consideration, is to obtain additional observations, the more the better. The

available techniques for orbit improvement are of two basic types

1. numerical integration of the equations of motion

2. fitting a precessing elliptical model

and both types are in use. Papers which publish orbital elements for satellites of

the outer planets from Voyager data using a precessing elliptical model fit to obser-

vations include Smith et al. (1981), Smith et al. (1982), Synnott et al. (1983), Owen

and Synnott (1987) and Owen et al. (1991). Others which numerically integrate

the equations of motion to fit to the observational data are Jacobson et al. (1991),

Jacobson et al. (1992) and Jacobson (1998a,1998b).


2.5.1 Integration of the full equations of motion for an entire system

The gravitational force, Fij , acting on a point mass mi due to a point mass mj is

given by Newton’s universal law of gravitation (Newton 1687),

Fij =Gmimj(rj − ri)

|(rj − ri)|3(2.49)

where ri and rj are the position vectors of mi and mj with respect to some non-

accelerating origin in inertial space. The order of the indices is important. We are

modeling the effect on the object denoted by the first index caused by the object

denoted by the second index. Eqn. 2.49 can be extended to include the effects of

more than one mass acting on mi,

Fij =n∑

j=1,j 6=i

Gmimj(rj − ri)

|(rj − ri)|3(2.50)

where n is the total number of masses in the system. Dealing with accelerations

instead of forces Eqn. 2.50 is,

ri =n∑

j=1,j 6=i

Gmj(rj − ri)

|(rj − ri)|3(2.51)

Eqn. 2.51 is the n-body problem which is not generally soluble analytically. It is

however soluble numerically. Given the states (positions and velocities) and masses

of the n bodies at a particular epoch it is possible to calculate their states at

any time in the past or future. Planets and satellites are not point masses and

the effects of their shapes on their gravitational fields has to be accounted for. A

common method is to model them as oblate spheroids which adds the terms Aij

and Aji into Eqn. 2.51 (e.g. Peters 1981).

ri =n∑

j=1,j 6=i

Gmj(rj − ri)

|(rj − ri)|3−Aji + Aij (2.52)

The set of equations made up from Eqn. 2.52 for the n bodies are the full equations

of motion of the system.

Eqn. 2.52 can be solved for the entire n-body system using iterative numerical

integration techniques. The values of ri and ri (i = 1, n) are calculated at each

iteration using the values from the previous step. A traditional scheme for integrat-

ing the equations of motion is the Runge-Kutta 4th order scheme and is extensively

described in the literature (e.g. Danby 1992).


In recent years much more accurate and powerful techniques have come into use.

The scheme that has been used at QMW is the Runge-Kutta-Nystrom 12th order

scheme of Dormand et al. (1987a, 1987b), specifically written to solve Eqn. 2.52.

Analysis has shown the Runge-Kutta-Nystrom 12th order technique to be extremely

accurate (Hadjifotinou and Harper 1995).

2.5.2 The precessing ellipse model

The orbit of an object is modeled as a simple precessing ellipse. The following

treatment of a precessing elliptical model for a satellite orbit is taken from Taylor

(1998). When inclinations and eccentricities are small the pericentres and nodes of

an orbit are difficult to determine. In such cases in place of the eccentricity, e and

the longitude of pericentre, it is easier to use h and k where

h = e sin (2.53)

k = e cos (2.54)

and p and q instead of the inclination, i and the longitude of ascending node, Ω

where

p = sin i sin Ω (2.55)

q = sin i cos Ω (2.56)

The six orbital elements at anytime are given by

a = a0 (2.57)

λ = λ0 + nt (2.58)

h = h0 cos βt+ k0 sin βt (2.59)

k = k0 cosβt− h0 sin βt (2.60)

p = p0 cos γt+ q0 sin γt (2.61)

q = q0 cos γt− p0 sin γt (2.62)

where a is the orbital semi-major axis, λ is the mean longitude, n is the sidereal

mean motion and β, γ are the apsidal and nodal precession rates respectively. The

terms a0, λ0, h0, k0, p0 and q0 are the values of the orbital elements at epoch.


Where there are resonant relationships between the mean motions of two or more

satellites additional periodic terms have to be included in the mean longitude. We

neglected these periodic terms for Atlas and Prometheus as they have no significant

resonances with any other detected saturnian satellite. Pandora however, is very

close to a resonance with Mimas and this is taken into account (Chapter 7). It

must be stressed that these are not osculating but ‘geometric’ orbital elements

which include the effects of the primary’s oblateness (Greenberg 1981).

The apsidal and nodal precession rates are calculated directly from a, n the

equatorial radius of the primary, R, and the first two even zonal gravitational

harmonics J2 and J4 which (for small e and i) gives (Murray and Dermott 1999),

β = n

[

3

2J2

(

R

a

)2

− 9

8J2

2

(

R

a

)4

− 15

4J4

(

R

a

)4]

(2.63)

γ = −n[

3

2J2

(

R

a

)2

− 27

8J2

2

(

R

a

)4

− 15

4J4

(

R

a

)4]

(2.64)

The semi-major axis is determined in two ways. In the differential correction process

(see section 2.6) a, λ, n, h, k, p and q are free parameters. The a from the fitting

process is afitted and can be considered to be a scale factor. The a that is actually

used is acalc and is derived from the sidereal mean motion n using (Murray and

Dermott 1999)

n2 =GM

a3calc

(

1 +3

2J2

(

R

acalc

)2

− 15

8J4

(

R

acalc

)4)

+ . . . (2.65)

It is acalc that is used to calculate β and γ in Eqns. 2.63 and 2.64.

Let the position vector of the satellite in the planetocentric frame be O where

(to O(e2) and O(sin2 i)) from Taylor (1998)

OX = a

[

−3k

2+(

1 − 3

8k2 − 5

8h2 − 1

2p2)

cosλ

+(

1

4hk +

1

2pq)

sinλ+1

2k cos 2λ+

1

2h sin 2λ

+3

8

(

k2 − h2)

cos 3λ+3

4hk sin 3λ

]

(2.66)

OY = a[

−3

2h +

(

1

4hk +

1

2pq)

cosλ

+(

1 − 5

8k2 − 3

8h2 − 1

2q2)

sin λ− 1

2h cos 2λ

+1

2k sin 2λ− 3

4hk cos 3λ+

3

8

(

k2 − h2)

sin 3λ]

(2.67)


OZ = a[

−3

2hq +

3

2kp− p cosλ+ q sin λ

−1

2(hq + kp) cos 2λ+

1

2(kq − hp) sin 2λ

]

(2.68)

where OX points along the intersection of the ascending node of the primary’s

equator at epoch with the Earth mean equator at J2000 and OZ lies parallel to the

primary’s spin angular momentum vector at epoch. Finally, OY is in the plane of

the primary’s equator such that X, Y , Z form a right handed triad. This frame is

illustrated in Fig. 2.8.

planetary equator plane at epoch

mean Earth equatorplane at J2000

ascending node of planetary equator

planet

on mean Earth equator

centre of planet

sense of rotation

O

OO

x

yz

Figure 2.8: Reference frame for precessing elliptical orbit.

The apparent pointing direction at time t is

P (t) = O (t− τ) − S (t− τ) (2.69)

where

τ =| O(t− τ) − S(t) |

c(2.70)

τ is the light travel time, c is the speed of light and S is the planetocentric position

vector of the spacecraft. Eqn. 2.70 has to be solved iteratively to obtain a value

for τ . The apparent pointing direction calculated using Eqn. 2.69 allows for both

light travel time and classical stellar aberration effects (e.g. Hohenkerk et al. 1992).


We have ignored the effects of gravitational light bending and relativistic stellar

aberration as these are much smaller than the accuracy of our observations.

2.5.3 Generation of an ephemeris

Both techniques for the improvement of orbits have similarities. A set of starting

parameters is needed. These are the seven orbital elements at epoch for the pre-

cessing ellipse and the state vector at epoch for all the bodies in the system along

with their masses (usually G × Mass) for the numerical integration. Additional

parameters can be added to each model. The precessing ellipse can have β and γ

as variables instead of calculating them from Eqns. 2.63 and 2.64. The masses and

J2 and J4 values (J2 and J4 from Campbell and Anderson 1989) can be allowed to

vary for any or all of the bodies under consideration in the numerical model.

Whatever model is being used, a trajectory for an object is calculated using that

model and then the pointing vector to the object from the observer’s location at

the observation times is determined. The calculated trajectory, positions at given

times for the precessing ellipse and state vectors at given times for the numerical

model, constitutes an ephemeris. The calculated pointing vectors are compared to

the actual observations of the object and the residuals calculated. The residual of

an observation is given by

residual = |Pobserved − Pcalculated| (2.71)

where P is the unit pointing vector from the observer to the object. The starting

parameters of the model are changed to reduce the size of the residual. Once the

residual has been minimised the modeled trajectory is the ‘best’ dynamical fit to the

observations given the restrictions and limitations of the model used. The residual

is minimised using a least-squares differential correction technique (section 2.6).

2.6 The differential correction of elements

An orbit is fitted to the observations using a differential correction process. The

following treatment of the differential correction of orbits is from Danby (1992).

It is assumed that the trajectory of a body can be described by some model e.g


Newtonian Gravitation, Keplerian ellipses etc. All the parameters which are used

in this model are contained in the vector X. Using the model and a given set of

parameters X, the values of a set of observations can be calculated. This set of

observations calculated using X and the model are contained in the vector Y. So

Y is some function of X

Yc = f (X) (2.72)

with f () representing the model. When X is input into the model Y is output. The

subscript c means a value calculated using the model whereas the subscript o means

the actual observed value. By initially assuming that there are no observational

errors, to find Yo given X we need to solve

Yo = f (X) (2.73)

taking an initial estimate for X, call it Xe Eqn. 2.72 gives

Yc = f (Xe) (2.74)

the residual in the observations, y is then

y = Yo −Yc (2.75)

If the model used is accurate and there are no observational errors a correct value

of X exists where

X = Xe + x (2.76)

where x is the difference between the estimated and actual value of X. Using

Eqn. 2.76 in Eqn. 2.73 gives

Yo = f (Xe + x) (2.77)

Now letting the Jacobian matrix J be

J =dY

dX(2.78)

where J, with elements Ji,j, is determined by varying the input parameters X by a

small amount δXj

dYi

dXj=f (Xj + δXj) − f (Xj − δXj)

2δXj(2.79)


with i being the number in the observations set and j the number of parameters.

Expanding Eqn. 2.77 using Newton’s method and neglecting terms of order x2 or

higher gives

f (Xe + x) ≃ f (Xe) + Jx (2.80)

or

Jx ≃ f (Xe + x) − f (Xe) (2.81)

using Eqns. 2.75, 2.77 and 2.74 and taking Eqn. 2.81 to be precise gives

Jx = y (2.82)

Eqn. 2.82 is solved to give x and x is added to Xe to give an improved estimate for

X as per Eqn. 2.76. This improved value for X is inserted back into Eqn. 2.81 as

Xe and the procedure repeated iteratively until convergence in X achieved. At this

point X = Xe and the differential correction process is completed.

The precessing ellipse, for satellite orbits, is described in a planetocentric carte-

sian reference frame. This frame is defined in a very similar way to the J2000

reference frame (Section 2.1.2). The X-axis is parallel to the vector from the centre

of the planet pointing towards the intersection of the ascending node of the plane-

tary equator at epoch on the mean Earth equator at J2000 ( 2000 JAN 01 12:00:00.0

TDB). The Z-axis is parallel to the spin angular momentum vector of the planet

at epoch. The Y-axis being defined by the vector cross-product Y = Z × X. This

is illustrated in Fig. 2.9. The orbital elements are defined as in Fig. 2.5.

The actual observations have to be converted into the reference frame that is

used for the precessing elliptical model for the differential correction process to

work. If α and δ are the right ascension and declination of the primary’s pole in the

co-ordinate system of the observations (J2000 reference frame for our data) then

PX

PY

PZ

=

cosN − sinN cos J sinN sin J

sinN cosN cos J − cosN sin J

0 sin J cos J

T

px

py

pz

(2.83)

where

N = α + 90

J = 90 − δ


mean Earth equatorplane at J2000

ascending node of planetary equator

planet

on mean Earth equator

centre of planet

sense of rotation

α

J2000z

J2000 y

J2000x

planetary equatorplane at J2000

X

YZ

90 − δ

Figure 2.9: Reference frame for orbital elements of a satellite.

2.7 Discussion

2.7.1 Integration of the full equations of motion

Provided all the masses that influence a system, along with their state vectors

at epoch, are included and accurate a numerical integration gives a dynamically

correct representation of any system under study. The trajectory of an object from

a numerical integration is as close to the actual trajectory of that object as it is

possible to calculate.

A correctly executed numerical integration models the dynamical characteristics

of an entire system at once. However this means that a lot of information is required

to begin with, accurate masses and states for all the objects at epoch.

It is very difficult to get an accurate feel for an orbit just from a state vector.

It is of course possible to convert the state vector into a set of osculating elements

(section 2.2) but as previously discussed osculating elements are not necessarily a


good representation of an orbit’s actual shape. The subtle effects on the orbit of a

satellite due to the perturbations of another, while accurately modeled, are difficult

to determine from state vectors.

An approach used by a number of authors is to fit an orbit to observations using

a numerical integration technique and then generate an ephemeris. A precessing

elliptical model is then fitted to this generated ephemeris (e.g. Harper 1987, Harper

and Taylor 1993 and Taylor 1998). The elliptical model can easily be modified to

include terms representing the effects of other bodies in the system. In this the way

subtle, long term effects (e.g. of resonances) can be extracted from the numerically

generated ephemeris.

2.7.2 The precessing ellipse model

The precessing ellipse model is very simple to implement, it requires no great pro-

gramming skill and little computing time. The best fit orbit can be easily visualised

from the seven orbital elements at epoch corresponding to the ‘best’ fit solution. It

can provide a highly accurate fit to an orbit (e.g. see the fits to the JPL ephemerides

in Chapters 5, 6 and 7). Its great disadvantage is that it does not include the ef-

fects of perturbations due to other objects. In many cases these perturbations are

small and average out over a long period of time. Where this is the case the use

of a precessing ellipse model is perfectly valid. Where a resonance occurs however,

the effects of perturbations are significant and periodic, and the precessing ellipse

model can produce inaccurate results. Modifications can be made to account for

perturbations (section 7.8) but only when the magnitudes of the relevant terms are

known beforehand.

2.8 Conclusions

Fitting an orbit to observations using a numerical integration technique provides

the most accurate dynamical representation of the actual trajectory of an object.

It includes subtle effects due to perturbations from other objects in the system.

The precessing ellipse model is easy to program and fit to observations. It does


not include the effects due to other objects in a system. Where these effects are

small the inaccuracies introduced are also small. In such cases a precessing ellipse

provides an accurate model for the actual dynamics of a system.

For Prometheus, which has no low order resonances with any object in the

saturnian system, a precessing ellipse model is an accurate method of orbit deter-

mination. Since a precessing ellipse is an uncomplicated, valid and accurate way

for modeling the orbit of Prometheus it is used in this work.

The same model is also used for determining the orbits of Atlas and Pandora

as an extension to the primary work on Prometheus. While valid for Atlas the

precessing ellipse requires some modification to accurately model Pandora’s motion

(section 7.8).

Chapter 3

Voyager Images

3.1 Introduction

The majority of the work presented in this thesis uses Voyager images, therefore

a discussion of the Voyager spacecraft and the quirks of Voyager images would be

useful. In this chapter information regarding the Voyager spacecraft in general, and

the Imaging Science Subsystem (ISS) in particular, is presented and discussed. The

techniques for the processing of Voyager ISS images are described in detail.

3.2 Voyager spacecraft and instruments

The two Voyager spacecraft were launched in 1977, Voyager 1 bound for Jupiter

and Saturn and Voyager 2 for Jupiter, Saturn, Uranus and Neptune in the “Grand

Tour”. To date the Voyager project, which is still on-going, is the most ambitious

program of outer Solar System exploration ever undertaken.

Each spacecraft carries an array of instruments which are described in detail in

the published literature (e.g. Danielson et al. 1981, Smith et al 1977 and Thompson

1990). The only instruments of interest for this work are the two cameras of the

Imaging Science Subsystem (ISS).

A Voyager spacecraft carries two cameras, a narrow angle camera (NAC) and

a wide angle camera (WAC). The NAC has a focal length of ∼ 1500 mm and the

WAC a focal length of ∼ 200mm. The corresponding fields of view are 7.4 × 7.4

53

CHAPTER 3. VOYAGER IMAGES 54

mrad for the NACs and 56 × 56 mrad for the WACs. The cameras and other ISS

instrumentation are mounted on a scan platform which allowed them to be pointed

in directions independent of the orientation of the spacecraft bus during planetary

encounters. The cameras could be shuttered simultaneously (called a BOTSIM) or

separately. Since the cameras are boresighted with each other, in BOTSIM mode

the NAC image is a high resolution blowup of the central part of the WAC image.

Each Voyager camera consists of an eight position, off-axis filter wheel, a lens

system, a shutter and a magnetically focused slow-scan vidicon tube with its sup-

plementary electronics. The Voyager camera optics were state of the art in 1977

and indeed compare favourably with modern systems. In fact the optics from one

of the flight spare WACs are now on their way to Saturn as part of the Cassini

wide angle camera. The seventies vintage vidicon tube however does not compare

favourably with modern CCDs.

The lens system brings the image to a focus on the face plate of the vidicon

tube. The face plate is a layer of photosensitive selenium sulphide one inch square.

Charge builds up where photons hit the face plate and the two dimensional image is

stored on the plate as a pattern of electrostatic charge. The vidicon electron beam

then scans the plate line by line, converting the image into a sequence of electrical

signals. In many ways the vidicon system is very similar to a standard photocopier.

The face plate of the vidicon tube is divided into a 800×800 square grid for readout

purposes. However, it is important to remember that the face plate is not actually

physically divided into such a grid in the way that a CCD is. Each element of this

800 × 800 grid is square in shape and is referred to as a pixel.

All Voyager images are subject to radiometric responsiveness induced errors,

geometric distortions and the introduction of artifacts. The artifacts include, but

are not limited to, “hot pixels” and the famous dust “donuts”. To make matters

more complicated the geometric distortions introduced vary from one image to the

next. The errors introduced are due to a combination of the camera optics and

the processes of building up the image and then reading it off the face plate of the

vidicon tube.


3.3 Voyager images

Each image as it is read from the vidicon tube is an 800×800 array, giving 640,000

square pixels. Each pixel has a brightness level, called a DN (Digital number),

between 0-255 i.e. 256 brightness (DN) levels. Since 256 is 28 or an eight digit

binary number the images are often referred to as eight bit images. Such 800× 800

images which haven’t been processed in any way to remove radiometric or geometric

distortions are called ‘raw’, or ‘image-space’, images and will be referred to as such

from now on.

The coordinate system used for Voyager images has its origin in the upper

left corner of the image. The positive x direction, traditionally referred to as the

sample direction, is to the right. The positive y direction, traditionally called the

line direction, is downwards. The Voyager image coordinate system is illustrated

in Fig. 3.1. It is usual to give coordinates as (line,sample) not (sample,line) and

so we will follow common usage and give coordinates in the (line,sample) format.

Each pixel can have integer line and sample coordinates in the range 0-799.

Therefore, the pixel in the lower left hand corner has coordinates (799,0); the upper

right hand corner (0,799); the lower right (799,799) etc.

The fact that each pixel has integer coordinates can cause problems when co-

ordinates to sub-pixel accuracy are required. The integer coordinates of a pixel,

which varies from 0-799 in both sample and line, are the coordinates of the exact

centre of the square pixel. In this ‘pixel coordinate’ system the coordinates of the

top left, top right, bottom left and bottom right corners of the raw image are (-0.5,-

0.5), (-0.5,799.5), (799.5,-0.5) and (799.5,799.5) respectively. Whenever coordinates

are given to sub-pixel accuracy in the ‘pixel coordinate’ system, this is the frame

of reference used. We also used another coordinate system we called ‘continuous

coordinates’, which can take all values in the range 0.0-800.0 for raw images. The

coordinates in the ‘pixel’ system are converted to continuous coordinates by

linecontinuous = linepixel + 0.5 (3.1)

samplecontinuous = samplepixel + 0.5 (3.2)

The very top left of the image is (0.0,0.0); top right (0.0,800.0); top left pixel


(0,0) (0,799)

(799,799)(799,0)

sample

line

Figure 3.1: Raw Voyager image orientation.

(0.5,0.5); bottom right pixel (799.5,799.5) etc. With continuous coordinates it is

clear that a sample value of 456.7 means that the raw image location is 456.7/800.0

of the way along the length of the x axis. Unless it is specifically stated otherwise

the ‘pixel coordinate’ system is used.

3.4 The geometrical correction of Voyager im-

ages

As previously discussed, the images from the Voyager NAC and WAC cameras are

subject to geometrical distortions. These distortions have to be removed thus pro-

ducing a geometrically corrected image. During the geometrical correction process


1000x1000 pixel object-space image

Transformed 800x800 pixelimage-space image

Figure 3.2: Typical shape of 800× 800 image-space pixel grid mapped onto 1000×1000 pixel grid of object-space image.

the 800×800 pixel raw image is mapped onto a 1000×1000 pixel grid. The 800×800

grid of the raw image is called ‘image-space’, while the 1000× 1000 pixel grid onto

which the geometrically corrected raw image is mapped is referred to as ‘object-

space’. Of course the 640,000 pixels of the raw image cannot fill the 1,000,000 pixel

grid of the object-space image. The geometrical distortion introduced into Voyager

images by the imaging system is of ‘barrel’ type. Imagine the raw image as being

a square piece of rubber. Take it and stretch the corners diagonally away from

the centre. The resulting shape is roughly what the geometrically corrected raw,

‘image-space’, image looks like when mapped onto the 1000 × 1000 pixel object-


space image as illustrated by Fig. 3.2. The distortion not only varies from one

image to the next but also from one region of the image to another, making a single

global correction for the image impossible. Since the introduced distortion varies

even within a single image it is unfeasible to calculate a single set of transformation

parameters that will geometrically correct more than a small region of a single raw

image. A different set of parameters must be calculated for every image shuttered

and they can only be determined after the image has been taken.

3.4.1 Overview of the geometrical correction process

The process of transforming each 800×800 raw Voyager image into a geometrically

corrected 1000 × 1000 image has the following steps:

• Locate the reseau marks on the raw image.

• Generate the 85 pseudo-reseaus.

• Calculate the transformation parameters for each of the 506 triangular areas an

image is divided into.

• For each of the 1,000,000 pixels in the object-space image determine which trian-

gular area it falls into. Use the appropriate transformation parameters to transform

the object-space pixel into image-space.

• Determine the DN value of the object-space pixel by either the nearest neighbour

or pixel interpolation technique.

3.4.2 Reseau marks

The exact geometrical distortion in each image must be determined separately,

using only information contained within the image itself and knowledge of the char-

acteristics of each individual vidicon tube. The necessary information is contained

in a grid of points called ‘reseau’ marks. Reseau is taken from the French for ‘web’

and the reseau marks are a grid of points actually etched onto the face of the vidi-

con tube itself. Where a reseau has been etched the selenium sulphide layer is

destroyed and the face plate becomes insensitive to incident light i.e. there is no

electrostatic charge build up in that location. A pixel in the location of a reseau

mark has a DN of 0. The reseau marks can be clearly seen in raw Voyager im-


ages, see Fig. 3.3. Published images usually have these reseau marks removed for

Figure 3.3: A raw Voyager image showing reseau marks

an improved aesthetic effect, although they contain no more information and can

indeed be misleading. Reseau marks are removed by averaging the DN values of

the surrounding pixels and replacing the ∼ 0 DN pixels within the reseau by this

averaged value. We retained the reseau marks in all the images used.

Each vidicon tube has a slightly different pattern of reseau marks etched onto

its face-plate. The actual physical locations of the reseau marks on the face-plates

are known to sub-pixel accuracy. After manufacture each vidicon tube was put

through an exhaustive series of laboratory tests. The location of the reseau marks


on the vidicon face-plates was measured using a theodolite, in conjunction with

accurate surveying techniques, to accuracies of ±0.002 mm. When these positions

are converted to line and sample coordinates the locations of the reseaus in the

object-space image is known to sub-pixel accuracy. These object-space reseau lo-

cations served to define a geometrically correct space which would result if, say

a grid target, were imaged through a geometrically perfect camera system. The

object-space reseau locations for a particular vidicon tube remain constant. Com-

parison of the actual positions of the reseaus in image-space on the raw image with

their object-space locations enables the determination of a set of ‘distortion pa-

rameters’. These distortion parameters map the object-space reseau locations onto

their image-space locations and vice versa. By taking large numbers of test images

in the laboratory a set of nominal image-space reseau coordinates was obtained for

each vidicon tube. A reseau’s nominal location is the average of its actual locations

in the test images. For every image taken using that vidicon tube the actual image-

space location of the reseau marks in the raw image should be near their nominal

location, say within 10 pixels or so. Both the nominal image-space reseau locations

and their corresponding constant object-space locations are used in the geometrical

correction of the raw image.

3.4.3 Location of reseau marks

Each of the vidicon tubes manufactured for the Voyager project has 202 reseau

marks etched onto the face-plate, Fig. 3.4 shows the object-space reseau mark lo-

cations for the Voyager 2 narrow angle camera. The object-space reseau locations

for all the other cameras are similar. Fig. 3.4 clearly shows that the reseaus are

concentrated around the edge of the image with fewer in the middle. Presumably

this is because the centre of the image is the region of greatest importance and

reseaus here might obscure items of interest.

Clearly the greater the number of reseau marks the better any distortion correc-

tions will be. However the more reseaus the greater the area of the image obscured,

202 reseaus seemed a reasonable compromise.


1 2 3 4 5 6 7 8 9 10 11 12

1314 15 16 17 18 19 20 21 22

23

24 25 26 27 28 29 30 31 32 33 34 35

36 37 38 39 40 41 42 43 44 45 46

47 48 49 50

51 52 53 54 55 56 57 58 59 60 61

62 63 64 65

66 67 68 69 70 71 72 73 74 75 76

77 78 79 80

81 82 83 84 85 86 87 88 89 90 91

92 93 94 95

96 97 98 99 100 101 102 103 104 105 106

107 108 109 110

111 112 113 114 115 116 117 118 119 120 121

122 123 124 125

126 127 128 129 130 131 132 133 134 135 136

137 138 139 140

141 142 143 144 145 146 147 148 149 150 151

152 153 154 155

156 157 158 159 160 161 162 163 164 165 166

167 168 169 170 171 172 173 174 175 176 177 178

179180 181 182 183 184 185 186 187 188

189

190 191 192 193 194 195 196 197 198 199 200 201

202

Figure 3.4: Object-space image reseau mark locations for vidicon tube number 5,the tube used in the Voyager 2 narrow angle camera (NAC). The reseau ID numbersare included above the corresponding reseau mark.


3.4.4 The detection of reseaus in a raw image

The region of an image around the nominal image-space reseau location of each

reseau mark is searched. If an actual physical reseau mark is located its position is

used, if a physical mark isn’t located within a fixed distance of its nominal image-

space location then the nominal image-space location is used. In this way the actual

image-space space location of the 202 reseau marks on an image are determined.

3.4.5 The generation of ‘pseudo-reseau’ marks

For the purposes of geometrical correction Voyager images are divided up into 506

triangular areas. The reseau grid illustrated in Fig. 3.4 clearly cannot be divided

into 506 triangular areas with any great ease. The solution is to generate extra

reseau marks. These extra reseaus, called ‘pseudo-reseau’ marks, are generated

from already existing ‘real’ reseau mark locations. In this way 85 pseudo-reseau

marks are created making a total of 287 marks in total, both real and pseudo. The

positions of the extra pseudo-reseau marks are illustrated in Fig. 3.5.

The image is then divided into triangular areas with reseau marks, real and

pseudo-, at the vertices as illustrated in Fig. 3.6. The extra pseudo-reseau marks

are used to ensure that the triangular areas the image is divided into are of roughly

equal size. This ensures that the transformation parameters are reasonably contin-

uous between one triangle and its immediate neighbours.

The image-space locations of the reseau marks and triangular areas for the

Voyager 2 NAC image FDS43686.55 are shown in Fig. 3.7 for comparison with the

object-space locations of the same features in Fig. 3.6.

3.4.6 Mapping image-space locations into object-space

As described previously a raw image is divided into 506 triangular areas, with

the vertices of the triangles denoted by the position of reseau marks. Similarly

the geometrically correct image in object-space is also divided into 506 triangular

areas with the vertices being co-incident with the object-space locations of the same

reseau marks that denote the vertices of the triangles in image-space.

The locations of reseau marks in image-space transform exactly to the reseau


1 2 3 4 5 6 7 8 9 10 11 12

1314 15 16 17 18 19 20 21 22

23

24 25 26 27 28 29 30 31 32 33 34 35

36 37 38 39 40 41 42 43 44 45 46

47 48 49 50

51 52 53 54 55 56 57 58 59 60 61

62 63 64 65

66 67 68 69 70 71 72 73 74 75 76

77 78 79 80

81 82 83 84 85 86 87 88 89 90 91

92 93 94 95

96 97 98 99 100 101 102 103 104 105 106

107 108 109 110

111 112 113 114 115 116 117 118 119 120 121

122 123 124 125

126 127 128 129 130 131 132 133 134 135 136

137 138 139 140

141 142 143 144 145 146 147 148 149 150 151

152 153 154 155

156 157 158 159 160 161 162 163 164 165 166

167 168 169 170 171 172 173 174 175 176 177 178

179180 181 182 183 184 185 186 187 188

189

190 191 192 193 194 195 196 197 198 199 200 201

202

Figure 3.5: Generated object-space pseudo-reseau marks locations using the object-space image reseau mark locations for vidicon tube 5. The pseudo-reseaus are thered dots.

mark locations in object-space. This of course means that the vertices of a particular

triangular area in image-space transform exactly to the vertices of the corresponding

triangle in object-space. We make the assumption that every point that lies within

the boundary of a image-space triangle transforms into object-space in exactly the

same way. This of course is not necessarily true but since the area of the triangle is

small when compared to the total area of the image any variations in transformation

parameters across the triangle are also likely to be small.

For each triangle there are six variables, the coordinates of the vertices, in image-


1 2 3 4 5 6 7 8 9 10 11 12

1314 15 16 17 18 19 20 21 22

23

24 25 26 27 28 29 30 31 32 33 34 35

36 37 38 39 40 41 42 43 44 45 46

47 48 49 50

51 52 53 54 55 56 57 58 59 60 61

62 63 64 65

66 67 68 69 70 71 72 73 74 75 76

77 78 79 80

81 82 83 84 85 86 87 88 89 90 91

92 93 94 95

96 97 98 99 100 101 102 103 104 105 106

107 108 109 110

111 112 113 114 115 116 117 118 119 120 121

122 123 124 125

126 127 128 129 130 131 132 133 134 135 136

137 138 139 140

141 142 143 144 145 146 147 148 149 150 151

152 153 154 155

156 157 158 159 160 161 162 163 164 165 166

167 168 169 170 171 172 173 174 175 176 177 178

179180 181 182 183 184 185 186 187 188

189

190 191 192 193 194 195 196 197 198 199 200 201

202

Figure 3.6: Object-space image divided into grid of 506 triangles using reseau marklocations for vidicon tube 5, the tube in the Voyager 2 narrow angle camera (NAC).


12 3 4 5 6 7 8 9 10 11

12

13 14 15 16 17 18 19 20 21 2223

24 25 26 27 28 29 30 31 32 33 34 35

36 37 38 39 40 41 42 43 44 45 46

47 4849 50

51 52 53 54 55 56 57 58 59 60 61

62 6364 65

66 67 68 69 70 71 72 73 74 75 76

77 7879 80

81 82 83 84 85 86 87 88 89 90 91

92 9394 95

96 97 98 99 100 101 102 103 104 105 106

107 108109 110

111 112 113 114 115 116 117 118 119 120 121

122 123124 125

126 127 128 129 130 131 132 133 134 135 136

137 138139 140

141 142 143 144 145 146 147 148 149 150 151

152 153154 155

156 157 158 159 160 161 162 163 164 165 166

167 168 169 170 171 172 173 174 175 176 177 178

179180 181 182 183 184 185 186 187 188 189

190191 192 193 194 195 196 197 198 199 200

201

202

Figure 3.7: Image-space locations of reseau and pseudo-reseau marks and triangulargrid for the Voyager 2 NAC image FDS43686.55.


space that transform into six variables in object-space. Since all twelve variables are

known the simultaneous linear equations describing the transformation from image-

space to object-space for the triangle are soluble. The simultaneous equations for

transforming from object-space to image-space are

limage = a + b× lobject + c× lobjectsobject (3.3)

simage = d+ e× lobject + f × lobjectsobject (3.4)

and those for transforming from image-space to object-space are

lobject = g + h× limage + i× limagesimage (3.5)

sobject = j + k × limage +m× limagesimage (3.6)

For each triangle the simultaneous equations are solved giving the transformation

parameters a, b, c, d, e, f , g, h, i, j, k and m. Any point within the boundary

of the triangle can be transformed from an image-space to object-space location or

vice versa.

The transformation parameters for all 506 triangles are obtained in turn. For

those parts of the image that lie outside a defined triangular area, basically around

the very edge of the image, the transformation parameters of the closest triangular

area are utilised.

Each pixel in a image is checked to see which triangular area it falls into, or

which area is closest. The pixel is then mapped from its image-space location

to its object-space location using the transformation parameters for the identified

triangular area. Of course pixels can also be mapped from object-space to image-

space in the same way.

It is highly unlikely that the image-space position of a pixel will map exactly

onto a valid pixel location in the 1000 × 1000 object-space grid. It is much more

likely that the image-space location of a pixel will map across four object-space

pixels as illustrated in Fig. 3.8, although in this figure an object-space pixel is

being mapped onto an image-space grid the principle is the same. In Fig. 3.8 what

should the intensity levels, i.e. DN values, of the image-space pixels 1, 2, 3 and 4

be? The situation becomes even more complicated when additional object-space

pixels are mapped onto the same area in image-space. To avoid this problem when


a raw image is geometrically corrected the image-space pixels are not transformed

into object-space. A slightly different technique is used.

A 1000 × 1000 pixel grid is generated, each pixel in the grid being initially

assigned a DN of 0. This blank grid will become the geometrically corrected image in

object-space. The transformation parameters for each of the triangular areas in the

image-space image are calculated as previously described. Each pixel in the object-

space grid is then stepped through in turn and transformed into its corresponding

location on the image-space raw image leading to a situation illustrated in Fig. 3.8

1 2

3 4

P

Figure 3.8: Object-space pixel, P, mapped onto image-space pixel grid.

In Voyager image processing there are two main ways that the DN value of

an object-space pixel are determined, both involve transforming the pixel into its

corresponding image-space position. If the object-space pixel is mapped to an

invalid image-space location the DN of the object-space pixel is set equal to 0. An

invalid image-space location is one that is not on the raw image e.g. image-space

location (801,200).

When a raw image-space image is transformed to a geometrically corrected

object-space image the technique used involves mapping object-space pixels to


image-space locations. The DN value of each object-space pixel is then calculated

using the nearest image-space pixels to its mapped image-space location.

3.4.7 The nearest neighbour pixel mapping technique

The nearest neighbour technique involves mapping an object-space pixel onto its

image-space location and then determining which image-space pixel is closest to

that mapped position. In Fig. 3.8 the largest fraction of the mapped object-space

pixel lies within image-space pixel 1. Therefore, image-space pixel 1 is closest to

the mapped position of the object-space pixel. The DN of the object-space pixel is

set equal to the value of image-space pixel 1. This procedure is performed on all

1,000,000 object-space pixels.

3.4.8 Pixel interpolation method

Again an object-space pixel is mapped onto its image-space location as in Fig. 3.9.

In this case the intensity level of all the pixels 1-4 is used to find an average for the

DN value of the object-space pixel,

DNobject = (x1y1DN1) + (x2y1DN2) + (x1y2DN3) + (x2y2DN4) (3.7)

where DNi denotes the intensity of image-space pixel i.

3.4.9 Comparison of the two methods

The nearest neighbour method is very simple to use and any loss of fine detail in an

image due to complicated interpolation of DN values is kept to a minimum. With

pixel interpolation it is possible to get DN values in the geometrically corrected

image that don’t occur in the raw image. Averaging over four image-space pixels

can lead to loss of fine detail. Due to the nearest neighbour method’s superior

ability to retain fine detail this was use throughout this work.

One slight drawback of the nearest neighbour method is that due to the trans-

formation technique, spatial features can be offset by up to half a pixel in the

geometrically corrected object-space image. This offset can lead to to a ‘wavy’

appearance of a straight feature in the corrected image.


xx

y

y

1 2

1

2

1 2

3 4

P

Figure 3.9: Object-space pixel mapped onto image-space pixel grid for pixel inter-polation.

3.5 VICAR

Traditionally, Voyager images have been analysed using a software package called

VICAR (Video Image Communication and Retrieval). The VICAR software was

originally written in the early 1970’s, to run on hardware that is long since obsolete

and under operating systems that have not been in widespread use for many years.

The software had very little associated documentation, either within the code itself

or in the users manual, the VICAR Users Guide (circa. 1977). Images are processed

by VICAR mainly in an interactive mode, requiring a great deal of input by the

operator. Batch processing of multiple images is difficult. Generally VICAR is

still only in use by scientists who learned its use during the active portion of the

Voyager project, which effectively ended with Voyager 2’s encounter with Neptune

in 1987. Because of its age, and various shortcomings that became apparent to the

community during its use, VICAR has been modified many times. These modifica-

tions were made in no apparent coherent, centralised or controlled manner. Each

institution using VICAR has made its own modifications to suit its needs at the


time. In some cases the modifications are quite major. Imperial College has a mod-

ified version of VICAR, called BISHOP, which was designed to run on a DEC/VAX

11/780 (Thompson 1990). The image format used by BISHOP is incompatible with

the standard VICAR format. It is unknown whether BISHOP is still functioning.

In many cases institutions have lost the ability to use VICAR as the machines it

was tuned to run on have been scrapped. Indeed the machine the OPNAV team

at the Jet Propulsion Laboratory (JPL) used to run VICAR has been retired and

they have lost the ability to process raw Voyager images. The machine used to run

VICAR at the the Lunar and Planetary Laboratory at the University of Arizona

is nine years old. It is probable that any copies of VICAR still running are totally

incompatible with each other. The shortcomings of VICAR are a direct result of it

being designed for a specific set of hardware and operating system over twenty years

ago. The limitations of computer hardware and software in the 1970’s, portability

across systems being totally unknown, meant that software had to be tuned for each

individual machine it was used on. Because of the great difficulty of using VICAR

at an institution where it wasn’t originally installed in the 1980’s for the Voyager

project, some authors have abandoned it entirely for Voyager image processing. A

combination of a single global transformation for geometrical correction and SPICE

FORTRAN routines for image navigation was used by Gordon (1994) and Gordon

et al. (1995). The only part of VICAR that has survived intact, and is still in

widespread use, is the VICAR image format. The images from the Cassini Imaging

Science Subsystem (ISS) cameras are in the VICAR image format.

3.6 MINAS

To avoid the problems that have been described with VICAR, and to take advantage

of the advances in computer hardware and software since the 1970’s, the Cassini

ISS team decided that a completely new software package should be written for

use with Cassini ISS images. One of the criteria was that the software should

be easily installable across a wide variety of hardware and operating systems. This

portability was achieved by programming in the commercial language IDL, which is


widely used for graphical applications. IDL commands are identical no matter what

computer or operating system is used. The extensive task of writing IDL compilers

for various systems, and ensuring compatibility across them, is the responsibility of

IDL’s commercial suppliers. Since IDL is in general use in the scientific, medical

and business communities there is a wide range of documentation available. This

widespread commercial use is likely to ensure that IDL will be supported for many

years to come, with compilers being written for new hardware/operating systems as

they are released. Backwards compatibility with earlier versions of IDL is assured

by the supplier.

The image processing and navigation package being developed at CICLOPS

(Cassini Imaging Central Laboratory for Operations) at the University of Arizona,

under the direction of Carolyn Porco, for the Cassini ISS team is called MINAS

(Modular Image Navigation and Analysis Software). Since MINAS is written in

IDL, the package is system independent, the same code is installed on Sun ma-

chines running Solaris and PCs running Linux for example. Although specifically

written for Cassini, MINAS was designed so that it could be used with images taken

by any imaging system be it ground based telescope or as yet undesigned spacecraft.

This of course means that MINAS can be used to process, geometrically correct and

navigate Voyager images. MINAS wasn’t specifically written for Voyager images

and so lacks the routines necessary for reseau detection and geometrical correction.

However, the modular nature of MINAS means that IDL routines for the processing

of Voyager images can be easily integrated. Indeed, one of MINAS’ design philoso-

phies is that individual members of the ISS team should write MINAS compatible

IDL procedures to perform specific tasks and then CICLOPS will include these

procedures in MINAS itself.

3.7 Routines for reseau location and geometrical

correction

The FORTRAN77 subroutines within VICAR that carry out reseau location and

geometrical correction are called RESLOC and GEOMA respectively. We wrote


MINAS compatible IDL procedures that perform the same functions as VICAR’s

RESLOC and GEOMA. The only available guides to exactly how RESLOC and

GEOMA perform their tasks is the undocumented source code for the subroutines

and brief descriptions of the subroutines in the VICAR Users Guide (circa 1977).

The source code was available as a legacy from an unsuccessful attempt to install

VICAR at Queen Mary in the early 1990’s. The source code was written in the

1970’s for the computers of the day and various tortuous programming tricks are

used because of the limited memory available to such computers. The lack of

documentation, and the programming style used, makes the source code for the

subroutines very difficult to follow.

The two main MINAS compatible IDL procedures written were resloc.pro and

geoma.pro. They perform identical tasks to the FORTRAN77 VICAR subroutines

RESLOC and GEOMA. Other lower level procedures were written but are not

discussed individually since they are called by resloc.pro and geoma.pro. They

are covered in the descriptions of those top-level procedures.

All the IDL procedures written for the geometrical correction of Voyager images

were written solely by the author, based on the FORTRAN77 VICAR subrou-

tines RESLOC and GEOMA. Where appropriate pre-existing MINAS procedures

were utilised instead of writing brand new code. The procedures written are fully

compatible with MINAS and will be included in future MINAS distributions by

CICLOPS.

3.7.1 resloc.pro: locating reseau marks

Each raw image should contain 202 reseau marks. In real images the geometrical

distortion introduced by the imaging system pushes some of the reseaus off the

800 × 800 pixel grid of the raw image. This results in each raw image having a

variable number of reseaus with a maximum of 202.

The nominal image-space reseau locations for each camera are known. Each

reseau mark has a physical size of 0.040×0.040 mm on the face-plate of the vidicon

tube. Fig. 3.10 shows a 9×9 pixel region of a raw image around the actual location

of a reseau mark. Within resloc.pro a reseau is modeled by a two dimensional


Figure 3.10: 9 × 9 pixel region around the location of a reseau mark.

shape function. This function is constant for all reseau marks and is taken to be a

Gaussian function in two dimensions with the form

F(x, y) = 255(1 − e(x2+y2)/2) (3.8)

where x and y are sample and line coordinates respectively. The line and sample

coordinates take integer values from −4 to +4. F(x, y) takes integer values between

0-255 so 255(1 − e(x2+y2)/2) is rounded to the nearest integer. The reseau is thus

modeled as a 9 × 9 pixel square region.

The reseau shape function, F(x, y), is then scanned over the raw image area

with line coordinates between 4-795 and sample coordinates between 4-795. At

each pixel location within the defined image area the cross-correlation term of the

image data within the 9 × 9 area centred on the pixel location and the Gaussian

shape reseau profile, F(x, y), is calculated. The locations of local maxima in the

grid of correlation coefficients are then identified. If the value of the correlation

coefficient at an identified local maxima is less than some minimum set value, ccmin,

then the point is discarded. The default value of ccmin is 0.75 and we used this value

throughout our image processing. The correlation maxima can only be located to

the nearest line and sample at this stage.

Each of the 202 reseau marks is stepped through in turn. The closest local


maximum in the correlation coefficient to the nominal image-space location of the

reseau mark is assumed to be the actual location of the reseau mark in the raw

image. If this assumed location is more then a certain number of pixels from the

nominal image-space location then the reseau is flagged as NOT LOCATED. If the

assumed location is within the specified number of pixels than the reseau is flagged

as LOCATED. The maximum distance an assumed pixel location can be from its

nominal image-space location is an input variable to resloc.pro. The default value

is 5.0 pixels, and this has been used throughout this work.

The next step is to determine the image-space locations of all LOCATED reseaus

to sub-pixel accuracy. If the assumed location of a LOCATED reseau is (i, j) then

the value of the correlation coefficient at that point is ρ(i, j). Sub-pixel accuracy

is obtained by fitting a paraboloid to the 2-dimensional correlation function in the

vicinity of pixel (i, j). We used a 3 × 3 pixel area centred on pixel (i, j).

The image-space locations of the LOCATED reseau marks are known at this

point. The locations of the NOT LOCATED reseaus are determined next. Each

of the 202 reseau marks has a number of near-neighbour reseau marks, varying

between 3 and 6 depending on the location of the reseau mark on the grid. For

example reseau mark 1 has reseaus 2, 13 and 24 as near-neighbours while reseau 120

has reseaus 109, 119, 121 and 124 see Fig. 3.4. There is a list of near-neighbours for

each reseau mark embedded within VICAR. This is not a straightforward list of the

adjacent reseau marks, sometimes a reseau is listed as a near-neighbour twice or an

adjacent reseau isn’t listed at all. In VICAR the list is generated using an involved

algorithm, we ran the algorithm to generate a simple list of near-neighbours for

each reseau.

Each NOT LOCATED reseau is stepped through in turn and its near-neighbour

reseaus examined. If all the near-neighbour reseaus are also NOT LOCATED then

the program moves on the next NOT LOCATED reseau. If one or more of the

near-neighbour reseaus are LOCATED the average displacement of the LOCATED

near-neighbour reseaus from their nominal image-space locations is calculated. The

NOT LOCATED reseau is then displaced from its nominal image-space location by

this averaged amount and then marked as LOCATED. However, this reseau remains


NOT LOCATED when it is used as a near-neighbour. All the NOT LOCATED

reseaus are stepped through, at the end of the iteration all the newly located re-

seaus are also marked as LOCATED when used as near-neighbours. The process is

repeated until all 202 reseaus are LOCATED.

Once all 202 ‘real’ reseau marks have been located, the 85 ‘pseudo-reseau’ marks

are generated from specific real reseaus. The list of exactly which ‘real’ reseaus are

used in the generation of each pseudo-reseau was recreated from VICAR via software

archeology. In general, the sample coordinate of a pseudo-reseau is the average of

the sample coordinates of the two nearest real reseau marks on the previous line

of real reseaus. The line coordinate being the average of the line coordinates of

the nearest two real reseau marks on the same line. This averaging technique

is only used for the object-space pseudo-reseau coordinates. The corresponding

coordinates in image-space are calculated using a bilinear mapping technique using

nearest neighbour real reseau marks, four in the centre of the image and three

around the edges. For a particular pseudo-reseau mark, near the image edge, its

image-space coordinates are given by

lineimage = d+ e× sampleobject + f × lineobject (3.9)

sampleimage = a + b× sampleobject + c× lineobject (3.10)

where (lineimage, sampleimage) and (lineobject, sampleobject) are the coordinates of the

pseudo-reseau mark in image-space and object-space respectively. The constants a,

b, c, d, e and f are obtained by solving

sampleimageX

sampleimageY

sampleimageZ

=

1.0 sampleobjectX lineobjectX

1.0 sampleobjectY lineobjectY

1.0 sampleobjectZ lineobjectZ

a

b

c

(3.11)

and

lineimageX

lineimageY

lineimageZ

=

1.0 sampleobjectX lineobjectX

1.0 sampleobjectY lineobjectY

1.0 sampleobjectZ lineobjectZ

d

e

f

(3.12)

where (lineimageI, sampleimageI) and (lineobjectI, sampleobjectI) are the coordinates

are the near-neighbour reseau marks in image-space and object-space respectively.


Here the variable I takes the values X, Y and Z which are the ID numbers of the

three nearest neighbour real reseau marks.

A similar scheme is followed for pseudo-reseau creation in the centre of the image

where four instead of three near-neighbour real reseaus are used

line image = e+ f × sampleobject + g × lineobject

+h× sampleobject × lineobject (3.13)

sample image = a+ b× sampleobject + c× lineobject

+d× sampleobject × lineobject (3.14)

with a, b, c, d, e, f , g and h being the solutions to a set of simultaneous linear

equations similar to Eqns. 3.11 and 3.12 but for four points instead of three.

The coordinates of all 287 reseau marks, both real and pseudo, in both image-

space and object-space are now known. The vertices of each of the 506 triangu-

lar areas correspond to the locations of specific reseau marks. The output from

resloc.pro is a set of parameters, called the geoma parameters, which contains

the coordinates of the vertices of the 506 triangular areas in both image-space and

object-space.

3.7.2 geoma.pro: Performing the Geometrical Correction

The actual transformation of a raw image into geometrically correct object-space

image is performed by the routine geoma.pro. Input variables are the raw image

and the geoma parameters for a particular raw image output by resloc.pro. The

output variable is the 1000×1000 object-space image. The pixels are mapped using

the nearest neighbour technique as the default although pixel interpolation can also

be used.

The transformation parameters are calculated for each of the 506 triangular

areas that the image is divided into. For each area the transformation parameters

a, b, c, d, e, f , g, h, i, j, k and m are the solutions to

limage = a+ b× lobject + c× sobject

simage = d+ e× lobject + f × sobject (3.15)


and

lobject = g + h× limage + i× simage

sobject = j + k × limage +m× simage (3.16)

where l and s are the line and sample coordinates of the vertices of the triangular

area. Since there are three vertices, and thus 12 equations, they are soluble for the

12 transformation parameters. The sets of Eqns. 3.15 and 3.16 are solved for each

of the 506 areas in turn using a using a ‘back-substitution’ with Singular Value

Decomposition technique (the IDL procedure svsol).

Each of the 1,000,000 pixels in the object-space image initially has its DN set to

0. The 506 triangular areas are then stepped through and the index numbers of the

pixels contained within the area determined. The index numbers run sequentially

from 0 to 999,999. Pixel 0 is the pixel at the extreme left of the top row in the

object-space image. The index number then increases to the right along the row,

when the end of a row is reached the next index number is given to the extreme

left pixel of the next row down and the index again increases to the right along the

row. Pixel 999,999 is at the extreme right of the bottom row. Fig 3.11 illustrates

the scheme for allocating index numbers for a 4× 4 pixel image. An index number

1 2 3 45 6 7 89 10111213141516

Figure 3.11: Index numbers for a 4 × 4 pixel image.


is converted into integer pixel coordinates using

l = int

(

index

Nline

)

s = index− l ×Nline (3.17)

where Nline is the number of pixels in each line of the image, index is the index

number and int() gives the nearest integer rounded down. As already discussed we

assumed that the transformation parameters for all the pixels within a particular

triangular area are the same as the transformation parameters calculated for the

vertices of the area.

Taking the 506 areas in turn, each of the identified pixels within the area is

transformed into its raw-space location. The DN value of the object space pixel

under consideration is determined from its 4 nearest neighbour raw-space pixels

using the nearest neighbour or pixel interpolation technique. The actual DN of that

object-space pixel in the 1000 × 1000 geometrically corrected image is then set to

this determined value. For pixels lying outside a triangular area the transformation

parameters for the nearest area are used.

In this way a geometrically corrected version of the raw image is built up on the

1000 × 1000 grid of the object-space image. The output from geoma.pro is this

1000 × 1000 pixel geometrically corrected object-space image.

3.8 Comparison of VICAR and MINAS results

Images corrected using MINAS were compared to the same images corrected using

VICAR. The VICAR corrected images were provided by Vance Haemmerle at CI-

CLOPS. Also provided were the reseau locations in each image as determined by

VICAR. The MINAS and VICAR images were compared by subtracting one from

the other. Identical images resulting in each pixel in the 1000× 1000 grid having a

DN of 0.

When the reseau locations, as determined by MINAS and VICAR, were com-

pared the mean residual in the line and sample identified positions of the reseaus

was on the order of a few hundredths of a pixel. Occasional pixels had residuals of

up to 0.5 pixels or so. These slight differences in the identified reseau mark locations


are consistent with slight differences in the code used to locate them. The pixels

with the large residuals being those where the position had to be interpolated.

When the VICAR reseau locations were used in the MINAS geoma.pro routine

the MINAS corrected image was essentially identical to the VICAR version. There

were occasional limited areas of the image where the DN levels differed by 1 DN

or so. When an image processed entirely with MINAS was compared to a VICAR

image, again they were practically identical. In some small areas of the image the

DN values differed by up to 3 DN levels. All features, ring edges, satellites were

however in the same location in both images.

The differences between the VICAR and MINAS geometrically corrected images

can be explained by slight differences in some of the parameters used in the cor-

rection process along with minor differences in algorithms. VICAR itself does not

produce 100% accurate corrected images and can only realistically locate reseaus to

within ∼ 0.1 pixels (VICAR users guide circa. 1977). Differences in located reseau

positions of ∼0.03 pixels or so between VICAR and MINAS and the differences in

the corrected image resulting from them are insignificant. We were therefore confi-

dent that the routines written for MINAS were producing geometrically corrected

images that were as accurate as those produced by VICAR.

Chapter 4

Image Navigation

4.1 Introduction to image navigation

As discussed in the description of methods of orbit determination (Chapter 2) the

data used, the observations, consists of sets of unit pointing vectors along with

their corresponding observation times. The raw data, images taken by telescopes or

spacecraft cameras, are simply pictures of areas of the sky. While the observation

time associated with an image is known, the pointing vector to an object in an

image is not immediately apparent. The process of determining the actual pointing

direction of the centre of an image, along with the orientation of the image, is called

image navigation. Once an image has been navigated the pointing vector to any

object in the image can be easily calculated.

Image navigation is necessary because while the approximate pointing of the

telescope or camera when the image was shuttered is known, the information gen-

erally contains inaccuracies. For example, the camera pointing information in the

Voyager SEDRs includes errors due to inaccuracies in the positions of the Voyager

spacecraft (on the order of 10km) and uncertainties in the orientation of the scan

platform on which the cameras are mounted. These combine to give an uncertainty

in the pointing direction of the cameras of up to ∼ 2.5 × 10−3 radians (Showalter

1991), which is approximately one third of the field of view of the Narrow Angle

Camera (NAC).

80

CHAPTER 4. IMAGE NAVIGATION 81

4.2 Methods of image navigation

While not strictly an image navigation technique, angular separation of satellite

pairs is used for orbit determination purposes. Large numbers of observations of

the satellites of the outer planets are in the form of angular separations between

satellite pairs (see for example Harper and Taylor 1993). Measuring such angular

separations with micrometers was the only practical accurate method before the

widespread use of photographic plates. Even with ground based photographs it is

often very difficult to locate a single satellite accurately in terms of right ascension

and declination. The difficulty stems from the need for two or more reference stars

(with accurately known positions) to navigate an image. Two such relative positions

enables the absolute inertial position of a satellite to be determined. Often there

are not two such reference stars in an image and other methods have to be used.

Often the positions of a satellite relative to the known, and assumed accurate,

ephemeris positions of other satellites or the centre of the primary planet are used

(e.g. Nicholson et al. 1996, Poulet et al. 2000) in image navigation. The disadvan-

tage of these methods is that the positions of other useful reference satellites are

not always known to the required accuracy and that the centre of the image of an

oblate planet, possibly obscured by a ring system, is difficult to fix. Synnott et

al. (1983) used the positions of reference stars and the satellites Mimas, Enceladus,

Tethys, Dione and Rhea as fiducial points (Smith et al. 1981). This requirement for

two or more fiducial points limited the number of images that Synnott et al. (1983)

could navigate and therefore use.

The advantage of using the angular separation of satellite pairs is that it is easy

to accurately measure and requires no a priori knowledge about the actual inertial

position of either satellite. All the information necessary for orbit determination

purposes can be obtained directly from an image. Often sets of angular separation

measurements of various combinations of satellite pairs are obtained from the same

image. The great disadvantage is that the orbits of both satellites (or a whole

system if a combination of pairs is used) has to be solved simultaneously. Also

you need to have the same two satellites in every image, this criterion is usually

satisfied for ground based and HST images but is rarely the case for most of the


higher resolution Voyager images of Atlas, Prometheus and Pandora. Because of

this the satellite pair separation method is unsuitable for determining the orbits of

these satellites.

For the planet Saturn, the locations of circular, or near circular, features in

the ring system are known to high accuracy, ∼ 100 metres or so (Porco et al. 1984,

Nicholson et al. 1990 and French et al. 1993). Such features can be used to navigate

images and thus determine the pointing direction to an object, such as a satellite,

providing there is sufficient curvature of any feature used in a particular image

(Gordon 1994, Gordon et al. 1996 and Murray et al. 1997). We used circular

features in the outer ring system of Saturn along with the position of the planetary

limb to navigate the Atlas, Prometheus and Pandora images used in this work.

4.3 Software for image navigation, MINAS

We used the software package MINAS (see section 3.6) for the purposes of image

navigation. MINAS has routines that allow geometrically correct images (see Chap-

ter 3) to be navigated interactively using planetary limbs and circular features in

ring systems.

For a particular image the MINAS routines require information on the charac-

teristics of the planet and ring system, the pixel scale of the image, the state of the

observer with respect to the planet, and the approximate orientation of the image

at the observation time. Since the required data is general, MINAS can navigate

images taken by any camera or telescope.

For the purpose of navigating Voyager images we wrote routines that extracted

the state of the spacecraft with respect to Saturn from a NAIF SP kernel (Ac-

ton 1990) and the approximate pointing of the camera from a NAIF C kernel.

NAIF is the Navigation and Ancillary Information Facility node of the Planetary

Data System (PDS). It is responsible for the implementation and operation of the

SPICE information system. SPICE stands for Spacecraft, Planet, Instrument,

“C-matrix” and Events and it is a way of providing ancillary observation geome-

try data and related tools used in the interpretation and planning of observations


made by spacecraft. NAIF provides a toolkit of FORTRAN 77 and C subroutines

along with ‘kernels’ containing information on spacecraft trajectories, planetary

ephemerides, pointing directions for instruments and so forth. The SP kernels used

were vg1 sat.bsp and vg2 sat.bsp for Voyager 1 and Voyager 2 respectively. These

kernels are available via ftp at

ftp:naif.jpl.nasa.gov:/pub/naif/voyager/spk.

The C kernels were vg1 sat qmw na.bc, vg1 sat qmw wa.bc, vg2 sat qmw na.bc and

vg2 sat qmw wa.bc which were created from Voyager SEDRs at QMW for use with

Voyager Saturn images (Gordon 1994 and Gordon et al. 1996). SP kernels con-

tain ephemerides for spacecraft, planets and satellites etc. while C kernels contain

pointing information for instruments. SEDRs are Supplementary Experiment Data

Records and contain ancillary information for an image like the state of the space-

craft, temperature of the cameras, pointing direction and so on. The extracted

information is passed directly to the MINAS navigation procedures.

4.4 Navigating an image

From the information supplied, the MINAS routines present the locations of two

ring features and the planetary limb, calculated from the supplied geometry, su-

perimposed on the image. MINAS draws in the position of these features based on

the information it has been given. Due to inaccuracies in the pointing direction the

drawn in locations are unlikely to coincide with the actual features in the image.

Fig. 4.1 illustrates a Voyager image with drawn in positions for ring features based

on the initially supplied information.

The position and orientation of the dotted lines superimposed on the image are

then interactively modified. When the lines indicating the calculated positions of

the features exactly coincide with the actual positions of the features in the image,

the pointing direction assumed by the MINAS procedure at that point is the actual

pointing direction of the instrument (see Fig. 4.2). The process of matching the

calculated and actual positions of features can be performed automatically, but we

found from experience that visually matching the lines with the corresponding fea-


Figure 4.1: Example of initial image displayed by MINAS. The white dotted linesindicate the locations of the outer edge of the A ring and the inner edge of theEncke gap based on the information supplied to the program. The image is ageometrically corrected FDS43686.55 which has been stretched to bring out thedotted lines against the background of the rings. The pointing information suppliedis clearly inaccurate.


tures provided better results. The now accurate pointing direction used to calculate

Figure 4.2: Example of final image displayed by MINAS. The white dotted linesindicating the locations of the outer edge of the A ring and the inner edge of theEncke gap now coincide with the features’ actual location in the image. The pointingdirection assumed for the instrument is now accurate.

the feature locations is out-putted, along with the image ID number and image time

for future use. The pointing direction of the camera/telescope at the image time

as determined by MINAS is out-putted as a SPICE C-matrix. Each image must

be navigated separately using this software, batch processing is not possible at the

current stage in the MINAS’ development.


x

y

z

image plane

origin of imagecoordinate system

y

x

centre ofimage

x,y,z instrument coordinate

x,y image coordinate frame

frameinstrument boresightpointing to target

Figure 4.3: Instrument frame for Voyager cameras

4.5 Calculating the pointing vector to an object

in an image

The line,sample (l, s) coordinates of an object in a navigated image have to be

converted into a pointing vector in inertial space. The first step is to convert the line,

sample coordinates into a pointing vector in the instrument frame, the frame that is

fixed with respect to the camera/telescope. The origin of the instrument frame is at

the centre of the image plane with the z-axis pointing along the instrument boresight

vector. The x- and y-axes are orthogonal to each other in the plane perpendicular

to the z-axis. The orientation of the x- and y-axes may be defined differently for

different instruments, the instrument frame for the Voyager cameras is illustrated

in Fig. 4.3. This figure illustrates the relationship between the instrument frame

and image coordinate system, it is not an accurate representation of the actual path

followed by a light ray traveling through the imaging system.

The origin of the image frame has to be mapped onto the origin of the instrument

frame. If the origin of the instrument frame has coordinates lcentre, scentre in the


image frame then

linstrument = limage − lcentre (4.1)

sinstrument = simage − scentre (4.2)

gives the line(l) and sample(s) coordinates of a point in the image in instrument

coordinates. For space probe images it is usual to refer to coordinates in the x and

y directions as line and sample respectively and we will follow convention and refer

to line and sample coordinates. The pointing vector to an object in the instrument

frame, Q, is defined by (VICAR users guide circa 1977)

Qx =1

σsinstrument (4.3)

Qy =1

σlinstrument (4.4)

Qz = f (4.5)

where f is the focal length of the camera in mm and σ is a scale factor measured in

pixels per mm. The values of f and σ for the Voyager cameras is given in Danielson

et al (1981) and Table 5.1.

The unit pointing vector in the instrument frame, Q, is related to the unit

pointing vector, P, in the reference frame co-moving with the instrument, the J2000

frame (Earth mean equator, dynamical equinox of J2000) throughout this work,

through the transpose of C, the SPICE C-matrix for that image.

P = CTQ (4.6)

In this way, P, the unit vector pointing from the instrument towards the position

of the target object in the J2000 frame co-moving with the instrument, can be

calculated.

Chapter 5

Atlas

5.1 Introduction

The small saturnian satellite Atlas orbits just exterior to the outer edge of the A

ring (see Fig. 5.1). It was discovered during the Voyager 1 encounter in 1980 and

its orbital elements determined using only 8 images (Smith et al. 1981). Atlas was

also imaged by the Voyager 2 cameras in 1981 but no orbital elements have been

published that utilise these Voyager 2 images. The Voyager 2 science report simply

reproduces the Voyager 1 elements (Smith et al. 1982).

The pre-1995 orbits of Prometheus and Pandora calculated by Synnott et al. -

(1983) have been checked using the original data (Jacobson private communication)

but no independent check has been made on Atlas’ orbit. In light of the changes

in Prometheus’ orbit (Nicholson et al. 1996, Bosh and Rivkin 1996), indications

of undetected mass in the region between the outer edge of the A ring and the F

ring and the fact that Atlas’ orbit is unchecked and only uses Voyager 1 data an

investigation of the orbit of Atlas is warranted.

If significant changes in the orbit of Atlas are detectable between the Voyager

1 and 2 epochs, constraints could then be placed on possible mechanisms for the

observed changes in Prometheus’ motion.

88

CHAPTER 5. ATLAS 89

Figure 5.1: Atlas in FDS43917.52. The F ring is in the bottom right of the imageand the outer edge of the A ring at the top left. Atlas is the circled object. Onlya small section of the complete image is shown. This is the raw image with theintensity enhanced somewhat

CHAPTER 5. ATLAS 90

5.2 Search methodology

Software was written by the author in FORTRAN77 using the SPICE library (Ac-

ton 1990). The program, findimage.f, searched through the Voyager dataset for

candidate images that could include the object of interest, in this case Atlas.

In order to locate candidate images, an orbit had to be assumed for the object.

We used the orbit within the NAIF SP kernels vg1 sat.bsp and vg2 sat.bsp for

the Voyager 1 and Voyager 2 images respectively. Since no great accuracy in the

assumed orbit is required a circular orbit with the elements from Smith et al. (1981)

would have sufficed.

Using the ephemerides of the Voyager spacecraft and Atlas contained within

the utilised SP kernels, the pointing direction from the spacecraft to Atlas at the

exposure time of each image, P, is easily calculated. The effects of the light time,

and planetary aberration effects were allowed for.

Each of the images in the Voyager dataset for Saturn was stepped through in

turn. The pointing direction to Atlas from the spacecraft was calculated at the

image mid-time. The image mid-time is simply the shutter opening time plus half

the length of the exposure. This information is given in the headers of the Voyager

images.

The assumed pointing direction of the camera is then obtained from a SPICE

C-kernel. We used the kernels vg1 sat qmw na.bc, vg1 sat qmw wa.bc,

vg2 sat qmw na.bc and vg2 sat qmw wa.bc for the Voyager 1 NAC and WAC and

Voyager 2 NAC and WAC respectively. As stated in Chapter 4 these kernels were

prepared at QMW from Voyager SEDRs (Gordon 1994). A rotation matrix, called a

C-matrix , is obtained from SPICE C-kernels. Using Eqn. 4.6 the pointing direction

of the camera, BJ2000, is then simply the transpose of the C-matrix times the

boresight vector of the camera in the instrument frame, Binstrument

BJ2000 = CTBinstrument (5.1)

where BJ2000 is in the J2000 reference frame (section 2.1.2).

Due to Atlas’ small size, 18.5 × 17.2 × 13.5 km (Davies et al. 1994), and corre-

sponding faintness when compared to the much larger Prometheus, images where

CHAPTER 5. ATLAS 91

the calculated resolution was worse than 150 km per pixel were instantly rejected.

These images were excluded from the next stage in the process.

The angular separation between the calculated pointing direction to the obj-

ect, P, and the boresight vector of the camera, BJ2000, is then determined. If this

angular separation is greater than a set value it is assumed that Atlas is not in the

image and the image is then rejected. The image is marked for further examination

if the angular separation is less than the set value, as Atlas could possible be in

the image. The set value used in this work was 12(FOV of the camera)+ 2.5× 10−3

radians. The error in the pointing direction of the camera can be up to 2.5 × 10−3

radians (Showalter 1991).

The output from findimage.f is a list of images that have been marked for

further examination. Each of these images is then viewed using the XV image tool

and Atlas searched for. If an image includes an object that could be Atlas it is

marked as an identified image. False identifications of Atlas are removed at the

orbit determination stage 5.6.

5.3 Geometrical correction

All identified images were geometrically corrected using MINAS and the procedures

resloc.pro and geoma.pro as described in Chapter 3. The 800 × 800 pixel raw

image-space images being transformed into 1000 × 1000 pixel object-space images.

5.4 Image navigation

After geometrical correction, as described in Chapter 3, all the identified images

were navigated using MINAS, as described in Chapter 4. The images were navigated

using the outer edge of the A Ring, the inner edge of the Encke gap and the limb

of Saturn. The inner edge of the Encke Gap is circular (Showalter 1991), but the

outer edge of the A Ring is not. Its shape is consistent with the seven lobed pattern

expected due to the 7:6 inner Linblad resonance with the co-orbital satellites Janus

and Epimetheus (Porco et al. 1984). However, the amplitude of the radial distortion

in the A Ring edge is only 6.7 ± 1.5 km (Porco et al. 1984), so for our purposes

CHAPTER 5. ATLAS 92

treating the ring edge as circular is a reasonable approximation. The parameters

used for Saturn, the ring features, and the Voyager cameras in the image navigation

and orbit fitting processes are listed in Table 5.1 Any images for which navigation

Table 5.1: Constants used in the image navigation and orbit determination process.

Where f and σ are the focal length and scale factor of the camera as used in

Eqns. 4.3-4.5

Parameter Value

Radius Outer Edge A-Ring 136774.4kma

Radius Inner Edge Encke Gap 133423.53kmb

Equatorial Radius of Saturnc 60330.0kmd

Radii of Saturne 60268.0 × 60268.0 × 54364.0kmf

J2 0.016298d

J4 −0.000915d

f (Vgr 1 NAC) 1500.19mmg

f (Vgr 1 WAC) 200.29mmg

f (Vgr 2 NAC) 1503.49mmg

f (Vgr 2 WAC) 200.77mmg

σ 84.8214 pixels/mmg

aPorco et al. (1984)bNicholson et al. (1990)cused to calculate spherical harmonicsdCampbell and Anderson (1989)eused to calculate location of limbfDavies et al. (1994)gDanielson et al. (1981)

was not possible, mainly due to the absence of ring features, were discarded. The

SPICE C-matrices for the navigated images, along with the image FDS (Flight

Data System) ID numbers are stored in a text file.

CHAPTER 5. ATLAS 93

5.5 Generating a set of observations

The centroided line,sample (l, s) coordinates of Atlas on the raw image-space images

were transformed into object-space line,sample coordinates using the appropriate

transformation parameters and geoma.pro (section 3.7.2). For each image the

object space line,sample (l, s) coordinates were mapped to instrument frame coor-

dinates using Eqns. 4.1 and 4.2

linstrument = lobject−space − 499.5

sinstrument = sobject−space − 499.5

where (499.5,499.5) are the pixel-coordinates of the centre of the image in object-

space and the origin of the line and sample directions in the instrument frame. The

pointing direction to Atlas, Q, in the instrument frame using Eqns. 4.3, 4.4 and 4.5

is

Qx =1

σsinstrument

Qy =1

σlinstrument

Qz = f

with σ and f from Table 5.1. The unit pointing vector to Atlas in the instrument

frame is simply Q. This is then transformed into a unit pointing vector, P, in

the J2000 reference frame with the origin at the Voyager spacecraft at the image

mid-time using Eqn. 4.6

P = CT Q

where C is the C-matrix determined for the image in the image navigation process

(section 5.4).

Once P had been calculated for each image the final result of the image naviga-

tion process was a text file containing the image mid-time and unit pointing vector

to Atlas for each image. This text file was the set of observations used for the

process of orbit determination.

CHAPTER 5. ATLAS 94

5.6 Orbit determination

A FORTRAN77 program was written by the author for the purpose of orbit deter-

mination utilising the SPICE library (Acton 1990) and SPICE library extensions

written by Gordon (1994). This program, diffcorr.f, fits a set of observations to

the precessing elliptical model of Taylor (1998) which is described in Section 2.5.2.

The program takes as input an ephemeris for the observer and a set of initial

orbital elements at epoch. Seven parameters are fitted a0, λ0, n0, e0, 0, i0 and Ω0.

It is assumed that a, n, e and i have no time dependency. The effects of any body

other than Saturn are ignored

Based on the initially assumed orbital elements a set of expected pointing vec-

tors to Atlas at the observations mid-times are calculated. Full allowance is made

for light time and planetary aberration effects. Differences between the expected

pointing vectors and the actual observations are used, along with a differential cor-

rection technique, to make a ‘better’ estimate of the initial orbital elements (section

2.6). A ‘better’ estimate of the initial orbital elements is one where the root mean

square (rms) of the difference between the expected pointing vector and the actual

observations is reduced.

The initial orbital elements used to calculate the expected pointing vectors are

then updated to this ‘better’ value, the differences recalculated and the differential

correction algorithm run again.

The process is repeated iteratively, with the rms of the differences being reduced

at each iteration, until the program has converged. When convergence occurs the

difference between the initial orbital elements and the ‘better’ elements is smaller

then the formal errors from the fitting process associated with each parameter.

The final orbital elements at epoch after convergence provide the least squares

best fit to the supplied observations.

A complementary program to diffcorr.f was also written by the author. Again

written in FORTRAN 77 using the SPICE libraries the program, findlinesample.f,

calculates line and sample co-ordinates of an object in an image. The input variables

are the 7 elements of the object’s orbit and the navigated C-matrix for the image.

In some cases the identification of a satellite is ambiguous, there may be several

CHAPTER 5. ATLAS 95

objects in approximately the right location which could be the satellite. An orbit

is first determined using the unambiguous detections and then findlinesample.f

used to calculate the co-ordinates the object should be at in all the other images.

The other candidate images can then be searched near the identified co-ordinates.

Experience has shown that if the identified co-ordinates are on the image there is

usually a satellite candidate object within 2-3 pixels. This technique allows the

identification of faint satellite detections that are missed during the initial visual

examination of an image.

During the early stages of the research the orbits were fitted using a method

involving the numerical integration of the full equations of motion (section 2.5.1).

The Runge-Kutta-Nystrom 12th order scheme of Dormand et al. (1987a, 1987b) was

used to integrate the equations. As detailed in Chapter 2, the end result from a

numerical integration is the state at epoch for all the bodies in the system. A state is

of little utility when a determined orbit is to be compared with the orbital elements

obtained by other authors. At the time I was unable to fit a precessing elliptical

model to an ephemeris generated by numerical integration. All the elliptical models

used failed to converge during the differential correction process.

The precessing elliptical model of Taylor (1998) was first used in the final year

of the research, and was the first such model to successfully converge during the

differential correction process. Time constraints prevented its use to fit an orbit to

a numerically generated ephemeris.

5.7 Identified images

The 7 Voyager 1 and 18 Voyager 2 navigable images of Atlas that we identified

are listed in Tables 5.2 and 5.3 respectively. Of these images the 4 Voyager 1 and

7 Voyager 2 which were marked in the data of Synnott et al. (1983) as used by

Jacobson (private communication) as including Atlas are indicated.

CHAPTER 5. ATLAS 96

Table 5.2: Voyager 1 Images of Atlas

FDS Image Mid-time Used by Line Sample Phase Res. a

No. UTC Synnott lcont. scont. Angle() km

34667.32 Nov. 3 18:34:58.7 • 687.5 588.4 12.4 99.2

34775.24 Nov. 7 08:52:36.9 • 558.7 111.5 12.4 61.4

34785.21 Nov. 7 16:50:12.9 • 167.6 784.5 14.5 57.9

34785.25 Nov. 7 16:53:24.9 • 627.4 303.6 14.5 57.8

34830.20 Nov. 9 04:49:22.5 235.6 232.5 12.5 42.4

34930.48 Nov. 12 13:11:46.7 0.5 592.4 39.7 5.7

34930.59 Nov. 12 13:20:34.7 388.6 165.6 40.0 55.3

aThe year of all images is 1980. All images are taken with the Narrow Angle Camera (NAC)

except 34930.59 which is a Wide Angle Camera (WAC) image. The resolution and sun phase

angle information is calculated using the derived Voyager 1 orbit in Table 5.4 .

CHAPTER 5. ATLAS 97

Table 5.3: Voyager 2 Images of Atlas.

FDS Image Mid-time Used by Line Sample Phase Res.a


43655.12 Aug. 14 11:04:00.5 • 303.5 257.5 7.9 92.5

43655.28 Aug. 14 11:16:48.5 • 413.5 175.6 7.9 92.3

43655.44 Aug. 14 11:29:36.5 • 342.5 112.5 7.9 92.1

43656.00 Aug. 14 11:42:24.5 • 391.2 290.4 7.9 92.0

43656.16 Aug. 14 11:55:12.5 • 465.4 153.9 7.9 91.8

43684.35 Aug. 15 10:34:24.7 201.5 297.5 6.8 85.6

43703.18 Aug. 16 01:32:48.5 164.5 418.8 6.9 81.1

43737.47 Aug. 17 05:08:00.5 677.4 533.5 6.9 72.0

43752.38 Aug. 17 17:00:49.4 425.8 98.6 7.3 67.2

43779.59 Aug. 18 14:53:35.5 505.5 643.7 8.6 61.8

43810.56 Aug. 19 15:39:11.5 12.4 273.5 7.5 54.0

43818.37 Aug. 19 21:47:59.5 175.5 31.6 9.6 51.2

43835.06 Aug. 20 10:59:11.5 380.6 604.5 9.8 47.6

43843.48 Aug. 20 17:56:47.5 614.3 133.5 7.8 44.5

43854.20 Aug. 21 02:22:23.5 100.9 47.1 10.5 42.3

43861.50 Aug. 21 08:22:23.5 500.2 362.3 8.1 39.9

43917.52 Aug. 23 05:11:59.5 • 626.2 431.7 9.2 26.0

43938.11 Aug. 23 21:27:13.9 • 263.2 6.4 10.6 21.3

aThe year of all images is 1981. All images are taken with the Narrow Angle Camera (NAC).

The resolution and sun phase angle information is calculated using the derived Voyager 2 orbit in

Table 5.5 .

CHAPTER 5. ATLAS 98

An additional Voyager 1 image was identified in the Synnott et al. (1983) data,

FDS 34667.36 taken at 1980 Nov. 3 18:38:08.5 UTC, which we were unable to

confirm actually existed let alone locate. Smith et al. (1981) state that 8 Voyager

1 images were used, we are currently unable to account for the extra 3 images.

Possibly they are the additional 3 images that we identified.

5.8 Results

We determined orbits at the Voyager 1 and Voyager 2 epochs separately using

the 7 Voyager 1 and 18 Voyager 2 images respectively. A determination using the

combined data was also calculated. The orbital elements are shown in Table 5.4.

The Voyager 1 elements of Smith et al. (1981) are shown for comparison.

Table 5.4: The orbital elements of Atlas. The epoch for the elements from thiswork is 2444839.6682. The Smith et al. (1981) elements are epoch 2444513.5 withλ advanced forwards to 2444839.6682 using n = 598.08/day.

para- Smith Voyager 1 Voyager 2 Combined

metera et al. (1981) only only Voyager 1 & 2

a 137670 137621± 62 137714 ± 14 137704 ± 12

acalc - 137669 ± 7 137667.0± 0.5 137666.37 ± 0.03

λ 247 ± 16 312 ± 13 317.43 ± 0.02 317.45 ± 0.01

n 598.08 ± 0.05 598.29 ± 0.05 598.302 ± 0.003 598.3069± 0.0002

e (2 ± 3) × 10−3 (1.2 ± 1.4) × 10−3 (9 ± 2) × 10−4 (6 ± 1) × 10−4

i 0.0 ± 0.2 0.17 ± 0.06 0.02 ± 0.01 0.017 ± 0.007

- 198 ± 18 243 ± 8 233 ± 8

Ω - 82 ± 5 318 ± 16 317 ± 19

aDistances are in km, longitudes in degrees and rates in degrees/day. Errors for elements

from this work are the formal errors from the fitting process. All longitudes measured from the

ascending node of Saturn’s equator at epoch on the Earth mean equator at J2000, i measured

from Saturn’s equatorial plane at epoch.

CHAPTER 5. ATLAS 99

5.9 Discussion

The mean motions derived from the Voyager 1 & 2 data separately are within 1σ

of each other. These results may be misleading due to the large error associated

with the Voyager 1 mean motion. This large error is a direct consequence of the

low number (7) of images utilised.

The difference between the Voyager 2 only and combined mean motions may be

a better value to use, they differ by 1.7σ. A difference which is not large enough to

be significant.

The fit to the Voyager 1 data is clearly much worse than the fit to the Voyager

2 data. The error in the Voyager 1 value of e is actually larger than the derived

value.

It is reasonable to conclude that there was no significant change in the orbit of

Atlas between the Voyager encounters. This being the case, fitting a single orbit to

the combined dataset gives the most accurate determination. As a result we adopt

the combined orbit fit for the orbital elements of Atlas (see Table 5.8).

The adopted combined fit differs significantly from that of Smith et al. (1981),

the mean motion being ∼ 0.22/day higher. Taking the mean longitude at JD

2444513.5 of Smith et al. (1981) and precessing it forwards to JD 244839.6682

using the n from our combined fit, 598.3069± 0.0002/day, gives a mean longitude

at epoch, λ, of 321, close to the combined fit value of 317. It appears that the

mean motion of Smith et al. (1981) is too low by ∼ 0.22/day. To allow a direct

comparison to be made between our results and those of Smith et al. (1981) we

derived a fit to the Voyager 1 dataset at the Voyager 1 epoch, JD 2444513.5, see

Table 5.5.

Table 5.5 clearly shows that the mean longitude at JD 2444513.5 of Smith et

al. (1981) is comparable with that of our Voyager 1 fit at JD 2444513.5. The large

difference in λs in table is therefore primarily a result of the low mean motion of

Smith et al. (1981).

CHAPTER 5. ATLAS 100

Table 5.5: Voyager 1 orbital elements of Atlas. The epoch is JD 2444513.5. TheSmith et al. (1981) elements have been changed into the reference frame used forthis work.

para- Smith et al. (1981) Voyager 1

metera only

a 137670 137621 ± 62

acalc - 137669 ± 7

λ 292 ± 1.5 289 ± 2

n 598.08 ± 0.05 598.29 ± 0.05

e (2 ± 3) × 10−3 (1.2 ± 1.1) × 10−3

i 0.0 ± 0.2 0.17 ± 0.05

- 339 ± 50

Ω - 296 ± 10





5.10 The distribution of longitudes at observa-

tion mid-times

Figs. 5.2 and 5.3 show the x-y coordinates of Atlas, during the Voyager 1 and

Voyager 2 encounters, in the planetary reference frame used for Saturn, at the

observation mid-times. The x-y plane is the equator plane of Saturn at epoch. The

positions are numbered in ascending chronological order, the same order as they

are presented in Tables 5.2 and 5.3.


1

2

34

5

6

7

mean direction to Sun

Figure 5.2: The x-y plane coordinates of Atlas at the Voyager 1 observation mid-times calculated from the combined Voyager fit in Table 5.4


1

2

3

4

5

6

7

8

9

10

1112

13

14

15

16

17

18


Figure 5.3: The x-y plane coordinates of Atlas at the Voyager 2 observation mid-times calculated from the combined Voyager fit in Table 5.4


Examination of Figs. 5.2 and 5.3 clearly shows that none of the images used has

Atlas either transiting Saturn or in the shadow of the planet. This lack of observa-

tions in these longitudes ranges is unsurprising, Atlas being effectively invisible in

the visible light range in the shadow and swamped by light from the planetary disc

when transiting.

Locations which are marked in the same colour (except black) indicate observa-

tions which are close together in time, all occurring within 1/2 an orbital period.

Ideally the entire longitude range for the orbit should be sampled, this is especially

important for accurate determination of e (Synnott et al. 1983). The Voyager 1

data does not sample the available longitude range very well at all, not unsurprising

considering there are only 7 images available.

The Voyager 2 observations (Fig. 5.2) sample the longitude range reasonably

well. Although there is some clustering of observations, it is not enough to cause

concern over the accuracy of the orbital fit.

5.11 The orbit of Atlas in JPL Ephemerides

Since there is only one published orbit for Atlas, that of Smith et al. (1981), an

analysis was performed on the JPL Ephemerides to investigate the orbit of Atlas

as presented in them.

A program was written which fits the freely precessing ellipse model of Taylor

(1998), detailed in section 2.5.2, to a set of planetocentric position vectors obtained

from an ephemeris. The perturbations due other bodies in the system, resonances

etc. were ignored.

The reference frame used is a Cartesian coordinate system frame. The z-axis

is parallel to the spin angular momentum vector of Saturn at epoch. The x-axis

points along the direction to the intersection at epoch of the ascending node of the

saturnian equator with the mean Earth Equator at J2000. The y-axis lies in the

plane of the saturnian equator orthogonal to both the x- and z-axes. This is the

planetary reference frame (for Saturn) as illustrated in Fig. 2.9.

The fitting program, FitOrb2Ephm.f, generates a specified number of plan-


etocentric position vectors for the object at given times. The precessing ellipse is

then fitted to the set of position vectors using a least squares differential correction

technique. The process is identical to that described in sections 2.5.2 and 2.6 except

that the observations are now positions not pointing vectors and the rms error is a

distance in km not an angle.

For Atlas, 160 position vectors at 3600.0 second intervals were obtained from an

ephemeris. A precessing ellipse was fitted to the data with the starting values for

the parameters of the ellipse being taken from the adopted combined orbit fit from

Table 5.5. A second fit was also performed with the apsidal and nodal precession

rates, β and γ set equal to 0.0/day. Position vectors were obtained from the SP-

kernels vg2 sat.bsp (the Voyager 2 ephemeris) and sat081.4.bsp (the current Cassini

small satellites ephemeris). The results obtained are presented in Tables 5.6 and

5.7. In both cases the North Pole orientation of Saturn is taken from French et

al. (1993).

The fitting process was very robust, a wide range of starting values for the

parameters of the ellipse converged on the exact values given in Tables 5.6 and 5.7.

The only strong correlations were between the mean longitude at epoch (λ) and the

mean motion (n).

Examination of the data presented in Tables 5.6 and 5.7 indicates that a simple

ellipse is a very good model for the ephemerides of Atlas, at least over several tens

of days. Somewhat surprisingly the non-precessing ellipse is a better fit to the

ephemerides than the precessing model. The very low eccentricies suggest that it

was assumed that Atlas was in a circular orbit for the generation of the ephemerides.

The JPL ephemerides are generated by numerical integration of the full equa-

tions of motion for an object in a many body system (Standish 1990). Our analysis

of the ephemerides of Atlas in vg2 sat.bsp and sat081.4.bsp suggests that they are

so consistent with a non-precessing elliptical model that such a model may well

have been used in their generation and not a numerical integration.

It is possible that the very limited available data (12 images in the Synnott et

al. (1983) data supplied to Jacobson (private communication)) prevented reliable

numerical determination of Atlas’ orbit and forced the use of ellipses.


Table 5.6: Orbital elements for Atlas from a fit to the SP kernel vg2 sat.bsp: Epoch2444839.6682 JED. 160 positions at 3600.0 sec. intervals starting at 244824.6682JED.

para- Precessing Ellipse Model Non-Precessing Ellipse Model

metera β = γ = 0.0

a 137640.00± 0.10 137639.99998± 0.00003

acalc 137666.512± 0.003 137666.5118296± 0.0000009

λ 317.6214 ± 0.0002 317.62142221± 0.00000007

n 598.30600 ± 0.00002 598.306000000± 0.000000006

e (0.0 ± 4.5) × 10−7 (2.6 ± 0.1) × 10−10

i (1.2752 ± 0.006) × 10−2 (1.279067 ± 0.000002) × 10−2

224 ± 134 191 ± 3

Ω 322.9 ± 0.2 356 ± 7

rms 0.17 0.00005





The fits to vg2 sat.bsp and sat081.4.1.bsp differ slightly. Some of the difference

is almost certainly due to the fact the the ephemerides cover epochs 20 years apart.

Since the only observations of Atlas is the small Voyager set, none of the differences

can be due to additional data on Atlas. Other factors must be responsible.

In numerical integrations of the full equations of motion of a body the positions,

masses and velocities of other bodies are factors. Sat081.4.bsp is more recent than

vg2 sat.bsp and presumably uses more reliable data on the positions, masses and

velocities of Saturn, the saturnian satellites, Jupiter, Uranus, the Sun etc. As such

the orbital elements derived from sat081.4.1.bsp have a higher degree of confidence

than those from vg2 sat.bsp.

The mean motion derived from sat081.4.bsp, n = 598.30666 ± 0.00002/day,

is within 1σ of our derived value from the combined Voyager 1 & 2 dataset of

n = 598.3069 ± 0.0002/day. Interestingly the mean motions derived from both


Table 5.7: Orbital elements for Atlas from a fit to the SP kernel sat081.4.bsp: Epoch2453371.00 JED. 160 positions at 3600.0 sec. intervals starting at 2453360.00 JED.



a 137666.47± 0.09 137666.47 ± 0.02

acalc 137666.412± 0.003 137666.4117± 0.0006

λ 230.0435 ± 0.0002 230.04348± 0.00003

n 598.30666 ± 0.00002 598.306656± 0.00004

e (1.3 ± 0.4) × 10−6 (1.29 ± 0.09) × 10−6

i (1.141 ± 0.005) × 10−2 (1.147 ± 0.001) × 10−2

143 ± 16 121 ± 2

Ω 338.8 ± 0.2 0.97 ± 0.06

rms 0.15 0.03





ephemerides are much higher than the published value of n = 598.08 ± 0.05/day

(Smith et al. 1981) but much closer to all our derived values (see Table 5.4).

5.12 Conclusions

There is no evidence that the orbit of Atlas changed between the Voyager 1 and

Voyager 2 encounters. As such Atlas appears to be behaving in a completely pre-

dictable manner. The elements of this moon’s orbit have been determined to much

greater accuracy then previously, enabling predictions of its position when first de-

tected by Cassini to be made with higher confidence. We have also been able to

place much tighter constraints on Atlas’ eccentricity and inclination. The orbit we

have adopted is from the combined Voyager data and summarised in Table 5.8.

Our adopted value for the mean motion, n = 598.3069 ± 0.0002/day is ∼0.22/day higher than the published value (Smith et al. 1980), but within 1σ of the


Table 5.8: The adopted orbital elements of Atlas: epoch2444839.6682

parametersa Adopted Orbit

a 137704 ± 12

acalc 137666.37± 0.03

λ 317.45 ± 0.01

n 598.3069 ± 0.0002

e (6 ± 1) × 10−4

i 0.017 ± 0.007

233 ± 8

Ω 317 ± 19

aDistances are in km, longitudes in degrees and rates in degrees/day.

Errors for elements from this work are the formal errors from the fitting

process. All longitudes measured from the ascending node of Saturn’s

equator at epoch on the Earth mean equator at J2000, i measured from

Saturn’s equatorial plane at epoch.

value derived from the JPL Ephemeris sat081.4.bsp. The similarity between n from

this work and from the JPL Ephemerides suggests that unpublished data may be

included in the JPL Ephemerides.

Chapter 6

Prometheus

6.1 Introduction

The saturnian satellite Prometheus, provisionally named S1980S27, was discovered

during the Voyager 1 encounter with Saturn in 1980 (Smith et al. 1981). Like Atlas

(see Chapter 5), Prometheus orbits between the outer edge of the A ring and the

F ring. The first published orbit for Prometheus was derived using 18 Voyager 1

images (Smith et al. 1981). Additional images of Prometheus were obtained during

the Voyager 2 encounter in 1981 and 27 of them were used by Synnott et al. (1983)

to derive a new orbit. Fig. 6.1 shows the highest resolution image of Prometheus

currently available. Fig. 6.2 is a WAC image shuttered at the same time as the

NAC image in Fig. 6.1, clearly illustrating Prometheus’ location with respect to the

F ring and the outer edge of the A ring.

Synnott et al. (1983) derived separate values for Prometheus’ eccentricity, e, and

longitude of pericentre, , at each encounter using the 18 Voyager 1 and 27 Voyager

2 images, while the mean motion, n, and semi-major axis, a, were derived using

combined images from both encounters. The value for the inclination, i, was derived

from Voyager 2 data only. Synnott et al. (1983) stated that derived solutions for

e, i and using the combined data set were consistent with the Voyager 2 values.

The full combined solution and a separate value for the Voyager 2 mean motion

were not published.

108

CHAPTER 6. PROMETHEUS 109

Figure 6.1: Prometheus in FDS43998.29. This is a region of the highest resolution,∼ 6km per pixel, narrow angle camera image of Prometheus. This is a raw imagewith the intensity slightly enhanced.


Figure 6.2: Prometheus in FDS43998.32. This is a raw wide angle camera imageshuttered at the same time as FDS43998.29. The Encke and Keeler Gaps in theA ring are clearly visible as is the narrow F ring running diagonally through thecentre of the image. Prometheus is the circled object.


The commonly quoted values for Prometheus’ orbital elements (see, for example,

Burns 1986) are those from Synnott et al. (1983), with e, and i derived only from

the 27 Voyager 2 images. Only the determination of n, and therefore a, also includes

data from the 18 Voyager 1 images.

In determining an orbit it is standard practice to combine data sets obtained at

widely separate epochs to produce a single solution (see, for example, Harper and

Taylor 1994). This combination of observations, often separated by time intervals

as long as decades, usually leads to highly accurate determinations. There was no

reason at the time to think that combining Voyager 1 and Voyager 2 results would

not lead to a more accurate determination of Prometheus’ orbit.

As described in section 1.1, in 1995 Prometheus was observed to be lagging

behind its expected location based on the Voyager ephemerides. We discussed

possible explanations for this lag in section 1.3. We re-examined the Voyager dataset

in an attempt to locate additional images to those used by Synnott et al. (1983).

All the located images were used to determine an orbit for Prometheus. Three

separate orbit determinations were performed

1. Using only the Voyager 1 images

2. Using only the Voyager 2 images

3. Using the combined Voyager 1 and Voyager 2 images

in an attempt to ascertain if the orbit of Prometheus’ changed significantly in the

9 months between the two Voyager encounters. At the very least, use of images

in addition to those of Synnott et al. (1983) would allow an improvement in the

determined elements. In addition, an independent check on Prometheus’ published

orbital elements (Jacobson used Synnott et al.’s (1983) data) would immediately

indicate if the elements of Synnott et al. (1983) are grossly incorrect.

6.2 Search methodology and orbit determination

The search methodology, geometrical correction, image navigation and orbit deter-

mination techniques used for Prometheus were almost identical to those already


detailed for Atlas (sections 5.2, 5.3, 5.4 and 5.6 respectively). The only differences

being that Prometheus not Atlas was being searched for and images where the

resolution was worse then 320 km per pixel (not 150 km per pixel) were instantly

rejected. This figure was chosen because the worst resolution image used by Synnott

et al. (1983) has a resolution of 320 km per pixel.

Since these techniques have already been described in Chapter 5 we shall not

describe them again here.


The 56 Voyager 1 and 66 Voyager 2 navigable images of Prometheus that we iden-

tified are listed in Tables 6.1 and 6.2 respectively. Also indicated in the tables is

whether a particular image was used by Synnott et al. (1983) for their Prometheus

orbit determination. Synnott et al. ’s (1983) data was also used by Jacobson (private

communication) for an independent determination of Prometheus’ orbital elements.

The line and sample coordinates are given in continuous and not integer pixel coor-

dinates (see section 3.3). The resolutions and solar phase angle information included

for each image were calculated using the individually determined orbits from this

work for Prometheus at the Voyager 1 and Voyager 2 epochs (see Tables 6.3 and

6.4 respectively).


Table 6.1: Voyager 1 Images.



34151.07 Oct. 17 13:26:60 • 293.6 200.4 13.3 278.5

34271.15 Oct. 21 13:33:23 539.6 742.4 12.9 235.8

34388.24 Oct. 25 11:16:35 146.8 463.5 13.0 196.9

34388.30 Oct. 25 11:21:23 100.6 366.4 13.0 196.9

34388.36 Oct. 25 11:26:11 100.6 385.5 13.1 196.9

34388.42 Oct. 25 11:30:59 92.6 451.5 13.1 196.8

34388.48 Oct. 25 11:35:47 115.6 438.5 13.1 196.7

34388.54 Oct. 25 11:40:35 189.3 278.2 13.1 196.7

34389.00 Oct. 25 11:45:23 217.9 274.3 13.1 196.6

34389.06 Oct. 25 11:50:11 181.6 445.7 13.1 196.6

34389.12 Oct. 25 11:54:59 202.7 394.3 13.1 196.5

34389.18 Oct. 25 11:59:47 192.4 286.4 13.1 196.5

34389.24 Oct. 25 12:04:35 47.6 397.4 13.1 196.4

34389.30 Oct. 25 12:09:23 127.9 317.7 13.1 196.4

34389.36 Oct. 25 12:14:11 • 245.8 274.3 13.1 196.3

34389.42 Oct. 25 12:18:59 120.7 274.3 13.2 196.3

34389.48 Oct. 25 12:23:47 90.9 254.9 13.2 196.2

34389.54 Oct. 25 12:28:35 51.8 374.6 13.2 196.1

34390.00 Oct. 25 12:33:23 95.8 303.4 13.2 196.1

34390.06 Oct. 25 12:38:11 122.6 292.9 13.2 196.0

34390.12 Oct. 25 12:42:59 132.1 371.7 13.2 196.0

34390.18 Oct. 25 12:47:47 • 190.5 261.5 13.2 195.9

34390.24 Oct. 25 12:52:35 • 220.8 240.6 13.2 195.8

34390.30 Oct. 25 12:57:23 221.7 316.5 13.2 195.8



Table 6.3




34390.36 Oct. 25 13:02:11 215.5 407.3 13.2 195.7

34390.42 Oct. 25 13:06:59 197.7 318.4 13.2 195.6

34390.54 Oct. 25 13:16:35 83.2 375.0 13.2 195.5

34391.00 Oct. 25 13:21:23 150.6 282.5 13.2 195.4

34391.06 Oct. 25 13:26:11 • 178.6 325.7 13.2 195.4

34391.12 Oct. 25 13:30:59 • 201.7 370.8 13.3 195.3

34391.18 Oct. 25 13:35:47 • 256.5 289.6 13.3 195.2

34391.24 Oct. 25 13:40:35 • 257.5 392.7 13.3 195.2

34391.30 Oct. 25 13:45:23 • 313.6 280.8 13.3 195.1

34391.36 Oct. 25 13:50:11 287.6 454.7 13.3 195.0

34400.10 Oct. 25 20:41:24 762.2 752.5 12.7 191.3

34444.00 Oct. 27 07:45:22 • 700.7 392.5 13.1 177.6

34637.32 Nov. 02 18:34:59 • 552.6 599.5 12.8 108.7

34637.36 Nov. 02 18:38:11 571.7 530.3 12.7 108.6

34637.40 Nov. 02 18:41:23 576.7 488.5 12.7 108.6

34667.16 Nov. 03 18:22:10 • 224.7 260.4 13.6 99.6

34667.20 Nov. 03 18:25:23 • 221.6 350.2 13.6 99.6

34667.24 Nov. 03 18:28:35 • 251.2 337.4 13.6 99.5

34701.59 Nov. 04 22:08:35 458.5 19.5 13.4 88.1

34702.07 Nov. 04 22:14:59 404.6 48.4 13.4 88.0

34702.11 Nov. 04 22:18:11 171.9 352.4 13.5 88.0

34702.15 Nov. 04 22:21:23 216.2 268.5 13.5 88.0

34702.19 Nov. 04 22:24:35 141.1 431.3 13.5 87.9

34727.25 Nov. 05 18:29:23 • 33.4 155.8 13.3 77.3

34727.35 Nov. 05 18:37:23 • 104.5 22.3 13.3 77.2

34757.08 Nov. 06 18:15:47 113.6 269.4 13.7 68.9



Table 6.3




34779.22 Nov. 07 12:02:59 98.2 76.6 14.4 59.9

34785.36 Nov. 07 17:02:13 • 203.5 621.6 12.7 57.3

34795.50 Nov. 08 01:13:25 685.7 478.4 14.5 54.8

34802.52 Nov. 08 06:50:59 • 758.6 139.8 13.1 51.0

34832.01 Nov. 09 06:10:13 681.9 376.9 15.1 42.3

34832.05 Nov. 09 06:13:25 83.6 712.9 15.2 42.2



Table 6.3


Table 6.2: Voyager 2 Images



42728.56 Jul. 14 14:03:11 • 665.5 720.5 6.9 319.4

42893.36 Jul. 20 01:47:11 • 45.5 38.4 6.8 278.9

43023.52 Jul. 24 09:59:59 • 50.5 113.7 6.7 247.7

43393.15 Aug. 05 17:30:29 149.8 489.6 6.6 157.7

43398.41 Aug. 05 21:51:16 • 704.9 567.4 7.2 156.0

43401.17 Aug. 05 23:56:01 • 316.6 102.6 7.2 154.4

43479.29 Aug. 08 14:29:40 290.3 748.7 6.7 134.8

43491.09 Aug. 08 23:49:38 • 778.3 720.8 7.3 133.1

43494.02 Aug. 09 02:08:01 • 444.2 184.5 7.3 131.4

43536.26 Aug. 10 12:03:13 • 442.6 344.5 6.6 121.2

43544.23 Aug. 10 18:24:49 • 485.8 638.4 7.3 120.6

43585.25 Aug. 12 03:14:25 356.7 267.7 7.6 109.0

43585.29 Aug. 12 03:17:37 403.6 213.3 7.5 109.0

43626.58 Aug. 13 12:28:48 • 628.5 193.3 6.9 98.4

43627.30 Aug. 13 12:54:24 • 538.3 349.5 6.8 98.4

43628.34 Aug. 13 13:45:36 • 548.7 576.5 6.7 98.5

43629.38 Aug. 13 14:36:48 • 500.0 595.5 6.6 98.6

43630.42 Aug. 13 15:28:00 • 316.6 593.5 6.6 98.7

43635.50 Aug. 13 19:34:24 • 271.2 687.6 7.5 98.0

43646.06 Aug. 14 03:47:12 • 637.6 501.7 6.8 93.9

43655.12 Aug. 14 11:04:00 • 184.8 573.2 7.7 93.0

43655.28 Aug. 14 11:16:48 • 286.7 430.5 7.7 92.9

43655.44 Aug. 14 11:29:36 • 209.5 303.6 7.8 92.7

43656.00 Aug. 14 11:42:24 • 253.6 416.6 7.8 92.6

43656.16 Aug. 14 11:55:12 • 324.6 212.3 7.8 92.4



Table 6.4




43657.20 Aug. 14 12:46:24 • 437.2 236.9 7.9 91.8

43657.52 Aug. 14 13:12:00 • 431.8 54.3 7.9 91.5

43663.24 Aug. 14 17:37:36 • 741.6 90.6 7.1 89.3

43663.56 Aug. 14 18:03:12 • 644.4 244.6 7.0 89.3

43684.29 Aug. 15 10:29:37 559.9 377.9 6.7 84.9

43684.35 Aug. 15 10:34:25 552.3 273.0 6.7 84.9

43684.41 Aug. 15 10:39:13 467.9 259.0 6.7 84.9

43686.55 Aug. 15 12:26:25 385.6 416.6 6.8 85.1

43686.59 Aug. 15 12:29:37 376.4 517.9 6.9 85.1

43695.16 Aug. 15 19:07:12 253.4 115.5 8.1 81.9

43695.20 Aug. 15 19:10:25 260.3 187.6 8.1 81.9

43695.24 Aug. 15 19:13:37 318.6 123.2 8.1 81.9

43703.18 Aug. 16 01:32:49 752.9 504.0 6.8 80.3

43703.22 Aug. 16 01:36:01 732.3 537.1 6.8 80.3

43710.08 Aug. 16 07:00:48 • 252.8 140.7 8.0 79.4

43723.20 Aug. 16 17:34:25 • 158.6 216.5 6.9 75.9

43732.18 Aug. 17 00:44:49 253.4 125.3 8.4 72.6

43732.22 Aug. 17 00:48:01 250.6 193.5 8.4 72.6

43732.26 Aug. 17 00:51:13 335.4 98.7 8.4 72.5

43737.47 Aug. 17 05:08:01 503.0 58.7 7.2 71.0

43737.51 Aug. 17 05:11:13 512.2 136.7 7.2 71.0

43737.55 Aug. 17 05:14:26 519.1 219.6 7.1 71.0

43774.21 Aug. 18 10:23:12 480.0 57.5 7.1 74.1

43785.17 Aug. 18 19:08:00 313.7 346.6 7.4 61.7

43785.21 Aug. 18 19:11:12 124.5 145.8 9.0 60.1

43796.05 Aug. 19 03:46:25 782.5 575.3 9.0 60.1



Table 6.4




43796.13 Aug. 19 03:52:48 366.5 311.3 7.2 57.4

43805.50 Aug. 19 11:34:24 502.4 84.4 7.2 57.4

43843.36 Aug. 20 17:47:12 749.2 170.5 9.3 54.2

43849.10 Aug. 20 22:14:24 700.7 241.9 9.9 44.3

43858.21 Aug. 21 05:35:12 237.3 604.6 7.9 43.2

43858.25 Aug. 21 05:38:24 68.9 411.1 10.3 41.8

43859.55 Aug. 21 06:50:24 631.1 98.8 10.4 41.7

43932.27 Aug. 23 16:52:00 110.5 754.6 10.6 40.8

43932.37 Aug. 23 17:00:00 180.1 558.8 14.7 22.4

43932.47 Aug. 23 17:08:00 80.8 544.5 14.8 22.2

43933.07 Aug. 23 17:24:07 86.7 110.4 15.0 22.0

43933.17 Aug. 23 17:32:07 133.6 35.2 15.0 21.9

43933.27 Aug. 23 17:40:07 100.6 38.6 15.0 21.8

44253.58 Sep. 03 10:04:48 114.5 326.5 92.9 67.8

44254.31 Sep. 03 10:31:12 8.6 624.5 92.7 67.9



Table 6.4


6.4 Results

The derived orbit fits for Prometheus, using Voyager 1 and Voyager 2 data sepa-

rately are shown in Tables 6.3 and 6.4 respectively. All errors for our fits are the

formal errors from the fitting process.

For comparison, the fits of Synnott et al. (1983) and Jacobson (private com-

munication) are also shown. No errors have been included for Jacobson’s elements

since none are given in the source material. Jacobson’s elements are quoted to the

same number of significant figures as our own. The orbits were derived using 56

Voyager 1 and 66 Voyager 2 images. Synnott et al. (1983) used 18 Voyager 1 and 27

Voyager 2 images. Jacobson’s (private communication) fits were performed using

the data of Synnott et al. (1983). The longitudes of Synnott et al. (1983) have been

transformed into the same reference frame, and where necessary the same epoch,

as used in both this work and that of Jacobson (private communication). Synnott

et al. (1983) quote rms residuals of better than 0.5 pixels while we achieved rms

residuals of 0.20 NAC pixels for the Voyager 1 and 0.22 NAC pixels for the Voyager

2 fits.

In order to directly compare the methods of image navigation and orbit determi-

nation used in this work with those of Synnott et al. (1983) and Jacobson (private)

Tables 6.5 and 6.6 show derived fits using only the 18 Voyager 1 and 27 Voyager

2 images used by Synnott et al. (1983). We achieved rms residuals of 0.24 NAC

pixels (Voyager 1) and 0.33 NAC pixels (Voyager 2) fits.

Since all fits are at the Voyager 2 epoch, Julian Ephemeris Date 2444839.6682,

the error associated with the Voyager 1 mean longitude at epoch is an order of

magnitude higher than the Voyager 2 mean longitude at epoch error. Fits were

also performed for the Voyager 1 data at the Voyager 1 epoch, Julian Ephemeris

Date 2444513.5. The results for these orbit fits are shown in Tables 6.7 and 6.8, for

our 57 images and the 18 used by Synnott et al. (1983) respectively.

The errors in the Voyager 1 mean longitudes at the Voyager 1 epoch are com-

parable to the Voyager 2 mean longitude at epoch errors at the the Voyager 2

epoch. Although the aim of this work was to derive separate orbital elements for

Prometheus at the Voyager 1 and Voyager 2 epochs, we have included fits to the


Table 6.3: Voyager 1 Results: epoch 2444839.6682 JED.

element This Work Synnott et al. Jacobson Units

a 139393 ± 15 139353a 139377.4 km

acalc 139375.5± 0.4 – – km

λ 192.0 ± 0.8 181b ± 7 186 deg

e (1.4 ± 0.2) × 10−3 (3.0 ± 0.3) × 10−3 2.6 × 10−3 –

n 587.3010± 0.0028 587.28c ± 0.2 587.2833 deg/day

195 ± 5 218d ± 20 218 deg

i 0.028 ± 0.006 0.0 ± 0.15c 0.0 deg

Ω 13 ± 22 – – deg

avalue quoted in Smith et al. (1981) for a as no Voyager 1 value given by Synnott et al. (1983).bprecessed and transformed into the reference frame and epoch of this work from Smith et

al. (1981) as no Voyager 1 value given in Synnott et al. (1983)cfrom Smith et al. (1981) as no Voyager 1 value given in Synnott et al. (1983).dprecessed and transformed into the reference frame and epoch of this work.



a 139287 ± 12 139353 139377.4 km

acalc 139377.5± 0.2 – km

λ 188.53 ± 0.01 – 188.54 deg

e (2.6 ± 0.1) × 10−3 (2.4 ± 0.2) × 10−3 2.3 × 10−3 –

n 587.2886± 0.0013 – 587.2896 deg/day

245 ± 1 213a ± 20 213 deg

i 0.025 ± 0.008 0.0 ± 0.1 0.0 deg

Ω 25 ± 13 – – deg

atransformed into the reference frame of this work.


Table 6.5: Voyager 1 Results using Synnott et al. ’s 18 images: epoch 2444839.6682JED


a 139498 ± 21 139353a 139377.4 km

acalc 139376.9± 0.7 – – km

λ 189 ± 1 181b ± 7 186 deg

e (1.1 ± 0.3) × 10−3 (3.0 ± 0.3) × 10−3 2.6 × 10−3 –

n 587.2922± 0.0044 587.28c ± 0.2 587.2833 deg/day

170 ± 12 218d ± 20 218 deg

i 0.016 ± 0.013 0.0c ± 0.15 0.0 deg

Ω 296 ± 65 – – deg

avalue quoted in Smith et al. (1981) for a as no Voyager 1 value given by Synnott et al. (1983).bprecessed and transformed into the reference frame and epoch of this work from Smith et

al. (1981) as no Voyager 1 value given in Synnott et al. (1983)cfrom Smith et al. (1981) as no Voyager 1 value given in Synnott et al. (1983).dprecessed and transformed into the reference frame and epoch of this work.

Table 6.6: Voyager 2 Results using Synnott et al.’s 27 images: epoch 2444839.6682JED


a 139362 ± 31 139353 139377.4 km

acalc 139378.7± 1.0 – km

λ 188.44 ± 0.06 – 188.54 deg

e (3.2 ± 0.3) × 10−3 (2.4 ± 0.2) × 10−3 2.3 × 10−3 –

n 587.2808± 0.0062 – 587.2896 deg/day

256 ± 2 213a ± 20 213 deg

i 0.033 ± 0.013 0.0 ± 0.1 0.0 deg

Ω 334 ± 34 – – deg

atransformed into the reference frame of this work.


Table 6.7: Voyager 1 Results: epoch 2444513.5 JED


a 139393 ± 15 139353a 139377.4 km

acalc 139375.5 ± 0.4 – – km

λ 153.07 ± 0.10 149.0b ± 0.5 153 deg

e (1.4 ± 0.2) × 10−3 (3.0 ± 0.3) × 10−3 2.58 × 10−3 –

n 587.3010 ± 0.0028 587.28c ± 0.2 587.2833 deg/day

16 ± 5 – 8d deg

i 0.028 ± 0.007 0.0c ± 0.15 0.0 deg

Ω 187 ± 21 – – deg

avalue quoted in Smith et al. (1981) for a as no Voyager 1 value given by Synnott et al. (1983).btransformed into the reference frame of this work from Smith et al. (1981) as no Voyager 1

value given in Synnott et al. (1983)cfrom Smith et al. (1981) as no Voyager 1 value given in Synnott et al. (1983).dtransformed into the reference frame of this work.

Table 6.8: Voyager 1 Results using Synnott et al.’s 18 images: epoch 2444513.5JED


a 139498 ± 21 139353a 139377.4 km

acalc 139376.9± 0.7 – – km

λ 153.4 ± 0.2 149.0b ± 0.5 153 deg

e (1.1 ± 0.3) × 10−3 (3.0 ± 0.3) × 10−3 2.6 × 10−3 –

n 587.2922± 0.0044 587.28c ± 0.2 587.2833 deg/day

352 ± 12 8d deg

i 0.016 ± 0.013 0.0c ± 0.15 0.0 deg

Ω 110 ± 60 – – deg

avalue quoted in Smith et al. (1981) for a as no Voyager 1 value given by Synnott et al. (1983).btransformed into the reference frame of this work from Smith et al. (1981) as no Voyager 1

value given in Synnott et al. (1983)cfrom Smith et al. (1981) as no Voyager 1 value given in Synnott et al. (1983).dtransformed into the reference frame of this work.


Table 6.9: Combined Voyager 1 & 2 Results: epoch 2444839.6682 JED


a 139337 ± 10 139353 139377.43 km

acalc 139377.33± 0.01 −− km

λ 188.526 ± 0.009 – 188.54 deg

e (1.92 ± 0.09) × 10−3 – 2.3 × 10−3 –

n 587.28942 ± 0.00007 587.2890 ± 0.0005 587.28917 deg/day

228 ± 2 – 213 deg

i 0.030 ± 0.005 – 0.0 deg

Ω 53 ± 7 – – deg

Table 6.10: Combined Synnott et al.’s (1983) Voyager 1 & 2 Results: epoch2444839.6682 JED


a 139394 ± 11 139353 139377.43 km

acalc 139377.31± 0.02 – km

λ 188.51 ± 0.02 – 188.54 deg

e (1.94 ± 0.23) × 10−3 – 2.3 × 10−3 –

n 587.28958 ± 0.00014 587.2890 ± 0.0005 587.28917 deg/day

237 ± 4 – 213 deg

i 0.016 ± 0.010 – 0.0 deg

Ω 337 ± 52 – – deg


combined Voyager 1 and Voyager 2 dataset for the sake of completeness. Table 6.9

has the fit for our 123 images while Table 6.10 has the fit just for the 45 images of

Synnott et al. (1983). The rms errors were 0.21 NAC pixels (our 123 images) and

0.31 NAC pixels (the 45 images of Synnott et al. (1983)).

6.5 The distribution of longitudes at observation

mid-times

Figs. 6.3 and 6.4 show the x-y coordinates of Prometheus during the Voyager 1

and Voyager 2 encounters, in the planetary reference frame used for Saturn at the





1

23

45

67

89

1011

1213

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

3536

37

383940

4142

4344

45

46474849

50

51

52

53

54

5556


Figure 6.3: The x-y plane coordinates of Prometheus at the observation mid-timescalculated from the Voyager 1 fit in Table 6.3


1

2

3

4

5

6

7

8

9

10

11

1213

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

3334

353637

3839

40

41

424344

454647

48

4950

51

5253

54

55

5657

58

59

60

61

62

63

64

65

66




Examination of Figs. 6.3 and 6.4 clearly shows that none of the images used has

Prometheus either transiting Saturn or in the shadow of the planet. This lack of

observations in these longitudes ranges is unsurprising, Prometheus being effectively

invisible in the visible light range in the shadow and swamped by light from the

planetary disc when transiting.





data does not sample the available longitude range completely. There is a clustering

of observations around the West ring ansa in both longitude and observation times.

This clustering could effect the accuracy of the orbital fit. In Chapter 7 the effect

of clustering of observations is investigated. It is concluded that it does not affect

the orbit fit to any great extent. The Voyager 2 observations (Fig. 7.2) sample the

longitude range very well. Although there is some clustering of observations, it is

not enough to cause concern over the accuracy of the orbital fit.

6.6 Discussion

The derived orbital elements from this Chapter show some deviation from those

of Jacobson (private communication) and Synnott et al. (1983). Most noticeably

our mean motions at both the Voyager 1 and Voyager 2 epochs are higher than

the previously derived values. Importantly our rms residuals are only half as large

as those of Synnott et al. (1983). Prometheus’ inclination is in the range 0.026 −0.028 deg with a mean of 0.027 ± 0.006 deg, the value of Synnott et al. (1983) is

0.0 ± 0.1.

The new derivations of Prometheus’ orbit using a larger dataset than that of

Synnott et al. (1983) show a difference in mean motions between the Voyager 1 and

Voyager 2 epochs of 3σ. It therefore appears that Prometheus’ orbit underwent a

change between the two Voyager encounters.

Any explanation for possible causes of such a change in the orbit, and for the


∼ 19 lag seen in 1995, must account for the observed mean longitudes in 1980, 1981

and 1994-2000. Table 6.11 shows the calculated mean longitudes in 1995 based on

the derived 1980 and 1981 mean longitudes and mean motions from our Voyager 1,

Voyager 2 and combined fits in Tables 6.3, 6.4 and 6.9. Also included is the mean

longitude of Prometheus in 1995 from French et al. (2000) and the observed mean

longitude during the August 1995 ring plane crossing (Nicholson et al. 1996).

Table 6.11: Mean longitudes for Prometheus from vari-ous authors. Epoch: 2449940.0 JED

Fit Mean longitudea

Nicholson et al. (1996) 339.23 ± 0.10

French et al. (2000) 339.12 ± 0.05

Voyager 1 fit 62 ± 14

Voyager 2 fit 16 ± 6

Combined fit 359.0 ± 0.4

Jacobson (private communication) 358.32 ± 2.56

ain degrees


Clearly none of the derived mean longitudes at epoch and corresponding mean

motions give the observed mean longitude during the August 1995 ring plane cross-

ing event. The ∼ 19 longitude lag observed in 1995 is still present with our derived

values for the combined Voyager mean motion. The lag in 1995 based on the indi-

vidual Voyager 1 or Voyager 2 is much larger.

6.7 The orbit of Prometheus in JPL Ephemerides

The program FitOrb2Ephm.f, (section 5.6), was used to fit a precessing ellipse to

the ephemerides for Prometheus in the SP-kernels vg2 sat.bsp and sat081.4.bsp. For

Prometheus, 160 position vectors at 3600.0 second intervals were obtained from the

relevant ephemeris. A precessing ellipse was fitted to the data with starting values

for the parameters of the ellipse being taking from the combined orbit fit from Table

6.9. A second fit was also performed with the apsidal and nodal precession rates,

β and γ set equal to 0.0/day. Position vectors were obtained from the SP-kernels

vg2 sat.bsp (the Voyager 2 ephemeris) and sat081.4.bsp (the current Cassini small

satellites ephemeris). The results obtained are presented in Tables 6.12 and 6.13. In

both cases the North Pole orientation of Saturn is taken from French et al. (1993).

As with the fits for Atlas, see Chapter 5, the fitting process was very robust.

For a wide range of starting values for the parameters of the ellipse convergence to

the exact values given in Tables 6.12 and 6.13 was achieved. There were no strong

correlations between any of the fitted parameters.

Examination of the data presented in Tables 6.12 and 6.13 shows that a simple

ellipse is a very good model for the ephemerides of Prometheus, at least over several

tens of days. For the more recent ephemeris, sat081.4.bsp, a precessing ellipse

is a better fit to the ephemerides than the non-precessing model. While for the

older ephemeris, vg2 sat.bsp, the non-precessing ellipse fits the data better then

the precessing model.

The non-precessing ellipse fit to vg2 sat.bsp in Table 6.12 has identical mean

motion, n and eccentricity, e to that of Synnott et al. (1983), which has n =

587.2890±0.0005/day and e = (2.4±0.6)×10−3, while the other orbital parameters


Table 6.12: Orbital elements for Prometheus from a fit to the SP kernel vg2 sat.bsp:Epoch 2444839.6682 JED. 160 positions at 3600.0 sec. intervals starting at2444836.6682 JED.



a 139352 ± 2 139353.0000± 0.0001

acalc 139377.47± 0.08 139377.399474± 0.000004

λ 188.5080 ± 0.0009 188.50835287± 0.00000005

n 587.2886 ± 0.0005 587.28900046± 0.00000003

e (2.39 ± 0.01) × 10−3 (2.3999977 ± 0.0000006) × 10−3

i (1.3 ± 0.1) × 10−2 (1.27906 ± 0.00008) × 10−3

216.7 ± 0.2 217.52230± 0.00001

Ω 357 ± 6 356.3949± 0.0003

rms 3.84 0.00022

aDistances are in km (rms is a distance), longitudes in degrees and rates in degrees/day. Errors

for elements from this work are the formal errors from the fitting process. All longitudes measured

from the ascending node of Saturn’s equator at epoch on the Earth mean equator at J2000, i

measured from Saturn’s equatorial plane at epoch.

are well within the error bars. The non-precessing ellipse is such a good fit that

it suggests that the ephemeris of Prometheus in vg2 sat.bsp may well have been

generated using a non-precessing elliptical model with the parameters of Synnott

et al. (1983).

The precessing ellipse fit to sat081.4.bsp in Table 6.13 has identical mean mo-

tion, n and eccentricity, e to the combined Voyager 1 and Voyager 2 fit of Jacobson

(private communication), see Table 6.9, while the other orbital parameters are well

within the error bars. The precessing ellipse is such a good fit that it suggests that

the ephemeris of Prometheus in sat081.4.bsp may well have been generated using

a precessing elliptical model with the parameters of Jacobson (private communica-

tion).


Table 6.13: Orbital elements for Prometheus from a fit to the SP kernel sat081.4.bsp:Epoch 2453371.00 JED. 160 positions at 3600.0 sec. intervals starting at 2453368.00JED.



a 139377.42± 0.09 139377 ± 2

acalc 139377.372± 0.003 139377.38 ± 0.08

λ 67.31334± 0.00004 67.3122 ± 0.0009

n 587.28917 ± 0.00002 587.2891± 0.0005

e (2.2890 ± 0.0004) × 10−3 (2.277 ± 0.010) × 10−3

i (1.140 ± 0.005) × 10−2 (1.1 ± 0.1) × 10−3

336.762 ± 0.009 337.6 ± 0.2

Ω 1.9 ± 0.3 0 ± 6

rms 0.15 3.72





6.8 Comparison of the derived orbits with expla-

nations for the origins of Prometheus’ lag

We will now examine the possible explanations for the observed lag of Prometheus,

presented in section 1.3, in light of the obtained results.

6.8.1 A co-orbital companion

If an orbital switch between Prometheus and a hypothetical co-orbital occurred

between the two Voyager encounters, Prometheus’ two distinct mean motions are

587.3010 ± 0.0028 deg /year and 587.2927 ± 0.0011 deg /year. Since both of these

are higher than the Synnott et al. (1983) value of 587.2890 ± 0.0005 deg /year any

combination of them would lead to a lag greater than ∼ 19 deg in August 1995.

A companion satellite being maintained in a horseshoe orbit with an orbital


switch occurring between the two Voyager encounters can explain the mean motion

difference between the Voyager 1 and Voyager 2 epochs. However, such a companion

cannot explain the observed lag in 1995.

Of course, a co-orbital satellite, with the orbital switch occurring at a time other

than coincident with the Voyager encounters, can still explain the observed lag in

1995. Nicholson et al. (1996) considered an object the size of the saturnian satellite

Atlas with a libration period of ∼ 30 years. As Nicholson et al. (1996) point out

such a large object should have been seen by the Voyager spacecraft. However,

the difference between the derived Voyager 1 and Voyager 2 mean motions is not

explained by this hypothetical co-orbital of Nicholson et al. (1996).

6.8.2 Periodic encounters with the F ring

The problem is to account for both the observed lag and the observed mean motion.

Orbital fits to HST observations from 1994-2000 give a mean motion of 587.287555±0.000048 deg /day (French et al. 2000) with the lag increasing by 0.57 deg /year.

Even if Prometheus underwent a collision, or collisions, that reduced its mean mo-

tion from 587.2890 deg /day (Synnott et al. 1983) to 587.287555 deg /day immedi-

ately after the Voyager 2 encounter this would still only account for about 10 deg of

the ∼ 19 deg accumulated lag observed in 1995. Furthermore, it cannot account for

the differences in the mean motions between the Voyager 1 and Voyager 2 epochs.

6.8.3 Cometary impact

The probability of such an impact with a 0.2 km diameter object has been estimated

as 107±1year−1 (Nicholson et al. 1996). The argument is the same as for a collision

with an F ring object, a collision reducing the mean motion to that of French

et al. (2000) can only account for a 10 deg lag in 1995 and cannot explain the

difference in Voyager 1 and Voyager 2 mean longitudes. To be consistent with all

data would require at least three impacts. One occurring between the Voyager

1 and Voyager 2 encounters which reduces the mean motion, another between the

Voyager encounters and the 1995 ring plane crossing events which reduces the mean

motion even further and a final one just before the 1995 ring plane crossings which


raises the mean motion to that of French et al. (2000). Such a high frequency of

impacts with large objects seems highly implausible.

6.8.4 Gravitational interaction with the F ring

This theory requires the mass contained within the F ring to be quite high, of the

order of the mass of Prometheus itself. Due to the random, chaotic, nature of

the gravitational interactions this theory can explain the Voyager 1 and Voyager 2

mean motion differences, the 1995 lag and the 1994-2000 HST mean motion and

lags. The main problem is explaining how the F ring can maintain its well defined

structure under the influence of these random gravitational forces.

6.8.5 Other mechanisms

Gross errors in the Voyager ephemeris, the predicted lag due to back-reaction from

density wave torques and lag resulting from secular effects are all considered by

Nicholson et al. (1996) and rejected as being too small.

6.8.6 Comments

If the mean motion of Prometheus has indeed decreased from its Voyager 1 value of

587.3010±0.0028 deg /day (this work) to 587.287555±0.000048 deg /day (French et

al. 2000) then there is a corresponding increase in semi-major axis from 139375.5 km

to 139377.6 km respectively, an increase of ∼ 120 metres a year. If this rate of expan-

sion continued unchecked the semi-major axis of Prometheus’ orbit would double in

∼ 106 years. This would imply an unfeasibly short lifetime for the Prometheus, F

ring and Pandora system. Clearly this rate of change in the mean motion cannot be

maintained. If it was a one off event, occurring sometime between 1981 and 1995,

the probability that it would occur just as it was able to be observed is infinitesi-

mally small. This implies that it was not an isolated occurrence but simply one in

a sequence of such events. Every one of these events leading to a decrease in mean

motion leads to the afore-mention short lifetime for the Prometheus, F ring and

Pandora system. It is therefore reasonable to assume that for each event increasing

the mean motion there is, on average, one that decreases it by a similar amount.


Of course over a long enough period of time a slow increase or decrease due to a

net imbalance in mean motion increasing/decreasing events is perfectly possible.

So this change in Prometheus’ mean motion is unlikely to be an isolated event. It

would be interesting to know if the other objects in the region, Atlas and Pandora,

exhibit similar changes in mean motion.

6.9 Conclusion

Not only did Prometheus’ orbit demonstrably change between the Voyager encoun-

ters and the ring plane crossing events in 1995, but change is also apparent between

the Voyager 1 and Voyager 2 encounters. There is currently no single theory that

satisfactorily accounts for the changes in Prometheus’ orbit yet also reproduces the

observed mean longitudes at various epochs. The orbit of Prometheus should con-

tinue to be monitored at every opportunity in order to detect further changes and

thus provide more information enabling the exact dynamical mechanism behind

them to be explained.

Chapter 7

Pandora

7.1 Introduction

Like Atlas and Prometheus, the saturnian satellite Pandora was discovered during

the Voyager 1 encounter with Saturn in 1980 (Smith et al. 1981). Pandora orbits

in the region just exterior to the narrow F ring. The orbit initially published for

Pandora was derived using 32 Voyager 1 images (Smith et al. 1981). The Voyager 2

encounter with Saturn in 1981 provided another opportunity for imaging Pandora

and Synnott et al. (1983) used 39 images to improve Pandora’s orbit. Fig 7.1 shows

a typical late encounter image of Pandora shuttered by the Voyager 2 narrow angle

camera. Pandora’s location just exterior to the F ring is clearly illustrated.

7.2 Search methodology and orbit determination

The search methodology, geometrical correction, image navigation and orbit de-

termination techniques used for Pandora were almost identical to those already

detailed for Atlas (sections 5.2, 5.3, 5.4 and 5.6 respectively). The only differences

being that Pandora, not Atlas, was being searched for and images where the res-

olution was worse then 320 km per pixel (not 150 km per pixel) were instantly

rejected. This figure was chosen because the worst resolution image used by Syn-

nott et al. (1983) for the determination of the orbit of Prometheus has a resolution

of 320 km per pixel (Chapter 6). Pandora is smaller, and fainter, than Prometheus

135

CHAPTER 7. PANDORA 136

Figure 7.1: Pandora in FDS43854.11. The Encke Gap and F ring are clearly visible,Pandora is circled.


so this figure for the minimum was deemed to be more than adequate.

Since these techniques have already been described in Chapter 5 we shall not

describe them again here.


The 38 Voyager 1 and 49 Voyager 2 navigable images of Pandora that we identified

are listed in Tables 7.1 and 7.2 respectively. Also indicated is whether a particular

image was used by Synnott et al. (1983) for their Pandora orbit determination. The

data of Synnott et al. (1983) was also used by Jacobson (private communication) for

an independent determination of Pandora’s orbital elements. Of the images we used,

28 of the Voyager 1 and 37 of the Voyager 2 were also used by Synnott et al. (1983).

The line and sample coordinates are given in continuous and not integer pixel

coordinates (see section 3.3). The resolutions and solar phase angle information

included for each image were calculated using the individually determined orbits

for Pandora at the Voyager 1 and Voyager 2 epochs (see Tables 7.3 and 7.4).


Table 7.1: Voyager 1 Images of Pandora

FDS Image Mid-time Used by Line Sample Phase Res.


34300.41 Oct. 22 13:06:10.5 573.5 418.2 13.2 225.4

34389.18 Oct. 25 11:59:47.0 432.6 510.5 13.1 196.5

34389.36 Oct. 25 12:14:11.0 446.6 480.3 13.1 196.4

34389.54 Oct. 25 12:28:35.0 208.7 558.4 13.2 196.2

34390.12 Oct. 25 12:42:59.0 246.4 532.5 13.2 196.0

34390.24 Oct. 25 12:52:35.0 • 304.6 385.6 13.2 195.9

34390.30 Oct. 25 12:57:23.0 • 289.6 453.5 13.2 195.9

34390.36 Oct. 25 13:02:11.0 • 267.7 535.4 13.2 195.8

34390.42 Oct. 25 13:06:59.0 • 234.6 437.4 13.2 195.7

34390.48 Oct. 25 13:11:47.0 126.6 321.4 13.2 195.7

34391.06 Oct. 25 13:26:11.0 • 151.8 408.6 13.2 195.5

34391.12 Oct. 25 13:30:59.0 • 159.5 443.0 13.2 195.4

34391.18 Oct. 25 13:35:47.0 • 197.6 352.5 13.2 195.3

34391.24 Oct. 25 13:40:35.0 • 182.5 446.2 13.3 195.3

34391.30 Oct. 25 13:45:23.0 • 223.4 323.5 13.3 195.2

34391.36 Oct. 25 13:50:11.0 • 181.6 487.6 13.3 195.1

34391.42 Oct. 25 13:54:59.0 • 214.2 369.5 13.3 195.1

34391.48 Oct. 25 13:59:47.0 • 202.6 396.4 13.3 195.0

34391.54 Oct. 25 14:04:35.0 • 163.5 375.8 13.3 194.9

34392.00 Oct. 25 14:09:23.0 97.5 339.3 13.3 194.9

34392.06 Oct. 25 14:14:11.0 • 17.0 363.2 13.3 194.8

34392.12 Oct. 25 14:18:59.0 • 79.4 259.4 13.3 194.7

34392.18 Oct. 25 14:23:47.0 • 81.1 400.3 13.3 194.6

34392.30 Oct. 25 14:33:23.0 • 177.7 306.4 13.3 194.5

34392.36 Oct. 25 14:38:11.0 212.0 293.9 13.3 194.4

34392.42 Oct. 25 14:42:59.0 • 131.6 426.7 13.3 194.4




34392.48 Oct. 25 14:47:47.0 • 65.5 272.6 13.3 194.3

34392.54 Oct. 25 14:52:35.0 • 9.8 426.6 13.3 194.2

34393.12 Oct. 25 15:06:59.0 • 57.5 332.8 13.3 194.0

34404.34 Oct. 26 00:12:34.5 • 593.5 658.3 12.8 191.1

34449.06 Oct. 27 11:50:10.5 • 707.5 338.3 13.3 174.9

34457.52 Oct. 27 18:50:58.8 • 556.6 531.7 12.6 171.6

34467.01 Oct. 28 02:10:10.5 • 700.6 411.2 13.3 169.1

34701.45 Nov. 4 21:57:22.5 • 523.4 25.6 12.6 86.6

34704.23 Nov. 5 00:03:46.5 781.4 170.6 12.4 86.6

34704.35 Nov. 5 00:13:22.5 503.5 629.0 12.4 86.6

34817.30 Nov. 8 18:33:22.5 • 632.0 277.5 12.5 47.0

34835.25 Nov. 9 08:53:22.5 • 174.4 510.7 12.5 40.3



Table .


Table 7.2: Voyager 2 Images of Pandora

FDS Image Mid-time Used by Line Sample Phase Res.


42420.55 Jul. 4 07:38:23.5 • 41.5 50.4 7.2 394.1

42514.35 Jul. 7 10:34:23.5 • 209.5 219.3 7.1 371.1

42686.10 Jul. 13 03:50:23.5 • 100.5 129.8 6.9 330.0

42741.34 Jul. 15 00:09:35.5 • 239.6 9.6 6.9 316.0

43023.52 Jul. 24 09:59:59.5 • 90.5 77.3 6.7 247.0

43439.54 Aug. 7 06:49:37.5 • 595.2 654.9 6.5 145.6

43507.17 Aug. 9 12:44:01.5 • 311.9 318.3 7.3 128.2

43536.26 Aug. 10 12:03:13.5 • 110.6 412.4 6.7 122.5

43544.23 Aug. 10 18:24:49.5 • 346.1 268.7 7.5 119.3

43596.54 Aug. 12 12:25:37.4 428.5 722.8 7.4 107.6

43625.54 Aug. 13 11:37:36.5 • 670.8 393.0 6.8 98.8

43626.26 Aug. 13 12:03:12.5 • 563.8 456.9 6.7 98.8

43626.58 Aug. 13 12:28:48.5 • 538.2 631.2 6.7 98.9

43627.30 Aug. 13 12:54:24.5 • 432.2 681.6 6.6 98.9

43628.34 Aug. 13 13:45:36.5 • 423.8 673.2 6.6 99.0

43629.38 Aug. 13 14:36:48.5 • 371.2 451.6 6.7 99.2

43630.42 Aug. 13 15:28:00.5 • 201.9 231.6 6.8 99.2

43631.10 Aug. 13 15:50:24.5 • 189.0 483.4 6.9 99.2

43634.46 Aug. 13 18:43:12.5 • 219.5 471.8 7.5 98.2

43635.50 Aug. 13 19:34:24.5 • 339.4 208.1 7.7 97.7

43646.06 Aug. 14 03:47:12.5 • 555.5 761.6 6.7 94.2

43649.50 Aug. 14 06:46:24.5 • 317.5 238.5 6.9 94.5

43653.54 Aug. 14 10:01:36.5 • 280.5 427.7 7.7 93.4

43655.12 Aug. 14 11:04:00.5 • 252.6 302.5 7.8 92.7

43655.28 Aug. 14 11:16:48.5 • 359.4 194.2 7.9 92.5

43655.44 Aug. 14 11:29:36.5 • 286.6 104.6 7.9 92.4




43656.00 Aug. 14 11:42:24.5 • 333.8 255.1 7.9 92.2

43656.16 Aug. 14 11:55:12.5 • 407.6 90.6 7.9 92.1

43657.20 Aug. 14 12:46:24.5 • 523.3 272.3 7.9 91.4

43657.52 Aug. 14 13:12:00.5 • 515.4 163.9 7.9 91.1

43662.52 Aug. 14 17:12:00.5 • 781.6 169.2 7.0 89.5

43663.24 Aug. 14 17:37:36.5 • 711.7 387.2 6.9 89.5

43663.56 Aug. 14 18:03:12.5 • 603.7 501.3 6.8 89.5

43684.35 Aug. 15 10:34:24.7 564.6 307.5 6.7 85.0

43686.55 Aug. 15 12:26:24.5 325.0 403.2 6.9 85.1

43695.24 Aug. 15 19:13:37.0 306.7 105.2 8.1 81.8

43703.18 Aug. 16 01:32:48.5 776.5 496.3 6.8 80.2

43710.08 Aug. 16 07:00:48.5 • 328.6 223.5 7.9 79.5

43715.52 Aug. 16 11:36:00.5 706.3 297.9 8.0 76.3

43723.20 Aug. 16 17:34:24.5 • 345.0 247.3 6.8 75.7

43724.41 Aug. 16 18:39:13.4 196.7 407.1 7.0 75.7

43732.22 Aug. 17 00:48:01.4 51.7 146.1 8.4 72.9

43752.26 Aug. 17 16:51:13.4 502.3 191.5 8.5 67.4

43780.11 Aug. 18 15:03:11.86 • 717.6 653.6 7.1 61.5

43796.09 Aug. 19 03:49:35.5 666.9 87.8 7.4 56.5

43805.50 Aug. 19 11:34:23.5 266.5 388.2 9.2 55.1

43818.49 Aug. 19 21:57:35.5 • 174.4 565.5 7.5 52.1

43854.11 Aug. 21 02:15:11.5 • 464.0 178.5 7.7 42.3

43854.36 Aug. 21 02:35:11.5 587.2 389.9 7.7 42.3



Table .


7.4 Results

The derived orbit fits for Pandora using Voyager 1 and Voyager 2 data separately

are shown in Table 7.3 and 7.4 respectively. All errors for our fits are the formal

errors from the fitting process.

For comparison, the fits of Synnott et al. (1983) and Jacobson (private com-

munication) are also shown. No errors have been included for Jacobson’s elements

since none are given in the source material. Jacobson’s elements are quoted to the

same number of significant figures as our own. The orbits were derived using 38

Voyager 1 and 49 Voyager 2 images. Synnott et al. (1983) used 32 Voyager 1 and

39 Voyager 2 images. Jacobson’s fits were performed using the observational data

of Synnott et al. (1983). The longitudes of Synnott et al. (1983) have been trans-

formed into the same reference frame, and where necessary the same epoch, as used

in both this work and by Jacobson (private communication). Synnott et al. (1983)

quote rms residuals of better than 0.5 pixels while we achieved rms residuals of 0.18

NAC pixels for the Voyager 1 and 0.24 NAC pixels for the Voyager 2 fits.

In order to directly compare the methods of image navigation and orbit determi-

nation used in this work with those of Synnott et al. (1983) and Jacobson (private

communication), Tables 7.5 and 7.6 show derived fits using the 28 of the 32 Voyager

1 and the 37 of the 39 Voyager 2 images used by Synnott et al. (1983) that we could

identify and navigate. We achieved rms residuals of 0.19 NAC pixels (Voyager 1)

and 0.29 NAC pixels (Voyager 2) fits.

A fit to the combined Voyager 1 and Voyager 2 dataset was also performed.

Table 7.7 has the fit for our 87 images while Table 7.8 has the fit just for the

65 images we could identify or navigate of Synnott et al.’s (1983) 71 images. We

achieved rms residuals of 0.21 NAC pixels and 0.24 NAC for our combined images

and the 65 images we used of the 71 of Synnott et al. (1983) respectively.




a 141869 ± 32 – 141712.61 km

acalc 141712.5 ± 0.8 141700a – km

λ 82 ± 1 71b ± 7 82 deg

e (5.7 ± 0.4) × 10−3 (4.4 ± 0.6) × 10−3 3.98 × 10−3

n 572.7895 ± 0.0046 572.77a ± 0.02 572.7877 deg/day

55 ± 3 70c ± 30 67 deg

i 0.09 ± 0.01 0.05a ± 0.15 0.0 deg

Ω 263 ± 1 – – deg

aThis is the value quoted in Smith et al. (1981) as there is no value in Synnott et al. (1983).bValue from Smith et al. (1981) precessed and transformed into the reference frame and epoch

used in this work.ctransformed into reference frame used in this work



a 141670 ± 13 – 141712.61 km

acalc 141712.2 ± 0.5 141700 – km

λ 82.13 ± 0.02 – 82.19 deg

e (3.8 ± 0.1) × 10−3 (4.2 ± 0.6) × 10−3 4.73 × 10−3

n 572.7911 ± 0.0028 – 572.7927 deg/day

53 ± 1 62a ± 30 70 deg

i 0.050 ± 0.007 0.0 ± 0.1 0.0 deg

Ω 222 ± 6 – – deg

atransformed into reference frame used in this work


Table 7.5: Voyager 1 Results using 28 of Synnott et al. ’s 32 images: epoch2444839.6682 JED.


a 141863 ± 32 – 141712.61 km

acalc 141712.0 ± 0.9 141700a – km

λ 83 ± 2 71b ± 7 82 deg

e (5.7 ± 0.4) × 10−3 (4.4 ± 0.6) × 10−3 3.98 × 10−3

n 572.7924 ± 0.0053 572.77a ± 0.02 572.7877 deg/day

57 ± 3 70c ± 30 67 deg

i 0.08 ± 0.01 0.05a ± 0.15 0.0 deg

Ω 251 ± 4 – – deg

aThis is the value quoted in Smith et al. (1981) as there is no value in Synnott et al. (1983).bValue from Smith et al. (1981) precessed and transformed into the reference frame and epoch

used in this work.ctransformed into reference frame used in this work

Table 7.6: Voyager 2 Results using 37 of Synnott et al. ’s 39 images: epoch2444839.6682 JED.


a 141713 ± 19 – 141712.61 km

acalc 141712.4 ± 0.6 141700 – km

λ 82.11 ± 0.03 – 82.19 deg

e (3.6 ± 0.2) × 10−3 (4.2 ± 0.6) × 10−3 4.73 × 10−3

n 572.7900 ± 0.0037 – 572.7927 deg/day

50 ± 2 62a ± 30 70 deg

i 0.06 ± 0.1 0.0 ± 0.1 0.0 deg

Ω 222 ± 8 – – deg

atransformed into reference frame used in this work


Table 7.7: Combined Voyager 1 & 2 Results: epoch 2444839.6682 JED.


a 141731 ± 13 – 141712.61 km

acalc 141712.63 ± 0.01 141700 km

λ 82.12 ± 0.01 – 82.15 deg

e (4.5 ± 0.1) × 10−3 – 4.37 × 10−3

n 572.78859± 0.00009 572.7891 ± 0.0005 572.78891 deg/day

57 ± 1 – 68 deg

i 0.053 ± 0.007 – 0.0 deg

Ω 245 ± 4 – – deg

Table 7.8: Combined 65 of Synnott et al. ’s(1983) Voyager 1 & 2 Results: epoch2444839.6682 JED.


a 141744 ± 18 – 141712.61 km

acalc 141712.62± 0.02 141700 – km

λ 82.11 ± 0.01 – 82.15 deg

e (4.4 ± 0.2) × 10−3 – 4.37 × 10−3

n 572.78869 ± 0.00011 572.7891 ± 0.005 572.78891 deg/day

56 ± 1 – 68 deg

i 0.06 ± 0.01 – 0.0 deg

Ω 241 ± 5 – – deg


7.5 The distribution of longitudes at observation

mid-times

Figs. 7.2 and 7.3 show the x-y coordinates of Pandora, during the Voyager 1 and

Voyager 2 encounters, in the planetary reference frame used for Saturn, at the




1

234

56

78

910

1112

1314

1516

1718

1920

2122

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38




1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49




Examination of Figs. 7.2 and 7.3 clearly shows that none of the images used

has Pandora either transiting Saturn or in the shadow of the planet. Position 18 in

Fig. 7.3 only appears to be in Saturn’s shadow because the mean region is shown

not the location of the shadow at the observation times. The lack of observations

in these longitudes ranges is unsurprising, Pandora being effectively invisible in the

visible light range in the shadow and swamped by light from the planetary disc

when transiting.





data does not sample the available longitude range completely. In addition 28 of the

38 Voyager 1 observations were made within 3 hours of each other, when Pandora

was in the region of the West ring ansa. This closeness of the vast majority of the

observations, along with the limited sampling of the longitude range leads to the

fit to the Voyager 1 data being of lower confidence than would otherwise be the

case. A fit to the Voyager 1 data was performed with observations 3-14 and 16-28

inclusive removed, leaving just the first, middle and last of the cluster of images.

The determined elements were consistent with those obtained using the full 38 ob-

servations. So the cluster of 28 images does not seem to effect the accuracy of the

fit in any meaningful way.

The Voyager 2 observations (Fig. 7.2) sample the longitude range very well.

Although there is some clustering of observations, it is not enough to cause concern

over the accuracy of the orbital fit.

7.6 Discussion of the orbit fits

The mean motions from the three fits are all within 1σ of each other. The mean

longitudes and longitudes of pericentre are comparable. There is no indication that

Pandora’s orbit changed between the Voyager 1 and Voyager 2 encounters. The

number of images used (87) is not significantly greater than the 71 of Synnott et


al. (1983). Our combined orbit (Table 7.7) is consistent with the combined orbit

of Synnott et al. (1983) and Jacobson (private communication), the mean motions

differing by only ∼ 1σ.

7.7 The orbit of Pandora in JPL Ephemerides

The program FitOrb2Ephm.f, described in Chapter 5, was used to fit a pre-

cessing ellipse to the ephemerides for Pandora in the SP-kernels vg2 sat.bsp and

sat081.4.bsp. For Pandora, 160 position vectors at 3600.0 second intervals were ob-

tained from the relevant ephemeris. A precessing ellipse was fitted to the data with

starting values for the parameters of the ellipse being taking from the combined or-

bit fit from Table 7.7. A second fit was also performed with the apsidal and nodal

precession rates, β and γ set equal to 0.0/day. Position vectors were obtained

from the SP-kernels vg2 sat.bsp (the Voyager 2 ephemeris) and sat081.4.bsp (the

current Cassini small satellites ephemeris). The results obtained are presented in

Tables 7.9 and 7.10. In both cases the North Pole orientation of Saturn is taken

from French et al. (1993).

As with the fits for Atlas, see Chapter 5, and Prometheus, Chapter 6, the fitting

process was very robust. For a wide range of starting values, the parameters of the

ellipse convergence to the exact values given in Tables 7.9 and 7.10 was achieved.

There were no strong correlations between any of the fitted parameters.

Examination of the data presented in Tables 7.9 and 7.10 shows that a simple

ellipse is a very good model for the ephemerides of Pandora, at least over several

tens of days. For the more recent ephemeris, sat081.4.bsp, a precessing ellipse

is a better fit to the ephemerides than the non-precessing model. While for the

older ephemeris, vg2 sat.bsp, the non-precessing ellipse fits the data better than

the precessing model.

The non-precessing ellipse fit to vg2 sat.bsp in Table 7.9 has identical mean

motion, n, and eccentricity, e, to that of Synnott et al. (1983), see Tables 7.7 and

7.4, while the other orbital parameters are well within the error bars. The non-

precessing ellipse is such a good fit that it suggests that the ephemeris of Pandora


Table 7.9: Orbital elements for Pandora from a fit to the SP kernel vg2 sat.bsp:Epoch 2444839.6682 JED. 160 positions at 3600.0 sec. intervals starting at2444836.6682 JED.



a 141699 ± 4 141700.0001± 0.0003

acalc 141712.8 ± 0.1 141712.547715± 0.000009

λ 82.203 ± 0.002 82.2023529± 0.0000001

n 572.7873 ± 0.0008 572.78910045± 0.00000006

e (4.19 ± 0.02) × 10−3 (4.199996 ± 0.000001) × 10−3

i (1.3 ± 0.2) × 10−2 (1.27906 ± 0.00002) × 10−2

65.74 ± 0.09 66.522367± 0.000006

Ω 357 ± 9 356.3948± 0.0007

rms 6.43 0.00046





in vg2 sat.bsp may well have been generated using a non-precessing elliptical model

with the parameters of Synnott et al. (1983).

The precessing ellipse fit to sat081.4.bsp in Table 7.10 has identical mean motion,

n, eccentricity, e, and semi-major axis, a, to the combined Voyager 1 and Voyager

2 fit of Jacobson (private communication), see Table 7.7. The precessing ellipse is

such a good fit that it suggests that the ephemeris of Pandora in sat081.4.bsp may

well have been generated using a precessing elliptical model with the parameters of

Jacobson (private communication). Also worth noting in the precessing ellipse fit

to sat081.4.bsp is that the semi-major axis from the fitting process, a, and the semi-

major axis derived from the mean motion, acalc, are within ∼ 200 metres of each

other. This close agreement between a and acalc is also apparent in the precessing

ellipse fit for Pandora from sat081.4.bsp (see Table 6.13) where it is ∼ 500 metres.

In all other cases of fitting ellipses, both to real observations and to ephemerides,


Table 7.10: Orbital elements for Pandora from a fit to the SP kernel sat081.4.bsp:Epoch 2453371.00 JED. 160 positions at 3600.0 sec. intervals starting at 2453368.00JED.



a 141712.60± 0.09 141712 ± 4

acalc 141712.579± 0.003 141712.9 ± 0.1

λ 94.39098± 0.00004 94.391 ± 0.002

n 572.78891 ± 0.00002 572.7870± 0.0008

e (4.3710 ± 0.0004) × 10−3 (4.35 ± 0.02) × 10−3

i (1.144 ± 0.005) × 10−2 (1.1 ± 0.2) × 10−2

284.381 ± 0.001 285.22 ± 0.06

Ω 1.8 ± 0.2 1 ± 10

rms 0.15 6.68





the difference in on the order of 10’s of kilometres.

7.8 The 3:2 near-resonance with Mimas

Up to this point no consideration had been given to perturbations on the orbit of

Pandora due to other satellites. The precessing ellipse model of Taylor (1998) used

to model the orbits of Atlas, Prometheus and Pandora is only valid if there are no

significant perturbations due to other satellites. Synnott et al. ( 1983) and Jacobson

(private communication) made no allowance for the proximity of Pandora to the

3:2 co-rotation eccentricity resonance (CER) with Mimas. Pandora’s semi-major

axis is ∼ 60 km from the location of the exact resonance. For details of resonances

in general and the Pandora-Mimas 3:2 CER in particular see Murray and Dermott

(1999).


Allowance can be made for this resonance by modifying the mean longitude

term, Eqn. 2.59 to

λ = λ0 + nt + A sin ((2π/T )t+ θ) (7.1)

where A sin ((2π/T )t+ θ) is a periodic term introducing the effect of the resonance

with A as the amplitude, T the period and θ the phase of the periodic effect of the

resonance.

French et al. (2000) fitted data from HST observations in the period 1994-2000

and found that Pandora was lagging behind its position expected from Synnott

et al. (1983) by 1.27/year. In addition there was a periodic term with amplitude

0.78 and period ∼ 600 days. French et al. (2000) stated that the period, amplitude

and phase of this periodic term were entirely consistent with the effects of the

Pandora-Mimas 3:2 CER when calculated from theory. The relevant theories are

the planetary disturbing function and Lagrange’s planetary equations.

7.8.1 The planetary disturbing function

While the general three (and indeed n > 2) body problem is analytically insoluble,

it is possible to express the equations of motion of planets that are mutually inter-

acting in term of their orbital elements, thus obtaining a set of variational equations.

This technique was developed independently by Laplace (1772) and Lagrange (1776)

and the variational equations developed are called Lagrange’s planetary equations.

Consider a mass m orbiting a central mass M in an elliptical orbit. If these were

the only two bodies in the system, and treating the masses as point-masses, the

analytical solution is an ellipse with constant orbital elements a, λ, n, e, , i and Ω

(section 2.2). Adding a third mass m′ to the system, orbiting M in an elliptical orbit

exterior to that of m, introduces additional gravitational accelerations in addition

to the two-body accelerations due to M (see Fig 7.4).

This is the three-body problem and is not analytically soluble (section 2.2).

Now if both M ≫ m and M ≫ m′ then these additional accelerations due to the

introduction of m′ can be treated as perturbations of the two-body orbit. The ad-

ditional accelerations of m and m′ relative to M are derived from the gradient of

the perturbing potential, usually called the disturbing function, R. The disturb-


r

r

r − rM

m

m/

/

/

Figure 7.4: The position vectors r and r′ of masses m and m′ with respect to thecentral mass M .

ing function can be derived from Universal Gravitation and Newton’s three laws

of motion, although we shall only give a brief summary complete derivations are

available in Murray and Dermott (1999) and Ellis and Murray (2000) for example.

The disturbing function for the inner mass can be written as

R =µ′

|r′ − r| − µ′ r · r′r′3

(7.2)

where µ′ = Gm′. Note that primed (′) quantities always denote the outer mass and

r′ > r at all times. The disturbing function for the inner mass is

R′ =µ

|r− r′| − µr · r′r3

(7.3)

where µ = Gm. While Eqns. 7.2 and 7.3 are for two orbiting bodies a similar anal-

ysis can be performed for any number of bodies. The accelerations associated with

the disturbing function can arise from sources other the point-mass gravitational

forces e.g. effects due to the oblateness of M .

The disturbing function can be analysed in several ways, but we shall follow

Murray and Dermott (1999) and consider the series expansion of the disturbing

function in Legendre polynomials in terms of the orbital elements. Murray and

Harper (1993) give the complete expansion of the disturbing function to eighth

order in e and i.

The orbital elements used are a, λ, e, , i and Ω for the inner body, mass m,

and similar primed (′) quantities for the outer body, mass m′. R is the disturbing


function for perturbations on m due to m′ and R′ is the disturbing function for

perturbations onm′ due to m. Murray and Dermott (1999) show that the expansion

of R has the form

R = µ′∑

S (a, a′, e, e′, i, i′) cosφ (7.4)

where φ is a linear combination with the general form

φ = j1λ′ + j2λ+ j3

′ + j4 + j5Ω′ + j6Ω (7.5)

where the ji’s are integers and6∑

i=1

ji = 0 (7.6)

(Eqn. 7.6 only applies if the angles are referred to a fixed reference direction). The

S term in Eqn. 7.4 has the form

S ≈ f (α)

a′e|j4|e′|j3|i|j6|i′|j5| (7.7)

where α = a/a′ and so α < 1.0. Eqn. 7.7 is the D’Alembert relation. The function

f(α) can be expressed as a function of Laplace coefficients and their derivatives.

The Laplace coefficient b(j)s (α) is defined by

1

2b(j)s (α) =

1

2π

∫ 2π

0

cos jψdψ

(1 − 2α cosψ + α2)2 (7.8)

The terms in Eqn. 7.4 for which j1 and j2 are non-zero, i.e. those which involve

the mean longitudes, produce periodic perturbations while those where j1 = j2 = 0

are called ‘secular’ perturbations. Over timescales of 100-1000s of years these secular

perturbations can appear to be monotonically rising or falling. The expansions of

Eqn. 7.4 and also expansions for R′ can be found in Murray and Harper (1993) to

eighth order and Murray and Dermott (1999) to fourth order in the appropriate

orbital elements.

7.8.2 Lagrange’s planetary equations

Expanding the disturbing function provides the disturbing potential given the or-

bital elements. The orbital variations of the perturbed body can be analysed by

using the Lagrange planetary equations. These equations require an additional an-

gle, ǫ, where

λ = nt + ǫ (7.9)


Lagrange’s equations for the variations of the orbital elements are (Murray and

Dermott 1999) to lowest order in e and i

dn

dt= − 3

a2

∂R∂λ

(7.10)

de

dt= − 1

na2e

∂R∂

(7.11)

di

dt= − 1

na2 sin i

∂R∂Ω

(7.12)

d

dt=

1

na2e

∂R∂e

+sin 1

2i

na2

∂R∂i

(7.13)

dΩ

dt=

1

na2 sin i

∂R∂i

(7.14)

dǫ

dt=

e

2na2e

∂R∂e

(7.15)

For a full derivation of Lagrange’s planetary equations see Brouwer and Clemence

(1961) and Roy (1988). In Eqn. 7.10, a has been substituted for by n = −32a/a

(Kepler’s third law) and partial derivatives involving ǫ by those involving λ (Murray

and Dermott 1999).

7.8.3 The effects of the Pandora-Mimas 3:2 CER from theory

If the resonance is sufficiently ‘shallow’ the simplified approximation developed later

in Eqn. 7.20 may be used. The resonance is shallow if the ratio

3nMimas − 2nPandora

nMimas

is much larger than MassMimas/MassSaturn. From Harper and Taylor (1993) the

mean motion ratio is ∼ 3 × 10−4 and the mass ratio ∼ 1 × 10−7. This three orders

of magnitude difference indicates that the resonance is shallow and the derived

Eqn. 7.20 is valid.

Using Eqn. 7.10, the time variability of the mean motion, n is given by

n = − 3

a2

∂R∂λ

(7.16)

If a term of the averaged expansion of the disturbing function is of the form

〈R〉 = µ′S cosφ (7.17)

then∂〈R〉∂λ

= −j2µ′S sinφ, (7.18)


using the definition of φ from Eqn. 7.5. Substituting Eqn. 7.18 into Eqn. 7.10 gives

n =3j2a2µ′S sinφ (7.19)

We now assume, that at least to lowest order, the orbital elements on the right-

hand side of Eqn. 7.19, with the exception of λ and λ′ remain constant (i.e. are time

independent). We further assume that Eqn. 2.12 is valid i.e. λ ≈ n∆t + constant1

and λ′ ≈ n′∆t+ constant2. The solution for n from Eqn. 7.19 then becomes

n = n0 −1

j1n′ + j2n

3j2a2µ′S cosφ (7.20)

Both de/dt and di/dt will have similar solutions since they also involve ∂〈R〉/∂λ.

Taking starting values for the orbital elements at epoch of Pandora and Mimas

we numerically integrated Eqn. 7.20 using a fourth order Runge-Kutta technique

to determined the value of φ, λ and n for Pandora at any time. We used a stepsize

of 0.002 days ∼ 0.01 of Pandora’s orbital period. The resonant argument for the

Pandora-Mimas 3:2 CER is

φ = 2λ′ − 3λ−′ (7.21)

The value of φ was calculated at each iteration. The elements for Mimas were

calculated using the theory of Mimas of Harper and Taylor (1993), which takes into

account the effects of the 4:2 Mimas-Tethys resonance. The elements at epoch for

Pandora were taken from the combined Voyager 1 and Voyager 2 fit in Table 7.7.

The mean longitude for Pandora at each iteration was obtained using a modified

form of Eqn. 2.59

λi = λi−1 + ni−1∆t (7.22)

where i is the number of the iteration step and ∆t is the stepsize. The value of ∆λ

was also calculated at each step where

∆λi = λi − (λ0 + n0t) (7.23)

Therefore ∆λ is the difference between the longitude with the perturbation due to

the resonance included and the unperturbed mean longitude. The results obtained

for φ, n and ∆λ are shown in Figs. 7.5, 7.6 and 7.7 respectively


Figure 7.5: Time variation of the resonant argument of Pandora. Epoch:2444839.6682 JED

Figure 7.6: Time variation of the mean motion of Pandora. Epoch: 2444839.6682JED


Figure 7.7: Time variation of ∆Λ for Pandora. Epoch: 2444839.6682 JED

The amplitude, A, period, T and phase, θ, of ∆λ are

A = 0.85

T = 633 days

θ = 42.7

when Eqn. 7.1 is used with these values in fitting a precessing ellipse to the data

instead of Eqn. 2.59, we obtained the fit to the full set of Voyager 1 and Voyager 2

observations shown in Table 7.11.

When the values of A and T derived from theory was used but the phase, θ was

fitted as well the best fit orbit to the data is shown in Table 7.12 with the ‘best’ θ

being θ = 28.7.

7.8.4 Discussion

When the effects of the 3:2 CER with Mimas are included in the orbit fit the mean

motion drops from 572.78859 ± 0.00009/day (Table 7.7) to 572.78439 ± 0.00009

(Table 7.11) or 572.78549± 0.00009 (Table 7.12). There is a corresponding change

in the mean longitude at epoch from 82.12±0.01 to 114.63±0.01 or 114.82±0.01.

The effects of the 3:2 resonance clearly have a significant effect on the motion of


Table 7.11: The orbital elements of Pandora: epoch2444839.6682 JED, including the calculated effects ofthe 3:2 CER with Mimas.

parametersa Orbit Fit

a 141731 ± 13

acalc 141713.32± 0.01

λ 82.13 ± 0.01

n 572.78439 ± 0.00009

e (4.4 ± 0.1) × 10−3

i 0.054 ± 0.007

56 ± 1

Ω 243 ± 4

rms error 0.2976

aDistances are in km, longitudes in degrees, rates in de-

grees/day and the rms error in arcseconds. Errors for elements

from this work are the formal errors from the fitting process. All

longitudes measured from the ascending node of Saturn’s equator

at epoch on the Earth mean equator at J2000, i measured from


Pandora and must be included in any orbit fitting process. The fits for Pandora of

Synnott et al. (1983) and Jacobson (private communication), which do not include

the effects of this resonance, are unsatisfactory.

The derived orbit of Pandora of French et al. (2000), which does include the

effects of the resonance is shown in Table 7.13.

The mean motion and eccentricity of the fit using the best fit phase (Table 7.12)

are consistent (within ∼ 1σ) with those of French et al. (2000) (Table 7.13).


Table 7.12: The orbital elements of Pandora: epoch2444839.6682 JED, including the calculated effects ofthe 3:2 CER with Mimas. The best fit value of thephase is used.


a 141731 ± 13

acalc 141713.14± 0.01

λ 82.15 ± 0.01

n 572.78549 ± 0.00009

e (4.4 ± 0.1) × 10−3

i 0.053 ± 0.007

56 ± 1

Ω 243 ± 4

rms error 0.2971


grees/day and the rms error in arcseconds. Errors for elements

from this work are the formal errors from the fitting process. All

longitudes measured from the ascending node of Saturn’s equator

at epoch on the Earth mean equator at J2000, i measured from



Table 7.13: The orbital elements of Pandora of Frenchet al. (2000): epoch 2449940.0 JED, including the cal-culated effects of the 3:2 CER with Mimas.


acalc 141713.1119± 0.0084

λ 96.03 ± 0.05

n 572.785574± 0.000051

e (4.53 ± 0.36) × 10−3

395.8 ± 6.3

rms error ∼ 0.5


grees/day and the rms error in pixels. The quoted errors are three

times the formal errors from the fitting process (i.e. 3σ). All lon-

gitudes measured from the ascending node of Saturn’s equator at

epoch on the Earth mean equator at J2000.

Chapter 8

Planning Future Observations

8.1 Introduction

A great deal of time and effort goes into the planning of the observations to be

made by a spacecraft on a mission to a Solar System object. A wide variety of

instrument packages are usually carried. The Cassini spacecraft has the following

instruments

• Composite Infrared spectrometer (CIRS).

• Imaging Science Subsystem (ISS).

• Ultraviolet Imaging Spectrograph (UVIS).

• Visual and Infrared Mapping Spectrometer (VIMS).

• Cassini Radar (RADAR).

• Radio Science Subsystem (RSS).

• Cassini Plasma Spectrometer (CAPS).

• Cosmic Dust Analyser (CDA).

• Ion and Neutral Mass Spectrometer (INMS).

• Dual Technique Magnetometer (MAG).

• Magnetospheric Imaging Instrument (MIMI).

162

CHAPTER 8. PLANNING FUTURE OBSERVATIONS 163

• Radio and Plasma Wave Science (RPWS).

Each of these science packages has its own team of investigators and instrumen-

tation on the spacecraft (RADAR and RSS use some elements of the spacecraft’s

communications links as well).

Data gathered by the science packages is stored on the spacecraft Solid State

Recorders (SSRs) before being regularly down-linked to one of the receivers of

the Deep Space Network (DSN). The instruments are capable of generating vast

amounts of data which could very quickly exceed the memory capacity of the SSRs,

this places constraints on the amount of data that can be gathered. Hence all the

instruments cannot be gathering data all of the time. Another restriction is the

orientation of the spacecraft. All the instruments are fixed to the spacecraft bus

and are pointed at the desired target by actually changing the spacecraft orienta-

tion. Some instruments are mounted orthogonally to others, e.g. ISS and RADAR,

so it is impossible to take simultaneous ISS images and RADAR observations of

the same target. So it is possible that different teams could want the spacecraft

pointing in entirely different directions at the same time.

Space on the SSRs between successive downlinks is allocated to individual in-

struments. Even if no other team wants to take data simultaneously and the space-

craft can be pointed in the direction a particular team wants the space restrictions

on the SSRs may mean that data cannot be collected. Also, down-linking data

costs money, for use of the DSN etc., so financial constraints require the number

of downlinks to be minimised. The amount of data collected should be kept to the

minimum needed to achieve the required scientific goals.

Each team puts in bids, stating what observations they want made, when they

want them, how much SSR space is required and how critical the observations

are. Each bid has to be fully supported by a scientific argument. The bids of all

the teams are compared and time and SSR space allocated, bearing in mind the

aforementioned financial constraints on the number of downlinks. Time and SSR

space is allocated years before the actual time of the observations. At the time of

writing (Oct. 2000) this process for Saturn for Cassini is nearing completion even

though Saturn Orbit Insertion (SOI) is not until July 1st 2004.


To ensure that a bid stands the best chance of being accepted it must require

the minimum number of observations to achieve the stated scientific goals. An

advantage is also to schedule observations when no other team is likely to want to

take data. If such scheduling is achieved the bid is then accepted or rejected solely

on the strength of the scientific case and the amount of data required.

In this chapter we investigate the minimum number of observations of a satellite

that are required to achieve a specific accuracy in the determination of its orbital

elements. Any derived relationships between number of observations and accuracy

in orbit determination can then be used to justify observations requested during

the bidding process.

8.2 The error in e from geometry

Consider an observer, O, making observations of a satellite, S, orbiting a planet,

P. The observer is a distance, d, from the planet while the satellite is a distance, r,

from the planet. The geometry of the situation is shown in Fig. 8.1.

O

P

S

d

r

θ

Figure 8.1: Geometry of observation 1


The angle at O between the lines OS and OP is θ. Now for θ ≤ 0.08 radians

we have

r ≃ dθ (8.1)

r and θ are then increased by small amounts, δr and δθ, as shown in Fig. 8.2

O

P

S

d

r

θ

δr

δθ

Figure 8.2: Geometry of observations 2

r + δr ≃ d(θ + δθ) (8.2)

Which with a little rearranging becomes

δr ≃ dθ + dδθ − r (8.3)

putting Eqn. 8.2 into Eqn. 8.3 gives

δr ≃ dθ + dδθ − dθ (8.4)

leading to

δr ≃ dδθ (8.5)

For an ellipse

rapocentre = a(1 + e) (8.6)


making a constant and differentiating Eqn. 8.6 gives

δrapocentre = aδe (8.7)

or

δe =δrapocentre

a(8.8)

for e≪ 1 we can say rapocentre ≃ r, hence

δrapocentre ≃ δr (8.9)

thus

δe =δr

a(8.10)

Putting Eqn. 8.2 into Eqn. 8.10 gives

δe ≃ dδθ

a(8.11)

Now what if we have N observations where δθ is the observational error. Further-

more, assume that δθ has a Gaussian distribution. Eqn. 8.11 becomes

δe ≃ dδθ

a√N

(8.12)

Eqn. 8.12 gives the relationship between the absolute error in e for the orbit de-

termination of a satellite using N images taken at a distance d from a planet with

the mean observational error being δθ. So from geometrical arguments we have an

expression for the absolute error that can be expected in the determination of the

eccentricity of a satellite.

∆e ≃ d∆θ

a√N

(8.13)

It is important to note that ∆e does not depend on e at all.

Due to the assumptions that have been made, Eqn. 8.12 is only valid if e ≪ 1

and θ ≤ 0.08 radians. Using this last restriction in Eqn. 8.2 means that d ≥ 12.5r.

So far no consideration has been given to angle between the observer-planet line

and the orbital plane of the satellite. For a satellite orbiting in, or near to, the

equator plane of the planet this is the sub-spacecraft latitude (SSL). When the SSL

is taken into account Eqn. 8.13 is expected to be of the form

∆e ≃ d∆θ

a√NF(SSL) (8.14)

where F(SSL) is a function of the sub-spacecraft latitude.


8.2.1 Determining F(SSL)

Using the precessing elliptical model of Taylor (1998), sets of observations of objects

orbiting Saturn were generated. It was assumed that the observations were evenly

spaced in time along one orbital period of the body. The observation point was

stationary with respect to the center of Saturn with

x = d cosSSl

y = 0

z = d sinSSL

The parameters used to generate the sets of observations were a, e, d, SSL, N and

∆θ. At all times it was ensured that d ≥ 12.5r. For the purpose of this analysis the

longitudes giving the position of the body in its orbit at the start of the observation

period were unimportant.

The sets of observations were then processed though the orbit determination

software outlined in Chapter 5 to derive a fitted orbit for the object corresponding

to each set of test observations. A record was made of the formal errors in e and i

in each case.

A least squares polynomial fit was performed on the data and this is shown in

Fig. 8.3 . The least squares fit to the data has the form

F(SSL) = 1.3481678−1.852335 |SSl|+1.098612 SSL2−0.1946722 |SSL|3 (8.15)

where SSL is measured in radians. Therefore Eqns. 8.14 and 8.15 together can be

used to estimate the number of images needed to achieved a required accuracy in

e.

8.2.2 Other orbital parameters

A similar analysis can be performed on the other orbital elements to determine the

relationship between number of images and accuracy achieved.


Figure 8.3: Relationship between F(SSL) and SSL

8.2.3 Limitations

The analysis was performed using the assumption that the input parameters d,∆θ, a

and SSL remained constant throughout. This of course is not the case when the

images are being taken from a moving spacecraft. During an observation run the

spacecraft-planet distance, d, can change quite dramatically as can the SSL. When

this is the case Eqn. 8.14 , with F(SSL) given by Eqn. 8.15, is invalid. For actual

observations from a spacecraft orbiting a planet a different approach must be taken.

8.3 Expected accuracy of realistic observations

It has been noted that when observations are made from a spacecraft orbiting

a planet, parameters effecting the accuracy of orbit determinations don’t remain

constant but can be subject to quite large variations. The expected accuracy of de-

termined orbital parameters cannot be obtained via simple analysis as in Eqn. 8.14.

In such more realistic cases a different approach was used.


The technique employed requires an accurate ephemeris for the observing space-

craft and approximate elements for the orbit of the target object. The number of

observations to be made and the time of each must be decided before any calcula-

tions can be made.

The position of the target object at any time is calculated using Eqns. 2.66, 2.67

and 2.68. The position of the spacecraft at the observation time is obtained from

the spacecraft ephemeris. The apparent position of the target object as seen from

the spacecraft is obtained from the positions of each object with allowances being

made for planetary aberration and light travel time, given by Eqn. 2.69.

In this way a set of theoretical observations is calculated. Random errors are

introduced into this observation set with a mean error of ∆θ. Observations are then

processed through the orbit determination process outlined in Chapter 5 . The orbit

determination includes formal errors for all the (assumed) orbital parameters of the

target object. The number of observations and the observation times are changed

until the required accuracy in the determination of a particular orbital parameter

is reached.

Using this method it is possible to plan a series of observations for a forthcoming

space mission, provided the accuracy requirements are known

8.4 Planning Cassini observations of the small

saturnian satellites.

A technique similar to that just outlined was employed to plan Cassini observa-

tions of the small saturnian satellites. Using an accurate ephemeris for the Cassini

spacecraft and a generated ephemeris for the satellite of interest, a series of syn-

thetic observations of the satellite were generated. The set of synthetic observations

also included synthetic ‘observational errors’ with mean value σ arcseconds. The

ephemeris for the satellite was generated using the precessing ellipse model of Taylor

(1998) (section 2.5.2).

Having an accurate ephemeris for Cassini and the elements of the satellite’s

orbit enabled the generation of a series of synthetic observations of the satellite,


P(ti), at times ti. The number of observations made, i, the times at which they

are made, t(i) and the mean observational error to be introduced, σ, are the only

input variables. The varying observational geometry and positions of Cassini and

the satellite depend on t(i).

8.4.1 Introducing synthetic ‘observational errors’

For inclusion of generated ‘observational errors’ the unit pointing vectors, P(ti)

were used instead of the pointing vectors P(ti). The unit pointing vectors were

converted into sets of Right Ascension and Declination coordinates, RA(i) and

DEC(i). From the mean error, σ, a set of random right ascension and declination

errors were calculated

δRA(i) = σ cos(xi) (8.16)

δDEC(i) = σ sin(xi) (8.17)

where xi is a random number with values between 0 and 2π. Since the mean error,

σ, is small we assumed that

RAobserved(i) = RA(i) + δRA(i) (8.18)

DECobserved(i) = DEC(i) + δDEC(i) (8.19)

and then converted the sets of RAobserved(i) and DECobserved(i) coordinates back

into unit pointing vectors, Pobserved(i) which now included synthetic observational

errors.

8.4.2 Fitting an orbit

The set of observations Pobserved(i), along with their associated observation times

t(i), were then processed through the orbit determination software as described

in section 5.6. The formal errors in each of the orbital elements as determined

by the fitting process were used as the accuracy of the orbit determination. For

observations of a particular satellite the dependency of the accuracy of the fit on i,

t(i) and σ could be determined.


8.4.3 Examples of calculated orbit determination accuracy for Cassini

The following examples are all sets of observations of small saturnian satellites

which were proposed to the Cassini ISS team by the author. The assumed and

determined orbital elements are fully listed.

Table 8.1: 90 observations of Prometheus starting at 2005 APR 29 10:00:00.0 JED,each observation at intervals of 12 mins. The epoch of the orbits is 2005 APR 2910:00:00.0 JED

parametera Assumed orbit Determined Orbit

a 139377.41 139376± 1

acalc 139377.37 139377.1± 0.4

λ 65.78 65.780 ± 0.001

n 587.28916 587.2906± 0.0022

e 0.00229 (2.267 ± 0.010) × 10−3

304.71 304.7 ± 0.2

i 0.011 0.0117 ± 0.0006

Ω 1.98 6 ± 3

σ 2.0




from Saturn’s equatorial plane at epoch. σ is in arcseconds.


Table 8.2: 90 observations of Pandora starting at 2005 APR 30 10:00:00.0 JED,each observation at intervals of 12 mins. The epoch of the orbits is 2005 APR 3010:00:00.0 JED


a 141712.578 141713± 1

acalc 141712.578 141713.1± 0.3

λ 102.9610 102.9630± 0.0008

n 572.78891 572.7858± 0.0016

e 0.004370 (4.368 ± 0.007) × 10−3

228.320 228.02 ± 0.06

i 0.01143 0.0124 ± 0.0005

Ω 17.4 19 ± 2

σ 2.0






Table 8.3: 90 observations of Epimetheus starting at 2005 MAY 17 15:00:00.0 JED,each observation at intervals of 12 mins. The epoch of the orbits is 2005 APR 1715:00:00.0 JED


a 151412.76 151412± 1

acalc 151412.76 151412.4± 0.7

λ 236.3553 236.355 ± 0.001

n 518.49171 518.493 ± 0.003

e 0.009868 (9.845 ± 0.008) × 10−3

344.024 344.10 ± 0.05

i 0.3349 0.3353 ± 0.0006

Ω 143.94 143.93 ± 0.08

σ 2.0






8.4.4 The effects of varying the number of observations and assumed observational

error

The following tables show the effects of changing the number of observations and

the assumed observational error on the accuracy of the orbit determination. The

assumed orbits are the same as in section 8.4.3. The orbital elements of both the

assumed orbit and the determined orbit are not listed in the tables. The formal

error from the fitting process for each of the determined elements is listed as a

fraction of that element e.g. ∆e = σe/edetermined

Table 8.4: Observations of Prometheus starting at 2005 APR 29 10:00:00.0 JED,ending at 2005 APR 30 04:00:00.0 JED. Constant number of images. The epoch ofthe orbits is 2005 APR 29 10:00:00.0 JED

parametera

N 160 160 160 160

σ 1 2 3 4

∆a 5.2 × 10−6 1.0 × 10−5 1.6 × 10−5 2.1 × 10−5

∆acalc 1.3 × 10−6 2.5 × 10−6 3.8 × 10−6 5.1 × 10−6

∆λ 7.9 × 10−6 1.6 × 10−5 2.4 × 10−5 3.2 × 10−5

∆n 1.9 × 10−6 3.8 × 10−6 5.7 × 10−6 7.7 × 10−6

∆e 2.2 × 10−3 4.3 × 10−3 6.5 × 10−3 8.7 × 10−3

∆ 2.5 × 10−4 4.9 × 10−4 7.3 × 10−4 9.9 × 10−4

∆i 3.0 × 10−2 5.7 × 10−2 8.9 × 10−2 0.11

∆Ω 0.5 0.8 1.5 × 10−2 0.65

rms 0.103 0.204 0.309 0.414

aDistances are in km, longitudes in degrees and rates in degrees/day. N is the number of

observations. All longitudes measured from the ascending node of Saturn’s equator at epoch on

the Earth mean equator at J2000, i measured from Saturn’s equatorial plane at epoch. σ and rms

are in arcseconds.

Table 8.4 clearly illustrates that the formal errors in a, acalc, λ, n, e, and i are

linearly dependent on the assumed mean error, σ, in the observations. This is in

agreement with Eqn. 8.14 for ∆e derived from elementary geometrical arguments

(where ∆θ = σ). It appears that the geometrical argument for the dependency on


Table 8.5: Observations of Prometheus starting at 2005 APR 29 10:00:00.0 JED,ending at 2005 APR 30 04:00:00.0 JED. Constant σ. The epoch of the orbits is2005 APR 29 10:00:00.0 JED

parametera

N 20 40 80 160

σ 2.5 2.5 2.5 2.5

∆a 2.4 × 10−5 1.8 × 10−5 1.3 × 10−5 8.8 × 10−6

∆acalc 4.1 × 10−6 2.9 × 10−6 2.0 × 10−6 1.4 × 10−6

∆λ 4.2 × 10−5 3.1 × 10−5 2.2 × 10−5 1.5 × 10−5

∆n 6.2 × 10−6 4.4 × 10−6 3.0 × 10−6 2.1 × 10−6

∆e 9.4 × 10−3 7.2 × 10−3 5.1 × 10−3 3.5 × 10−3

∆ 1.1 × 10−3 8.2 × 10−4 5.7 × 10−4 4.0 × 10−4

∆i 0.19 9.7 × 10−2 6.6 × 10−2 5.1 × 10−2

∆Ω 0.69 0.51 0.54 8.8 × 10−3

rms 0.492 0.400 0.272 0.191

aDistances are in km, longitudes in degrees and rates in degrees/day. N is the number of

observations. All longitudes measured from the ascending node of Saturn’s equator at epoch on

the Earth mean equator at J2000, i measured from Saturn’s equatorial plane at epoch. σ and rms

are in arcseconds.

∆θ holds for all the orbital elements. No conclusions can be reached regarding ∆Ω’s

dependency on σ since Ω ≈ 0 (Table 8.1). This makes accurate estimation of ∆Ω

problematic since a small change in Ω has a disproportionally large effect on ∆Ω.

If the error in the determination of an element is indeed proportional to 1/√

(N)

i.e.

∆ ∝ 1√N

(8.20)

then a graph of ∆ plotted against 1/√N should be a straight line. In Fig. 8.4

the values from Table 8.5 are plotted. Fig. 8.4 shows that for all the elements,

∆ ∝ 1/√N , is an accurate description. The value of i deviates slightly from this

relationship but it is accurate enough to make an estimate.

This indicates that Eqn. 8.14 is valid for predicting the rms errors in the deter-


Figure 8.4: Sets of plots of the ∆ values from Table 8.5 on the y-axis against thecorresponding 1/

√N values on the x-axis. In the semi-major axis plot the bottom

line is acalc.

mined values of all the orbital elements, not just e. Hence,

∆ ≃ d∆θ

a√NF(SSL) (8.21)

where ∆ is the error in any orbital element. Care should be taken since Eqn. 8.21

is probably only valid for fixed F(SSL). The value of the sub-spacecraft latitude

(SSl) remained fairly constant for the synthetic observations generated in Table

8.5.

Eqn. 8.21 can be used to support requests for specified numbers of images of

satellites. It indicates the requirements for achieving the accuracy desired in the

determination of an orbit.

Chapter 9

Summary and Discussion

9.1 Summary

This work was an attempt to derive orbits for the saturnian satellites Atlas, Prome-

theus and Pandora at the Voyager 1 and Voyager 2 epochs. The Voyager Saturn

data set was searched with the aim of identifying images of these satellites not used

by Smith et al. (1981) and Synnott et al. (1983). A number of explanations for the

∼ 19 mean longitude lag of Prometheus observed in 1995 were examined in light

of the the derived Voyager 1 and Voyager 2 orbits.

The orbits for Prometheus and Pandora derived by Synnott et al. (1983) and

Jacobson (private communication) were approximately the same as derived in this

work using the same image set. The Prometheus orbital elements from this work

were determined using three times as many images as those of Synnott et al. (1983)

and Jacobson (private communication). The rms error of the fit to the combined

Voyager data set was half that achieved by Synnott et al. (1983).

There was a difference in the determined mean motion of Prometheus at the two

Voyager epochs of ∼ 3σ, while the eccentricities differed by ∼ 6σ. We concluded

that not only did Prometheus’ orbit change between the 1980’s and 1995 but also

underwent a change between the two Voyager encounters. None of the derived

orbits were compatible with the observed mean longitude in 1995. There was no

theory that adequately explained the changes in Prometheus’ orbit and produced

the observed mean longitudes at various epochs.

177

CHAPTER 9. SUMMARY AND DISCUSSION 178

For Pandora the number of images used in this work was not significantly greater

than used by Synnott et al. (1983). There were no indications of any change in

Pandora’s orbit between the two Voyager encounters. When the effects of the 3:2

CER with Mimas were included the mean motion and mean longitude at epoch

for the derived orbits changed significantly. The simple precessing ellipse model

was not representative of the actual dynamics of Pandora’s orbit. An additional

periodic term had to be included in the calculation of the mean longitude.

For Atlas we used three times as many images as in the determination of Smith et

al. (1981). Our derived orbit differed significantly from that of Smith et al. (1981)

but was comparable with that in the current JPL ephemerides. There was no

indication that Atlas’ orbit changed between the two Voyager encounters.

The orbits of Pandora and Atlas did not show significant change between the

two Voyager epochs while that of Prometheus did. This indicated that the change

in Prometheus’ orbit was real and not a flaw in the whole image navigation/orbit

determination process.

9.2 Accuracy of determinations

The simulations showed that the formal error from the fitting process for each of

the orbital elements is modeled by the equation

∆ ≃ d∆θ

a√NF(SSL)

This can be used to support request for imaging sequences of satellites during the

Four year Cassini tour of the Saturn system.

9.3 Future work

Prometheus’ anomalous motion indicates that there is undetected mass in the F

ring region, either small satellites or within the F ring itself. In the course of deter-

mining the orbits of Atlas, Prometheus and Pandora in excess of 150 navigated and

geometrically corrected images of the F ring region were produced. These images

are ideal for use in a search for previously undetected small satellites. Additional

CHAPTER 9. SUMMARY AND DISCUSSION 179

images of the Pioneer Gap region could be easily located, geometrically corrected

and navigated.

Combining the precessing ellipse model of Taylor (1998) with numerical inte-

gration of the full equations of motion along the lines used by Harper (1987) would

prove useful. The trajectory of the satellite would be fit to the observations using

numerical integration of the equations of motion with all bodies in the system being

included. From the derived state at epoch the numerical integration can be used

to produce an ephemeris. The precessing ellipse model can then be fitted to the

ephemeris. All the subtle effects due to the perturbations of other objects would

be included in the ephemeris while the precessing ellipse model produces easily

understandable orbital elements.

Bibliography

Acton, C. H. 1990. The SPICE concept: An approach to providing geometric and

other ancillary information need for interpretation of data returned from space sci-

ence instruments. Internal memo, Jet Propulsion Laboratory, Cal Tech, Pasadena,

Ca.

Bosh, A. S., and A. S. Rivkin 1996. Observations of Saturn’s inner satellites during

the May 1995 ring–plane crossing. Science 272, 519–521.

Brouwer, D., and G. M. Clemence 1961. Methods of Celestial Mechanics. Academic

Press, Orlando.

Burns, J. A. 1986. Some background about satellites. In Satellites (J. A. Burns,

and M. S. Matthews, Eds.), pp. 1–38. Univ. of Arizona Press, Tucson.

Campbell, J. K., and J. D. Anderson 1989. Gravity field of the saturnian system

from Pioneer and Voyager tracking data. Astron. J. 97, 1485–1495.

Danby, J. M. A. 1992. Fundamentals of Celestial Mechanics, 2nd edition. Willmann-

Bell, Richmond.

Danielson, G. E., P. N. Kupferman, T. V. Johnson, and L. A. Soderblom 1981.

Radiometric performance of the Voyager cameras. J. Geophys. Res. 86, 8683–8689.

Davies, M. E., V. K. Abalakin, M. Bursa, J. H. Lieske, B. Morando, D. Morrison,

P. K. Seidelmann, A. T. Sinclair, B. Yallop, and Y. S. Tjuflin 1994. Report of the

IAU/COSPAR working group on cartographic coordinates and rotational elements

of the planets and satellites: 1994. Celest. Mech. and Dynam. Astron. 63, 127–148.

180

BIBLIOGRAPHY 181

Dermott, S. F., and C. D. Murray 1981a. The dynamics of tadpole and horseshoe

orbits. I. Theory, Icarus 48, 1–11.

Dermott, S. F., and C. D. Murray 1981b. The dynamics of tadpole and horseshoe

orbits. II. The coorbital satellites of Saturn, Icarus 48, 12–22.

Dormand, J. R., M. E. A. El-Mikkawy, and P. J. Prince, 1987a. Families of em-

bedded Runge-Kutta-Nystrom formulae. IMA J. Num. Analysis 7, 235–250.

Dormand, J. R., M. E. A. El-Mikkawy, and P. J. Prince 1987b. High-order embed-

ded Runge-Kutta-Nystrom formulae. IMA J. Num. Analysis 7, 423–430.

Ellis, K. M., and C. D. Murray 2000. The disturbing function in solar system

dynamics. Icarus 147, 129–144.

Escobal, P. R. 1965. Methods of Orbit Determination. John Wiley, New York.

French, R. G., and 17 colleagues 1993. Geometry of the Saturn system from the 3

July 1989 occultation of 28 Sgr and Voyager observations. Icarus 103, 163–214.

French, R. G., K. J. Hall, C. A. McGhee, P. D. Nicholson, J. Cuzzi, L. Dones, and

J. Lissauer 1998. The peregrinations of Prometheus. B. A. A. S. 30, 1141.

French, R. G., C. McGhee, L. Dones, and J. Lissauer 2000. The peripatetics of

Prometheus and Pandora. B. A. A. S. 32, 1090.

Gordon, M. K. 1994. A Search for Small Satellites and Coorbital Material in the

Saturnian System. Ph.D. Thesis, University of London.

Gordon, M. K., C. D. Murray, and K. Beurle 1996. Further evidence for the exis-

tence of additional small satellites of Saturn. Icarus 121, 114–125.

Graps, A. L., A. L. Lane, L. J. Horn, and K. E. Simmons 1984. Evidence for

material between Saturn’s A and F rings from the Voyager 2 photopolarimeter

experiment. Icarus 60, 409–415.

Greenberg, R. 1981. Apsidal precession of orbits about an oblate planet. Astron.

J. 86, 912–914.

BIBLIOGRAPHY 182

Hadjifotinou, K. G., and D. Harper 1995. Numerical integration of orbits of plan-

etary satellites. Astron. Astrophys. 303, 940–944.

Harper, D. 1987. The Dynamics of the Outer Satellites of Saturn. Ph.D. Thesis,

University of Liverpool.

Harper, D., and D. B. Taylor 1993. The orbits of the major satellites of Saturn. A

& A 268, 326–349.

Harper, D., and D. B. Taylor 1994. Analysis of ground–based observations of the

satellites of Saturn 1874–1988. A & A 284, 619–628.

Herget, P. 1948. Computation of preliminary orbits. Astron. J. 70, 1–3.

Hog, E., C. Fabricus, V. V. Makarov, S. E. Urban, T. E. Corbin, G. L. Wycoff, U.

Bastian, P. Schwekendiek, and A. Wicenec 2000a. The Tycho-2 catalogue of the

2.5 million brightest stars. A & A, submitted.

Hog, E., C. Fabricus, V. V. Makarov, U. Bastian, P. Schwekendiek, A. Wicenec,

S. E. Urban, T. E. Corbin, and G. L. Wycoff 2000b. Construction and verification

of the Tycho-2 catalogue . A & A, submitted.

Hohenkerk, C. Y., B. D. Yallop, C. A. Smith, and A. T. Sinclair 1992. Celestial

reference systems. In The Explanatory Supplement to the Astronomical Almanac

(P. K. Seidelmann, Ed. ). pp. 133–134. University Science Books, Mill Valley.

Jacobson, R. A. 1998a. The orbits of the inner uranian satellites from Hubble Space

Telescope and Voyager 2 observations. Astron. J. 115, 1195–1199.

Jacobson, R. A. 1998b. The orbit of Phoebe from Earthbased and Voyager obser-

vations. Astron. Astrophys. Suppl. Ser. 128, 7–17.

Jacobson, R. A., J. K. Campbell, A. H. Taylor, and S. P. Synnott 1992. The

masses of Uranus and its major satellites from Voyager tracking data and Earth-

based uranian satellite data. Astron. J. 103, 2068–2078.

BIBLIOGRAPHY 183

Jacobson, R. A., J. E. Reidel, and A. H. Taylor 1991. The orbits of Triton and

Nereid from spacecraft and Earthbased observations. astron. astrophys 247, 565–

575.

Lagrange, J. L. 1776. Sur l’alteration des moyens movements des planetes.

Mem. Acad. Sci. Berlin, 199, oeuvres completes 6, 255, Gauthier-Villars (1869),

Paris.

Laplace, P. S. 1772. Memoire sur les solutions particulieres des equations differen-

tielles et sur les inegalities seculaires des planetes. Mem. Acad. Royale des Sciences

de Paris, Oeuvres completes 9, 325, Gauthier-Villars (1895), Paris

Marsden, B. G. 1985. Initial orbit determination: the pragmatist’s point of view.

Astron. J. 90, 1541–1547.

Murray, C. D., and S. F. Dermott 1999. Solar System Dynamics. Cambridge

Univ. Press, Cambridge.

Murray, C. D., and S. M . Giuliatti-Winter 1996. Periodic collisions between the

moon Prometheus and Saturn’s F ring. Nature 380, 139–141.

Murray, C. D., and D. A. Harper 1993. Expansion of the planetary disturbing

function to eighth order in the individual orbital elements, QMW Math Notes

No. 15. Queen Mary and Westfield College, London.

Newton, I. 1687. Philosophiae Naturalis Principia Mathematica. Royal Society,

London.

Nicholson, P. D., M. L. Cooke, and E. Pelton 1990. An absolute radius scale for

Saturn’s rings. Astron. J. 100, 1339–1362

Nicholson, P. D., M. R. Showalter, L. Dones, R. G. French, S. M. Larson, J. J.

Lissauer, C. A. McGhee, P. Seitzer, B. Sicardy, and G. E. Danielson 1996. Ob-

servations of Saturn’s ring–plane crossings in August and November 1995. Science

272, 509–515.

Owen, W. M., R. M. Vaughn, and S. P. Synnott 1991. Orbits of the six new

satellites of Neptune. Astron. J. 101, 1511–1515.

BIBLIOGRAPHY 184

Peters, C. F. 1981. Numerical integration of the satellites of the outer planets.

Astron. Astrophys. 104, 37–41.

Poincare, H. 1892–1899. Les methods nouvelles de la Mechanique celeste. Gauthier-

Villars I-III,Paris.

Porco, C. C., G. E. Danielson, P. Goldreich, J. B. Holberg, and A. L. Lane 1984.

Saturn’s nonaxisymmetric ring edges at 1.95Rs and 2.27Rs. Icarus 60, 17–28.

Porco, C. C., and V. Haemmerle 2000. Resolution of Masursky pointing issues,

Cassini ISS internal report, CICLOPS, LPL, Univ. of Arizona.

Poulet, F., B. Sicardy, P. D. Nicholson, E. Karkoschka, and J. Caldwell 2000a.

Saturn’s ring-plane crossings of August and November 1995. A model for the new

F ring objects. Icarus 144, 135–148.

Roy, A. E. 1988. Orbital Motion, third edition. Adam Hilger, Bristol.

Showalter, M. R. 1991. Visual detection of S1981S13, Saturn’s eighteenth satellite,

and its role in the Encke Gap. Nature 351, 709–713.

Showalter, M. R., L. Dones, and J. J. Lissauer 1999. Revenge of the sheep: effects

of saturn’s F ring on the Orbit of Prometheus. B. A. A. S. 31, 1141.

Smith, B. A. 1977. The Voyager imaging experiment. Space Sci. Rev. 21, 103–127.

Smith, B. A., and 26 colleagues 1981. Encounter with Saturn - Voyager 1 imaging

science results. Science 212, 163–191.

Smith, B. A., and 28 colleagues 1982. A new look at the Saturn system: the Voyager

2 images . Science 215, 504–537.

Squires, G. L. 1976. Practical physics. McGraw-Hill, London.

Standish, E. M. 1990. The observational basiss for the JPL’s DE200, the planetary

ephemerides of the Astronomical Almanac. Astron. Astrophys. 233, 252-271.

Synnott, S. P., R. J. Terrile, R. A. Jacobson, and B. A. Smith 1983. Orbits of

Saturn’s F ring and its shepherding satellites. Icarus 53, 156–158.

BIBLIOGRAPHY 185

Taff, L. G. 1984. On initial orbit determination. astron. J. 89, 1426–1428.

Taylor, D. B. 1998. Determination of starting conditions for ephemerides of the

Uranian satellites I–V. Nautical Almanac Office Technical Note 72.

Thompson, R. P. 1990. The Rings of Saturn and Uranus. Ph.D. Thesis, University

of London.

VICAR Users Guide circa 1977. Jet Propulsion Laboratory, Cal. Tech., Pasadena.

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