Statistical Orbit determination I Fall 2012 Professor George H. Born

CCARColorado Center for

Astrodynamics Research

University of ColoradoBoulder 1

ASEN 5070Statistical Orbit determination I

Fall 2012

Professor George H. BornProfessor Jeffrey S. Parker

Lecture 2: Background




Monday is Labor Day!

D2L issue CAETE issue

Quiz - Now 24 hours per quiz (1pm – 1pm)

Homework 1

Office Hours

Announcements




Quiz Results




Had to shift Eduardo’s office hours because there was a distinct lack of awesomeness on Tuesdays.

Office Hours Changes




Astrodynamics background◦ Review orbital elements a bit more◦ Talk about perturbations in dynamical model

J2, Drag, SRP, etc.◦ Talk about partials

Coordinate Frames and Time Systems

Coding hints and tricks (mostly next Tuesday)◦ LaTex: intro◦ MATLAB: ways to speed up your code◦ Python: intro

Notes about laptops and phones, etc.

Today’s Lecture




Representing a satellite’s state

Cartesian Coordinates◦ x, y, z, vx, vy, vz in some coordinate frame

Spherical Elements◦ Lat, Lon, Alt, V, FPA, FPAz in some coordinate frame (or similar

set)

Keplerian Orbital Elements◦ a, e, i, Ω, ω, ν in some coordinate frame (or similar set)

When are each of these useful?

Astrodynamics Background




Keplerian Orbital Elements

Consider an ellipse

Periapse/perifocus/periapsis◦ Perigee, perihelion

r = radius rp = radius of periapse ra = radius of apoapse

a = semi-major axis e = eccentricity = (ra-rp)/(ra+rp) rp = a(1-e) ra = a(1+e)

ω = argument of periapse f/υ = true anomaly






Orientation of ellipse requires a reference frame◦ Typically “Earth Mean Equator and

Equinox of J2000.0” (EME2000, or just J2000).

◦ Or Earth Mean Ecliptic of J2000.

The obliquity of the Earth’s spin axis is the angle between the equatorial and ecliptic planes.◦ ~23.5 deg at present.






Orientation of ellipse requires a reference frame◦ Typically “Earth Mean Equator

and Equinox of J2000.0” (EME2000, or just J2000).

◦ Or Earth Mean Ecliptic of J2000.

i = inclination Ω = Right ascension of ascending

node (= longitude of ascending node in an inertial J2000 coordinate frame)





Shape and Size◦ a, e, rp, ra, Period

Orientation◦ i, Ω, ω

Position◦ f/υ, E, M, (t-tp)

Advantages◦ Visualization.◦ In a 2-body world, they don’t change

with time (except the position).

In the real world, do they change?





Shape and Size◦ a, e, rp, ra, Period

Orientation◦ i, Ω, ω

Position◦ f/υ, E, M, (t-tp)

Advantages◦ Visualization.◦ In a 2-body world, they don’t change

with time (except the position).

In the real world, do they change?


Earth’s orbit Moon’s orbit

Yes, but usually not much, and we can use perturbation theory to model the variations.




Coordinate Frames

Ascending node is the point where the Sun crosses the equator moving from the southern hemisphere to the northern hemisphere: vernal equinox (~ March 21)

The descending node is autumnal equinox (~Sept 21)




Coordinate Frames

Choose X-axis (coinciding with vernal equinox) as inertial direction; Z-axis coincident with Earth angular velocity vector (ωe), period of rotation = 86164 sec, “sidereal” period

GMST=αG = ωe(t – t0) + αG0




Coordinate Frames (XYZ) represents a nonrotating coordinate system with X

directed to the vernal equinox, and origin coinciding with Earth center (geometric center of the spherical Earth, or more precisely, the Earth center of mass)

In reality, the location of the equinoxes change with time (use the equinox of a particular date as reference, e.g., January 1, 2000, 12:00 or more specifically, mean equator and vernal equinox of J2000)

(xyz) is an Earth-fixed frame (ECF) and rotates with it, with x coincident with the intersection of the Greenwich meridian and the equator




Coordinate Frames

Define xyz reference frame (Earth centered, Earth fixed; ECEF or ECF), fixed in the solid (and rigid) Earth and rotates with it

Longitude λ measured from Greenwich Meridian

0≤ λ < 360° E; or measure λ East (+) or West (-)

Latitude (geocentric latitude) measured from equator (φ is North (+) or South (-))◦ At the poles, φ = + 90° N or

φ = -90° S




I’d like us to think about the effects of small variations in coordinates, and how these impact future states.

Effects of Small Variations

Initial State:(x0, y0, z0, vx0, vy0, vz0)

Final State:(xf, yf, zf, vxf, vyf, vzf)

Example: Propagating a state in the presence of NO forces




What happens if we perturb the value of x0?




Force model: 0

Initial State:(x0+Δx, y0, z0, vx0, vy0, vz0)




What happens if we perturb the value of x0?




Force model: 0

Initial State:(x0+Δx, y0, z0, vx0, vy0, vz0)

Final State:(xf+Δx, yf, zf, vxf, vyf, vzf)




What happens if we perturb the position?



Force model: 0

Initial State:(x0+Δx, y0+Δy, z0+Δz,

vx0, vy0, vz0)

Final State:(xf+Δx, yf+Δy, zf+Δz,

vxf, vyf, vzf)




What happens if we perturb the value of vx0?




Force model: 0

Initial State:(x0, y0, z0, vx0-Δvx, vy0, vz0)




What happens if we perturb the value of vx0?




Force model: 0

Final State:(xf+tΔvx, yf, zf, vxf+Δvx, vyf, vzf)

Initial State:(x0, y0, z0, vx0+Δvx, vy0, vz0)




What happens if we perturb the position and velocity?


Force model: 0




We could have arrived at this easily enough from the equations of motion.


Force model: 0




This becomes more challenging with nonlinear dynamics


Force model: two-body









The partial of one Cartesian parameter wrt the partial of another Cartesian parameter is ugly.






Final State:(xf, yf, zf, vxf, vyf, vzf)Force model: two-body




Represent variations as functions of Keplerian orbital elements


Final State:Force model: two-body

Then, what is ?













If

Then, what is ?







If

(Remember, af=a0)







If




The point is that we can relate small perturbations from one element to another easier using Keplerian orbital element than Cartesian.

Other brain teasers





How does Rf vary if V0 is increased?


Final State(Rf, Vf)


A: Increases

B: Decreases

C: Stays the same

D: Not enough information

Initial State(R0, V0)




How does Rf vary if V0 is increased?


Final State(Rf, Vf)


A: Increases

B: Decreases

C: Stays the same






How does Rf vary if R0 is increased?


Final State(Rf, Vf)


Hint:

A: Increases

B: Decreases

C: Stays the same






How does Rf vary if R0 is increased?


Final State(Rf, Vf)


Hint:

A: Increases

B: Decreases

C: Stays the same






Any Questions?

(quick break)

Intermission




The Not-So-Short Guide to LaTex

LaTex




Everyday Orbital Motion

Ballistic motion is orbital motion

Solid Earth prevents a body in ballistic motion from reaching perigee

A body dropped from rest, at the equator, is shown◦ Perigee: 11.2 km◦ Eccentricity: 0.9965




Perturbed Motion

The 2-body problem provides us with a foundation of orbital motion

In reality, other forces exist which arise from gravitational and nongravitational sources

In the general equation of satellite motion, f is the perturbing force (causes the actual motion to deviate from exact 2-body)




Perturbed Motion: Planetary Mass Distribution

Sphere of constant mass density is not an accurate representation for planets

Define gravitational potential, U, such that the gravitational force is




Gravitational Potential

The commonly used expression for the gravitational potential is given in terms of mass distribution coefficients Jn, Cnm, Snm

n is degree, m is order Coordinates of external

mass are given in spherical coordinates: r, geocentric latitude φ, longitude




Gravity Coefficients

The gravity coefficients (Jn, Cnm, Snm) are also known as Stokes Coefficients and Spherical Harmonic Coefficients

Jn:◦ Gravitational potential represented in zones of latitude;

referred to as zonal coefficients Cnm, Snm:

◦ If n=m, referred to as sectoral coefficients◦ If n≠m, referred to as tesseral coefficients



University of ColoradoBoulder

Earth J2 (Degree 2 Zonal Harmonic)

J2 represents a dominant characteristic of the shape of the planet◦ Positive J2: oblate

spheroid ◦ Negative J2: prolate

spheroid Scientific controversy

in 1735: was Earth oblate or prolate?

Oblate spheroid Prolate spheroid




Resolution of Controversy In 1735, one view of the shape of the Earth was based

on work of Newton, who had argued for the oblate shape (centrifugal forces)

Another view was based on measurements of the length of 1° of latitude in France, supported a prolate spheroid

French Academy of Sciences funded two expeditions to make measurements of 1° of latitude near the Arctic Circle (northern Scandinavia) and near the equator (now Ecuador)

It took ~10 years for the equator team to complete, so the first results were from Scandinavia, and equator verified it: the Earth was an oblate spheroid, J2 is +




Vanguard

Determination of Earth gravity coefficients resulted from Vanguard-I (NRL project)

First network of tracking stations, known as Minitrack, was deployed to support objectives: “determine atmospheric density and the shape of the Earth”

To achieve objectives, all basic elements of orbit determination were involved and a state of the art IBM 704 computer was used to determine the orbit




Shape of Earth: J2, J3

U.S. Vanguard satellite launched in 1958, used to determine J2 and J3

J2 represents most of the oblateness; J3 represents a pear shape

J2 = 1.08264 x 10-3

J3 = - 2.5324 x 10-6




J2 and Orbit Design

As altitude increases, J2 perturbation diminishes (from a great distance the Earth is equivalent to a point mass)

Use J2 perturbation in orbit design, e.g., solar synchronous satellite◦ If dΩ/dt = +360°/365.25 days, the line of nodes will keep

a fixed (in an average sense) orientation with respect to the Earth-Sun direction

◦ Must be retrograde; for 600 km altitude, i=98°




Perturbations from Spherical Harmonics

Mean Ω, ω, M exhibit secular variation (caused by even degree Jn)

Mean a, e, i are constant Odd degree Jn cause long period perturbations (period of

argument of perigee motion) All harmonic coefficients cause short period

perturbations (period is 1, ½, 1/3, etc multiple of the orbital period)

m≠0 harmonic coefficients cause m-daily perturbations (i.e., 1, ½, 1/3, etc multiple of one day)

Special category: resonant perturbations (e.g., geosynchronous, GPS, …)




Secular Variations

Secular variations of Ω (positive J2)◦ 0° < i <90° : dΩ/dt < 0◦ i = 90°: dΩ/dt = 0◦ 90° < i < 180°: dΩ/dt > 0

Secular variations of ω (positive J2)◦ i=63.4° or 116.6°, dω/dt = 0 (critical i)◦ See Table 2.3.3 for more details

Secular variations produced by all even-degree zonal harmonics




Influence of J2 on Satellite Motion

Oblateness produces linear (secular) changes in Ω, ω, M

Periodic variations in all elements; e.g., semimajor axis exhibits a twice per orbital revolution variation

Approximate equations for variations in semimajor axis shown at left




Example

GPS known as PRN 05 (p. 70) Observed elements and rates:

◦ a=26560.5 km, e=0.0015, i=54.5°◦ dΩ/dt = -0.04109°/day

Contributions from analytical rates:◦ J2: -0.03927°/day◦ Moon: -0.00097°/day◦ Sun: -0.00045°/day◦ Total: -0.04069°/day (difference with observed is

0.0004°/day, or 1%)




J2 and Orbit Design

As altitude increases, J2 perturbation diminishes (from a great distance the Earth is equivalent to a point mass)

Use J2 perturbation in orbit design, e.g., solar synchronous satellite◦ If dΩ/dt = +360°/365.25 days, the line of nodes will keep

a fixed (in an average sense) orientation with respect to the Earth-Sun direction

◦ Must be retrograde; for 600 km altitude, i=98°




Atmospheric Drag

Atmospheric drag is the dominant nongravitational force at low altitudes if the celestial body has an atmosphere

Depending on nature of the satellite, lift force may exist

Drag removes energy from the orbit and results in da/dt < 0, de/dt < 0

Orbital lifetime of satellite strongly influenced by drag

From D. King-Hele, 1964, Theory of Satellite Orbits in an Atmosphere




Other Forces

The Earth (and all planets) are not rigid bodies◦ Gravitationally induced deformation (tides), both in fluid parts of

the planet and solid◦ Section 2.3.6 provides more detail◦ Earth ΔJ2 from luni-solar tides is ~ 10-8

Relativity (small effect on motion of perigee) Nongravitational forces

◦ Atmospheric drag (dependent on CD A/m) Responsible for orbit decay, da/dt < 0

◦ Solar radiation pressure, SRP (dependent on CR A/m)◦ Earth radiation pressure◦ Other (including thermal radiation)◦ Unknown or not well understood forces




Any Questions?

HW Due in one week

Quizzes with 24-hour timeframe

MATLAB and Python next Tuesday◦ Materials to be posted online.

Hopefully CAETE and D2L will cooperate!

Don’t forget Labor Day

Final Thoughts




(Start of Lecture 3…)




Coordinate Systems and Time: I

The transformation between ECI and ECF is required in the equations of motion

ECI is represented by ICRF (International Celestial Reference Frame, usually close to J2000)

ECF is represented by ITRF (International Terrestrial Reference Frame), e.g., ITRF-2000 which gives coordinates of international space geodetic global sites




Coordinate Systems and Time: II

Equinox location is function of time◦ Sun and Moon interact with

Earth J2 to produce Precession of equinox (ψ) Nutation (ε)

Newtonian time (independent variable of equations of motion) is represented by atomic time scales (dependent on Cesium Clock)




Precession / Nutation

Precession

Nutation (main term):




Earth Rotation

The angular velocity vector ωE is not constant in direction or magnitude◦ Direction: polar motion

Chandler period: 430 days Solar period: 365 days

◦ Magnitude: related to length of day (LOD) LOD dependent on

atmospheric winds Components of ωE

depend on observations; difficult to predict over long periodsPolar Motion: 1987 Jan 4 to 1988 Dec. 29




Earth Rotation and Time Sidereal rate of rotation: ~2π/86164 rad/day Variations exist in magnitude of ωE, from upper

atmospheric winds, tides, etc. UT1 is used to represent such variations

◦ UTC is kept within 0.9 sec of UT1 (leap second) Polar motion and UT1 observed quantities Different time scales: GPS-Time, TAI, UTC, TDT Time is independent variable in satellite equations of

motion; relates observations to equations of motion (TDT is usually taken to represent independent variable in equations of motion)




Transformation Between ECI and ICF

Transformation between ECI and ECF

P is the precession matrix (~50 arcsec/yr)

N is the nutation matrix (main term is 9 arcsec with 18.6 yr period)

S’ is sidereal rotation (depends on changes in angular velocity magnitude; UT1)

W is polar motion Caution: small effects may be

important in particular application

Documents

Statistical Orbit determination I Fall 2012 Professor George H. Born