1. Statistical Orbit determination I Fall 2012 Professor George H. Born
ASEN 5070
Statistical Orbit determination I
Fall 2012
Professor George H. Born
Professor Jeffrey S. Parker
Lecture 2: Background

2. Announcements Monday is Labor Day! D2L issue CAETE issue
Quiz - Now 24 hours per quiz (1pm – 1pm)
Homework 1
Office Hours

3. Quiz Results

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

5. Today’s Lecture 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.

6. Astrodynamics Background
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?

7. 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

8. Astrodynamics Background
Keplerian Orbital Elements
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.

9. Astrodynamics Background
Keplerian Orbital Elements
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)

10. Astrodynamics Background
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?

11. Astrodynamics Background
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.

12. 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)

13. Coordinate Frames
Choose X-axis (coinciding with vernal equinox) as inertial direction; Z- axis coincident with Earth angular velocity vector (ωe), period of rotation = sec, “sidereal” period
GMST=αG = ωe(t – t0) + αG0

14. 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

15. 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

16. Effects of Small Variations
I’d like us to think about the effects of small variations in coordinates, and how these impact future states.
Example: Propagating a state in the presence of NO forces
Final State:
(xf, yf, zf, vxf, vyf, vzf)
Initial State:
(x0, y0, z0, vx0, vy0, vz0)

17. Effects of Small Variations
What happens if we perturb the value of x0?
Force model: 0
Initial State:
(x0+Δx, y0, z0, vx0, vy0, vz0)
Final State:
(xf, yf, zf, vxf, vyf, vzf)
Initial State:
(x0, y0, z0, vx0, vy0, vz0)

18. Effects of Small Variations
What happens if we perturb the value of x0?
Force model: 0
Final State:
(xf+Δx, yf, zf, vxf, vyf, vzf)
Initial State:
(x0+Δx, y0, z0, vx0, vy0, vz0)
Final State:
(xf, yf, zf, vxf, vyf, vzf)
Initial State:
(x0, y0, z0, vx0, vy0, vz0)

19. Effects of Small Variations
What happens if we perturb the position?
Force model: 0
Final State:
(xf+Δx, yf+Δy, zf+Δz, vxf, vyf, vzf)
Initial State:
(x0+Δx, y0+Δy, z0+Δz, vx0, vy0, vz0)
Initial State:
(x0, y0, z0, vx0, vy0, vz0)

20. Effects of Small Variations
What happens if we perturb the value of vx0?
Force model: 0
Initial State:
(x0, y0, z0, vx0-Δvx, vy0, vz0)
Final State:
(xf, yf, zf, vxf, vyf, vzf)
Initial State:
(x0, y0, z0, vx0, vy0, vz0)

21. Effects of Small Variations
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)
Final State:
(xf, yf, zf, vxf, vyf, vzf)
Initial State:
(x0, y0, z0, vx0, vy0, vz0)

22. Effects of Small Variations
What happens if we perturb the position and velocity?
Force model: 0

23. Effects of Small Variations
We could have arrived at this easily enough from the equations of motion.
Force model: 0

24. Effects of Small Variations
This becomes more challenging with nonlinear dynamics
Force model: two-body

25. Effects of Small Variations
This becomes more challenging with nonlinear dynamics
Final State:
(xf, yf, zf, vxf, vyf, vzf)
Force model: two-body
Initial State:
(x0, y0, z0, vx0, vy0, vz0)
The partial of one Cartesian parameter wrt the partial of another Cartesian parameter is ugly.

26. Effects of Small Variations
This becomes more challenging with nonlinear dynamics
Final State:
(xf, yf, zf, vxf, vyf, vzf)
Force model: two-body

27. Effects of Small Variations
Represent variations as functions of Keplerian orbital elements
Force model: two-body
Final State:
Then, what is ?

28. Effects of Small Variations
Represent variations as functions of Keplerian orbital elements
Force model: two-body
Final State:

29. Effects of Small Variations
Represent variations as functions of Keplerian orbital elements
Force model: two-body If
Final State:
Then, what is ?

30. Effects of Small Variations
Represent variations as functions of Keplerian orbital elements
Force model: two-body If
Final State:
(Remember, af=a0)

31. Effects of Small Variations
Represent variations as functions of Keplerian orbital elements
Force model: two-body If
Final State:

32. Effects of Small Variations
The point is that we can relate small perturbations from one element to another easier using Keplerian orbital element than Cartesian.
Other brain teasers

33. Effects of Small Variations
How does Rf vary if V0 is increased?
Force model: two-body
A: Increases
B: Decreases
C: Stays the same
D: Not enough information
Final State
(Rf, Vf)
Initial State
(R0, V0)

34. Effects of Small Variations
How does Rf vary if V0 is increased?
Force model: two-body
A: Increases
B: Decreases
C: Stays the same
D: Not enough information
Final State
(Rf, Vf)
Initial State
(R0, V0)

35. Effects of Small Variations
How does Rf vary if R0 is increased?
Force model: two-body
Hint:
A: Increases
B: Decreases
C: Stays the same
D: Not enough information
Final State
(Rf, Vf)
Initial State
(R0, V0)

36. Effects of Small Variations
How does Rf vary if R0 is increased?
Force model: two-body
Hint:
A: Increases
B: Decreases
C: Stays the same
D: Not enough information
Final State
(Rf, Vf)
Initial State
(R0, V0)

37. Intermission
Any Questions?
(quick break)

38. LaTex
The Not-So-Short Guide to LaTex

39. 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:

40. 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)

41. 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

42. 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 

43. 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

44. 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

45. 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 +

46. 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

47. 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 = x 10-3
J3 = x 10-6

48. 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°/ 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°

49. 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, …)

50. 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 for more details
Secular variations produced by all even-degree zonal harmonics

51. 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

52. Observed elements and rates:
Example
GPS known as PRN 05 (p. 70)
Observed elements and rates:
a= km, e=0.0015, i=54.5°
dΩ/dt = °/day
Contributions from analytical rates:
J2: °/day
Moon: °/day
Sun: °/day
Total: °/day (difference with observed is °/day, or 1%)

53. 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°/ 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°

54. 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

55. 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 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

56. Final Thoughts 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 

57. (Start of Lecture 3…)

58. 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 which gives coordinates of international space geodetic global sites

59. 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)

60. Precession / Nutation
Precession
Nutation (main term):

61. 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 periods
Polar Motion: 1987 Jan 4 to 1988 Dec. 29

62. 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)

63. 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 periód)
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