1. ASEN 5070: Statistical Orbit Determination I Fall 2014
Professor Brandon A. Jones
Lecture 2: Basics of Orbit Propagation

2. Announcements Monday is Labor Day!
Homework 0 – You could already have it finished
Homework 1 – Due September 5
Syllabus questions?

3. Today’s Lecture The orbit propagation problem Orbital elements
Perturbed Satellite Motion
Coordinate and Time Systems

4. Brief Review of Astrodynamics

5. Review of Astrodynamics
What’s μ (other than a greek letter)?

6. Review of Astrodynamics
What’s μ (other than a greek letter)?
μ is the gravitational parameter of a massive body
μ = GM

7. Review of Astrodynamics
What’s μ?
μ is the gravitational parameter of a massive body
μ = GM
What’s G?
What’s M?

8. Review of Astrodynamics
What’s μ?
μ is the gravitational parameter of a massive body
μ = GM
What’s G? Universal Gravitational Constant
What’s M? The mass of the body

9. Review of Astrodynamics
What’s μ?
μ = GM
G = ± × km3/kg/s2
MEarth ~ × 1024 kg
or × 1024 kg
or × 1024 kg
Use a value and cite where you found it!
μEarth = 398, ± km3/s2
(Tapley, Schutz, and Born, 2004)
How do we measure the value of μEarth?

10. Review of Astrodynamics
Problem of Two Bodies
µ = G(M1 + M2)
XYZ is nonrotating, with zero acceleration; an inertial reference frame

11. Integrals of Motion
Center of mass of two bodies moves in straight line with constant velocity
Angular momentum per unit mass (h) is constant, h = r x V = constant, where V is velocity of M2 with respect to M1, V= dr/dt
Consequence: motion is planar
Energy per unit mass (scalar) is constant

12. Orbit Plane in Space

13. Equations of Motion in the Orbit Plane
The uθ component yields:
which is simply h = constant

14. Solution of ur Equations of Motion
The solution of the ur equation is (as function of θ instead of t):
where e and ω are constants of integration.

15. The Conic Equation Constants of integration: e and ω
e = ( ξ h2/µ2 )1/2
ω corresponds to θ where r is minima
Let f = θ – ω, then
r = p/(1 + e cos f)
which is “conic equation” from
analytical geometry (e is conic “eccentricity”,
p is “semi-latus rectum” or “semi-parameter”,
and f is the “true anomaly”)
Conclude that motion of M2 with respect to M1 is a “conic section”
Circle (e=0), ellipse (0<e<1), parabola (e=1), hyperbola (e>1)

16. Types of Orbital Motion

17. Orbit Elements

18. Six Orbit Elements
The six orbit elements (or Kepler elements) are constant in the problem of two bodies (two gravitationally attracting spheres, or point masses)
Define shape of the orbit
a: semimajor axis
e: eccentricity
Define the orientation of the orbit in space
i: inclination
Ω: angle defining location of ascending node (AN)
: angle from AN to perifocus; argument of perifocus
Reference time:
tp: time of perifocus (or mean anomaly at specified time)

19. Orbit Orientation
i - Inclination
Ω - RAAN
ω – Arg. of Perigee

20. 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

21. Satellite State Representations
Cartesian Coordinates
x, y, z, vx, vy, vz in some coordinate frame
Keplerian Orbital Elements
a, e, i, Ω, ω, ν (or similar set)
Topocentric Elements
Right ascension, declination, radius, and time rates of each
When are each of these useful?

22. Keplerian Orbital Elements
Shape:
a = semi-major axis
e = eccentricity
Orientation:
i = inclination
Ω = right ascension of ascending node
ω = argument of periapse
Position:
ν = true anomaly
What if i=0?
If orbit is equatorial, i = 0 and Ω is undefined.
In that case we can use the “True Longitude of Periapsis”

23. Keplerian Orbital Elements
Shape:
a = semi-major axis
e = eccentricity
Orientation:
i = inclination
Ω = right ascension of ascending node
ω = argument of periapse
Position:
ν = true anomaly
What if e=0?
If orbit is circular, e = 0 and ω is undefined.
In that case we can use the “Argument of Latitude”

24. Keplerian Orbital Elements
Shape:
a = semi-major axis
e = eccentricity
Orientation:
i = inclination
Ω = right ascension of ascending node
ω = argument of periapse
Position:
ν = true anomaly
What if i=0 and e=0?
If orbit is circular and equatorial, neither ω nor Ω are defined
In that case we can use the “True Longitude”

25. Keplerian Orbital Elements
Shape:
a = semi-major axis
e = eccentricity
Orientation:
i = inclination
Ω = right ascension of ascending node
ω = argument of periapse
Position:
ν = true anomaly
M = mean anomaly
Special Cases:
If orbit is circular, e = 0 and ω is undefined.
In that case we can use the “Argument of Latitude” ( u = ω+ν )
If orbit is equatorial, i = 0 and Ω is undefined.
In that case we can use the “True Longitude of Periapsis”
If orbit is circular and equatorial, neither ω nor Ω are defined
In that case we can use the “True Longitude”

26. Cartesian to Keplerian Conversion
Handout offers one conversion.
We’ve coded up Vallado’s conversions
ASEN 5050 implements these
Check out the code RVtoKepler.m
Check errors and/or special cases when i or e are very small!

27. Perturbed Satellite Motion

28. 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, a is the perturbing force (causes the actual motion to deviate from exact 2- body)

29. 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

30. 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 evaluation point are given in spherical coordinates: r, geocentric latitude φ, longitude 

31. Gravity Coefficients
The gravity coefficients (Jn, Cnm, Snm) are also known as Stokes Coefficients and Spherical Harmonic Coefficients
Jn, Cn0:
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
These parameters may be used for orbit design (take ASEN 5050 for more details!)

32. 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

33. 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

34. Other Forces
What are the other forces that can perturb a satellite’s motion?
Solar Radiation Pressure (SRP)
Thrusters
N-body gravitation (Sun, Moon, etc.)
Electromagnetic
Solid and liquid body tides
Relativistic Effects
Reflected radiation (e.g., ERP)
Coordinate system errors
Spacecraft radiation

35. Coordinate and Time Frames

36. 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

37. Coordinate Systems and Time
The transformation between ECI and ECF is required in the equations of motion
Depends on the current time!
Thanks to Einstein, we know that time is not simple…

38. Time Systems
Countless systems exist to measure the passage of time. To varying degrees, each of the following types is important to the mission analyst:
Atomic Time
Unit of duration is defined based on an atomic clock.
Universal Time
Unit of duration is designed to represent a mean solar day as uniformly as possible.
Sidereal Time
Unit of duration is defined based on Earth’s rotation relative to distant stars.
Dynamical Time
Unit of duration is defined based on the orbital motion of the Solar System.

39. Time Systems: Time Scales

40. What are some issues with each of these?
Time Systems
Question: How do you quantify the passage of time?
Year
Month
Day
Second
Pendulums
Atoms
What are some issues with each of these?
Gravity
Earthquakes
Snooze alarms

41. Time Systems: The Year Definitions of a Year
Julian Year: days, where an SI “day” = SI “seconds”.
Sidereal Year: mean solar days
Duration of time required for Earth to traverse one revolution about the sun, measured via distant star.
Tropical Year: days
Duration of time for Sun’s ecliptic longitude to advance 360 deg. Shorter on account of Earth’s axial precession.
Anomalistic Year: days
Perihelion to perihelion.
Draconic Year: days
One ascending lunar node to the next (two lunar eclipse seasons)
Full Moon Cycle, Lunar Year, Vague Year, Heliacal Year, Sothic Year, Gaussian Year, Besselian Year

42. Coordinate Systems and Time
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)

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

44. 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)

45. Coordinate Frames Inertial: fixed orientation in space Rotating
Inertial coordinate frames are typically tied to hundreds of observations of quasars and other very distant near-fixed objects in the sky.
Rotating
Constant angular velocity: mean spin motion of a planet
Osculating angular velocity: accurate spin motion of a planet

46. Coordinate Systems Coordinate Systems = Frame + Origin
Inertial coordinate systems require that the system be non-accelerating.
Inertial frame + non-accelerating origin
“Inertial” coordinate systems are usually just non- rotating coordinate systems.

47. Coordinate System Transformations
Converting from ECI to 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
Earth Orientation Parameters
Caution: small effects may be important in particular application