ASEN 5070: Statistical Orbit Determination I Fall 2014 Professor Brandon A. Jones

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 = 6.67384 ± 0.00080 × 10-20 km3/kg/s2 • MEarth ~ 5.97219 × 1024 kg • or 5.9736 × 1024kg • or 5.9726 × 1024 kg • Use a value and cite where you found it! • μEarth = 398,600.4415 ± 0.0008 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 uθcomponent 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 = ( 1 + 2 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: semimajoraxis • 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 tesseralcoefficients • 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 = 1.08264 x 10-3 • J3 = - 2.5324 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. 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: 365.25 days, where an SI “day” = 86400 SI “seconds”. • Sidereal Year: 365.256 363 004 mean solar days • Duration of time required for Earth to traverse one revolution about the sun, measured via distant star. • Tropical Year: 365.242 19 days • Duration of time for Sun’s ecliptic longitude to advance 360 deg. Shorter on account of Earth’s axial precession. • Anomalistic Year: 365.259 636 days • Perihelion to perihelion. • Draconic Year: 365.620 075 883 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 • 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