1. University of Colorado Boulder ASEN 5070: Statistical Orbit Determination I Fall 2015 Professor Brandon A. Jones Lecture 2: Basics of Orbit Propagation

2. University of Colorado Boulder  Homework 0 ◦ You do not need to turn it in ◦ Intended to be a tutorial for those who have never used a numeric integrator (ode45()/ode113()/odeint())  Homework 1 – Due September 4 by start of lecture  Office Hours: ◦ Prof. Jones: W 3-4pm & Th 11-12noon  ECNT 420 ◦ Eduardo: W 2-3pm & Th 2:30-4:30pm  ECAE 1B44 (not CCAR meeting room!)  Syllabus questions? 2

3. University of Colorado Boulder  The orbit propagation problem (Chapter 2) ◦ Two-body problem ◦ Orbital elements ◦ Potential theory ◦ Perturbed Satellite Motion 3

4. University of Colorado Boulder 4 Two-Body Motion

5. University of Colorado Boulder  First, we start with Newton’s second law:  Combine that with Newton’s law of universal gravitation  To get the gravity acceleration 5

6. University of Colorado Boulder 6

7. University of Colorado Boulder  For unperturbed dynamics about a central body: 7

8. University of Colorado Boulder  What’s μ?  μ = GM  G = 6.67384 ± 0.00080 × 10 -20 km 3 /kg/s 2  M Earth ~ 5.97219 × 10 24 kg ◦ or 5.9736 × 10 24 kg ◦ or 5.9726 × 10 24 kg ◦ Use a value and cite where you found it!  μ Earth = 398,600.4415 ± 0.0008 km 3 /s 2 (Tapley, Schutz, and Born, 2004)  How do we measure the value of μ Earth ? 8

9. University of Colorado Boulder  We express orbit energy as a specific energy (energy per unit mass) instead of absolute energy  For two-body motion, the specific energy is 9

10. University of Colorado Boulder  Center of mass of two bodies moves in straight line with constant velocity  Specific angular momentum ◦ Consequence: motion is planar  Energy per unit mass (scalar) is constant 10

11. University of Colorado Boulder  The mass of the satellite is negligible relative to the primary body ◦ What are some possible exceptions to this?  The coordinate system is inertial ◦ What is the problem with this assumption?  The bodies are spherical with uniform mass density ◦ Is that true?  No other forces act on the system ◦ That isn’t true either… 11

12. University of Colorado Boulder 12 Orbit Elements

13. University of Colorado Boulder  The six orbit elements (or Kepler elements) are constant in the two-body problem ◦ 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 (different options):  t p : time of perifocus  Mean/true/eccentric anomaly at some time (requires two parameters) 13

14. University of Colorado Boulder  Consider an ellipse  Periapse/perifocus/periapsis ◦ Perigee, perihelion  r = radius  r p = radius of periapse  r a = radius of apoapse  a = semi-major axis  e = eccentricity = (r a -r p )/(r a +r p )  r p = a(1-e) r a = a(1+e)  ω = argument of periapse  f/υ = true anomaly 14

15. University of Colorado Boulder  i - Inclination  Ω - RAAN  ω – Arg. of Perigee 15

16. University of Colorado Boulder

17. University of Colorado Boulder  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? 17

18. University of Colorado Boulder  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”, i.e., the angle from the vernal equinox (inertial X-axis) to the perifocus 18

19. University of Colorado Boulder  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”, i.e., the angle from ascending node to satellite 19

20. University of Colorado Boulder  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”, i.e., the angle between the satellite position vector and the vernal equinox 20

21. University of Colorado Boulder  Shape: ◦ a = semi-major axis ◦ e = eccentricity  Orientation: ◦ i = inclination ◦ Ω = right ascension of ascending node ◦ ω = argument of periapse  Position (at time t): ◦ 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” ◦ 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” 21

22. University of Colorado Boulder  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! 22

23. University of Colorado Boulder 23 A Brief Introduction to Potential Theory

24. University of Colorado Boulder  Potential theory provides a means for studying harmonic functions  Given the potential energy V for a system, then 24

25. University of Colorado Boulder  In astrodynamics, we often write  This gives us  In HW 1, you will use potential theory to derive the two-body equation 25

26. University of Colorado Boulder 26 Homework 1

27. University of Colorado Boulder Homework # 1 Problem 1: Problem 2:

28. University of Colorado Boulder Homework # 1 Problem 3: Problem 4:

29. University of Colorado Boulder Homework # 1 Problem 5: Problem 6:

30. University of Colorado Boulder Homework # 1 Problem 7: Solution method discussed next week!

31. University of Colorado Boulder 31 Perturbed Satellite Motion

32. University of Colorado Boulder  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) 32

33. University of Colorado Boulder  Sphere of constant mass density is not an accurate representation for planets  Define gravitational potential function such that the gravitational acceleration is: 33

34. University of Colorado Boulder 34  The commonly used expression for the gravitational potential is given in terms of mass distribution coefficients J n, C nm, S nm  l is degree, m is order  Coordinates of evaluation point are given in spherical coordinates: r, geocentric latitude φ, longitude

35. University of Colorado Boulder  U.S. Vanguard satellite launched in 1958, used to determine J 2 and J 3  J 2 represents most of the oblateness; J 3 represents a pear shape  J 2 ≈ 1.08264 x 10 -3  J 3 ≈ - 2.5324 x 10 -6

36. University of Colorado Boulder  Our new orbit energy is  Is this constant over time? Why or why not?  What if we only include J 2 in U’ and pure rotation about Z-axis for Earth? 36

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38. University of Colorado Boulder 38  Atmospheric drag is the dominant nongravitational force at low altitudes if the celestial body has an atmosphere  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

39. University of Colorado Boulder  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 39