Methods of Orbit Propagation

1. Methods of Orbit Propagation Jim Woodburn

2. Why are you here? • You want to use space • You operate a satellite • You use a satellite • You want to avoid a satellite • You need to exchange data • You forgot to leave the room after the last talk

3. Motivation • Accurate orbit modeling is essential to analysis • Different orbit propagation models are required • Design, planning, analysis, operations • Fidelity: “Need vs. speed” • Orbit propagation makes great party conversation STK has been designed to support all levels of user need

4. Agenda • Analytical Methods • Exact solutions to simple approximating problems • Approximate solutions to approximating problems • Semi-analytical Methods • Better approximate solutions to realistic problems • Numerical Methods • Best solutions to most realistic problems

5. Analytical Methods Definition – Position and velocity at a requested time are computed directly from initial conditions in a single step • Allows for iteration on initial conditions (osculating to mean conversion)

6. Analytical Methods • Complete solutions • Two body • Vinti • General perturbations • Method of averaging Mean elements • Brouwer • Kozai

7. Two-Body • Spherically symmetric mass distribution • Gravity is only force • Many methods of solution • Two Body propagator in STK

8. Vinti’s Solution • Solved in spheroidal coordinates • Includes the effects of J2, J3 and part of J4 • But the J2 problem does not have an analytical solution • This is not a solution to the J2 problem • This is also not in STK

9. Interpolation with complete solutions • Standard formulations • Lagrangian interpolation, order 7 [8 sample pnts] • Position, Velocity computed separately • Hermitian interpolation, order 7 [4 sample pnts] • Position, Velocity computed together • Why interpolate? Just compute directly!

10. Fast Provide understanding Capture simple physics Serve as building blocks for more sophisticated methods Can be taught in undergraduate classes Not accurate Need something more difficult to teach in graduate classes Complete Soln Pros and Cons Cons Pros

11. General Perturbations • Use simplified equations which approximate perturbations to a known solution • Method of averaging • Analytically solve approximate equations • Using more approximations

12. GP – Central Body Gravity • Central Body Gravity • Defined by a potential function • Express U in terms of orbital elements • Average U over one orbit • Separate into secular and long term contributions • Analytically solve for each type of contribution

13. GP Mean Elements • Selection of orbit elements and method of averaging define mean elements • Only the averaged representation is truly mean • Brouwer • Kozai • It is common practice to “transform” mean elements to other representations

14. J2 and J4 propagators • J2 is dominant non-spherical term of Earth’s gravity field • Only model secular effects of orbital elements • Argument of Perigee • Right Ascension of the Ascending Node • Mean motion (ie orbital frequency) • Method • Escobal’s “Methods of Orbit Determination” • J2  First order J2 terms • J4  First & second order J2 terms; first order J4 terms • J4 produces a very small effect (takes a long time to see difference)

15. J2 and J4 equations • First-order J2 secular variations:

16. SGP4 • General perturbation algorithm • Developed in the 70’s, subsequently revised • Mean Keplerian elements in TEME frame • Incorporates both SGP4 and SDP4 • Uses TLEs (Two Line Elements) • Serves as the initial condition data for a space object • Continually updated by USSTRATCOM • They track 9000+ space objects, mostly debris • Updated files available from AGI’s website • Propagation valid for short durations (3-10 days)

17. Interpolation with GP • Standard formulations • Lagrangian interpolation, order 7 [8 sample pnts] • Position, Velocity computed separately • Should be safe • Hermitian interpolation, order 7 [4 sample pnts] • Position, Velocity computed together • Beware – Velocity is not precisely the derivative of position • Why interpolate? Just compute directly!

18. Fast Provide insight Useful in design Less accurate Difficult to code Difficult to extend Nuances Assumptions Force coupling GP Methods – Pros & Cons Cons Pros

19. Numerical Methods Definition – Orbit trajectories are computed via numerical integration of the equations of motion One must marry a formulation of the equations of motion with a numerical integration method

20. Cartesian Equations of Motion (CEM) • Conceptually simplest • Default EOM used by HPOP, Astrogator

21. Integration Methods for CEM • Multi-step Predictor–Corrector • Gauss-Jackson (2) • Adams (1) • Single step • Runge-Kutta • Bulirsch-Stoer

22. Numerical Integrators in STK • Gauss-Jackson (12th order multi-step) • Second order equations • Runge-Kutta (single step) • Fehlberg 7-8 • Verner 8-9 • 4th order • Bulirsch-Stoer (single step)

23. Pros Very fast Kick near circular butt Cons Special starting procedure Restart Fixed time steps Error control Pros Plug and play Change force modeling Change state Error control Cons Slower Not good party conversation Integrator Selection Multi-step Single step

24. Interpolation with CEM • Standard formulation • Lagrangian interpolation, order 7 [8 sample pnts] • Position, Velocity computed separately • Hermitian interpolation, order 5 [2 sample pnts] • Position, Velocity, Acceleration computed together • Integrator specific interpolation • Multi-step accelerations and sums

25. Simple to formulate the equations of motion Accuracy limited by acceleration models Lots of numerical integration options Physics is all in the force models Six fast variables CEM Pros and Cons Cons Pros

26. Variation of Parameters • Formulate the equations of motion in terms of orbital elements (first order) • Analytically remove the two body part of the problem VOP isNOTan approximation

27. VOP Process • Two/three step process • Integrate changes to initial orbit elements • Apply two body propagation • Rectification Integrate Propagate

28. VOP Process tk tk+1 tk+2 Time

29. VOP - Lagrange • Perturbations disturbing potential • Eq. of motion – Lagrange Planetary Equations

30. VOP - Poisson • Perturbations expressed in terms of Cartesian coordinates • Natural transition from CEM

31. VOP - Gauss • Perturbations expressed in terms of Radial (R), Transverse (S) and Normal (W) components • Provides insight into which perturbations affect which orbital elements (maneuvering)

32. VOP - Herrick • Uses Cartesian (universal) elements and Cartesian perturbations • Implementation in STK

33. Interpolation with VOP • Standard formulation • Lagrangian interpolation, order 7 [8 sample pnts] • Position, Velocity computed separately • Hermitian interpolation, order 7 [4 sample pnts] • Position, Velocity computed together • Danger due to potentially large time steps • Variation of Parameters • Special VOP interpolator, order 7 [8 sample pnts] • Deals well with large time steps in the ephemeris • Performs Lagrangian interpolation in VOP space

34. Fast when perturbations are small Share acceleration model with CEM (minus 2Body) Physics incorporated into formulation Errors at level of numerical precision for 2Body Additional code required Error control less effective Loses some advantages in a high frequency forcing environment VOP Pros & Cons Pros Cons

35. Encke’s Method • Complete solution generated by combining a reference solution with a numerically integrated deviation from that reference • Reference is usually a two body trajectory • Can choose to rectify • Not in STK (directly)

36. Encke Process tk tk+1 tk+2 Time

37. Encke Applications • Orbit propagation • Orbit correction • Fixing errors in numerical integration • Eclipse boundary crossings • AIAA 2000-4027, AAS 01-223 • Coupled attitude and orbit propagation • AAS 01-428 • Transitive partials

38. Semi-analytical Methods • Definition – Methods which are neither completely analytic or completely numerical. • Typically use a low order integrator to numerically integrate secular and long periodic effects • Periodic effects are added analytically • Use VOP formulation • Almost/Almost compromise

39. Semi-analytical Process • Convert initial osculating elements to mean elements • Integrate mean element rates at large step sizes • Convert mean elements to osculating elements as needed • Interpolation performed in mean elements

40. Semi-analytical Uses • Long term orbit propagation and studies • Constellation design • Formation design • Orbit maintenance

41. Semi-analytic in STK - LOP • Long Term Orbit Propagator • Developed at JPL • Arbitrary degree and order gravity field • Third body perturbations • Solar pressure • Drag – US Standard Atmosphere

42. Semi-analytic in STK - Lifetime • Developed as NASA Langley • Hard-coded to use 5th order zonals • Third body perturbations • Solar pressure • Atmospheric drag – selectable density model

43. DSST • Draper Semi-analytic Satellite Theory • Very complete semi-analytic theory • J2000 • Modern atmospheric density model • Tesseral resonances

44. Fast Provide insight Useful in design Orbit Constellations/Formations Closed Orbits Difficult to code Difficult to extend Nuances Assumptions Force coupling Semi-analytical Methods – Pros & Cons Cons Pros

45. Questions?