1. Methods of Orbit Propagation
Struggled certain places in GP, may need to extend up front material
Need to place talking points in notes areas
Carrico asked for chart on orbit accuracies from different prop types
Berry thought my Cowell definitions were incorrect, maybe just delete this
Need to add some humor
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 Semi-analytical Methods Numerical 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 General perturbations 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. Complete Soln Pros and Cons
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

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
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 Uses TLEs (Two Line Elements)
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 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. GP Methods – Pros & Cons Pros Cons Fast Provide insight
Useful in design
Less accurate
Difficult to code
Difficult to extend
Nuances
Assumptions
Force coupling
Insight – rates of orbit elements, frequency of effects on orbital elements
Design – critical inclination, frozen orbits
Assumptions – central body specific – relative periods of 3rd bodies, is J2 the dominant perturbation

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. Integrator Selection Multi-step Single step Pros Cons Pros Cons
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

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. CEM Pros and Cons Pros Cons
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

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 is NOT an 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. VOP Pros & Cons Pros Cons 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

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 , AAS
Coupled attitude and orbit propagation
AAS
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. Semi-analytical Methods – Pros & Cons
Fast
Provide insight
Useful in design
Orbit
Constellations/Formations
Closed Orbits
Difficult to code
Difficult to extend
Nuances
Assumptions
Force coupling
Insight – rates of orbit elements, frequency of effects on orbital elements
Design – critical inclination, frozen orbits
Assumptions – central body specific – relative periods of 3rd bodies, is J2 the dominant perturbation

45. Questions?

46. RAAN evolution comparison
Evolution of the Right Ascension of the Ascending Node
45.0
44.9
Two-Body: Constant
44.8
44.7
44.6
J2: Secular Only
44.5
44.4
HPOP 2x0:
Periodic and Secular
44.3
44.2
44.1
1 Jan :00:00.00
1 Jan :30:00.00
1 Jan :00:00.00
(UTCG)

47. SRP boundary mitigation
Crossing lighting boundaries
Penumbra event occurs over short time intervals
Discontinuity in force model sampling
Without mitigation, propagation is sensitive to sampling steps
STK’s Method: uses Encke correction to account for lighting changes
More efficient than re-starting integrator at boundary
AGI authored papers (Jim Woodburn)
AIAA , AAS

48. Numerical integrators (cont’d)
Bulirsch-Stoer
Uses first-order form of equations of motion
Handles discontinuities gracefully
Able to use VOP formulation
Richardson extrapolation with automatic step size control
2 steps
4 steps
6 steps
t+h t
extrapolate

49. Comparing integrators
LEO
24 hours
Nearly circular, 28 deg inclination
TwoBody vs. J2 vs. RK
RK vs. BS vs. GJ
Stepsizes: 60, 30, 15 sec
SRP Mitigation: On and Off

50. Choosing settings Table of ephemeris runs
Leo 1 day e=0.001, 740km altitude, 14 revs per day
Meo 3 days circular, 4 revs per day
Heo 7 days Molniya, 2 revs per day
Geo 14 days
7 different force model settings
3 integrators (RK78, BS, GJ)
4 formulations (standard, reg. time, vop, vop reg. time)
5 different error tolerances used for RK78 and BS

51. Choosing settings (cont’d)
1,344 cases total
Truth model
RK78, 10 sec max step size
Table of differences available from Help system

52. Take-aways MEO GEO LEO, HEO max difference is 2mm over all 336 cases
306 have less than 1 mm difference
GEO
max difference is 9mm over all 336 cases
289 have less than 1 mm difference
LEO, HEO
About half of LEO cases have less than 1 mm difference
HEO cases have the most diversity
Sensitive to SRP
VOP may not adequately sample large gravity fields
Boundary mitigation works better in real time, not reg. time

53. Cowell (based on Vallado)
Cowell’s Formulation – Specifies a formulation of the second order equations of motion in terms of Cartesian elements
Cowell’s Method – Specifies Cowell’s Formulation of equations of motion used with a numerical integration scheme based on finite differences (Gauss-Jackson)