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Methods of Orbit PropagationJim Woodburn
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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
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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
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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
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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)
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Analytical Methods
• Complete solutions– Two body– Vinti
• General perturbations– Method of averaging Mean elements– Brouwer– Kozai
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Two-Body
• Spherically symmetric mass distribution• Gravity is only force• Many methods of solution
• Two Body propagator in STK
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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
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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!
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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
Pros Cons
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General Perturbations
• Use simplified equations which approximate perturbations to a known solution
• Method of averaging• Analytically solve approximate equations
– Using more approximations
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• 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
GP – Central Body Gravity
SPLP UUUU sec
SPLPt 00
rU
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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
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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)
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J2 and J4 equations
• First-order J2 secular variations:
02
2
20 cos23 ttni
pRJ e
022
2
20 sin252
23 ttni
pRJ e
ie
pRJnn e 222
2
2 sin2311
231
00 ttnMM
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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)
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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!
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GP Methods – Pros & Cons
• Fast
• Provide insight
• Useful in design
• Less accurate
• Difficult to code
• Difficult to extend
• Nuances– Assumptions– Force coupling
Pros Cons
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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
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Cartesian Equations of Motion (CEM)
• Conceptually simplest• Default EOM used by HPOP, Astrogator
...33 srpdragrdBodiesaspherical aaaarra
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Integration Methods for CEM
• Multi-step Predictor–Corrector– Gauss-Jackson (2)– Adams (1)
• Single step– Runge-Kutta– Bulirsch-Stoer
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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)
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Integrator Selection
• 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
Multi-step Single step
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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
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CEM Pros and 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
Pros Cons
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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
perturbaM )(0
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VOP Process
• Two/three step process– Integrate changes to initial orbit elements– Apply two body propagation– Rectification
1kt tk
1kt tk perturbtkt aMt
k
)( Integrate
11 kt tk
Propagate
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VOP Process
Timetk tk+1 tk+2
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VOP - Lagrange
• Perturbations disturbing potential• Eq. of motion – Lagrange Planetary Equations
R
iR
iena
sin1
122
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VOP - Poisson
• Perturbations expressed in terms of Cartesian coordinates
• Natural transition from CEM
perturbar
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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)
Tnav2
2 RrSr
naa 2
2
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VOP - Herrick
• Uses Cartesian (universal) elements and Cartesian perturbations
• Implementation in STK
perturbaffrfrr ```0
perturbaggrgrr ```0
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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
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VOP 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
Pros Cons
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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)
Prrr
r
3
3
31
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Encke Process
Timetk tk+1 tk+2
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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
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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
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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
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Semi-analytical Uses
• Long term orbit propagation and studies• Constellation design• Formation design• Orbit maintenance
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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
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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
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DSST
• Draper Semi-analytic Satellite Theory• Very complete semi-analytic theory
– J2000– Modern atmospheric density model– Tesseral resonances
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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
Pros Cons
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Questions?