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### Methods of Orbit Propagation

Jim Woodburn

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

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

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

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)

Analytical Methods

- Complete solutions
- Two body
- Vinti
- General perturbations
- Method of averaging Mean elements
- Brouwer
- Kozai

Two-Body

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

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

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!

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 ConsCons

Pros

General Perturbations

- Use simplified equations which approximate perturbations to a known solution
- Method of averaging
- Analytically solve approximate equations
- Using more approximations

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

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

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)

J2 and J4 equations

- First-order J2 secular variations:

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)

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!

Fast

Provide insight

Useful in design

Less accurate

Difficult to code

Difficult to extend

Nuances

Assumptions

Force coupling

GP Methods – Pros & ConsCons

Pros

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

Cartesian Equations of Motion (CEM)

- Conceptually simplest
- Default EOM used by HPOP, Astrogator

Integration Methods for CEM

- Multi-step Predictor–Corrector
- Gauss-Jackson (2)
- Adams (1)
- Single step
- Runge-Kutta
- Bulirsch-Stoer

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)

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 SelectionMulti-step

Single step

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

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 ConsCons

Pros

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

VOP Process

- Two/three step process
- Integrate changes to initial orbit elements
- Apply two body propagation
- Rectification

Integrate

Propagate

VOP - Lagrange

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

VOP - Poisson

- Perturbations expressed in terms of Cartesian coordinates
- Natural transition from CEM

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)

VOP - Herrick

- Uses Cartesian (universal) elements and Cartesian perturbations
- Implementation in STK

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

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 & ConsPros

Cons

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)

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

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

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

Semi-analytical Uses

- Long term orbit propagation and studies
- Constellation design
- Formation design
- Orbit maintenance

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

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

DSST

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

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 & ConsCons

Pros

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