# Methods of Orbit Propagation

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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

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 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

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 General perturbations 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!

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

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
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 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)

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!

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

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)

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

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

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

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

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

VOP Process tk tk+1 tk+2 Time

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

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

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 Process tk tk+1 tk+2 Time

Encke Applications Orbit propagation Orbit correction
Fixing errors in numerical integration Eclipse boundary crossings AIAA , AAS Coupled attitude and orbit propagation AAS 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

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

Questions?

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)

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

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

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

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

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

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

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)