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Pg 1 of 45 AGI Methods of Orbit Propagation Jim Woodburn

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Pg 2 of 45 AGI 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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Pg 3 of 45 AGI 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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Pg 4 of 45 AGI 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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Pg 5 of 45 AGI 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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Pg 6 of 45 AGI Analytical Methods Complete solutions –Two body –Vinti General perturbations –Method of averaging Mean elements –Brouwer –Kozai

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Pg 7 of 45 AGI Two-Body Spherically symmetric mass distribution Gravity is only force Many methods of solution Two Body propagator in STK

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Pg 8 of 45 AGI Vinti’s Solution Solved in spheroidal coordinates Includes the effects of J 2, J 3 and part of J 4 But the J 2 problem does not have an analytical solution This is not a solution to the J 2 problem This is also not in STK

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Pg 9 of 45 AGI 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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Pg 10 of 45 AGI 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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Pg 11 of 45 AGI 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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Pg 12 of 45 AGI 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

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Pg 13 of 45 AGI 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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Pg 14 of 45 AGI 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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Pg 15 of 45 AGI J2 and J4 equations First-order J2 secular variations:

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Pg 16 of 45 AGI 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 space objects, mostly debris –Updated files available from AGI’s website –Propagation valid for short durations (3-10 days)

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Pg 17 of 45 AGI 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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Pg 18 of 45 AGI 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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Pg 19 of 45 AGI 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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Pg 20 of 45 AGI Cartesian Equations of Motion (CEM) Conceptually simplest Default EOM used by HPOP, Astrogator

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Pg 21 of 45 AGI Integration Methods for CEM Multi-step Predictor–Corrector –Gauss-Jackson (2) –Adams (1) Single step –Runge-Kutta –Bulirsch-Stoer

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Pg 22 of 45 AGI Numerical Integrators in STK Gauss-Jackson (12 th order multi-step) –Second order equations Runge-Kutta (single step) –Fehlberg 7-8 –Verner 8-9 –4 th order Bulirsch-Stoer (single step)

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Pg 23 of 45 AGI 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-stepSingle step

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Pg 24 of 45 AGI 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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Pg 25 of 45 AGI 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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Pg 26 of 45 AGI 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

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Pg 27 of 45 AGI VOP Process Two/three step process –Integrate changes to initial orbit elements –Apply two body propagation –Rectification Integrate Propagate

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Pg 28 of 45 AGI VOP Process Time tktk t k+1 t k+2

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Pg 29 of 45 AGI VOP - Lagrange Perturbations disturbing potential Eq. of motion – Lagrange Planetary Equations

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Pg 30 of 45 AGI VOP - Poisson Perturbations expressed in terms of Cartesian coordinates Natural transition from CEM

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Pg 31 of 45 AGI 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)

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Pg 32 of 45 AGI VOP - Herrick Uses Cartesian (universal) elements and Cartesian perturbations Implementation in STK

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Pg 33 of 45 AGI 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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Pg 34 of 45 AGI 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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Pg 35 of 45 AGI 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)

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Pg 36 of 45 AGI Encke Process Time tktk t k+1 t k+2

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Pg 37 of 45 AGI Encke Applications Orbit propagation Orbit correction –Fixing errors in numerical integration –Eclipse boundary crossings AIAA , AAS –Coupled attitude and orbit propagation AAS Transitive partials

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Pg 38 of 45 AGI 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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Pg 39 of 45 AGI 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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Pg 40 of 45 AGI Semi-analytical Uses Long term orbit propagation and studies Constellation design Formation design Orbit maintenance

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Pg 41 of 45 AGI 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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Pg 42 of 45 AGI Semi-analytic in STK - Lifetime Developed as NASA Langley Hard-coded to use 5 th order zonals Third body perturbations Solar pressure Atmospheric drag – selectable density model

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Pg 43 of 45 AGI DSST Draper Semi-analytic Satellite Theory Very complete semi-analytic theory –J2000 –Modern atmospheric density model –Tesseral resonances

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Pg 44 of 45 AGI 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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Pg 45 of 45 AGI Questions?

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Pg 46 of 45 AGI RAAN evolution comparison 1 Jan :00: Jan :00: Jan :30:00.00 (UTCG) Evolution of the Right Ascension of the Ascending Node Two-Body: Constant J2: Secular Only HPOP 2x0: Periodic and Secular

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Pg 47 of 45 AGI 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

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Pg 48 of 45 AGI 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

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Pg 49 of 45 AGI 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

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Pg 50 of 45 AGI Choosing settings Table of ephemeris runs –Leo1 daye=0.001, 740km altitude, 14 revs per day –Meo3 dayscircular, 4 revs per day –Heo7 daysMolniya, 2 revs per day –Geo14 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

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Pg 51 of 45 AGI Choosing settings (cont’d) 1,344 cases total Truth model –RK78, 10 sec max step size Table of differences available from Help system

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Pg 52 of 45 AGI Take-aways MEO –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

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Pg 53 of 45 AGI 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)

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