f and g taylor series solutions to the circular restricted three-body problem

F and G Taylor series solutions to the Circular Restricted Three-Body Problem

Etienne Pellegrini, and Ryan P. Russell

AAS/AIAA Spaceflight Mechanics Meeting

Santa Fe, NM, 1/27/14

Examples of three body trajectories,

propagated using the F&G CRTBP series

• Introduction

• The Circular Restricted Three Body Problem

• The Sundman Transformation

• Derivation of the F&G CRTBP series solutions

• Numerical results

– Three test scenarios

– Variable-Step Integration

– Fixed-Step Integration

• Conclusions and future work

Summary

Etienne Pellegrini - AAS/AIAA Spaceflight Mechanics Meeting - 1/27/14 - Santa Fe, NM

• F&G series:

– Taylor series of the f and g Lagrange functions

– Accurate way of propagating the 2-body problem and the Stark problem [Pellegrini2014]

Adapt the method to the CRTBP

Introduction


• Why the CRTBP?

– Applicable to many dynamical systems

– 3-body dynamics is of great interest for novel trajectory optimization software

– Power series solutions have been developed, but “F&G” type solutions have not been found in literature

r

v

r0

v0

• Use of the Sundman transformation

– Avoiding singularities and undesirable numerical behavior due to close approaches to the celestial bodies

– Efficient discretization schemes

Goal of this work

Apply the classic F&G technique to the CRTBP, and evaluate the resulting integration method

Main contributions

• Develop recursion formulas for the F&G CRTBP series & demonstrate their validity.

• Investigate the F&G CRTBP series’ performance

Introduction


• 2 masses in a circular orbit ; a massless particle is influenced by both.

• Equations of Motion (inertial frame):

The Circular Restricted 3-Body Problem


• Introduced in 1912 by Karl F. Sundman. Used in astrodynamics because it reduces instability, and helps removing collision singularities. [Szebehely67].

• The transformation slows time down as the particle gets close to a singularity

• Szebehely: “The introduction of a new independent variable, while regularizing the restricted problem, results in increased complexity of the equations of motion. […] The remedy is to increase the complexity of the regularizing transformations.”

Birkhoff, Thiele and Burrau, Levi-Civita, Lemaitre, Kustaanheimo-Stiefel,…

The Sundman transformation


• Sundman type transformation:

• Modifies the equations of motion:

The Sundman transformation


• Based on classic F&G derivation [Bate, Mueller,White 1971]

• Extra basis vector for out-of-plane motion ; chose to add 2!

• Extra series for keeping track of time

• Have to compute 𝐹𝑛, 𝐺𝑛 , 𝐴𝑛, 𝐵𝑛 , 𝑇𝑛

F&G CRTBP series


• Recursion

– Differentiate

– Identify with

– Requires to be able to repeatedly differentiate 𝐹𝑛, 𝐺𝑛 , 𝐴𝑛, 𝐵𝑛, 𝑇𝑛

F&G CRTBP series: recursion equations


• Plug in symbolic manipulation software generates

coefficient files

F&G CRTBP series: complete set of variables


• Benchmarks

– Variable-step integration

• (7)8th order Runge-Kutta-Fehlberg (RKF(7)8)

– Fixed-step integration

• 8th order Runge-Kutta-Fehlberg (RKF8)

– Inertial frame propagation (EOMs presented previously)

• Setup Details

– Software specifications

• Implemented in Fortran

• Compiler: gfortran v4.7.0

• F&G CRTBP coefficient files obtained using Matlab 17

– Hardware specifications

• Processor: quad-core Intel Xeon W3550

• 3.07GHz clock-speed

• 6GB RAM

Numerical Results


• Scenario 1: Orbit around 𝑚1 (𝑋0 = −1.915 0 0 0 1.044045197 0 𝑇) [Broucke68]

Results: Three test scenarios


Time (𝑠 = 1) 𝑠 = 𝑟1

𝑠 = 𝑟1𝑟2

• Scenario 2: Orbit around 𝑚2 (𝑋0 = 1. 713640573 0 0 0 − 0.633046910 0 𝑇)



Time (𝑠 = 1) 𝑠 = 𝑟2

𝑠 = 𝑟1𝑟2

• Scenario 3: From 𝑚2 to 𝑚1 (𝑋0 = 0.9 0 0 0 0.62 0.3 𝑇)



Time (𝑠 = 1) 𝑠 = 𝑟1

𝑠 = 𝑟1𝑟2 𝑠 = 𝑟2

• Compute truth using RKF(7)8 with quad precision and a low tolerance

Results: Comparison algorithm


Truth

Prescribed accuracy: 𝜖 = 0.001

• For each TS order, increase # of segments until accuracy is met



F&G, 5 segments

𝜖1 = 10

Truth


• For each TS order, increase # of segments until accuracy is met



F&G, 5 segments

F&G, 10 segments

𝜖1 = 10

𝜖2 = 3

Truth


• When specified accuracy is met, time the propagation



F&G, 5 segments

F&G, 10 segments

F&G, 50 segments Truth

𝜖1 = 10

𝜖2 = 3

𝜖3 = 0.001


Results: Speedups for variable-step integration


Scenario 1 Scenario 2

Results: Speedups for variable-step integration


Scenario 3

Results: Speedups for fixed-step integration



Results: Speedups for fixed-step integration


Scenario 3

Results: Number of steps necessary for convergence



Results: Number of steps necessary for convergence


Scenario 3

• F&G CRTBP series are developed and the validity of the solution is demonstrated.

• The method has comparable performance to that of RKF (w/up to 4 times speedups in some cases).

• The Sundman type transformations improve the fixed-step propagations Reduce the number of steps, better discretization

• The RKF benefits more from the Sundman transformation than the F&G CRTBP series (increased complexity) Decreases efficiency of the series

• Future work

– Development of a series solutions using a more complex regularization technique

Conclusions & Future work


Thank you for your attention! Any questions?


• Szebehely, V.G.: Theory of Orbits, The Circular Restricted Three-Body Problem. Academic Press, New York, NY (1967).

• Szebehely, V.G., Peters, F.: Complete Solution of a General Problem of Three Bodies. Astron. J. 72, 876 – 883 (1967).

• Broucke, R.: Periodic Orbits in the Restricted Three-Body Problem with Earth-Moon Masses. , Pasadena, California (1968).

• R. R. Bate, D. D. Mueller, and J. E. White, Fundamentals of Astrodynamics. New-York, NY. Dover Publications, 1971.

• Pellegrini, E., Russell, R.P., Vittaldev, V.: F and G Taylor Series Solutions to the Stark Problem with Sundman Transformations. Celestial Mechanics and Dynamical Astronomy (to appear)

References


f and g taylor series solutions to the circular restricted three-body problem

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