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
Etienne Pellegrini - AAS/AIAA Spaceflight Mechanics Meeting - 1/27/14 - Santa Fe, NM
• 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
Etienne Pellegrini - AAS/AIAA Spaceflight Mechanics Meeting - 1/27/14 - Santa Fe, NM
• 2 masses in a circular orbit ; a massless particle is influenced by both.
• Equations of Motion (inertial frame):
The Circular Restricted 3-Body Problem
Etienne Pellegrini - AAS/AIAA Spaceflight Mechanics Meeting - 1/27/14 - Santa Fe, NM
• 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
Etienne Pellegrini - AAS/AIAA Spaceflight Mechanics Meeting - 1/27/14 - Santa Fe, NM
• Sundman type transformation:
• Modifies the equations of motion:
The Sundman transformation
Etienne Pellegrini - AAS/AIAA Spaceflight Mechanics Meeting - 1/27/14 - Santa Fe, NM
• 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
Etienne Pellegrini - AAS/AIAA Spaceflight Mechanics Meeting - 1/27/14 - Santa Fe, NM
• Recursion
– Differentiate
– Identify with
– Requires to be able to repeatedly differentiate 𝐹𝑛, 𝐺𝑛 , 𝐴𝑛, 𝐵𝑛, 𝑇𝑛
F&G CRTBP series: recursion equations
Etienne Pellegrini - AAS/AIAA Spaceflight Mechanics Meeting - 1/27/14 - Santa Fe, NM
• Plug in symbolic manipulation software generates
coefficient files
F&G CRTBP series: complete set of variables
Etienne Pellegrini - AAS/AIAA Spaceflight Mechanics Meeting - 1/27/14 - Santa Fe, NM
• 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
Etienne Pellegrini - AAS/AIAA Spaceflight Mechanics Meeting - 1/27/14 - Santa Fe, NM
• Scenario 1: Orbit around 𝑚1 (𝑋0 = −1.915 0 0 0 1.044045197 0 𝑇) [Broucke68]
Results: Three test scenarios
Etienne Pellegrini - AAS/AIAA Spaceflight Mechanics Meeting - 1/27/14 - Santa Fe, NM
Time (𝑠 = 1) 𝑠 = 𝑟1
𝑠 = 𝑟1𝑟2
• Scenario 2: Orbit around 𝑚2 (𝑋0 = 1. 713640573 0 0 0 − 0.633046910 0 𝑇)
Results: Three test scenarios
Etienne Pellegrini - AAS/AIAA Spaceflight Mechanics Meeting - 1/27/14 - Santa Fe, NM
Time (𝑠 = 1) 𝑠 = 𝑟2
𝑠 = 𝑟1𝑟2
Results: Three test scenarios
Etienne Pellegrini - AAS/AIAA Spaceflight Mechanics Meeting - 1/27/14 - Santa Fe, NM
• Scenario 3: From 𝑚2 to 𝑚1 (𝑋0 = 0.9 0 0 0 0.62 0.3 𝑇)
Results: Three test scenarios
Etienne Pellegrini - AAS/AIAA Spaceflight Mechanics Meeting - 1/27/14 - Santa Fe, NM
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
Etienne Pellegrini - AAS/AIAA Spaceflight Mechanics Meeting - 1/27/14 - Santa Fe, NM
Truth
Prescribed accuracy: 𝜖 = 0.001
• For each TS order, increase # of segments until accuracy is met
Results: Comparison algorithm
Etienne Pellegrini - AAS/AIAA Spaceflight Mechanics Meeting - 1/27/14 - Santa Fe, NM
F&G, 5 segments
𝜖1 = 10
Truth
Prescribed accuracy: 𝜖 = 0.001
• For each TS order, increase # of segments until accuracy is met
Results: Comparison algorithm
Etienne Pellegrini - AAS/AIAA Spaceflight Mechanics Meeting - 1/27/14 - Santa Fe, NM
F&G, 5 segments
F&G, 10 segments
𝜖1 = 10
𝜖2 = 3
Truth
Prescribed accuracy: 𝜖 = 0.001
• When specified accuracy is met, time the propagation
Results: Comparison algorithm
Etienne Pellegrini - AAS/AIAA Spaceflight Mechanics Meeting - 1/27/14 - Santa Fe, NM
F&G, 5 segments
F&G, 10 segments
F&G, 50 segments Truth
𝜖1 = 10
𝜖2 = 3
𝜖3 = 0.001
Prescribed accuracy: 𝜖 = 0.001
Results: Speedups for variable-step integration
Etienne Pellegrini - AAS/AIAA Spaceflight Mechanics Meeting - 1/27/14 - Santa Fe, NM
Scenario 1 Scenario 2
Results: Speedups for variable-step integration
Etienne Pellegrini - AAS/AIAA Spaceflight Mechanics Meeting - 1/27/14 - Santa Fe, NM
Scenario 3
Results: Speedups for fixed-step integration
Etienne Pellegrini - AAS/AIAA Spaceflight Mechanics Meeting - 1/27/14 - Santa Fe, NM
Scenario 1 Scenario 2
Results: Speedups for fixed-step integration
Etienne Pellegrini - AAS/AIAA Spaceflight Mechanics Meeting - 1/27/14 - Santa Fe, NM
Scenario 3
Results: Number of steps necessary for convergence
Etienne Pellegrini - AAS/AIAA Spaceflight Mechanics Meeting - 1/27/14 - Santa Fe, NM
Scenario 1 Scenario 2
Results: Number of steps necessary for convergence
Etienne Pellegrini - AAS/AIAA Spaceflight Mechanics Meeting - 1/27/14 - Santa Fe, NM
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
Etienne Pellegrini - AAS/AIAA Spaceflight Mechanics Meeting - 1/27/14 - Santa Fe, NM
Thank you for your attention! Any questions?
Etienne Pellegrini - AAS/AIAA Spaceflight Mechanics Meeting - 1/27/14 - Santa Fe, NM
• 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
Etienne Pellegrini - AAS/AIAA Spaceflight Mechanics Meeting - 1/27/14 - Santa Fe, NM