Accuracy and Efficiency Comparison of Six Numerical Integrators for Propagating Perturbed Orbits

Abstract

We present the results of a comprehensive study in which the precision and efficiency of six numerical integration techniques, both implicit and explicit, are compared for solving the gravitationally perturbed two-body problem in astrodynamics. Solution of the perturbed two-body problem is fundamental for applications in space situational awareness, such as tracking orbit debris and maintaining a catalogue of over twenty thousand pieces of orbit debris greater than the size of a softball, as well as for prediction and prevention of future satellite collisions. The integrators used in the study are a 5th/4th and 8th/7th order Dormand-Prince, an 8th order Gauss-Jackson, a 12th/10th order Runga-Kutta-Nystrom, Variable-step Gauss Legendre Propagator and the Adaptive-Picard-Chebyshev methods. Four orbit test cases are considered, low Earth orbit, Sun-synchronous orbit, geosynchronous orbit, and a Molniya orbit. A set of tests are done using a high fidelity spherical-harmonic gravity (70 × 70) model with and without an exponential cannonball drag model. We present three metrics for quantifying the solution precision achieved by each integration method. These are conservation of the Hamiltonian for conservative systems, round-trip-closure, and the method of manufactured solutions. The efficiency of each integrator is determined by the number of function evaluations required for convergence to a solution with a prescribed accuracy. The present results show the region of applicability of the selected methods as well as their associated computational cost. Comparison results are concisely presented in several figures and are intended to provide the reader with useful information for selecting the best integrator for their purposes and problem specific requirements in astrodynamics.

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Notes

  1. 1.

    Given a set of n + 1 data points (x0,y0),…, (xn,yn), the Newton interpolating polynomial is written as N(x) = [y0] + [y0,y1](xx0) + ⋯ + [y0,…,yn](xx0)(xx1)⋯(xxn− 1), and [y0,…,yn] denotes divided differences, e.g. \([y_{0},y_{1}]= \frac {y_{1} - y_{0}}{x_{1} - x_{0}}\).

  2. 2.

    Generally, the gravitational potential is defined in ECEF coordinate system wherein the Earth rotation is accounted for.

  3. 3.

    RKN(12)10 was not tested with drag due to software limitations.

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Acknowledgements

The authors would like to thank Abhay Masher and Austin Probe for their help and technical advice. Additionally, we would like to thank Dr. Julie Moses, Dr. Stacie Williams (AFOSR) and Dr. Alok Das (AFRL) for their financial support. Finally, we would like thank the Egyptian Cultural and Educational Bureau in Washington, D.C. for supporting the first author’s visit to Texas A&M University in Summer 2016.

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Correspondence to Ahmed M. Atallah.

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This work was done as a private venture and not in Woollands’ capacity as an employee of the Jet Propulsion Laboratory, California Institute of Technology.

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Atallah, A.M., Woollands, R.M., Elgohary, T.A. et al. Accuracy and Efficiency Comparison of Six Numerical Integrators for Propagating Perturbed Orbits. J Astronaut Sci 67, 511–538 (2020). https://doi.org/10.1007/s40295-019-00167-2

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Keywords

  • Orbit propagation
  • Numerical integration
  • Picard-Chebyshev