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A new set of integrals of motion to propagate the perturbed two-body problem

Abstract

A formulation of the perturbed two-body problem that relies on a new set of orbital elements is presented. The proposed method represents a generalization of the special perturbation method published by Peláez et al. (Celest Mech Dyn Astron 97(2):131–150, 2007) for the case of a perturbing force that is partially or totally derivable from a potential. We accomplish this result by employing a generalized Sundman time transformation in the framework of the projective decomposition, which is a known approach for transforming the two-body problem into a set of linear and regular differential equations of motion. Numerical tests, carried out with examples extensively used in the literature, show the remarkable improvement of the performance of the new method for different kinds of perturbations and eccentricities. In particular, one notable result is that the quadratic dependence of the position error on the time-like argument exhibited by Peláez’s method for near-circular motion under the \(J_{2}\) perturbation is transformed into linear. Moreover, the method reveals to be competitive with two very popular element methods derived from the Kustaanheimo-Stiefel and Sperling-Burdet regularizations.

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Notes

  1. 1.

    The word dromo is derived from the old Greek word (dròmos) that means running.

  2. 2.

    Note for instance that setting \(\mathcal U \ne 0\) the resulting dependent variables \(\left(\zeta _{1},\ldots ,\zeta _{7}\right)\) are not constant in the unperturbed motion.

  3. 3.

    The possibility of introducing a time-element in our set of elements will be presented in a forthcoming paper.

  4. 4.

    This is the numerical integrator DOP853 which is described in Section II.5 of Hairer et al. (2009).

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Acknowledgments

The study has been supported by the research project “Dynamic Simulation of Complex Space Systems” supported by the Dirección General de Investigación of the (former) Spanish Ministry of Innovation and Science through contract AYA2010-18796.

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Correspondence to Giulio Baù.

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Appendix I

Appendix I

Matrices \(Q_\mathcal{RI }\) and \(Q_{0}\)

We write below the expressions of the elements of the matrix \(Q_\mathcal{RI }\):

$$\begin{aligned}&Q_\mathcal{RI }\left(1,1\right)=\frac{R_{X}}{r},\\&Q_\mathcal{RI }\left(2,1\right)=\frac{R_{Y}}{r},\\&Q_\mathcal{RI }\left(3,1\right)=\frac{R_{Z}}{r},\\&Q_\mathcal{RI }\left(1,2\right)=\frac{H_{Y}R_{Z}-H_{Z}R_{Y}}{hr},\\&Q_\mathcal{RI }\left(2,2\right)=\frac{H_{Z}R_{X}-H_{X}R_{Z}}{hr},\\&Q_\mathcal{RI }\left(3,2\right)=\frac{H_{X}R_{Y}-H_{Y}R_{X}}{hr},\\&Q_\mathcal{RI }\left(1,3\right)=\frac{H_{X}}{h},\\&Q_\mathcal{RI }\left(2,3\right)=\frac{H_{Y}}{h},\\&Q_\mathcal{RI }\left(3,3\right)=\frac{H_{Z}}{h}. \end{aligned}$$

We write below the expressions of the elements of the matrix \(Q_{0}\):

$$\begin{aligned}&Q_{0}\left(1,1\right)=\frac{R_{X}}{r}\cos \triangle \phi -\left(\frac{H_{Y}R_{Z}-H_{Z}R_{Y}}{hr}\right)\sin \triangle \phi ,\\&Q_{0}\left(2,1\right)=\frac{R_{Y}}{r}\cos \triangle \phi -\left(\frac{H_{Z}R_{X}-H_{X}R_{Z}}{hr}\right)\sin \triangle \phi ,\\&Q_{0}\left(3,1\right)=\frac{R_{Z}}{r}\cos \triangle \phi -\left(\frac{H_{X}R_{Y}-H_{Y}R_{X}}{hr}\right)\sin \triangle \phi ,\\&Q_{0}\left(1,2\right)=\frac{R_{X}}{r}\sin \triangle \phi +\left(\frac{H_{Y}R_{Z}-H_{Z}R_{Y}}{hr}\right)\cos \triangle \phi ,\\&Q_{0}\left(2,2\right)=\frac{R_{Y}}{r}\sin \triangle \phi +\left(\frac{H_{Z}R_{X}-H_{X}R_{Z}}{hr}\right)\cos \triangle \phi ,\\&Q_{0}\left(3,2\right)=\frac{R_{Z}}{r}\sin \triangle \phi +\left(\frac{H_{X}R_{Y}-H_{Y}R_{X}}{hr}\right)\cos \triangle \phi ,\\&Q_{0}\left(1,3\right)=\frac{H_{X}}{h},\\&Q_{0}\left(2,3\right)=\frac{H_{Y}}{h},\\&Q_{0}\left(3,3\right)=\frac{H_{Z}}{h}. \end{aligned}$$

Expressions of \(\zeta _{4}, \zeta _{5}\) and \(\zeta _{6}\) when \(\zeta _{7}=0\)

In the case that \(\zeta _{7}=0\) Eqs. (55)–(57) become singular and we can use instead:

$$\begin{aligned} \zeta _{4}&= \frac{Q_{0}\left(1,3\right)}{2\zeta _{6}}, \end{aligned}$$
(69)
$$\begin{aligned} \zeta _{5}&= \frac{Q_{0}\left(2,3\right)}{2\zeta _{6}}, \end{aligned}$$
(70)
$$\begin{aligned} \zeta _{6}&= \pm \sqrt{\frac{Q_{0}\left(3,3\right)+1}{2}}. \end{aligned}$$

If additionally \(\zeta _{6}=0\) Eqs. (69) and (70) are singular and we can use instead:

$$\begin{aligned} \zeta _{4}=\pm \sqrt{\frac{1-Q_{0}(2,2)}{2}}, \quad \quad \zeta _{5}=\frac{Q_{0}(1,2)}{2\zeta _{4}}. \end{aligned}$$

Finally, if also \(\zeta _{4}=0\), then we have \(\zeta _{5}=\pm 1\).

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Baù, G., Bombardelli, C. & Peláez, J. A new set of integrals of motion to propagate the perturbed two-body problem. Celest Mech Dyn Astr 116, 53–78 (2013). https://doi.org/10.1007/s10569-013-9475-x

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Keywords

  • Perturbed two-body problem
  • Regularization
  • Generalized orbital elements
  • Orbit propagation
  • Linearization