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
The word dromo is derived from the old Greek word (dròmos) that means running.
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.
The possibility of introducing a time-element in our set of elements will be presented in a forthcoming paper.
This is the numerical integrator DOP853 which is described in Section II.5 of Hairer et al. (2009).
References
Arakida, H., Fukushima, T.: Long-term integration error of Kustaanheimo-Stiefel regularized orbital motion. Astron. J. 120(6), 3333–3339 (2000)
Arakida, H., Fukushima, T.: Long-term integration error of Kustaanheimo-Stiefel regularized orbital motion. II. Method of variation of parameters. Astron. J. 121(3), 1764–1767 (2001)
Battin, R.H.: An Introduction to the Mathematics and Methods of Astrodynamics. AIAA Education Series, AIAA, Reston, VA (1999)
Bombardelli, C., Baù, G., Peláez, J.: Asymptotic solution for the two-body problem with constant tangential thrust acceleration. Celest. Mech. Dyn. Astron 110(3), 239–256 (2011)
Bond, V.R.: The uniform, regular differential equations of the KS transformed perturbed two-body problem. Celest Mech 10(3), 303–318 (1974)
Bond, V.R.: Error propagation in the numerical solutions of the differential equations of orbital mechanics. Celest. Mech. 27, 65–77 (1982)
Bond, V.R., Allman, M.C.: Modern Astrodynamics: Fundamentals and Perturbation Methods. Princeton University Press, Princeton (1996)
Burdet, C.A.: Regularization of the two body problem. Zeitschrift für angewandte Mathematik und Physik 18(3), 434–438 (1967)
Burdet, C.A.: Theory of Kepler motion: the general perturbed two body problem. Zeitschrift für angewandte Mathematik und Physik 19(2), 345–368 (1968)
Burdet, C.A.: Le mouvement Keplerien et les oscillateurs harmoniques. Journal für die reine und angewandte Mathematik 238, 71–84 (1969)
Chelnokov, Y.N.: Application of quaternions in the theory of orbital motion of a satellite. I. Cosmic Res 30(6), 612–621 (1993)
Deprit, A.: Ideal elements for perturbed Keplerian motions. J. Res. Natl. Bur Standards 79B(1–2), 1–15 (1975)
Deprit, A., Elipe, A., Ferrer, S.: Linearization: Laplace vs. Stiefel. Celest. Mech. Dyn. Astron. 58(2), 151–201 (1994)
Ferrándiz, J.M.: A general canonical transformation increasing the number of variables with application in the two-body problem. Celest. Mech. 41(1–4), 343–357. (1987/1988)
Fukushima, T.: New two-body regularization. Astron. J. 133(1), 1–10 (2007a)
Fukushima, T.: Numerical comparison of two-body regularizations. Astron. J. 133(6), 2815–2824 (2007b)
Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I: Nonstiff Problems, vol. 1. Springer, Berlin (2009)
Hansen, P.A.: Auseinandersetzung einer Zweckmässigen Methode zur Berechnung der absoluten Störungen der Kleinen Planeten. Abh der Math-Phys Cl der Kon Sachs Ges der Wissensch 5, 41–218 (1857)
Kustaanheimo, P., Stiefel, E.L.: Perturbation theory of Kepler motion based on spinor regularization. Journal für die reine und angewandte Mathematik 218, 204–219 (1965)
Levi-Civita, T.: Questioni di meccanica classica e relativista. Zanichelli, Bologna (1924)
Peláez, J., Hedo, J.M., de Andrés, P.R.: A special perturbation method in orbital dynamics. In: Proceedings of the AAS/AIAA Space Flight Mechanics Meeting held January 23–27, 2005, Copper Mountain, Colorado, American Astronautical Society, Univelt, Inc., San Diego, CA, pp 1061–1078 (2005)
Peláez, J., Hedo, J.M., de Andrés, P.R.: A special perturbation method in orbital dynamics. Celest. Mech. Dyn. Astron. 97(2), 131–150 (2007)
Sharaf, M.A., Awad, M.E., Najmuldeen, S.A.A.: Motion of artificial satellites in the set of Eulerian redundant parameters (III). Earth Moon Planets 56(2), 141–164 (1992)
Sperling, H.: Computation of Keplerian Conic sections. Am. Rocket Soc. J. 31(5), 660–661 (1961)
Stiefel, E.L., Scheifele, G.: Linear and Regular Celestial Mechanics. Springer, New York (1971)
Stiefel, E., Rössler, M., Waldvogel, J., Burdet, C.A.: Methods of Regularization for Computing Orbits in Celestial Mechanics. Technical report, NASA CR-769, Washington, DC (1967)
Sundman, K.F.: Recherches sur le problème des trois corps. Acta Societatis Scientiarum Fennicae 34(6), 1–43 (1907)
Sundman, K.F.: Mémoire sur le problème des trois corps. Acta Mathematica 36(1), 105–179 (1912)
Szebehely, V.: Regularization in celestial mechanics. In: Oden J (ed) Computational Mechanics, Lecture Notes in Mathematics, vol 461, Springer, Berlin, pp 257–263, http://dx.doi.org/10.1007/BFb0074156 (1975)
Szebehely, V., Bond, V.: Transformations of the perturbed two-body problem to unperturbed harmonic oscillators. Celest. Mech. 30, 59–69 (1983)
Vallado, D.A.: Fundamentals of Astrodynamics and Applications, 2nd edn. Kluwer Academic Publishers, Dordrecht (2001)
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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Appendix I
Appendix I
1.1 Matrices \(Q_\mathcal{RI }\) and \(Q_{0}\)
We write below the expressions of the elements of the matrix \(Q_\mathcal{RI }\):
We write below the expressions of the elements of the matrix \(Q_{0}\):
1.2 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:
If additionally \(\zeta _{6}=0\) Eqs. (69) and (70) are singular and we can use instead:
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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DOI: https://doi.org/10.1007/s10569-013-9475-x