An analytical solution of the Lagrange equations valid also for very low eccentricities: influence of a central potential.

*(English)*Zbl 1145.70014Summary: This paper presents an analytic solution of the equations of motion of an artificial satellite, obtained using non singular elements for eccentricity. The satellite is under the influence of the gravity field of a central body, expanded in spherical harmonics up to an arbitrary degree and order. We discuss in details the solution we give for the components of the eccentricity vector. For each element, we have divided the Lagrange equations into two parts: the first part is integrated exactly, and the second part is integrated with a perturbation method. The complete solution is the sum of the so-called “main” solution and of the so-called “complementary” solution. To test the accuracy of our method, we compare it to numerical integration and to the method developed in W. M. Kaula [Theory of satellite geodesy, New York: Blaisdell Publ. Co. (1966), reprint Dover (2000; Zbl 0973.86001)], expressed in classical orbital elements. For eccentricities which are not very small, the two analytical methods are almost equivalent. For low eccentricities, our method is much more accurate.

##### Keywords:

analytic integration; artificial satellites; eccentricity vector; short and long period terms; spherical harmonics; zero eccentricities
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\textit{F. Deleflie} et al., Celest. Mech. Dyn. Astron. 94, No. 1, 105--134 (2006; Zbl 1145.70014)

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##### References:

[2] | Deleflie F., Métris G., Exertier P. (2006). Long-period variations of the eccentricity vector valid also for near-circular orbits around a non-spherical body, Celest. Mech. Dynam. Astron., this issue. · Zbl 1145.70013 |

[3] | Demailly J.-P. (1996). Analyse numérique et équations différentielles, EDP Sciences. |

[10] | Tisserand F. 1890, Traité de Mécanique Céleste, Gauthiers Villards, reprinted. 1960–1962. |

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