Periodic orbits about an oblate spheroid
1910; American Mathematical Society; Volume: 11; Issue: 1 Linguagem: Inglês
10.1090/s0002-9947-1910-1500856-2
ISSN1088-6850
Autores Tópico(s)Geophysics and Gravity Measurements
ResumoThe orbit of a particle about an oblate spheroid is not in general closed geoznetrically. The motion of the particle is not, therefore, in general, periodic from a geometric point of view. But if we consider the orbits as desclibed by the particle in a revolving meridian plane which passes constantly through the particle several classes of closed orbits can be found in which the motion is periodic. The failure of these orbits to close in space arises flom the incommensurability of the period of rotation of the line of nodes with the period of motion in the revolving plane. WNrhen these periods happen to be commensurable the orbits are closed in space and the motion is therefole peliodic, though the period may be very great. Indeed, it seems that snost of the difEculty in giving lnathematical expressions for the orbits about an oblate spheroid rests upon the questioll of incolnmensurability of peliods. The difficulty arising from the motion of the node can be overcome by the use of the revolving plane, but other incolnmensurabilities, such as tllat introduced by the eccentricity, can not be e]iminated in this manner. Orbits closed in the zevolving plane are considered most convelliently ill two general classes: (1) Those which reenter after one resrolution, (2) tllose which reenter after many revolutions. The existence of both classes is established in this paper and convenient methods for constructing the solutions are given. Orbits which reenter after the Srst revolution are naturally the simpler and will be considered in the first part of the paper. Those lying in the equatorial plane of the spheroid become straight lines ill the revolving p]ane, and it is shown that within the realm of convergence of the selies elnployed all orbits in the equatorial plane are periodic. WVhen the orbits do not lie in the equatorial plane there exists one, and only one, orbit for any arbitrarily assigned values of the inclination and the mean distance. These orbits reduce to cileles with the vanishing of the oblateness of the spheroid. In considering orbits which reenter only after sllany revolutions the differential equations are found to be very complex, and one would despair of proving the existence of these orbits by the ordinary direct computation of the necessary coefficients. However, a proof of their existence and a method for the construc55
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