General solution of the problem of the motion of an artificial satellite in the normal field of the earth's attraction

Aksenov, Ye. P.; Grebenikov, Ye.A.; Demin, V. G.

doi:10.1016/0032-0633(62)90052-1

Cited by 15 publications

(7 citation statements)

References 4 publications

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“…When this integrable problem is formulated with complexconjugate masses separated by an imaginary distance, Darboux found that the potential of attraction always assumes real values (cf. Lukyanov et al 2005); furthermore, this potential matches the potential of an oblate body by a suitable selection of the free parameters of the model (Aksenov et al 1961).…”

Section: Introductionmentioning

confidence: 78%

Satellite onboard orbit propagation using Deprit’s radial intermediary

Gurfil

Lara²

2014

Celest Mech Dyn Astr

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Short-term satellite onboard orbit propagation is required when GPS position measurements are unavailable due to an obstruction or a malfunction. In this paper, it is shown that natural intermediary orbits of the main problem provide a useful alternative for the implementation of short-term onboard orbit propagators instead of direct numerical integration. Among these intermediaries, Deprit's radial intermediary (DRI), obtained by the elimination of the parallax transformation, shows clear merits in terms of computational efficiency and accuracy. Indeed, this proposed analytical solution is free from elliptic integrals, as opposed to other intermediaries, thus speeding the evaluation of corresponding expressions. The only remaining equation to be solved by iterations is the Kepler equation, which in most of cases does not impact the total computation time. A comprehensive performance evaluation using Monte-Carlo simulations is performed for various orbital inclinations, showing that the analytical solution based on DRI outperforms a Dormand-Prince fixed-step Runge-Kutta integrator as the inclination grows.

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Section: Introductionmentioning

confidence: 78%

Satellite onboard orbit propagation using Deprit’s radial intermediary

Gurfil

Lara²

2014

Celest Mech Dyn Astr

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“…Newtonian systems can therefore serve as simple toy-models whose study as surrogates of Kerr and non-Kerr spacetimes offers valuable insights and opens interesting questions. For example, in Newtonian gravity, the quadratic constant is known to extend beyond equatorial plane symmetry (Aksenov, Grebenikov, & Demin 1963;Lukyanov et al 2005;Lynden-Bell 2003). Since the Kerr family is equatorially symmetric, this raises the question of whether other stationary axisymmetric solutions to the vacuum Einstein equations exist that still admit a quadratic constant of motion but are not equatorially symmetric.…”

Section: Discussionmentioning

confidence: 99%

“…In the oblate case, this is accomplished by consider-ing two Newtonian fixed centers with complex-conjugated masses located at constant imaginary distance. The resulting Darboux-Gredeaks potential has been used to approximate the gravitational field around other oblate planets (Aksenov, Grebenikov, & Demin 1963, Lukyanov et al 2005. Lynden-Bell (2003) has provided a simple and elegant derivation of the quadratic constant in the Euler problem, by noting that the kinetic part of the constant is the dot product of the angular momenta about the two fixed centers of attraction.…”

Section: Introductionmentioning

confidence: 99%

Constants of motion in stationary axisymmetric gravitational fields

Markakis

2014

Monthly Notices of the Royal Astronomical Society

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The motion of test particles in stationary axisymmetric gravitational fields is generally nonintegrable unless a nontrivial constant of motion, in addition to energy and angular momentum along the symmetry axis, exists. The Carter constant in Kerr-de Sitter spacetime is the only example known to date. Proposed astrophysical tests of the black-hole no-hair theorem have often involved integrable gravitational fields more general than the Kerr family, but the existence of such fields has been a matter of debate. To elucidate this problem, we treat its Newtonian analogue by systematically searching for nontrivial constants of motion polynomial in the momenta and obtain two theorems.First, solving a set of quadratic integrability conditions, we establish the existence and uniqueness of the family of stationary axisymmetric potentials admitting a quadratic constant. As in Kerr-de Sitter spacetime, the mass moments of this class satisfy a "no-hair" recursion relation M 2l+2 = a 2 M 2l , and the constant is Noetherrelated to a second-order Killing-Stäckel tensor. Second, solving a new set of quartic integrability conditions, we establish nonexistence of quartic constants. Remarkably, a subset of these conditions is satisfied when the mass moments obey a generalized "no-hair" recursion relation M 2l+4 = (a 2 + b 2 )M 2l+2 − a 2 b 2 M 2l . The full set of quartic integrability conditions, however, cannot be satisfied nontrivially by any stationary axisymmetric vacuum potential.

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“…Although for close artificial satellites, it gives a less accurate result than the Vinti problem does, nevertheless, because of its simplicity, it is applicable in constructing the theory of motion of distant satellites of the Earth and, possibly, with great success in the theory of motion of artificial satellites of planets. Proceeding from the classical problem of two fixed centers, E. P. Aksenov, E. A. Grebenikov, and V. G. Demin, as a result of the generalization of the problem mentioned above, clearly demonstrate that the integrable cases of Vinti, Barrara, and Kislik represent particular or limiting cases of a single general problem [3,4]. The generalized problem of two fixed centers (sometimes called the Euler problem) is at present the basis of the modern analytical theory of the motion of artificial satellites of the Earth [5,6].…”

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confidence: 88%