Remark on orbital precession due to central-force perturbations

Abstract

This is a comment on the recent paper by G. S. Adkins and J. McDonnell "Orbital precession due to central-force perturbations" published in Phys. Rev. D75 (2007), 082001 [arXiv:gr-qc/0702015]. We show that the main result of this paper, the formula for the precession of Keplerian orbits induced by central-force perturbations, can be obtained very simply by the use of Hamilton's vector.In the recent paper G. S. Adkins and J. McDonnell reconsidered the old problem of perihelion precession of Keplerian orbits under the influence of arbitrary central-force perturbations. Their main result is the formula for perihelion precession in the form of a one-dimensional integral convenient for numerical calculations.The reason why this well studied and essentially textbook problem [2] came into the focus of the current research is the recent usage of this classical effect to constrain hypothetical modifications of Newtonian gravity from higher dimensional models [3], as well as the density of dark matter in the solar system [4].Traditionally the simplest way to study the perihelion motion is the use of the Runge-Lenz vector [5,6]. The Runge-Lenz vectoris the extra constant of motion originated from the hidden symmetry of the Coulomb/Kepler problem [7]. Here α = GmM , L is the angular momentum vector and v is the relative velocity of a planet of mass m with respect to the Sun of mass M . Geometrically the Runge-Lenz vector points towards the perihelion. Therefore its precession rate is just the precession rate of the perihelion [8].However, we will use not the Runge-Lenz vector but its less known cousin, the Hamilton vector [9,10,11,12] where ϕ is the polar angle in the orbit plane. This very useful vector constant of motion of the Kepler problem was well known in the past, but mysteriously disappeared from textbooks after the first decade of the twentieth century [9,11,12,13]. Of course, A and u are not independent constants of motion. The relation between them isRemembering that the magnitude of the Runge-Lenz vector is A = α e, where e is the eccentricity of the orbit [5], we get from (3) the magnitude of the Hamilton vectorIf the potential U (r) contains a small central-force perturbation V (r) besides the Coulomb binding potential,the Hamilton vector (as well as the Runge-Lenz vector) ceases to be conserved and begins to precess with the same rate as the Runge-Lenz vector, because according to (3) the two vectors are perpendicular.To calculate the precession rate of the Hamilton vector we first find its time derivativėwhere µ = mM m+M is the reduced mass. To get (5), we have used Newton's equation of motion for˙ v and the equatioṅ e ϕ = −φ e r .

Perihelion precession from power law central force and magnetic-like force

“…it is the 43 00 =cy for the planet Mercury. Accidently, (13) and thus (25) is exact when n ¼ À4, which can be verified by use of (9).…”

“…To numerically calculate the precession with this method, it is actually enough to know the perturbation force as a function of planet position. In [13], O. I. Chashchina and Z. K. Silagadze expressed a similar view and advocated the use of another constant vector for an unperturbed orbit, i.e. the Hamilton vector, with which they also obtained the key one-dimensional integral form of precession in [12].…”

“…After this work was done, the author got to know [12,13]. In [12] G. S. Adkins and J. McDonnell used the perturbed orbit differential equation to discuss precessions due to central force perturbation, the main result is the precession in the form of a one-dimensional integral (see Eq.…”

“…The author has benefitted from discussions with Edna Cheung, Yuran Chen, Anke Knauf, Lingfei Wang, Tianheng Wang, Youhua Xu and Yun Zhang. Thanks go to Z. K. Silagadze for bringing [12,13] to the attention of F. X. This work is supported by the National Science Foundation of China under Grant No.…”

Regularization in Astrodynamics: applications to relative motion, low-thrust missions, and orbit propagation = Regularización en Astrodinámica: aplicaciones al movimiento relativo, misiones de bajo empuje, y propagación de órbitas

“…184-201) of the radial distance, and the special case of the Yukawa potential (Yukawa, 1935). Chashchina and Silagadze (2008) relied on Hamilton's vector to simplify the analytic solutions found by Adkins and McDonnell (2007). More elaborated potentials have been explored for modeling the perihelion precession (Schmidt, 2008).…”