Investigation of the motion of test particles in the gravitational field of axially symmetric central body in classical physics
Keywords:
Gravitational potential, Poisson equation, Maclaurin spheroid, quasicleple problem, quadrupole moment, perihelion displacementAbstract
The article deals with an axially symmetric body and examines its internal and external gravitational field within the framework of classical theory of gravity. The Maclaurin spheroid is used as a deformed body to represent objects with homogeneous density and rigid rotation, hence, axially symmetric bodies. The gravitational potential is derived from the Poisson equation for the external and internal fields, satisfying the boundary conditions at the center, on the surface of the body, and at infinity. The Poisson equation is solved analytically and exactly, applying the Green's function and the expansion into spherical harmonics (spherical functions). Moreover, the matching on the surface of the body for small deformations is shown as an example. In addition, the quadrupole moment of a deformed central object is considered and its influence on the motion of test bodies (particles) in the field of a given object is investigated. The quasi-Kepler problem is solved numerically in the Wolfram Mathematica program. It was shown that numerical calculations correspond to the analytical solution of the quasi-Kepler problem in the equatorial plane of the orbit. The perihelion shift of the planets of the solar system were also analyzed.
The article pursues scientific, methodological and academic goals and is intended for a wide audience of students, graduates and doctoral students in the specialties of physics, mechanics and astronomy.
References
2 C.W. Misner, K.S. Thorne, and J.A. Wheeler, Gravitation ( San Francisco: W. H. Freeman Press, 1973), 1278 p.
3 H.C. Ohanian and R. Ruffini Gravitation and Spacetime, 3rd ed. (Cambridge University Press, Cambridge, England, 2013), 528 p.
4 L.D. Landau and E.M. Lifshitz, Teoreticheskaja fizika, tom II. Teorija polja. 8th ed., (Moscow, Fizmatlit, 2006), 536 s. (in Russ)
5 S. Shapiro and S. Tʹjukolski Chernye dyry, belye karliki i nejtronnye zvezdy, (Moscow, Mir, 1985), 300 s. (in Russ).
6 J. Lense and H.Thirring, Phys. Z. 19, 156 (1918).
7 V.A. Fok, Teorija prostranstva, vremeni i tjagotenija (Moscow, Nauka, 1961), 569 s. (in Russ)
8 R.M. Wald, General Relativity, (The University of Chicago Press, 1984), 473 p.
9 M.P. Hobson, U G.P. Efstathio and A.N. Lazenby, General Relativity, An Introduction for Physicists, (Cambridge University Press, 2006), 592 p.
10 L. Ryder Introduction to General Relativity, (Cambridge University Press, 2009), 460 p.
11 B.F. Schutz, A First course in General Relativity, (Cambridge University Press, 2009), 412 p.
12 H. Stephani, D. Kramer, M.A H. MacCallum, C. Hoenselaers, and E. Herlt Exact Solutions of Einstein’s Field Equations, (Cambridge University Press, Cambridge, UK, 2003).
13 H. Queverdo and B. Mashhoon, Phys Letters A, 109 (1, 2), 13-18 (1985).
14 H. Queverdo and B. Mashhoon, Phys Letters A, 148, 149-153 (1990).
15 L.A. Pachon, J.A. Rueda and J.D. Sanabria-Gomez, Phys.Rev.D, 73, 104038 (2006).
16 V.S. Manko, J.D. Sanabria-Gomez and O.V. Manko, Phys. Rev. D, 62, 044048 (2000).
17 Boshkayev K., Quevedo H. and Ruffini R. Gravitational Field of Compact Objects in General Relativity, Phys. Rev. D, 2012, Vol. 86, 064043.
18 S. Chandrasekhar, Ellipsoidal Figures of Equilibrium, (Yale University Press, New Haven, CT, 1967).
19 R. Meinel, M. Ansorg, A. Kleinwachter, G. Neugebauer and D. Petroff, Relativistic Figures of Equilibrium, (Cambridge University Press, Cambridge, England, 2008).
20 J.D. Jackson Classical Electrodynamics, 3rd ed., (New York: John Wiley & Sons, 1999).
21 A.N. Tihonov and A.A. Samarskij Uravnenija matematicheskoj fiziki, (Moscow: Nauka, 1977), 735 p. (in Russ).
22 M. Abramowitz and I.A. Stegun Handbook of Mathematical Functions, (Dover Publications, Inc. New York, NY, USA, 1974).
23 E. Poisson and C.M. Will Gravity: Newtonian, Post-Newtonian, relativistic, (Cambridge: Cambridge University Press, 2014).
24 L.D. Landau and E. M. Lifshitz, Teoreticheskaja fizika», tom I. Mehanika, 5th ed., (Moscow: Fizmatlit, 2012), 224 s. (in Russ)
25 M.M. Abdilʹdin, Problema dvizhenija tel v obshej teorii otnositelʹnosti, (Almaty: Qazaq Universitetі, 2006), 132 s. (in Russ)
26 K.A. Boshkayev, Zh.А. Kalymova, N.S. Abdualiyeva, Zh.N. Brisheva and А.S. Taukenova, Rec.Contr.Phys., 1 (64), 67-80 (2018). (in Kaz).
27 F.P. Pijpers, Mon. Not. R. Astron. Soc. 297, L76–L80 (1998).
28 S. Godier and J. Rozelot, Astron. Astrophys., 350, 310–317 (1999).
29 C.M. Will, Living Reviews in Relativity 17, 4 (2014).
30 A. Fienga, H. Manche, J. Laskar and M. Gastineau, Celestial Mechanics and Dynamical Astronomy, 123, 325 (2015).
31 R.S. Park, W.M. Folkner and A.S. Konopliv, et al., Astrophysical Journal, 153, 121 (2017).
32 Wolfram Mathematica: https://www.wolfram.com/mathematica/
33 M.M. Abdilʹdin, Mehanika teorii gravitacii Ejnshtejna, (Alma-Ata: Nauka, 1988), 200 s. (in Russ)
34 V.A. Brumberg Reljativistskaja nebesnaja mehanika, (Moscow: Nauka, 1972). (in Russ)
35 Orbital parameters of planets of solar system: http://www.sai.msu.ru/neb/rw/natsat/plaorbw.htm
36 C.M. Will, Theory and experiment in gravitational physics, Revised edition, (Cambridge University Press, 1993), 396 p.
37 C.M. Will, Living Reviews in Relativity 9, 3 (2006).
