Integrals of finite motion in the Schwarzschild gravitational field up to terms of order c^–2
Abstract
Up to terms containing c–2, simple analytical dependences are derived that describe the finite motion of a test particle in the Schwarzschild geometry. Such a motion is considered as a correction to the unperturbed Keplerian motion under the condition that the ratio of the Schwarzschild radius to the radial coordinate is small. In this approximation, conserved integrals are also found that characterise the orbital motion of the particle. For this, the equations of motion are presented in the Hamiltonian form, and a number of canonical transformations of the generalised coordinates and momenta are made, which make it possible to integrate these equations. Periodic and secular contributions are derived for the osculating elements of the test particle orbit: the mean anomaly, the periapsis argument, and the semi-major axis. An algorithm for calculating the position of a particle in the c–2 approximation is proposed, which is comparable in computational complexity to the algorithm for solving the standard Kepler problem. An estimate of the error of the obtained approximate solutions is made and the limits of their applicability are indicated.
References
2. Landau LD, Lifshits EM. Teoriya polya [The classical theory of fields]. Moscow: Nauka; 1973. 504 p. Russian.
3. Zel’manov AL, Agakov VG. Elementy obshchei teorii otnositel’nosti [Elements of general theory of relativity]. Moscow: Nauka; 1989. 240 p. Russian.
4. Hagihara Y. Theory of the relativistic trajectories in a gravitational field of Schwarzschild. Japanese Journal of Astronomy and Geophysics. 1931;8:67–176.
5. Duboshin GN. Nebesnaya mekhanika. Osnovnye zadachi i metody [Celestial mechanics. Basic problems and methods]. 2nd edition. Moscow: Nauka; 1968. 800 p. Russian.
6. Brouwer D. Solution of the problem of artificial satellite theory without drag. Astronomical Journal. 1959;64(1274):378–396.
Copyright (c) 2023 Journal of the Belarusian State University. Physics
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
The authors who are published in this journal agree to the following:
- The authors retain copyright on the work and provide the journal with the right of first publication of the work on condition of license Creative Commons Attribution-NonCommercial. 4.0 International (CC BY-NC 4.0).
- The authors retain the right to enter into certain contractual agreements relating to the non-exclusive distribution of the published version of the work (e.g. post it on the institutional repository, publication in the book), with the reference to its original publication in this journal.
- The authors have the right to post their work on the Internet (e.g. on the institutional store or personal website) prior to and during the review process, conducted by the journal, as this may lead to a productive discussion and a large number of references to this work. (See The Effect of Open Access.)