The nonlinear trajectory equation (Binet’s equation) for a particle in a relativistic force field can only be solved numerically or, alternatively, by using a perturbational solution scheme. The latter approach was successfully applied by Albert Einstein in 1915 to deduce the celebrated formula that explains the anomalous precession of the perihelion of Mercury. In this article, Binet’s equation for Mercury is solved numerically to a high degree of accuracy (16 decimal digits). This is a necessary comparison basis for the main goal of this work – to deduce a simple analytic formula that perfectly reproduces the real relativistic trajectory. Several analytical models are proposed, and the main goal has been indeed achieved. Moreover, the fitting parameters for model D described in Section 3 can be obtained independently of the solution of Binet’s equation. Thus, we can say that the highly accurate relativistic trajectory (the largest discrepancy being about 30 cm) can be obtained without actually solving the nonlinear differential equation for this trajectory.
1. Binet, J. M. Mémoire sur la détermination des orbites des planètes et des comètes (Memoir on the determination of the orbits ofplanets and comets). Journal de l’École Polytechnique, 1831, 13, 249–288.
2. Newton, Sir I. Mathematical Principles of Natural Philosophy and his System of the World (translated by Motte, A., 1729). 8th printing. University of California Press, 1974.
3. Whittaker, E. T. A Treatise on the Analytical Dynamics of Particles and Rigid Bodies. 2nd ed. Cambridge University Press, 1917.
4. Einstein, A. Erklärung der Perihelbewegung des Merkur aus der allgemeinen Relativitätstheorie (Explanation of the perihelion motion of Mercury from the general theory of relativity). König. Preuss. Akad. Wiss., 1915, XLVII, 831–839. An English translation is available at:
http://einsteinpapers.press.princeton.edu/vol6-trans/124
5. Saca, J. M. An exact solution to the relativistic advance of perihelion: correcting the Einstein approximation. Astrophys. Space Sci., 2008, 315, 365.
https://doi.org/10.1007/s10509-008-9785-8
6. Mercury facts.
https://science.nasa.gov/mercury/facts/ (accessed 2025-04-30).
7. D’Inverno, R. Introducing Einstein’s Relativity. Clarendon Press, Oxford, 1992.
8. Thornton, S. T. and Marion, J. B. Classical Dynamics of Particles and Systems. 5th ed. Brooks/Cole, 2004.
9. Shahid-Saless, B. and Yeomans, D. K. Relativistic effects on the motion of asteroids and comets. Astron. J., 1994, 107, 1885–1889.
https://doi.org/10.1086/116999
10. Schwarzschild, K. Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie (On the gravitational field of a point-mass according to Einstein’s theory). Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften, 1916, 7, 189–196.
11. Selg, M. An interesting class of exact solutions to Binet’s equation, with references to Newton’s and Einstein’s theories of gravity. Eur. J. Phys., 2025, 46, 015002.
https://doi.org/10.1088/1361-6404/ad8f7d
12. Andersen, P. B. What GR predicts for the perihelion advance of planets. 2023.
https://www.paulba.no/pdf/GRPerihelionAdvance.pdf
13. Simon, J. L., Bretagnon, P., Chapront, J., Chapront-Touze, M., Francou, G. and Laskar, J. Numerical expressions for precession formulae and mean elements for the Moon and the planets. Astron. Astrophys., 1994, 282, 663–683.
14. Cheng, T.-P. Relativity, Gravitation and Cosmology. Oxford University Press, 2005.
15. Runge–Kutta method.
https://keisan.casio.jp/exec/system/1548123555 (in Japanese) (accessed 2025-04-30).
16. Plaskota, L. Automatic integration using asymptotically optimal adaptive Simpson quadrature. Numer. Math., 2015, 131, 173–198.
https://doi.org/10.1007/s00211-014-0684-3
17. UBASIC. https://en.wikipedia.org/wiki/UBASIC
