We study the general relativistic orbital equation and using a straightforward perturbation method and a mathematical device first introduced by d'Alembert, we work out approximate expressions of a bound planetary orbit in the form of trigonometrical polynomials and the first three terms of the power series development of the perihelion advance. The results are applied to a more precise determination of the total mass of the double pulsar J0737-3039.

Spatial geometry of the rotating disk and its non-rotating counterpart Am. J. Phys. 80, 772 (2012) The Standard Model appearing in Minute Physics video vignettes on YouTube youtube.com/watch?v=HVO0HgMi6Lc Phys. Teach. 50, 253 (2012) Embeddings and time evolution of the Schwarzschild wormhole Am. J. Phys. 80, 203 (2012) Schwarzschild and de Sitter solutions from the argument by Lenz and Sommerfeld Am. J. Phys. 79, 662 (2011)Visualizing circular motion around a Schwarzschild black hole Am.We derive the first-order orbital equation employing a complex variable formalism. We then examine Newton's theorem on precessing orbits and apply it to the perihelion shift of an elliptic orbit in general relativity. It is found that corrections to the inverse-square gravitational force law formally similar to that required by general relativity were suggested by Clairaut in the 18th century.

Einstein's perihelion advance formula can be given a geometric interpretation in terms of the curvature of the ellipse. The formula can be obtained by splitting the constant term of an auxiliary polar equation for an elliptical orbit into two parts that, when combined, lead to the expression of this relativistic effect. Using this idea, we develop a general method for dealing with orbital precession in the presence of central perturbing forces, and apply the method to the determination of the total (relativistic plus Newtonian) secular perihelion advance of the planet Mercury. V C 2015 American Association of Physics Teachers.

It is shown how the generalized Laplace coefficients can be employed to deduce explicit formulas for ordinary and Newcomb derivatives of the Laplace coefficients.

The elliptical orbit of the classical gravitational two-body problem can be determined by studying the free oscillations about a circular motion or the small motions around a fixed point in a rotating reference frame. In this last schematization we approximate the differential equation of motion by a succession of simple equations we solve iteratively, obtaining a piecemeal determination of the position vector r formally expressed in terms of Laurent polynomials, from which we quickly deduce the explicit time-dependent expressions in the form of complex trigonometric polynomials. This approach can also be used in the presence of perturbing forces and, by way of illustration, we study the effects of a small linear repulsive force on the elliptical orbit.