An improved singular perturbation solution for bound geodesic orbits in the Schwarzschild metric (Q792647)

This paper is an interesting attempt to free the derivation of an approximate solution for bound geodesic orbits in the Schwarzschild spacetime from the secular terms proportional to the angular coordinate \(\Phi\). It is shown that the Lindstedt-Poincaré method of strained parameters (using a new angular variable \(\psi =(1+\epsilon \omega_ 1+\epsilon^ 2\omega_ 2+...)\Phi\), \(\epsilon =3m^ 2/h^ 2)\) still introduces secular terms limiting the range of validity of the solution. The improved method suggested in paragraph 3 eliminates secular terms by considering the expansion of the coefficient of u in the original equation: \(d^ 2u/d\Phi^ 2+u=1+\epsilon u^ 2\), according to \(1=\omega^ 2_ 0+\epsilon \omega_ 1+\epsilon^ 2\omega_ 2+..\).. This method seems to be more accurate than the existing perturbation methods, up to terms of order \(\epsilon^ 2\), for \(\epsilon>10^{-4}\) when one considers the perihelion precession.