Expanding the range of validity of the simplest computation of the perihelion precession in Schwarzschild spacetime

Among the several methods to compute the perihelion precession for bounded orbits in Schwarzschild spacetime, the simplest is to ignore a term in the equations of motion. This is currently justified under the assumption that the eccentricity of the orbit is small. For cases such as Mercury in our solar system, whose eccentricity is not small, this method seems not to be applicable. Yet it gives the right result, the reason being that the term that has been excluded, although responsible for first order—in the ratio of the Schwarzschild radius over the radial coordinate—corrections of the orbit, only produces completely negligible higher order corrections for the perihelion precession. We show this result by two different procedures. We claim, therefore, that as long as the aim of the computation is the perihelion precession, one can safely drop that term regardless of the magnitude of the eccentricity.