New series expansion for the periapsis shift

Abstract

We propose a prescription for a new series expansion of the periapsis shift. The prescription formulates the periapsis shift in various spacetimes analytically without using special functions and provides simple and highly accurate approximate formulae. We derive new series representations for the periapsis shift in the Kerr and the Chazy–Curzon spacetimes by using the prescription, where the expansion parameter is defined as the eccentricity divided by the non-dimensional quantity that vanishes in the limit of the innermost stable circular orbit (ISCO). That is to say, the expansion parameter characterizes both the eccentricity of the orbit and its proximity to the ISCO. The smaller the eccentricity, the higher the accuracy of the formulae that are obtained by truncating the new series representations up to a finite number of terms. If the eccentricity is sufficiently small, the truncated new representations have higher accuracy than the post-Newtonian (PN) expansion formulae even in strong gravitational fields where the convergence of the PN expansion formula is not guaranteed. On the other hand, even if the orbit is highly eccentric, the truncated new representations have comparable or higher accuracy than the PN expansion formulae if the semi-major axis is sufficiently large. An exact formula for the periapsis shift of the quasi-circular orbit in the Chazy–Curzon spacetime is also obtained as a special case of the new series representation.