Approximate Matching of Analytic and Numerical Solutions for Rapidly Rotating Neutron Stars

TL;DR

This paper compares an analytic solution for rapidly rotating neutron stars with numerical solutions, focusing on matching quadrupole moments and analyzing innermost stable circular orbits to evaluate the analytic model's accuracy.

Contribution

It introduces a method to approximate neutron star spacetimes by matching an analytic solution to numerical data via quadrupole moments, improving understanding of rapidly rotating stars.

Findings

Matching condition satisfied only for very rapid rotation.

Two solution branches: one close to numerical octupole moments, one similar to Kerr.

Analytic solution generally outperforms Shibata-Sasaki expansion in ISCO predictions.

Abstract

We investigate the properties of a closed-form analytic solution recently found by Manko et al. (2000) for the exterior spacetime of rapidly rotating neutron stars. For selected equations of state we numerically solve the full Einstein equations to determine the neutron star spacetime along constant rest mass sequences. The analytic solution is then matched to the numerical solutions by imposing the condition that the quadrupole moment of the numerical and analytic spacetimes be the same. For the analytic solution we consider, such a matching condition can be satisfied only for very rapidly rotating stars. When solutions to the matching condition exist, they belong to one of two branches. For one branch the current octupole moment of the analytic solution is very close to the current octupole moment of the numerical spacetime; the other branch is more similar to the Kerr solution. We present an extensive comparison of the radii of innermost stable circular orbits (ISCOs) obtained with a) the analytic solution, b) the Kerr metric, c) an analytic series expansion derived by Shibata and Sasaki (1998) and d) a highly accurate numerical code. In most cases where a corotating ISCO exists, the analytic solution has an accuracy consistently better than the Shibata-Sasaki expansion. The numerical code is used for tabulating the mass-quadrupole and current-octupole moments for several sequences of constant rest mass.