Approximate matching of analytic and numerical solutions for rapidly rotating neutron stars

We investigate the properties of a closed-form analytic solution recently found by Manko et al. for the exterior space-time of rapidly rotating neutron stars. For selected equations of state we numerically solve the full Einstein equations to determine the neutron star space-time 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 space-times 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 space-time; 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 (i) the analytic solution, (ii) the Kerr metric, (iii) an analytic series expansion derived by Shibata and Sasaki and (iv) a highly accurate numerical code. In most cases where a co-rotating 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.