Rotating neutron stars: an invariant comparison of approximate and numerical space-time models

We compare three different models of rotating neutron star space-times: (i) the Hartle-Thorne slow-rotation approximation, keeping terms up to second order in the stellar angular velocity; (ii) the exact analytic vacuum solution of Manko et al.; and (iii) a numerical solution of the full Einstein equations. In the first part of the paper we estimate the limits of validity of the slow-rotation expansion by computing relative errors in the quadrupole moment, Q, of the space-time and in the corotating and counter-rotating radii of innermost stable circular orbits (ISCOs), R-+/-. We integrate the Hartle-Thorne structure equations for five representative equations of state. Then we match these models to numerical solutions of the Einstein equations, imposing the condition that the gravitational mass and angular momentum of the models be the same. We find that the Hartle-Thorne approximation gives very good predictions for the ISCO radii, with R-+/- accurate to better than 1 per cent even for the fastest millisecond pulsars. At these rotational rates the accuracy on Q is similar to 20 per cent, and better for longer periods. In the second part of the paper we focus on the exterior vacuum space-times, comparing the Hartle-Thorne approximation and the Manko analytic solution to the numerical models using Newman-Penrose (NP) coordinate-independent quantities. For all three space-times we introduce a physically motivated 'quasi-Kinnersley' NP frame. In this frame we evaluate a quantity, the speciality index S, measuring the deviation of each stellar model from Petrov Type D. Deviations from speciality on the equatorial plane are smaller than 5 per cent at star radii for the faster rotating models, and rapidly decrease for slower rotation rates and with distance. We find that, to leading order, the deviation from Type D is proportional to (Q -Q(Kerr)). Our main conclusion is that the Hartle-Thorne approximation is very reliable for most astrophysical applications.