2.8 Analytic approximations to the exterior spacetime

The exterior metric of a rapidly rotating neutron star differs considerably from the Kerr metric. The two metrics agree only to lowest order in the rotational velocity [149]. At higher order, the multipole moments of the gravitational field created by a rapidly rotating compact star are different from the multipole moments of the Kerr field. There have been many attempts in the past to find analytic solutions to the Einstein equations in the stationary, axisymmetric case, that could describe a rapidly rotating neutron star. An interesting solution has been found recently by Manko et al. [219220]. For non-magnetized sources of zero net charge, the solution reduces to a 3-parameter solution, involving the mass, specific angular momentum, and a parameter that depends on the quadrupole moment of the source. Although this solution depends explicitly only on the quadrupole moment, it approximates the gravitational field of a rapidly rotating star with higher nonzero multipole moments. It would be interesting to determine whether this analytic quadrupole solution approximates the exterior field of a rapidly rotating star more accurately than the quadrupole, 𝒪(Ω2 ), slow rotation approximation.

The above analytic solution and an earlier one that was not represented in terms of rational functions [218] have been used in studies of energy release during disk accretion onto a rapidly rotating neutron star [278Jump To The Next Citation Point279Jump To The Next Citation Point]. In [276Jump To The Next Citation Point], a different approximation to the exterior spacetime, in the form of a multipole expansion far from the star, has been used to derive approximate analytic expressions for the location of the innermost stable circular orbit (ISCO). Even though the analytic solutions in [276Jump To The Next Citation Point] converge slowly to an exact numerical solution at the surface of the star, the analytic expressions for the location and angular velocity at the ISCO are in good agreement with numerical results.

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