### 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. [219, 220]. 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, , 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 [278, 279]. In [276], 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 [276] 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.