Approximate analytic expressions for circular orbits around rapidly rotating compact stars (Research Note)

Aims. We calculate stationary configurations of rapidly rotating compact stars in general relativity, to study the properties of circular orbits of test particles in the equatorial plane. We search for simple, but precise, analytical formulae for the orbital frequency, specific angular momentum, and binding energy of a test particle that are valid for any equation of state and any rotation frequency of the rigidly rotating compact star, up to the mass-shedding limit. Methods. Numerical calculations are performed using precise 2D codes based on multi-domain spectral methods. Models of rigidly rotating neutron stars and the space-time outside them are calculated for several equations of state of dense matter. Calculations are also performed for quark stars consisting of self-bound quark matter. Results. At the mass shedding limit, the rotational frequency converges to a Schwarzschildian orbital frequency at the equator. We show that the orbital frequency of any orbit outside equator can also be approximated by a Schwarzschildian formula. Using a simple approximation for the frame-dragging term, we obtain approximate expressions for the specific angular momentum and specific energy on the corotating circular orbits in the equatorial plane of neutron star, which are valid to the stellar equator. The formulae recover reference numerical values with typically 1% of accuracy for neutron stars with M greater than or similar to 0.5 M-circle dot. These values are less precise for quark stars consisting of self-bound quark matter.