1 Introduction

Much of what is known about the masses of astronomical objects has been derived from observations of orbital motion and its theory (dating back to Newton 1687), so an accurate knowledge of the laws of orbital motion is of exceptional interest. It seems ironic that only first corrections to orbital motion caused by oblateness have been discussed for rotating stars, even in Newtonian physics, while the formulae for orbital frequency of black holes are known for all allowed angular momenta of the hole (Bardeen et al. 1972).

Interest in the possible range of orbital frequencies around compact bodies has greatly increased after the discovery of millisecond variability (kHz QPOs) in several low-mass X-ray binaries, including X-ray bursters and black hole candidates (for a review see van der Klis 2000). The orbital frequency around spherical bodies is given by the same formula in the Schwarzschild metric as in Newtonian physics, $\Omega=\sqrt{GM/r^3}$, but there is a difference in the allowed range. There is no limit to how high this Keplerian frequency may become in Newtonian physics as the radius of the orbit around an ever smaller gravitating sphere decreases, while in Einstein's theory of gravitation an upper limit to the frequency in stable circular orbits is attained in the marginally stable orbit (of radius $6GM/c^2\approx 9\,{\rm km} \times M/M_\odot$in the Schwarzschild metric). It had been suggested that this property may be used to test general relativity in the strong field-regime around accreting neutron stars, or to measure the stellar mass, by directly comparing the highest frequency manifest in the X-ray flux with relativistic formulae for the orbital frequency in the marginally stable orbit (Kluzniak & Wagoner 1985; Kluzniak et al. 1990), and now several authors have indeed tried to carry out this program for neutron stars (Kaaret et al. 1997; Zhang et al. 1998; Kluzniak 1998), for quark stars (Bulik et al. 1999a,b; Zdunik et al. 2000a,b; Gondek-Rosinska et al. 2001a,b; Datta et al. 2000), and for black holes (Strohmayer 2001).

In this letter we discuss the Newtonian limit of the maximum orbital frequency around uniformly rotating bodies in equilibrium. We compare analytic formulae, derived in Newtonian physics for bodies of constant density, with the results of fully relativistic numerical calculations carried out for quark stars of very low mass.