Orbital and epicyclic frequencies of Maclaurin spheroids

Abstract

We present analytical formulae for the orbital and epicyclic frequencies in orbits around Maclaurin spheroids in Newtonian gravity. The Laplace equation for the gravitational potential implies that the orbital frequency squared is the arithmetic mean of the squares of the epicyclic frequencies, $\omega _r^2 + \omega _z^2 = 2\Omega _{\rm o}^2$. The radial epicyclic frequency has a maximum at radius $r=\sqrt{2}ae$ for spheroid ellipticities $e>1/\sqrt{2}$, while for e = 0.834 583 18 it vanishes at the stellar equator (at r = a). For still larger ellipticities the innermost stable circular orbit (ISCO) is separated from the surface of the spheroid by a gap and has radius r ms = 1.198 203 ae. The vertical epicyclic frequency is always larger than the orbital one, and always by a factor of $\sqrt{2}$ in the marginally stable orbit. The presence of periastron motion, nodal precession (whose sense is the same as in retrograde orbits in the Kerr metric) and of the ISCO makes the properties of orbital motion around Maclaurin spheroids analogous to those in the Kerr metric. We find that the condition for the existence of circular orbits in test-particle motion is $\omega _r^2 + \omega _z^2 >0$, equally for the Maclaurin spheroid and for the Kerr metric.