2 Maclaurin spheroids

In Newtonian gravity, Maclaurin spheroids describe bodies of uniform density in hydrostatic equilibrium. In addition to their theoretical importance, they should be an excellent approximation for planetoids of moderate mass, and - as we show below - for quark stars of planetary mass.

As noticed by Thomas Simpson in 1743 and by d'Alembert, the rotational frequency of a Maclaurin spheroid (Fig. 1) approaches zero not only for $e\rightarrow0$, a "spheroid which departs only slightly from a sphere'', but also for $e\rightarrow1$, a "highly flattened spheroid'':

$$\Omega^2 2\pi G\rho (1-e^2)^{1/2}e^{-3}$$

$$\times \left\{(3-2e^2)\arcsin e - 3e(1-e^2)^{1/2}\right\},$$ (1)

where $\rho$ is constant and e=(1-b²/a²)^1/2 is the eccentricity of the spheroid, with a the major and b the minor axis of the same (Chandrasekhar 1969).

By explicitly evaluating the integrals in the expression for the potential given in Theorem 7, Chapter 3 of Chandrasekhar (1969), we find that the orbital frequencies, $\omega(r)$, also go to zero in the two limits $e\rightarrow0$ and $e\rightarrow1$. Specifically, at the equator,

$${\omega^2(a)}= 2\pi G\rho(1-e^2)^{1/2}{e^{-3} } \left\{\arcsin e- e(1-e^2)^{1/2}\right\}\cdot\,$$ (2)

The corresponding circular orbits (of radius r=a) are stable only for eccentrities $e\le e_{\rm s}\equiv0.83458318$, and the innermost (marginally) stable circular orbit, present in the equatorial plane for all $e>e_{\rm s}$, has frequency

$$\omega_{\rm ms}^2\equiv\omega^2(r_{\rm ms}) =0.5276189\times 2\pi G\rho(1-e^2)^{1/2}{e^{-3} },$$ (3)

also going to zero for $e\rightarrow1$ (Kluzniak 2001). The expressions (2), (3), for the maximum frequency in circular orbits are plotted in Fig. 2, their numerical values for a specific choice of density can be read off Fig. 3.

We stress that these expressions have been derived in Newtonian physics. The marginally stable orbit is present (and stable circular orbits are absent for $r<r_{\rm ms}$) only because the distribution of mass is non-spherical.