On Maclaurin and Jacobi Objects Embedded in Halos: The Shift in the Point of Bifurcation

This paper concerns the equilibrium structure of Maclaurin spheroids and Jacobi ellipsoids embedded in nonrotating halos of uniform density. The halo is assumed unresponsive to the embedded object, whereas the embedded object is allowed to respond to the gravitational field of the halo. We also ignore the effects of the halo pressure field on the embedded object. It is shown how the halo modifies the classical Maclaurin and Jacobi sequences. In particular, we locate the intersection of these two sequences, i.e., the point of bifurcation, and present a formula for the eccentricity at bifurcation (e_b) as a function of the ratio of halo density to density of rotating matter (ρ_H/ρ_B). We find that the halo increases the eccentricity at bifurcation; thus, it has a stabilizing influence. However, secular instability is never entirely suppressed, since e_b→1 only for ρ_H/ρ_B→∞. It is seen that the Ostrike-Peebles conjecture does not apply to the case of ρ_H/ρ_B?1.