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On Maclaurin and Jacobi Objects Embedded in Halos: The Shift in the Point of Bifurcation
Author(s) -
Waxman Allen M.
Publication year - 1980
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1002/sapm1980623263
Subject(s) - halo , eccentricity (behavior) , bifurcation , ellipsoid , instability , physics , conjecture , mathematics , mathematical analysis , gravitational field , intersection (aeronautics) , classical mechanics , pure mathematics , mathematical physics , astrophysics , mechanics , quantum mechanics , nonlinear system , astronomy , engineering , galaxy , political science , law , aerospace engineering
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.