Quasi-static collapse of homogeneous spheroids within constrained halos

The virial theorem in tensor form for subsystems is used to determine equilibrium configurations of two-component, concentric and copolar, homogeneous spheroids, one completely lying within the other. Then the related total energy and angular momenta are expressed explicitly as functions of the mass ratio,m, the equatorial axis ratio,y, the polar to equatorial axis ratios, _U , the rotation parameters, _rotU , and the anisotropy parameters, _pecU . With assigned masses, angular momenta, total energy, rotation parameters, and anisotropy parameters, there remain three independent equations and four unknowns, i.e. equatorial semiaxes and (polar to equatorial) axis ratios. To get a unique solution, and then describe the quasi-static collapse (via energy dissipation) of the inner subsystem within the halo, a number of alternative additional conditions are taken into consideration. It is also argued that the true situation, involving neither homogeneous nor spheroidal halos, lies between two extreme cases, namely: (i) the two subsystems collapse together with coinciding boundaries, and (ii) halos depart only faintly, or not at all, from the related initial configurations. Starting from systems with coinciding boundaries and specific angular momenta, the related evolutionary sequences are derived by substantial improvement of previous attempts. Some limiting situations are also considered, where the mass of one subsystem tends to zero and/or the mass of the other one tends to infinite. An application of the theory to systems of galactic mass and plausible other parameters shows that (i) initial configurations depend on the total mass and final (with a flat inner component) configurations depend on the mass of the collapsing visible body; (ii) quasi-static collapse depends mainly on the mass ratio,m, for given total masses and angular momenta, while the action of different additional conditions is negligible; (iii) halos as massive as about ten visible bodies depart from the initial configuration by no more than a few percents, using different additional conditions, and yield (for fiduciary values related to the Galaxy, i.e. total mass 10¹² M , total angular momentum 10⁷⁵ g cm² s^–1) final configurations related to flat visible bodies with equatorial semiaxes of about 15 kpc, close to observations.