Two-component, ideal, self-gravitating fluids: the fractional virial potential energy

Two-component systems are conceived as macrogases, and the related equation of state is expressed using the virial theorem for subsystems, under the restriction of homeoidally striated density profiles. Explicit calculations are performed for a useful reference case and a few cases of astrophysical interest, both with and without truncation radius. Shallower density profiles are found to yield an equation of state, ? = ?(y, m), characterized (for assigned values of the fractional mass, m = Mj /Mi ) by the occurrence of two extremum points, a minimum and a maximum, as found in an earlier attempt. Steeper density profiles produce a similar equation of state, which implies that a special value of m is related to a critical curve where the above mentioned extremum points reduce to a single horizontal inflexion point, and curves below the critical one show no extremum points. The similarity of the isofractional mass curves to van der Waals' isothermal curves, suggests the possibility of a phase transition in a bell-shaped region of the (Oy?) plane, where the fractional truncation radius along a selected direction is y = Rj /Ri , and the fractional virial potential energy is ? = (Eji )vir /(Eij )vir . Further investigation is devoted to mass distributions described by Hernquist (1990) density profiles, for which an additional relation can be used to represent a sample of N = 16 elliptical galaxies (EGs) on the (Oy?) plane. Even if the evolution of elliptical galaxies and their hosting dark matter (DM) haloes, in the light of the model, has been characterized by equal fractional mass, m, and equal scaled truncation radius, or concentration, ?u = Ru /r? , u = i, j, still it cannot be considered as strictly homologous, due to different values of fractional truncation radii, y, or fractional scaling radii, y? = r? /r? , deduced from sample objects