Two-component, ideal, self-gravitating fluids: the fractional virial   potential energy
  Two-component, ideal, self-gravitating fluids 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. Shallower density profiles are found to yield an equation of state, \phi=\phi(y,m), characterized (for assigned values of the fractional mass, m=M_j/ M_i) by the occurrence of two extremum points, a minimum and a maximum. 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 (O y \phi) plane, where the fractional truncation radius along a selected direction is y=R_j/R_i, and the fractional virial potential energy is \phi=(E_{ji})_{vir}/(E_{ij})_{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 (O y \phi) plane, under the assumption that the fractional mass related to EGs and their hosting dark matter (DM) haloes, has a universal value. In the light of the model, the evolution of isolated EGs appears to be other than strictly homologous, due to different values of fractional truncation radii, y, or fractional scaling radii, y^\dagger, deduced from sample objects. 
R. Caimmi, T. Valentinuzzi