A Principle of Corresponding States for Two-Component, Self-Gravitating Fluids - Caimmi, R. (Serbian Astronomical Journal, 2010)
Macrogases are defined as two-component,large-scale celestial objects where the subsystems interact only via gravitation.The macrogas equation of state is formulated and compared to the van der Waals (VDW)equation of state for ordinary gases.By analogy, it is assumed that real macroisothermal curves in macrogases occur as real isothermal curves in ordinary gases, where a phase transition(vapour-liquid observed in ordinary gases and gas-stars assumed in macrogases) takesplace along a horizontal linein the macrovolume-macropressure{small $(sf OsX_mathrmVsX_mathrmp)$} plane.The intersections between real and theoretical(deduced from the equation of state) macro isothermalcurves, make two regions of equal surface as for ordinary gases obeying the VDW equation of state.A numerical algorithm is developed for determining the following points of a selected theoretical macroisothermal curve on the {small $(sf OsX_mathrmVsX_mathrmp)$} plane:the three intersections with the related real macroisothermal curve,and the two extremum points (one maximum and one minimum). Different kinds of macrogases are studied in detail: UU, where U density profiles are flat, to be conceived as a simple guidance case; HH, where H density profiles obey the Hernquist (1990) law, which satisfactorily fits the observed spheroidal components of galaxies; HN/NH, where N density profiles obey the Navarro-Frenk-White (1995,1996, 1997) law, which satisfactorily fits the simulated nonbaryonic dark matter haloes.A different trend is shown by theoretical macroisothermal curves on the{small $(sf OsX_mathrmVsX_mathrmp)$} plane,according to whether density profiles are sufficiently mild (UU) or sufficiently steep (HH, HN/NH).In the former alternative, no critical macroisothermal curve exists, below or above which the trend is monotonous. In the latter alternative, a critical macroisothermal curve exists, as shown by VDW gases, where the critical point may be defined as the horizontal inflexion point. In any case, by analogy with VDW gases, the first quadrant of the{small $(sf OsX_mathrmVsX_mathrmp)$} plane may be divided into three parts: (i) The G region, where only gas exists; (ii) The S region,where only stars exist; (iii) The GS region,where both gas and stars, exist. With regard to HH and HN/NH macrogases, an application is made to a subsample ({small $N=16$}) of elliptical galaxies extracted from larger samples {small $(N=25,~N=48)$}of early type galaxies investigated within the SAURON project (Cappellari et al. 2006, 2007).Under the simplifying assumption of universal mass ratio of the two subsystems, {small $m$}, different models characterized by different scaled truncation radii, i.e. concentrations, {small $Xi_i$},{small $Xi_j$}, are considered and the related position of sample objects on the{small $(sf OsX_mathrmVsX_mathrmp)$} plane is determined. Macrogases fitting to elliptical galaxies are expected to lie within the S region or slightly outside the boundary between the S and the GS region at most. Accordingly,models where sample objects lie outside the S region and far from its boundary, or cannot be positioned on the{small $(sf OsX_mathrmVsX_mathrmp)$} plane, are rejected. For each macrogas, twenty models are considered for different values of {small $(Xi_i,Xi_j, m)$}, namely {small $Xi_i, Xi_j=5,~ 10,~ 20,~+infty$} {small $(Xi_i, Xi_j,$} both either finite or infinite), and {small $m=10,~ 20$}. Acceptable models are {small $(10,~10,~20)$}, {small $(10,~20,~20)$},{small $(20,~10,~20)$}, {small $(20,~20,~20)$}, for HH macrogases,and {small $(10,~5,~10)$}, {small $(10,~10,~20)$}, {small $(20,~10,~20)$},for HN/NH macrogases. Tipically, fast rotators are found to lie within the S region, while slow rotators are close (from both sides) to the boundary between the S and the GS region. The net effect of the uncertainty affecting observed quantities, on the position of sample objects on the {small $(sf OsX_mathrmVsX_mathrmp)$} plane, is also investigated. Finally, a principle of corresponding states is formulated for macrogases with assigned density profiles and scaled truncation radii.