The specific entropy of elliptical galaxies: an explanation for profile-shape distance indicators?

Dynamical systems in equilibrium have a stationary entropy; we suggest that elliptical galaxies, as stellar systems in a stage of quasi-equilibrium, may have in principle a unique specific entropy. This uniqueness, a priori unknown, should be reflected in correlations between the fundamental parameters describing the mass (light) distribution in galaxies. Following recent photometrical work on elliptical galaxies by Caon et al., Graham & Colless and Prugniel & Simien, we use the Sérsic law to describe the light profile and an analytical approximation to its three-dimensional deprojection. The specific entropy is then calculated, supposing that the galaxy behaves as a spherical, isotropic, one-component system in hydrostatic equilibrium, obeying the ideal-gas equations of state. We predict a relation between the three parameters of the Sérsic law linked to the specific entropy, defining a surface in the parameter space, an `Entropic Plane', by analogy with the well-known Fundamental Plane. We have analysed elliptical galaxies in two rich clusters of galaxies (Coma and ABCG 85) and a group of galaxies (associated with NGC 4839, near Coma). We show that, for a given cluster, the galaxies follow closely a relation predicted by the constant specific entropy hypothesis with a typical dispersion (one standard deviation) of 9.5per cent around the mean value of the specific entropy. Moreover, assuming that the specific entropy is also the same for galaxies of different clusters, we are able to derive relative distances between Coma, ABGC 85, and the group of NGC 4839. If the errors are due only to the determination of the specific entropy (about 10per cent), then the error in the relative distance determination should be less than 20per cent for rich clusters. We suggest that the unique specific entropy may provide a physical explanation for the distance indicators based on the Sérsic profile put forward by Young & Currie and recently discussed by Binggeli & Jerjen.