On Mechanics and Thermodynamics of a stellar galaxy in a two-component virial system and the Fundamental Plane

The paper confirms the existence of a special configuration (among the infinitive number of a priori possible virial states) which a B stellar (Baryonic) component may assume inside a given D dark halo potential well. This satisfies the d’Alembert Principle of virtual works and its typical dimension works as a scale length (we call tidal radius) induced on the gravitational field of the bright component by the dark one. Its dynamic and thermodynamic properties are here analyzed in connection with the physical reason for the existence of the Fundamental Plane (FP) for ellipticals and, in general, for two-component virialized systems. The analysis is performed by using two-component models with two power-law density profiles and two homogeneous cores. The outputs of this kind of models, at the special configuration, are summarized and compared with some observable scaling relations for pressure supported ellipticals. The problem of extending the results to a general class of models with Zhao (1996) [MNRAS 278, 488] profiles, which are more suitable for an elliptical galaxy system, is also taken into account. The virial equilibrium stages of the two-component system have to occur after a previous violent relaxation phase. If the stellar B component is allowed to cool slowly its virial evolution consists of a sequence of contractions with enough time to rearrange the virial equilibrium after any step. The thermodynamic process during the dynamical evolution is so divided into a sequence of transformations which are irreversible but occur between two quasi-equilibrium stages. Then, it is possible to assign: a mean temperature to the whole B component during this quasi-static sequence and the entropy variation between two consecutive virial steps. The analysis allows the conclusion that the induced scale length is a real confinement for the stellar system. This follows from the application of the Io Thermodynamics Principle under the virial equilibrium constraint, by checking how larger configurations turn out to be forbidden, according to the IIo Thermodynamics Principle. The presence of this specific border on the space of the baryonic luminous component has to be regarded as the physical reason why a stellar galaxy belongs to the FP and why astrophysical objects, with a completely different history and formation, but characterized by a tidal radius (as the globular clusters are) lie on the same FP. An other problem addressed is how this special configuration may be reached. This is strictly connected with the problem of the end state of the collisionless stellar system after a violent relaxation phase. Even if degeneracy towards the initial conditions is present on the FP, the mechanic and thermodynamic properties of the special configuration suggest this state may be the best candidate for the beginning of the B component virial evolution, and also give a possible explanation for why an elliptical is not completely relaxed in respect to its dark halo.