An extension of Chandrasekhar's tensor virial theorem for one subsystem distorted by the tidal potential induced by another subsystem is formulated, with the possibility to extend the results to the tidal potential induced by any number of subsystems. To this aim, the self-energy tensor, the interaction-energy tensor, the tidal-energy tensor, and the residual-energy tensor are defined for each subsystem. The above mentioned quantities are evaluated for the special case of homogeneous, coaxial ellipsoids, one lying completely within the other. Concerning the special case of spheroids, a comparison is made with the results of previous approaches

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We present a panoramic review of several observational and theoretical aspects of the modern astrophysical research about the origin of the Fundamental Plane (FP) relation for Early-Type Galaxies (ETGs). The discussion is focused on the problem of the tilt and the tightness of the FP, and on the attempts to derive the luminosity evolution of ETGs with redshift. Finally, a number of observed features in the FP are interpreted from the standpoint of a new theoretical approach based on the two-component tensor virial theorem.

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The equation of state and the crystal structure evolution with pressure were determined for two single crystals of pure natural MgAl2O4 spinels with different degrees of order. The two samples studied were cut from a larger single crystal and one of them was experimentally disordered at high temperature. The two crystals, showing an inversion parameter x of 0.27 and 0.15 at ambient conditions, were loaded together in a diamond anvil cell and their unit-cell edge was measured up to about 7.5 GPa at 14 different pressures. The unit-cell volume, V0, the bulk modulus, KT0, and its first pressure derivative, K′, were simultaneously refined using a third-order Birch-Murnaghan equation of state, giving the following coefficients: V0 = 529.32(2) Å3, KT0 = 193(1) GPa, K′ = 5.6(3) for the ordered sample and V0 = 528.39(2) Å3, KT0 = 192(1) GPa, K′ = 5.4(3) for the disordered one. Complete intensity data were collected at 0, 0.44, 2.92, 7.34, and 8.03 GPa in a separate experiment. For the ordered and disordered samples the oxygen atomic coordinate u remains practically unchanged inside the investigated pressure range with an average value of 0.2633(5) and 0.2614(2), respectively. As a consequence, the polyhedral compressibilities are similar and are not influenced by the Mg/Al distribution over the two crystallographic sites. This also suggests that pressure has little or no influence on the degree of order in the MgAl2O4 spinel