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

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The results of the present paper confirm the overall dependence of the geometric parameters of olivines on the Mg/(Fe+Mn) ratio. M1 and M2 volumes linearly decrease with increasing forsterite (Fo) content but at different rates. The difference between the volumes of the octahedral sites decreases gradually from Fo 92% to Fo 50% and then remains virtually constant down to 17o 0.2%.Fe(M1)/Fe(M2) ratios indicate that Fe 2+ prefers the M1 site in fayalite (Fa)-rich olivines (Fa = 39% to 92%; average Fe(M1)/Fe(M2) = 1.04 -+ 0.02). Instead, in Mg-olivines (Fo = 80% to 92%) a Mg-M1 preference (average Mg(M1)/Mg(M2) = 0.96 + 0.05) may exist. Finally, Mn(M1)/Mn(M2) ratios do not indicate a definite Mn preference for M2 in the studied olivines (MnO = 0.44% to 4.5 wt.%; average Mn(M1)/Mn(M2) = 0.99 + 0.13).No relevant influence on geometric parameters was observed between Ca-free and Cabearing (maximum CaO = 0.66 wt.%) olivines. ZusammenfassungStruktur-Verfeinerung yon 13 Olivinen der Forsterit-Fayalit-Serie aus Vulkaniten und ultramafischen Einschliissen Die Ergebnisse dieser Untersuchung best~itigen die Abh~ingigkeit der geometrischen Parameter der Olivine vom Verh~ltnis Mg/(Fe+Mn). Die Volumina M1 und M2 nehmen linear, jedoch jeweils verschieden, mit zunehmenden Forsterit (Fo)-Gehalt ab. Der Unterschied zwischen den Volumina der oktaedrischen Pl~itze nimmt graduell yon Fo 92% bis Fo 50% ab und bleibt dann bis zu Fo 0,2% konstant.Fe(M1)/Fe(M2)-Verh~iltnisse zeigen, dal~ Fe 2+ vorzugsweise M1 Pl~itze in fayalitreichen Olivinen (Fa = 39% bis 92%; durchschrtittliches Fe(M1)/Fe(M2) = 1,04 + 0,02) besetzen. In Mg-Olivinen (Fo = 80% bis 92%) hingegen dtirfte eine Mg-M1 Pr~iferenz vorliegen (durchschnitfliches Mg(M1)/Mg(M2) = 0,96 -+ 0,05). Mn(M1)/Mn(M2)-Verh~iltnisse weisen nicht aufeine definitive Mn-Pr~iferenz ftir M2 in den untersuchten Olivinen hin (MnO = 0,444,5 wt.%; durchschnittliches Mn(M1)/Mn(M2)-Verh~iltnis = 0,99 -+ 0,13).Der Calcium-Gehalt (Maximum, CaO = 0,66 wt.%) beeinflut~t die geometrischen Parameter der Olivine nicht.

The basic theory of potential-energy tensors related to heterogeneous spheres is reviewed, and the special case of truncated, singular, isothermal spheres is examined in detail. Special effort is devoted to a system made of two isothermal spheres, one completely lying within the other, the mass and the radius of the embedded sphere being negligible with respect to the mass and radius of the embedding sphere, and the radius of the embedded sphere being negligible with respect to the distance between the centres. The potential-energy tensors related to the potential induced by the embedding sphere on the mass distribution of the embedded sphere, are expressed as the sum of two contributions: one, coming from the embedded sphere after collapse towards its centre, and one other, related to the actual mass distribution of the embedded sphere. Using the latter, both a global and a local criterion in defining the tidal radius of the embedded sphere, are formulated in connection with either the binding-energy tensor or the virial-energy tensor. In doing this, the tensor components along the axis joining the centre of the embedding and the embedded sphere, are considered. The global criterion is related to the whole, embedded sphere, while the local criterion is related to an infinitesimal mass element placed at the boundary of the embedded sphere, where the distance from the centre of the embedding sphere attains a maximum. The virial theorem in tensor form is splitted into two distinct expressions, related to orbital and intrinsic motions of the embedded sphere. Alternative criterions in defining the tidal radius of the embedded sphere, are formulated taking into consideration the centrifugal tensor potential and the tensor potential induced by orbital motions. With regard to a selected criterion, the tidal radius calculated with and without the inclusion of the centrifugal potential, exhibits a maximum variation by a factor of about two, related to circular orbits. An application is made, where the embedding and the embedded sphere are taken as representative of the Galaxy and a globular cluster, respectively. It is found that a stability region exists for both the global and the local criterion and, in addition, the instability first occurs at the perigalacticon, as expected in connection with instantaneous tidal radius. A power-law dependence of tidal radius from cluster mass and galactocentric distance, aC* ∝ MC1/3R02/3, is shown to be consistent with data from a sample of 16 objects investigated by te{bra99} (1999). No significant correlation is found between the ratio of cluster radius to tidal radius and the orbital ratio of apogalacticon to perigalacticon, similar to averaged tidal radii defined by te{bra99} (1999). An additional object, Pal 5, which is experiencing progressive disruption via tidal shocks during disk passages, is shown to be among the less bound (or more unbound) clusters, within the framework of the model. If the representation of globular clusters as isothermal spheres introduces only systemat...