A global and a local criterion in defining the tidal radius

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 systematic errors in the ratio of cluster radius to tidal radius, aC/aC*=γ/η, then at least one other cluster, NGC 5466 (which has the highest value between the sample objects), is inferred to undergo tidal disruption, in the model interpretation.