Double gravitational bound system in virial equilibrium:tidal scale length on one subsystem by another

When a gravitational bound system of two components one completely inside the other is considered and virial equilibrium holds Clausius' virial generalization of each subcomponent defined as that which balances twice the kinetic energy of the corresponding virial equation does not in general match the total potential energy of the same subsystem. When the substructures are modeled as two concentric homothetic and oblate spheroids of spheroidal similar and coaxial strata mass distributions the conditions for the existence of a minimum in Clausius' virial function related to the inner component are searched and discussed as a function of both the two power-law density profiles. The semimajor axis at which the minimum appears leads to the definition of a tidal scale length induced from the outer subsystem assumed frozen in size and shape on the the inner one. The special case of two homogeneous spheroidal subsystems is also considered in order to highlight the effect of different eccentricities between the inner and the outer component. The tidal interaction term between the two matter distributions is taken into account by using tensor virial theorem. Applications to the world of galaxies are considered and some possible consequences for galaxy dynamics are taken into account.