The gain of internal energy of a star cluster caused by the tidal attraction of our Galaxy is examined. Expressions are derived which include the effects of a two-body orbit and internal motions of the cluster. These formulae are compared with previous results based on (i) uniform rectilinear motion and (ii) neglect of internal motions induced by cluster gravitation(i.e., impulsive approximation), and it is found that these simplifying assumptions generally introduce significant uncertainties.

On the instantaneous tidal relaxation approximation, formulae are derived for the ellipticities and virial theorem of a slightly flattened homogeneous rotating cluster (the largest axis of the cluster is directed towards the Galactic center), in terms of the Galactic tidal force and the characteristic intrinsic plus orbital angular velocity. The expression for a purely tidally-determined ellipticity is identical to that for an incompressible fluid body of uniform density. Orbital motion generally contributes significantly to the shape of the cluster. The virial theorem is identical to that for an isolated cluster except that the gravitational potential energy is multiplied by (1- χ ), where χ is a positive tidal correction term. To obtain the actual mass of a cluster, the virial theorem mass based on an isolated cluster should be multiplied by the factor 1/(1- χ ). The formulae are applied to open star clusters, the globular cluster ω Centauri, and dwarf elliptical galaxies in the Local Group.