Analysis of adiabatic dipolar oscillations of stars

We theoretically discuss adiabatic dipolar oscillations of stars, fully taking into account the variation in the gravitational field. By suitably choosing the dependent variables, and using the first integral specific to dipolar oscillations, we derive the second-order system of ordinary differential equations, which is the same in form as that obtained by neglecting the Eulerian perturbation to the gravitational potential. The derived system suggests that the conventional expressions of the critical frequencies should be corrected. In addition, we present a rigorous scheme to classify the dipolar eigenmodes of linear adiabatic oscillations of spherically symmetric stars.