Momentum conservation and mode classification of the dipolar oscillations of stars

Despite their apparent simplicity, the dipole modes of stellar oscillations are theoretically less tractable than the other nonradial modes. This is because neglecting the Eulerian perturbation to the gravitational potential (the Cowling approximation), which greatly simplifies the governing equations, is not always very accurate in the case of dipole modes. In this paper, we first derive an integral that is specific to dipole-mode oscillations. It is then shown that the fourth-order differential equation, which governs the nonradial adiabatic oscillations of stars, is reduced to a (linear) second-order equation in the dipole case without assuming the Cowling approximation. Based on this second-order equation, we discuss how to classify the adiabatic dipole modes theoretically. We specifically propose a scheme that allocates a unique index to each of the dipole modes.