Asymptotic representation of low- and intermediate-degree p-modes in stars

A first-order asymptotic representation is developed for low- and intermediate-degree p-modes in stars for which the lower boundary of the resonant acoustic cavity is not located close to the starʹs centre. To this end, a fourth-order system of differential equations in the radial parts of the divergence and the radial component of the Lagrangian displacement is adopted. The lower boundary of the resonant acoustic cavity is considered to be a turning point for one of the differential equations. As in a previous asymptotic study of low-degree p-modes with high radial orders, asymptotic expansion procedures applying to self-adjoint second-order differential equations with a large parameter are used by extension of these methods. The main result is that, in contrast with the usual first-order asymptotic theory for low-degree p-modes of high radial orders, the present first-order asymptotic representation leads to small frequency separations D n,l different from zero. The validity of the asymptotic representation is tested for p-modes of the equilibrium sphere with uniform mass density, since the modes of this model are determined by means of exact analytical solutions.