Acoustic oscillations in rapidly rotating polytropic stars I. Effects of the centrifugal distortion

A new non-perturbative method to compute accurate oscillation modes in rapidly rotating stars is presented. In this paper, the effect of the centrifugal force is fully taken into account while the Coriolis force is neglected. This assumption is valid when the time scale of the oscillation is much shorter than the inverse of the rotation rate and is expected to be suitable for high radial order p-modes of $\delta$ Scuti stars. Axisymmetric p-modes have been computed in uniformly rotating polytropic models of stars. In the frequency and rotation range considered, we found that as rotation increases (i) the asymptotic structure of the non-rotating frequency spectrum is first destroyed then replaced by a new form of organization (ii) the mode amplitude tends to concentrate near the equator (iii) differences with perturbative methods become significant as soon as the rotation rate exceeds about fifteen percent of the Keplerian limit. The implications for the seismology of rapidly rotating stars are then discussed.