Radial and Nonradial Oscillation Modes in Rapidly Rotating Stars
  Radial and nonradial oscillations offer the opportunity to investigate the interior properties of stars. We use 2D stellar models and a 2D finite difference integration of the linearized pulsation equations to calculate non-radial oscillations. This approach allows us to directly calculate the pulsation modes for a distorted rotating star without treating the rotation as a perturbation. We are also able to express the finite difference solution in the horizontal direction as a sum of multiple spherical harmonics for any given mode. Using these methods, we have investigated the effects of increasing rotation and the number of spherical harmonics on the calculated eigenfrequencies and eigenfunctions and compared the results to perturbation theory. In slowly rotating stars, current methods work well, and we show that the eigenfunction can be accurately modelled using 2nd order perturbation theory and a single spherical harmonic. We use 10 Msun models with velocities ranging from 0 to 420 km/s (0.89 Omega_c) and examine low order p modes. We find that one spherical harmonic remains reasonable up to a rotation rate around 300km s^{-1} (0.69 Omega_c) for the radial fundamental mode, but can fail at rotation rates as low as 90 km/s (0.23 Omega_c) for the 2H mode or l = 2 p_2 mode, based on the eigenfrequencies alone. Depending on the mode in question, a single spherical harmonic may fail at lower rotation rates if the shape of the eigenfunction is taken into consideration. Perturbation theory, in contrast, remains valid up to relatively high rotation rates for most modes. We find the lowest failure surface equatorial velocity is 120 km/s (0.30 Omega_c) for the l = 2 p_2 mode, but failure velocities between 240 and 300 km/s (0.58-0.69 Omega_c)are more typical. 
C.C. Lovekin and R.G. Deupree