Normal Modes of Oscillation for Rotating Stars. V. A New Numerical Method for Computing Nonradial Eigenfunctions

Publisher Website
Google Scholar
Abstract
This paper is a progress report on my efforts to develop a robust numerical scheme for computing eigenmodes of arbitrary order at arbitrary rotation rates. The slowly pulsating B stars and the line-profile variables on the upper main sequence provide the main impetus for the work. Consequently, the focus here is on the gravity modes responsible for these variables rather than on pressure modes. Unfortunately, the g-modes are harder to compute in general because their oscillation periods can be longer than the rotation periods of high-mass stars, which makes them more sensitive to the effects of inertial forces. The standard technique for calculating the nonradial modes of spherical stellar models is generalized to two dimensions. The four dependent variables are the radial and latitudinal components of the Lagrangian displacement together with the Eulerian pressure and gravitational potential perturbations, all evaluated on level surfaces and normalized to have nonzero values on the boundaries. The integration in polar angle is replaced by numerical differentiation which is performed by various techniques involving Fourier, Legendre, or Chebyshev transforms. For the radial integrations, two numerical schemes are explored. One involves explicit Runge-Kutta integration inward and outward with fitting at an intermediate level surface. The other is an implicit, finite-difference method that does not require an initial trial solution. Each has advantages and disadvantages, but both ultimately fail for the same reason—small numerical errors in the θ-derivatives of eigenfunctions that are being distorted by growing high-order spherical harmonic components that are mixed in by rotation. Lack of convergence appears when the ratio Ω/σ of angular velocity and mode frequency exceeds about 0.5. This makes high-order g-modes (n > 15) in the presence of large rotation inaccessible with the methods proposed here. Nevertheless, it is possible to conclude that rapid rotation and large radial order suppress the pressure perturbations but enhance the horizontal motions especially near the surface and the equatorial plane. In fact, this trend is sufficiently clear and persistent that it should be possible to model the line profiles of the observationally relevant modes without actually computing them exactly.