Radial and Nonradial Oscillation Modes in Rapidly Rotating Stars

Abstract

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.Comment: accepted for publication in Ap

Rotation and convective core overshoot inθOphiuchi

“…As the velocity gradually increases, the resulting distortion increases, allowing the modes to be identified. However, this process becomes more difficult as the rotation rate increases, and is described in more detail in Lovekin & Deupree (2008). Similar problems have been encountered by for example, Reese et al (2009).…”

“…This method can result in missed frequencies if the frequencies are sufficiently close together, but this can usually be avoided by reducing the frequency step size. For further discussion of the solution mechanism, refer to Clement (1998); Lovekin & Deupree (2008).…”

“…The frequencies shown in this plot were scaled by a factor of √ (GM/R(40 • ) 3 ) to account for differences in the radii of the models. We have chosen the radius at a colatitude of 40 • as this radius has been found to be the radius most appropriate for calculating a pulsation constant in similar models (Lovekin & Deupree 2008). …”

“…In order to offset the effects of this difference, we have scaled the frequencies by a factor of √ (GM/R(40 • ) 3 ) when looking for the effects of rotation and convective core overshoot. We have chosen the radius at this colatitude as it has been found to produce the best pulsation constant (Lovekin & Deupree 2008), and hence should be the best choice for scaling the frequencies.…”

Asymptotic analysis of high-frequency acoustic modes in rapidly rotating stars

“…New numerical codes have been able to accurately compute oscillation frequencies in centrifugally distorted polytropic models of stars Reese et al 2006) and are now extended to more realistic models (Reese et al 2009). In particular the previous calculations based on perturbative methods have been shown to be inadequate for these stars Lovekin & Deupree 2008). Nevertheless, interpreting the data requires an understanding of the mode properties that goes far beyond an accurate computation of frequencies.…”