Third Order Effect of Rotation on Stellar Oscillations of a B Star

The present paper aims at investigating the effect of rotation up to third order in the angular velocity of a star on p and g modes, based on the formalism developed by Soufi et al. (1998). Our ultimate goal is the study of oscillations of $\beta$ Cephei stars which are often rapidly rotating stars. Our study shows that the third-order perturbation formalism presented by Soufi et al. (1998) must be corrected for some missing terms and some misprints in the equations. As a first step in our study of $\beta$ Cephei stars, we quantify by numerical calculations the effect of rotation on the oscillation frequencies of a uniformly rotating zero-age main-sequence star with 12~$M_\odot$. For an equatorial velocity of 100~$\rm km \, s^{-1}$, it is found that the second- and third-order corrections for $(l,m)=(2,2)$, for instance, are of order of 0.01$\%$ of the frequency for radial order $n=6$ and reaches up to 0.5$\%$ for $n=14$.