The Contour Method: a new approach to finding modes of non-adiabatic stellar pulsations

The contour method is a new approach to calculating the non-adiabatic pulsation frequencies of stars. These frequencies can be found by solving for the complex roots of a characteristic equation constructed from the linear non-adiabatic stellar pulsation equations. A complex-root solver requires an initial trial frequency for each non adiabatic root. A standard method for obtaining initial trial frequencies is to use a star's adiabatic pulsation frequencies, but this method can fail to converge to non-adiabatic roots, especially as the growth and/or damping rate of the pulsations becomes large. The contour method provides an alternative way for obtaining initial trial frequencies that robustly converges to non-adiabatic roots, even for stellar models with extremely non-adiabatic pulsations and thus large growth/damping rates. We describe the contour method implemented in the GYRE stellar pulsation code and use it to calculate the non-adiabatic pulsation frequencies of $10\,\rm{M_{\odot}}$ and $20\,\rm{M_{\odot}}$ $\beta$ Cephei star models, and of a $0.9\,\rm{M_{\odot}}$ extreme helium star model.