Axisymmetric second order perturbations of rotating main sequence stars

Abstract

We calculate the second order perturbations driven by oscillation modes of rotating stars. Assuming that the typical amplitude $a$ of oscillation modes is small, we expand the perturbed quantities as $Q=Q^{(0)}+Q^{(1)}+Q^{(2)}+\cdots$ where $Q^{(0)}$ represents the equilibrium state and $Q^{(1)}$ and $Q^{(2)}$ are the first order and second order perturbations in $a$, respectively. We assume that the first order perturbations are given by non-axisymmetric modes and the second order perturbations are axisymmetric. For the second order perturbations, we derive a set of linear partial differential equations, which have inhomogeneous terms due to the first order perturbations. For low frequency $g$- and $r$-modes and overstable convective (OsC) modes of main sequence stars, we calculate the second order velocity field $\mathbf{v}^{(2)}$ and find that prograde $g$-modes and OsC modes tend to accelerate and retrograde $r$-modes to decelerate $v_ϕ^{(2)}$ in the surface equatorial regions where $v_ϕ^{(2)}$ is the $ϕ$ component of $\mathbf{v}^{(2)}$. Using the angular momentum conservation equation derived for waves, we discuss that low frequency $g$- and $r$-modes transport angular momentum between the inner and outer parts of the envelope. For OsC modes in the core resonantly coupled with envelope prograde $g$-modes, we find that they can transport angular momentum from the core to the outer envelope so that they tend to brake the core rotation. We also suggest that the OsC modes provide the outer envelope of rotating stars with the torque enough to support a decretion disc.