Lagrangian perturbation theory for rotating magnetic stars

Motivated by the possibility of radiation driven instabilities in rotating magnetic stars, we study the stability properties of general linear perturbations of a stationary and axisymmetric, infinitely conducting perfect fluid configuration threaded by a magnetic field and surrounded by vacuum. We develop a Lagrangian perturbation framework which enables us to formulate a strict stability criterion based on the notion of a canonical energy (a functional of the fluid displacement $\xi$ and its first time derivative). For any given choice of $\{\xi ,{\mathrm{\partial }}_t\xi \}$, the sign of the canonical energy determines whether the configuration is stable or not at the linear level. Our analysis provides the first complete description of the stability problem for a magnetic star, allowing for both rotation and the presence of a magnetic field in the exterior vacuum region. A key feature of the Lagrangian formulation is the existence of so-called trivial' fluid displacements, which do not represent true physical perturbations. In order for the stability criterion to make rigorous sense one has to isolate these trivials and consider only the physical canonical’ displacements. We discuss this problem and formulate a condition which must be satisfied by all canonical displacements. Having obtained a well-defined stability criterion we provide examples which indicate that the magnetic field has a stabilising effect on radiation driven instabilities.