The intimate relation between the low T/W instability and the corotation point

Abstract

We study the low T/W instability associated with the f mode of differentially rotating stars, where T and W are, respectively, the kinetic and gravitational energy of the star. Our stellar models are described by a polytropic equation of state and the rotation profile is given by the standard j-constant law. The properties of the relevant oscillation modes, including the instability growth time, are determined from time evolutions of the linearized dynamical equations in Newtonian gravity. In order to analyse the instability we monitor also the canonical energy and angular momentum. Our results demonstrate that the l = m = 2 f mode becomes unstable as soon as a corotation point develops inside the star (i.e. whenever there is a point where the mode's pattern speed matches the bulk angular velocity). Considering various degrees of differential rotation, we show that the instability grows faster deep inside the corotation region and deduce an empirical relation that correlates the mode frequency and the star's parameters, which captures the main features of the l = m = 2 f-mode growth time. This function is proportional to the product of the kinetic to gravitational energy ratio and the gradient of the star's spin, strengthening further the relationship between the corotation point and the low T/W instability. We briefly consider also the l = m = 2 r mode and demonstrate that it never moves far inside the corotation region even for significant differential rotation.

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ORCID for Nils Andersson: orcid.org/0000-0001-8550-3843

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