The GSF instability and turbulence do not account for the relatively low rotation rate of pulsars

¹ Astrophysics Group, EPSAM Institute, University of Keele, Keele, ST5 5BG, UK e-mail: r.hirschi@epsam.keele.ac.uk
² Institute for the Physics and Mathematics of the Universe, University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa 277-8583, Japan
³ Geneva Observatory, Geneva University, 1290 Sauverny, Switzerland e-mail: andre.maeder@unige.ch

Received: 9 February 2010
Accepted: 28 April 2010

Abstract

Aims. We examine the effects of the horizontal turbulence in differentially rotating stars on the Goldreich-Schubert-Fricke (GSF) instability and apply our results to pre-supernova models.

Methods. We derive the expression for the GSF instability with account of the thermal transport and smoothing of the μ-gradient by the horizontal turbulence. We apply the new expressions in numerical models of a 20  star.

Results. We show that if < 0 the Rayleigh-Taylor instability cannot be killed by the stabilising thermal and μ-gradients, so that the GSF instability is always there and we derive the corresponding diffusion coefficient. The GSF instability grows toward the very latest stages of stellar evolution. Close to the deep convective zones in pre-supernova stages, the transport coefficient of elements and angular momentum by the GSF instability can, in small parts of the star, be larger than the shear instability and even as large as the thermal diffusivity. However, the zones in which the GSF instability is acting are extremely narrow and there is not enough time left before the supernova explosion for a significant mixing to occur. Thus, the GSF instability remains insignificant for the evolution even when the inhibiting effects of the μ-gradient are reduced by the horizontal turbulence.

Conclusions. We conclude that the GSF instability in pre-supernova stages is not responsible for the relatively low rotation rate of pulsars compared to the predictions of rotating star models.