Nuclear symmetry energy and ther-mode instability of neutron stars

Abstract

We analyze the role of the symmetry energy slope parameter $L$ on the $r$-mode instability of neutron stars. Our study is performed using both microscopic and phenomenological approaches of the nuclear equation of state. The microscopic ones include the Brueckner-Hartree-Fock approximation, the well known variational equation of state of Akmal, Pandharipande, and Ravenhall, and a parametrization of recent auxiliary field diffusion Monte Carlo calculations. For the phenomenological approaches, we use several Skyrme forces and relativistic mean-field models. Our results show that the $r$-mode instability region is smaller for those models which give larger values of $L$. The reason is that both bulk ($\ensuremath{\xi}$) and shear ($\ensuremath{\eta}$) viscosities increase with $L$ and, therefore, the damping of the mode is more efficient for the models with larger $L$. We show also that the dependence of both viscosities on $L$ can be described at each density by simple power-laws of the type $\ensuremath{\xi}={A}_{\ensuremath{\xi}}{L}^{{B}_{\ensuremath{\xi}}}$ and $\ensuremath{\eta}={A}_{\ensuremath{\eta}}{L}^{{B}_{\ensuremath{\eta}}}$. Using the measured spin frequency and the estimated core temperature of the pulsar in the low-mass x-ray binary 4U 1608-52, we conclude that observational data seem to favor values of $L$ larger than $\ensuremath{\sim}$50 MeV if this object is assumed to be outside the instability region, its radius is in the range 11.5--12 (11.5--13) km, and its mass $1.4{M}_{\ensuremath{\bigodot}}$ ($2{M}_{\ensuremath{\bigodot}}$). Outside this range it is not possible to draw any conclusion on $L$ from this pulsar.