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Nuclear symmetry energy and the r-mode instability of neutron stars

We analyze the role of the symmetry energy slope parameter $L$ on the {\it 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 relativisic mean field models. Our results show that the {\it r}-mode instability region is smaller for those models which give larger values of $L$. The reason is that both bulk ($ξ$) and shear ($η$) 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 $ξ=A_ξL^{B_ξ}$ and $η=A_ηL^{B_η}$. 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 $\sim 50$ 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 $1.4M_\odot$($2M_\odot$). Outside this range it is not possible to draw any conclusion on $L$ from this pulsar.