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We analyze the role of the symmetry energy slope parameter $L$ on the {\itr}-mode instability of neutron stars. Our study is performed using bothmicroscopic and phenomenological approaches of the nuclear equation of state.The microscopic ones include the Brueckner--Hartree--Fock approximation, thewell known variational equation of state of Akmal, Pandharipande and Ravenhall,and a parametrization of recent Auxiliary Field Diffusion Monte Carlocalculations. For the phenomenological approaches, we use several Skyrme forcesand relativisic mean field models. Our results show that the {\it r}-modeinstability region is smaller for those models which give larger values of $L$.The reason is that both bulk ($\xi$) and shear ($\eta$) viscosities increasewith $L$ and, therefore, the damping of the mode is more efficient for themodels 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$\xi=A_{\xi}L^{B_\xi}$ and $\eta=A_{\eta}L^{B_\eta}$. Using the measured spinfrequency and the estimated core temperature of the pulsar in the low-massX-ray binary 4U 1608-52, we conclude that observational data seem to favorvalues of $L$ larger than $\sim 50$ MeV if this object is assumed to be outsidethe instability region, its radius is in the range $11.5-12$($11.5-13$) km, andits mass $1.4M_\odot$($2M_\odot$). Outside this range it is not possible todraw any conclusion on $L$ from this pulsar.

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