Using a simple model of a neutron star with a perfectly rigid crust constructed with a set of crust and core equations of state that span the range of nuclear experimental uncertainty in the symmetry energy, we calculate the instability window for the onset of the Chandrasekhar-Friedmann-Schutz instability in $r$-mode oscillations for canonical neutron stars ($1.4M_{\odot }$) and massive neutron stars ($2.0M_{\odot }$). In these models the crust-core transition density, and thus crustal thickness, is calculated consistently with the core equation of state (EOS). The EOSs are calculated using a simple model for the energy density of nuclear matter and probe the dependence on the symmetry energy by varying the slope of the symmetry energy at saturation density $L$ from 25 MeV (soft symmetry energy and EOS) to 115 MeV (stiff symmetry energy and EOS) while keeping the EOS of symmetric nuclear matter fixed. For the canonical neutron star, the lower bound of the $r$-mode instability window is reduced in frequency by $\approx 150$ Hz from the softest to the stiffest symmetry energy used, independent of mass and temperature. The instability window also drops by $\approx 100$ Hz, independent of EOS when the mass is raised from $1.4M_{\odot }$ to $2.0M_{\odot }$. Where temperature estimates are available, the observed neutron stars in low-mass x-ray binaries (LMXBs) have frequencies below the instability window for the $1.4M_{\odot }$ models, while some LMXBs fall within the instability window for $2.0M_{\odot }$ stars if the symmetry energy is relatively stiff, indicating that a softer symmetry energy is more consistent with observations within this model. Thus we conclude that smaller values of $L$ help stabilize neutron stars against runaway $r$-mode oscillations. The critical temperature, below which no star can reach the instability window without exceeding its Kepler frequency, varies by nearly an order of magnitude from soft to stiff symmetry energies. When the crust thickness and core EOS are treated consistently, a thicker crust corresponds to a lower critical temperature, the opposite result to previous studies in which the transition density was independent of the core EOS.

• Received 27 October 2011

DOI:https://doi.org/10.1103/PhysRevC.85.025801