Thermal evolution of neutron stars with internal frictional heating

It has been suggested that the frictional interaction of neutron superfluids with normal matter in the inner crust of neutron stars dissipates rotational energy of superfluids and generates heat. Incorporating a general formula of internal heating into the detailed numerical codes of thermal evolution, we examine the effects of the internal heating on thermal evolution of neutron stars. We find that when a very stiff equation of state is used, it takes as long as about 2 x 10 exp 4 yr for the interior of a neutron star to reach the isothermal state, even if a strong heat source is placed in a thin layer of the inner crust. This time scale reduces to a few hundred years or less when medium to soft equations of state are used. A neutron star cools by neutrino emissions during the earlier stages referred to as the neutrino cooling era, while during the later photon cooling era it cools primarily by emission of photons from its surface. We show that heating rates expected in the current superfluid-crust interaction model can greatly increase the surface temperature in the photon cooling era, significantly changing the thermal evolution of relatively old neutron stars if a stiff equation of state is adopted.