Long-term temperature evolution of neutron stars undergoing episodic accretion outbursts

Context. Transient neutron star low-mass X-ray binaries undergo episodes of accretion, alternated with quiescent periods. During an accretion outburst, the neutron star heats up due to exothermic accretion-induced processes taking place in the crust. Besides the long-known deep crustal heating of nuclear origin, a likely non-nuclear source of heat, dubbed "shallow heating", is present at lower densities. Most of the accretion-induced heat slowly diffuses into the core on a timescale of years. Over many outburst cycles, a state of equilibrium is reached when the core temperature is high enough that the heating and cooling (photon and neutrino emission) processes are in balance. Aims. We investigate how stellar characteristics and outburst properties affect the long-term temperature evolution of a transiently accreting neutron star. For the first time the effects of crustal properties are considered, particularly that of shallow heating. Methods. Using our code NSCoo1, we tracked the thermal evolution of a neutron star undergoing outbursts over a period of 10(5) yr. The outburst sequence is based on the regular outbursts observed from the neutron star transient Aq1 X-1. For each model we calculated the timescale over which equilibrium was reached and we present these timescales along with the temperature and luminosity parameters of the equilibrium state. Results. We performed several simulations with scaled outburst accretion rates, to vary the amount of heating over the outburst cycles. The results of these models show that the equilibrium core temperature follows a logarithmic decay function with the equilibrium timescale. Secondly, we find that shallow heating significantly contributes to the equilibrium state. Increasing its strength raises the equilibrium core temperature. We find that if deep crustal heating is replaced by shallow heating alone, the core would still heat up, reaching only a 2% lower equilibrium core temperature. Deep crustal heating may therefore not be vital to the heating of the core. Additionally, shallow heating can increase the quiescent luminosity to values higher than previously expected. The thermal conductivity in the envelope and crust, including the potentially low-conductivity pasta layer at the bottom of the crust, is unable to significantly alter the long-term internal temperature evolution. Stellar compactness and nucleon pairing in the core change the specific heat and the total neutrino emission rate as a function of temperature, with the consequences for the properties of the equilibrium state depending on the exact details of the assumed pairing models. The presence of direct Urca emission leads to the lowest equilibrium core temperature and the shortest equilibrium timescale.