Quiescent luminosities of accreting neutron stars with different equation of states.

We model the quiescent luminosity of accreting neutron stars with several equation of states (EOSs), including the effect of pion condensation and superfluidity. As a consequence of comparison with the observations, we show that the results with Togashi EoS (the strong direct Urca process is forbidden) and TM1e EoS (mass at direct Urca process is $2.06 M_\odot$) can explain the observations well by considering pion condensation and the effect of superfluidity, while LS220 EoS and TM1 EoS can explain the observations well by considering the baryon direct Urca process and the effect of superfluidity. Besides, we compare the results with the observations of a neutron star RX J0812.4-3114 which has the low average mass accretion rate ($\langle\dot{M}\rangle\sim(4-15)\times 10^{-12}~M_\odot ~\rm yr^{-1}$) but high thermal luminosity ($L_q^\infty\sim(0.6-3)\times10^{33}~\rm erg ~ s^{-1}$), and we suggest that a low-mass neutron star ($<1M_\odot$) with minimum cooling can explain the lower limit of the observation of thermal luminosity of RX J0812.4-3114, which is qualitatively consistent with the previous work~\cite{Zhao2019}. However, to explain its upper limit, some other heating mechanisms besides standard deep crustal heating may be needed.