Constraints on axions from neutron star in HESS J1731-347

To constrain the allowed range for the axion decay constant $f_{a}$ or, equivalently, for the axion mass $m_{a}$, we consider the cooling of a neutron star with strong proton superfluidity and normal (non-superfluid) neutrons inside its core and without strong magnetic field, by analogy with the observed supernova remnant in HESS J1731-347. For this specific case, we demonstrate that after the thermal relaxation is over, the hydrostatic structure of the neutron star can be well described with the aid of solution of Einstein field equations, applied to a sphere of fluid in hydrostatic equilibrium, derived by Tolman. The internal temperature of the neutron star is calculated assuming that the cooling occurs dominantly due to production of neutrino pairs and axions in the nn-bremsstrahlung. To impose a constraint to the axion decay constant the fact is used that the currently observed neutron star surface temperature does not deviate from the neutrino cooling scenario. For the KSVZ-axion model we find that $f_{a}>1.9\times 10^{8}$ GeV, while for the DFSZ-axion model we obtain $f_{a}>4.7\times 10^{9}$ GeV.