Constraining superfluidity in dense matter from the cooling of isolated neutron stars

We present a quantitative analysis of superfluidity and superconductivity in dense matter from observations of isolated neutron stars in the context of the minimal cooling model. Our new approach produces the best fit neutron triplet superfluid critical temperature, the best fit proton singlet superconducting critical temperature, and their associated statistical uncertainties. We find that the neutron triplet critical temperature is likely $2.{09}_{-1.41}^{+4.37}\times {10}^8$ K and that the proton singlet critical temperature is $7.{59}_{-5.81}^{+2.48}\times {10}^9$ K. However, we also show that this result only holds if the Vela neutron star is not included in the data set. If Vela is included, the gaps increase significantly to attempt to reproduce Vela's lower temperature given its young age. Further including neutron stars believed to have carbon atmospheres increases the neutron critical temperature and decreases the proton critical temperature. Our method demonstrates that continued observations of isolated neutron stars can quantitatively constrain the nature of superfluidity in dense matter.

Figure 1

Theoretical cooling curves illustrating how temperature and luminosity decrease over time. The black boxes represent neutron star cooling data and three colored bands show the $\pm 1\sigma$ uncertainties on the cooling curves (from superfluidity and superconductivity). The three bands represent three different values of $\eta$, ${10}^{-7}$ (purple, \\  hatching), ${10}^{-12}$ (green, horizontal hatching), and ${10}^{-17}$ (red, // hatching). The temperature results [labeled (a), (c), and (e)] correspond to the parameter limits in Table 2, while the luminosity results [labeled (b), (d), and (f)] correspond to the parameter limits in Table 3.

Figure 2

Uncorrelated samples from the critical temperatures from the fit to the luminosities as a function of the Fermi momenta for protons [left panel, (a)] and neutrons [right panel, (b)] without Vela or the carbon stars. The left boundary in both panels represents the Fermi momentum at the crust-core transition (denoted “$k_f^p\min$” and “$k_f^n\min$”). The right boundary represents the Fermi momentum in the center (denoted “$k_f^p\max$” and “$k_f^n\max$”) of a $1.4{\mathrm{M}}_{\odot }$ neutron star. The uncertainty in the critical temperatures is large and there is a preference for proton superconductivity to peak at lower Fermi momenta.

Figure 3

Uncorrelated samples from the critical temperatures from the fit to the luminosities as in Fig. 2 but now with Vela. Similar to previous works, we find strong proton superconductivity (a) and slightly weaker neutron triplet superfluidity (b). The proton superconductivity moves to higher densities and the density dependence of the neutron superfluid gap broadens to maximize the cooling to match the low luminosity of Vela.

Figure 4

Uncorrelated samples from the critical temperatures from the fit to the luminosities as in Figs. 2 and 3 now having added both Vela ($\mathrm{B}0833–45$) and the carbon-atmosphere stars to the analysis [(a) displays proton critical temperatures; (b) displays neutron critical temperatures]. Comparing to Fig. 3, the proton critical temperatures are smaller. The smaller proton critical temperature ensures younger stars are warm enough to match the relatively large luminosities from these two stars.