Tests of the nuclear equation of state and superfluid and superconducting gaps using the Cassiopeia A neutron star

The observed rapid cooling of the Cassiopeia A neutron star can be interpreted as being caused by neutron and proton transitions from normal to superfluid and superconducting states in the stellar core. Here we present two new Chandra ACIS-S Graded observations of this neutron star and measurements of the neutron star mass $M$ and radius $R$ found from consistent fitting of both the x-ray spectra and cooling behavior. This comparison is only possible for individual nuclear equations of state. We test phenomenological superfluid and superconducting gap models which mimic many of the known theoretical models against the cooling behavior. Our best-fit solution to the Cassiopeia A data is one in which the $(M,R)=(1.44M_{\mathrm{Sun}},12.6\text{km})$ neutron star is built with the BSk21 equation of state, strong proton superconductor and moderate neutron triplet superfluid gap models, and a pure iron envelope or a thin carbon layer on top of an iron envelope, although there are still large observational and theoretical uncertainties.

Figure 1

Top: Neutron singlet gap energy (left axis) and critical temperature (right axis). Middle: Proton singlet gap energy and critical temperature. Bottom: Neutron triplet gap energy and critical temperature. Labels indicate particular gap models (see Table 2).

Figure 2

Neutron star mass versus radius for three nuclear EOSs: APR (solid), BSk20 (long-dashed), and BSk21 (short-dashed). Squares indicate $(M,R)$ values which produce good fits to Chandra ACIS-S Graded data at a 90% confidence level.

Figure 3

Best-fit to Chandra ACIS-S Graded data, as determined by $\Delta {\chi }^2$ as a function of NS mass (top) and radius (bottom) for three nuclear EOSs: APR (solid), BSk20 (long-dashed), and BSk21 (short-dashed). Dotted lines indicate the 90% confidence level.

Figure 4

Critical temperature $T_{\mathrm{c}}$ for neutron singlet superfluidity as a function of fractional radius of a NS constructed using the APR EOS $\left(M=1.4M_{\mathrm{Sun}},R=11.6\mathrm{km}\right)$. Different curves correspond to different gap models that are shown in Fig. 1. Vertical dotted lines denote the boundaries between the core, inner crust, and outer crust of the NS.

Figure 5

Redshifted surface temperature $T_{\mathrm{s}}^{\infty }$ as a function of age for a $1.4M_{\mathrm{Sun}}$ APR NS with an iron envelope. Different curves are cooling simulations using one corresponding neutron singlet gap model (see Fig. 1), while the curve labeled “no superfluid” is a simulation that does not include any superfluid components. Crosses are the observed temperatures of the Cas A NS.

Figure 6

Critical temperature $T_{\mathrm{c}}$ for proton superconductivity as a function of fractional radius of a NS constructed using the APR EOS $(M=1.4M_{\mathrm{Sun}},R=11.6\mathrm{km}$; top panel) and BSk20 EOS $(M=1.4M_{\mathrm{Sun}},R=11.7\mathrm{km}$; bottom panel). Different curves correspond to different proton singlet gap models that are shown in Fig. 1. Vertical dotted line denotes the boundary between the core and inner crust of the NS.

Figure 7

Critical temperature $T_{\mathrm{c}}$ for proton superfluidity as a function of mass density of a NS constructed using the APR EOS. Different curves correspond to different proton singlet gap models that are shown in Fig. 1. Vertical dotted lines denote the core density of NSs of different mass.

Figure 8

Redshifted surface temperature $T_{\mathrm{s}}^{\infty }$ as a function of age for a $1.4M_{\mathrm{Sun}}$ APR NS with an iron envelope. Different curves are cooling simulations using one corresponding proton gap model (see Fig. 1), while the curve labeled “no superconductor” is a simulation that does not include a superconductor component. Crosses are the observed temperatures of the Cas A NS.

Figure 9

Critical temperature $T_{\mathrm{c}}$ for neutron triplet superfluidity as a function of fractional radius of a NS constructed using the APR EOS $(M=1.4M_{\mathrm{Sun}},R=11.6\mathrm{km}$; top panel) and BSk21 EOS $(M=1.4M_{\mathrm{Sun}},R=12.6\mathrm{km}$; bottom panel). Different curves correspond to different gap models that are shown in Fig. 1. Vertical dotted line denotes the boundary between the core and inner crust of the NS.

Figure 10

Critical temperature $T_{\mathrm{c}}$ for neutron triplet superfluidity as a function of mass density of a NS constructed using the APR EOS. Different curves correspond to different gap models that are shown in Fig. 1. Vertical dotted lines denote the core density of NSs of different mass.

Figure 11

Redshifted surface temperature $T_{\mathrm{s}}^{\infty }$ as a function of age for a $1.4M_{\mathrm{Sun}}$ APR NS with an iron envelope. Different curves are cooling simulations using one corresponding neutron triplet gap model (see Fig. 1), while the curve labeled “EEHOr” is a simulation that effectively does not include any superfluid components since this gap model has no effect for the relevant ages shown. Crosses are the observed temperatures of the Cas A NS.

Figure 12

Redshifted surface temperature $T_{\mathrm{s}}^{\infty }$ as a function of age for a $1.4M_{\mathrm{Sun}}$ NS using the APR (left), BSk20 (center), and BSk21 (right) EOS with an iron envelope. Different curves are cooling simulations using the SFB neutron singlet, CCDK proton singlet, and one of various neutron triplet gap models (see Fig. 1). Crosses are the observed temperatures of the Cas A NS.

Figure 13

Redshifted surface temperature $T_{\mathrm{s}}^{\infty }$ as a function of age for a $1.4M_{\mathrm{Sun}}$ NS using the APR (left), BSk20 (center), and BSk21 (right) EOS with a carbon envelope $\left(\Delta M\approx {10}^{-8}{M}_{\mathrm{Sun}}\right)$. Different curves are cooling simulations using the SFB neutron singlet, CCDK proton singlet, and one of various neutron triplet gap models (see Fig. 1). Crosses are the observed temperatures of the Cas A NS.

Figure 14

Redshifted surface temperature $T_{\mathrm{s}}^{\infty }$ as a function of year, with redshift $1+z_{\mathrm{g}}=1.229$. Crosses and $1\sigma$ error bars are the observed Chandra ACIS-S Graded temperatures of the Cas A NS. Cooling curve is for a $M=1.441M_{\mathrm{Sun}}$ and $R=12.59\mathrm{km}$ NS built using the BSk21 EOS with an iron envelope and SFB neutron singlet, CCDK proton singlet, and TToa neutron triplet gap models. Inset: Expanded view of temperature evolution as a function of time.