Cooling of neutron stars in soft x-ray transients

Thermal states of neutron stars in soft x-ray transients (SXRTs) are thought to be determined by “deep crustal heating” in the accreted matter that drives the quiescent luminosity and cooling via emission of photons and neutrinos from the interior. In this study, we assume a global thermal steady state of the transient system and calculate the heating curves (quiescent surface luminosity as a function of mean accretion rate) predicted from theoretical models, taking into account variations in the equations of state, superfluidity gaps, thickness of the light element layer, and a phenomenological description of the direct Urca threshold. We further provide a statistical analysis on the uncertainties in these parameters, and compare the overall results with observations of several SXRTs, in particular the two sources containing the coldest (SAX J1808.4-3658) and the hottest (Aql X-1) neutron stars. Interpretation of the observational data indicates that the direct Urca process is required for the most massive stars and also suggests small superfluid gaps.

Figure 1

Radial profiles ($r=0$ at the core) of the neutrino emissivity from various processes for a $2.1{\mathrm{M}}_{\odot }$ APR neutron star at different core temperatures. Theoretical predictions from models “SFB-a-T73” are employed for neutron singlet, neutron triplet and proton singlet pairing gaps (see text).

Figure 2

(a) A series of selected heating curves ${\log }_{10}{L}_{\gamma }^{\infty }-{\log }_{10}\dot{M}$ based on EoSs in Table 2 with superfluidity gap models “SFB-a-T73” and observational data taken from Refs. [47, 48]; two dashed arrows represent old data for the coldest star in SAX J1808 [49] to be replaced by the most recent one (enlarged solid arrow) with distance estimate updated [16], and the line cross with highest luminosity stands for the hottest star in Aql X-1. (b) Similar figure for “SFB-0-T73” gap models.

Figure 3

Exemplary heating curves ${\log }_{10}{L}_{\gamma }^{\infty }-{\log }_{10}\dot{M}$ for APR/HHJ EoS in comparison with luminosity and accretion rate measurements of the hottest (line cross, Aql X-1) and coldest (arrow, SAX J1808) sources; light elements (H or He) are not taken into account. See Fig. 2 for more EoSs and observational data.

Figure 4

Heating curves ${\log }_{10}{L}_{\gamma }^{\infty }-{\log }_{10}\dot{M}$ for APR/HHJ EoS, with superfluidity models “SFB-0-T73” (vanishing $n^3{P}_2$ gap). Effects from light-element layer in the envelope [see Eq. (7)] are shown for two different masses (APR: $2.0{\mathrm{M}}_{\odot },2.05{\mathrm{M}}_{\odot }$; HHJ: $1.7{\mathrm{M}}_{\odot },1.9{\mathrm{M}}_{\odot }$) below and above the direct Urca onset.

Figure 5

Schematic plot of regulating the direct Urca emissivity near its threshold. See Eq. (8) and text.

Figure 6

Neutrino emissivities plotted for a $2.1{\mathrm{M}}_{\odot }$ star taking into account shifting and broadening of the direct Urca threshold (see Sec. 4b). LHS, logarithm scale; RHS, linear scale (direct Urca only).

Figure 7

Introduce shifting and broadening of direct Urca process and recalculate the heating curves ${\log }_{10}{L}_{\gamma }^{\infty }-{\log }_{10}\dot{M}$ as in Fig. 3. Double arrows represent the measurements of 1H 1905+000 which are also compelling. See discussions in the text.

Figure 8

Posterior probability distributions for direct Urca threshold with broadening and $p^1{S}_0,n^3{P}_2$ superfluid critical temperatures ${\log }_{10}{T}_{\mathrm{cps}}^{\max }-{\log }_{10}{T}_{\mathrm{cnt}}^{\max }$; the thin lines are $1\sigma$ contours.

Figure 9

Posterior probability distributions for light-element amount ${\log }_{10}{\eta }_{\mathrm{Aql}}$, deep crustal heating energy $Q$, high-density EoS and masses of neutron stars in 1808 and Aql X-1; the thin lines are $1\sigma$ contours.

Figure 10

Posterior probability distributions for the surface luminosity ${\log }_{10}{L}_{\gamma }^{\infty }$ and core temperature ${\log }_{10}{T}_{\mathrm{core}}$ of neutron stars in 1808 and Aql X-1; the thin lines are $1\sigma$ contours.