Evidence of Gapless Superfluidity in MXB 1659-29 With and Without Late Time Cooling

Abstract

The interpretation of the thermal relaxation of some transiently accreting neutron stars in quasipersistent soft X-ray transients, especially MXB 1659-29, has been found to be challenging within the traditional deep crustal heating paradigm. Due to the pinning of quantized vortices, the neutron superfluid is not expected to remain at rest in the crust, as was generally assumed. We have recently shown that for sufficiently large relative superflows, the neutron superfluid could become gapless. This dynamical phase could naturally explain the late-time cooling of MXB 1659-29. However, the interpretation of the last observation of MXB 1659-29 in 2013 before its second accretion phase in 2015 remains debated, with some spectral fits being consistent with no further temperature decline. Here, we revisit the cooling of this neutron star considering the different fits. New simulations of the crust cooling are performed, accounting for neutron diffusion and allowing for gapless superfluidity. In all cases, gapless superfluidity is found to provide the best fit to observations.

1. Introduction

Although most known neutron stars are observed as isolated radio pulsars, some are found in binary systems. In Low-Mass X-ray Binaries (LMXBs), a neutron star swallows matter from a stellar companion with a mass lower than or similar to 1${\mathrm{M}}_{\odot }$ (${\mathrm{M}}_{\odot }$ is the solar mass) [1]. Mass accretion occurs through Roche lobe overflow; the accreted matter forms an accretion disk before falling to the neutron star’s surface. As a result of thermal–viscous instability in the accretion disk [2,3,4], accretion in most LMXBs occurs sporadically; accretion phases (also called outbursts) are separated by quiescent phases during which little or no accretion occurs. These particular LMXBs form the specific class of Soft X-ray Transients (SXTs), and are characterized by an outburst duration varying from weeks to months. However, for a certain subclass of SXTs known as quasipersistent Soft X-ray Transients (referred to as qSXTs), the duration of each outburst can persist for years or even decades. The energy released by accretion-induced nuclear reactions during these long outbursts drives the crust out of thermal equilibrium with the core. During quiescence, the altered crust thermally relaxes towards equilibrium with the core [5,6]. Over the last two decades, the thermal relaxation of a dozen qSXTs in quiescence has been monitored up to ${10}^3$–${10}^4$ days after the end of their outburst (see [7] for a review), which is long enough to probe neutron superfluidity in the deepest layers of the crust [8,9].

In the standard cooling theory, the crust is heated due to compression-induced nuclear reactions (electron captures and mainly pycnonuclear fusions) [6]. These reactions release $Q_{\mathrm{dch}}$∼$1.5$ MeV/nucleon and take place at densities of $\rho$∼${10}^{12}-{10}^{13}$ g ${\mathrm{cm}}^{-3}$ [10,11,12,13] in the inner crust; thus, this scenario is called deep crustal heating. In these crustal layers, free neutrons are expected to form ¹S₀ Cooper pairs at temperatures below $T_c$∼${10}^{10}$ K [14], giving rise to a neutron superfluid phase. Superfluidity in neutron stars has found strong support from radio timing of frequency glitches in numerous pulsars [15]. These sudden variations of the spin frequency are interpreted as global dynamical redistributions of angular momenta between the neutron superfluid and the rest of the star [16,17]. More recently, neutron superfluidity has also been corroborated by the rapid luminosity decline of the neutron star located at the center of the Cassiopeia A supernova remnant [18,19,20,21,22,23,24,25,26,27] and by the crust cooling of neutron stars located in LMXBs [8,28,29,30,31,32,33,34,35,36,37,38,39,40].

The standard cooling theory has been recently challenged. First, analyses of the thermal relaxation of some qSXTs during the first months of quiescence have highlighted the importance of additional heat sources releasing $Q_{\mathrm{sh}}$∼1–10 MeV/nucleon in the shallowest layers of the crust (at densities $\rho \le {10}^{10}$ g ${\mathrm{cm}}^{-3}$) [30]. The physical origin of this shallow heating is still unknown (see, e.g., [41] for a summary of the different sources and proposed scenarios). Second, in deep crustal heating, nuclei are assumed to sink very slowly and together with the neutrons they emit or capture, neglecting the redistribution of free neutrons between the various layers of the crust. As first pointed out in [42,43], such an approach leads to a discontinuous change of the neutron chemical potential at the neutron-drip transition, indicating that the neutrons are not in diffusive equilibrium. The redistribution of free neutrons in the crust was specifically studied in [44]. Accreted crust models accounting for superfluid neutron hydrodynamics (nHD) have been developed in [45,46,47]. With neutron diffusion, the equation of state of accreted crusts turns out to be very similar to that of non-accreted crusts, and the heat release drops to $Q_{\mathrm{nHD}}$∼0.1–0.3 MeV/nucleon.

In both the traditional and thermodynamically consistent models of qSXTs, the neutron superfluid is implicitly assumed to remain at rest in the crust. However, the superfluid is expected to decouple from the crust as a result of the pinning of quantized neutron vortices [48,49]. The existence of a lag between the superfluid and the rest of star is crucial for the interpretation of pulsar glitches [15,50,51]. Although monitoring the spin frequency in accreting neutron stars is much more challenging than in isolated radio pulsars, glitches have been reported in these systems as well [52,53]. In LMXBs, neutron stars are spun up by the transfer of angular momentum from the companion star; this scenario has been confirmed by measurements of high spin frequencies [54,55,56]. Therefore, the neutron superfluid is expected to lag behind. We have recently shown that the superfluid becomes gapless when the relative superfluid velocity ${\mathbb{V}}_n$ exceeds Landau’s velocity ${\mathbb{V}}_{Ln}$ [57], with superfluidity being destroyed at higher velocities ${\mathbb{V}}_{cn}^{\left(0\right)}\simeq 1.359{\mathbb{V}}_{Ln}$. Gapless superfluidity can significantly impact the thermal relaxation of the crust of qSXTs, especially MXB 1659-29 [38,39].

MXB 1659-29 was first observed in outburst in 1976 [58], and accretion continued for three years. From 1999 to 2001, MXB 1659-29 underwent another accretion cycle [59,60]; this outburst lasted 2.5 years, and is referred to as outburst I. During this period, nearly coherent oscillations with a frequency of ∼567 Hz were detected, suggesting that the neutron star might be an accreting millisecond X-ray pulsar [61,62,63]. The subsequent decade of monitoring showed a decrease in surface temperature consistent with thermal relaxation of the crust [62,64,65,66]. The effective surface temperatures $T_{\mathrm{eff}}^{\infty }$ (as seen by an observer at infinity) inferred from observations at different times are summarized in

Table 1. Inferred effective surface temperature $T_{\mathrm{eff}}^{\infty }$ (as seen by an observer at infinity) of MXB 1659-29 in electronvolts (eV) at different times after its 1999–2001 outburst (referred to as outburst I). Time is given in terms of the Modified Julian Date (MJD). The outburst ended at time $t_0=$ 52,162 MJD. The first six points come from [68], while the last point is from the best spectral fits of [67]. Uncertainties are provided at the 90% confidence level.

	Observatory	Obs ID	MJD	${\mathit{k}}_{\mathbf{B}}{\mathit{T}}_{\mathbf{eff}}^{\mathit{\infty }}$ (eV)
1	Chandra	2688	52,197.7	111.1 ± 1.3
2	Chandra	3794	52,563.0	79.5 ± 1.6
3	XMM-Newton	0153190101	52,711.6	73.0 ± 1.9
4	Chandra	3795	52,768.7	67.8 ± 2.1
5	Chandra	5469/6337	53,566.4	55.5 ± 2.4
6	Chandra	8984	54,583.8	54.8 ± 3.2
7 (fit 1)	Chandra	13711/14453	56,113	55.0 ± 3.0
7 (fit 2)	Chandra	13711/14453	56,113	49.0 ± 2.0
7 (fit 3)	Chandra	13711/14453	56,113	45.0 ± 3.0
7 (fit 4)	Chandra	13711/14453	56,113	43.0 ± 5.0

. In 2009, the cooling curve of MXB 1659-29 was modeled by Brown and Cumming [30]. Their best model predicted that the crust of MXB 1659-29 reached thermal equilibrium with the core ∼1000 days after the end of outburst I (between 2003 and 2004). However, a Chandra observation in 2012 reported by Cackett et al. [67] (at 56,113 MJD, corresponding to ∼3951 days after the end of outburst I) showed an unexpected dimming of X-ray luminosity. Depending on the spectral fits, two distinct scenarios have been considered:

• Fitting the quiescent X-ray spectra with an absorbed neutron star atmosphere model and allowing for a variation of the hydrogen column density along the line of sight $N_H$, resulting in $N_H=(4.7\pm 1.3)\times {10}^{21}$ ${\mathrm{cm}}^{-2}$. The associated effective surface temperature $k_{\mathrm{B}}{T}_{\mathrm{eff}}^{\infty }=55.0\pm 3.0$ eV (referred to as fit 1 in Table 1 and the following results) is consistent with the previous 2008 Chandra observation by Cackett et al. [66], and suggests that the thermal equilibrium between the crust and core was already restored in 2003–2004. Cackett et al. [67] invoked a thickening of the accretion disk or its precession as possible explanations of the significant increase in $N_H$ compared to the value deduced from the 2008 Chandra observation [66] ($N_H=2.0\times {10}^{21}$ ${\mathrm{cm}}^{-2}$). However, the same authors emphasized that such a scenario is not expected from the standard accretion disk theory.
• Alternatively, the observed drop in X-ray luminosity might suggest a further decrease of the neutron star’s temperature. Keeping the hydrogen column density fixed to the value obtained from the 2008 Chandra observation [66] ($N_H=2.0\times {10}^{21}$ ${\mathrm{cm}}^{-2}$) and using an absorbed neutron star atmosphere model yielded $k_{\mathrm{B}}{T}_{\mathrm{eff}}^{\infty }=49.0\pm 2.0$ eV (referred to as fit 2). More refined spectral models including a neutron star atmosphere and a power-law component then lead to lower effective surface temperatures of $k_{\mathrm{B}}{T}_{\mathrm{eff}}^{\infty }=45.0\pm 3.0$ eV (for a photon power-law index of $\mathrm{\Gamma }=1.5$, referred to as fit 3) and $k_{\mathrm{B}}{T}_{\mathrm{eff}}^{\infty }=43.0\pm 5.0$ eV (for a photon power-law index of $\mathrm{\Gamma }=2$, labeled as fit 4).

In 2015, MXB 1659-29 underwent a second outburst phase [69] (referred to as outburst II). Its subsequent cooling phase was studied by Parikh et al. [68], who also revisited the post-outburst I phase, assuming implicitly that the crust reached thermal equilibrium with the core before the 2012 Chandra observation [67] (this puzzling observation was omitted). In fact, the spectral fits obtained by Parikh et al. throughout the post-outburst II cooling phase of MXB 1659-29 did not yield any significant change of $N_H$ (see footnote 9 in [68]) that would corroborate the scenario of a thickening or precession of the accretion disk1. Iaria et al. [70] suggested that additional absorption caused by ejected matter during the accretion phase, could be responsible for the significant change in $N_H$. However, these authors did not provide a timescale for testing this scenario. The post-outburst I cooling of MXB 1659-29 was also studied in [35,36] within the standard paradigm with shallow heating, and in [37,40] accounting for neutron diffusion as well. However, all of these studies adopted the inferred surface temperatures from Parikh et al. [68], ignoring the 2012 Chandra observation [67].

The scenario involving further cooling of the neutron star crust was considered by Horowitz et al. [32], who adopted the value $k_{\mathrm{B}}{T}_{\mathrm{eff}}^{\infty }=49.0\pm 2.0$ eV (fit 2) for the inferred (effective) surface temperature from the 2012 Chandra observation [67]. According to their classical molecular dynamics simulations, they suggested that the puzzling temperature drop could result from the presence of a layer with low thermal conductivity located in the deepest regions of the crust. However, later quantum molecular dynamics simulations did not corroborate this possibility [71]. Using the same effective surface temperature as Horowitz et al. (i.e., fit 2), Deibel et al. [33] were able to reproduce the observations by taking the neutron pairing gap that Gandolfi et al. [72] extracted from quantum Monte Carlo computations and extrapolating it to high densities. This pairing gap, labeled as QMC08 in Figure 1, suggests the destruction of neutron superfluidity at densities below the crust–core transition. However, the same group later refined their quantum Monte Carlo computations, leading to different results [73] in which the neutrons remain superfluid in the deepest layers of the crust and core (see QMC22 in Figure 1). These refined gaps are also consistent with those obtained from other many-body calculations [74,75].

We have recently revisited the cooling of MXB 1659-29 assuming the same inferred effective surface temperature as in [32] (fit 2) and making use of the microscopic calculations of neutron pairing gaps from [73]. We found that both the late-time cooling of MXB 1659-29 and the puzzling 2012 Chandra observation can be explained if the neutron superfluid is in the gapless phase [38]. More systematic calculations (varying the composition of the heat blanketing envelope, the global structure of the neutron star, and the time-averaged accretion rate) accounting for neutron diffusion were also performed in [39]. In these studies we adopted the value of $k_{\mathrm{B}}{T}_{\mathrm{eff}}^{\infty }=43.0\pm 5.0$ eV (fit 4), as it corresponds to the most challenging data within the standard cooling paradigm.

In this paper, we pursue our analysis of MXB 1659-29 by testing the robustness of our cooling model with gapless superfluidity against the different values of $k_{\mathrm{B}}{T}_{\mathrm{eff}}^{\infty }$ inferred by Cackett et al. [67] from the 2012 Chandra observation.

2. Cooling Model

We investigate the thermal evolution of MXB 1659-29 (after the end of outburst I) using the crustcool code of Andrew Cumming2. This code was previously employed in [30,32,33,34,38,39,67] to study the same source. This code is based on the crust composition provided in [10,11,12] and solves the heat diffusion equation along with the hydrostatic structure equation assuming a constant gravity up to a column depth $y_{\mathrm{b}}={10}^{12}$ g ${\mathrm{cm}}^{-2}$ (see [30] for more details). Below this value, a heat-blanketing envelope model is used, providing a relation between the (non-redshifted) temperature at $y_{\mathrm{b}}$ and the (non-redshifted) effective surface temperature $T_{\mathrm{eff}}$ (see, e.g., [76,77] for more details about envelope models). We have modified this code to account for neutron diffusion and gapless superfluidity, as described in [38,39].

We consider a neutron star with mass ${\mathrm{M}}_{\mathrm{NS}}=1.62{\mathrm{M}}_{\odot }$ and radius ${\mathrm{R}}_{\mathrm{NS}}=11.2$ km (as considered in the original paper of Brown and Cumming [30]). As in previous studies [30,32,33,38,39,78], the accretion rate is assumed to be constant and fixed to its time-averaged value $\dot{m}={10}^{17}$ g ${\mathrm{s}}^{-1}$. We ran simulations of neutron star cooling using the envelope model provided by the crustcool\envelope_data\grid_He9 file. This envelope (labeled as He9) is made of pure helium down to column depth $y_{\mathrm{He}}={10}^9$ g ${\mathrm{cm}}^{-2}$ and of pure iron down to the bottom of the envelope at $y_{\mathrm{b}}={10}^{12}$ g ${\mathrm{cm}}^{-2}$. This specific choice of envelope is motivated by the cooling models from [30,34] and the recent results obtained in [38,39]. As in our recent analyses, we use the results obtained from quantum Monte Carlo computations [73] (see QMC22 in Figure 1) for the neutron pairing gaps ${\mathrm{\Delta }}_n^{\left(0\right)}$.

The fitting parameters of the neutron star crust cooling model are as follows:

• The core temperature $T_{\mathrm{core}}$, which is constrained by the (late-time) observations.
• The temperature at the bottom of the envelope $T_{\mathrm{b}}$ (during the outburst), which takes into account the influence of shallow heating in an effective way (see [31,34,79] for more discussions).
• The impurity parameter $Q_{\mathrm{imp}}$, which controls the thermal conductivity of the crust. As in our previous investigations [38,39], this parameter is assumed to be the same throughout the crust.
• The normalized neutron effective superfluid velocity ${\mathbb{V}}_n/{\mathbb{V}}_{Ln}$, which accounts for modification of the specific heat of superfluid neutrons in presence of superflow (see Equations (10) and (21) of [39]). We assume a uniform value of ${\mathbb{V}}_n/{\mathbb{V}}_{Ln}$ throughout the crust. We have also carried out simulations within the standard cooling theory (with a static neutron superfluid in the crust frame) by setting ${\mathbb{V}}_n/{\mathbb{V}}_{Ln}=0$, which we refer to as the BCS superfluidity in what follows.

The values of these cooling parameters and their uncertainties were obtained using the Markov Chain Monte Carlo (MCMC) method. We have adapted the mcee.py Python 3 script to allow for gapless superfluidity. This script is provided with the crustcool code. It makes use of the Python emcee package3 and draws samples from the posterior probability distributions of the parameters through the generation of $n_{\mathrm{walkers}}$ Markov chains in the parameter space, each with a length of $n_{\mathrm{steps}}$ (see [80] for more details). The whole procedure of the MCMC simulation is described in [39]. In our cooling simulations, we chose $n_{\mathrm{walkers}}=25$ and $n_{\mathrm{steps}}=3000$ and assumed uniform prior probabilities for the fitting parameters (as prescribed in [30]): $T_{\mathrm{core}}\in \left[0;20\right]\times {10}^7$ K, $T_{\mathrm{b}}\in \left[0.2;9.0\right]\times {10}^8$ K, ${log}_{10}\left(Q_{\mathrm{imp}}\right)\in \left[-2;2\right]$, and ${\mathbb{V}}_n/{\mathbb{V}}_{Ln}\in \left[1;1.359\right]$ (or fixed to ${\mathbb{V}}_n=0$ when neutrons are assumed to be in BCS regime). We have burnt-in the first 10% of the samples of each chain. The marginalized posterior probability distributions of the parameters were plotted using the Python corner package4. Except for multimodal distributions, the best-fit parameters were obtained using the median values of the respective one-dimensional marginalized probability distributions. Errors were estimated at the 68% uncertainty level (corresponding to the 0.16–0.84 quantiles of the marginalized posterior distributions).

3. Results

3.1. With BCS Superfluidity

Figure 2, Figure 3, Figure 4 and Figure 5 show the marginalized posterior probability distributions of the model parameters for the cooling of MXB 1659-29 with superfluid neutrons in the classical BCS regime within the accreted crust model of [45,46] and for the various effective surface temperatures $k_{\mathrm{B}}{T}_{\mathrm{eff}}^{\infty }$ inferred from the spectral fits for the 2012 Chandra observation [67]. The associated cooling curves are shown in Figure 6, Figure 7, Figure 8 and Figure 9, with the optimal fit parameters provided in

Table 2. Parameters of our best cooling models of MXB 1659-29 (post-outburst I) within the accreted neutron star crust model of [45,46] considering superfluid neutrons in the BCS (i.e., ${\mathbb{V}}_n/{\mathbb{V}}_{Ln}=0$) and gapless (i.e., ${\mathbb{V}}_n/{\mathbb{V}}_{Ln}\ge 1$) regimes. Results were obtained with ${\mathrm{M}}_{\mathrm{NS}}=1.62{\mathrm{M}}_{\odot }$, ${\mathrm{R}}_{\mathrm{NS}}=11.2$ km using the He9 envelope model and are displayed with $68\%$ uncertainty level. See text for details.

2012 Chandra Observation	${\mathit{T}}_{\mathbf{core}}$ (107 K)	${\mathit{Q}}_{\mathbf{imp}}$	${\mathit{T}}_{\mathbf{b}}$ (108 K)	${\mathbb{V}}_{\mathit{n}}/{\mathbb{V}}_{\mathit{Ln}}$
fit 1	${3.04}_{-0.19}^{+0.20}$	${7.41}_{-0.69}^{+0.72}$	3.14 ± 0.13	0
fit 2	2.69 ± 0.15	${8.05}_{-0.61}^{+0.71}$	3.11 ± 0.13	0
fit 3	${2.70}_{-0.17}^{+0.19}$	${8.05}_{-0.66}^{+0.75}$	3.12 ± 0.13	0
fit 4	${2.86}_{-0.20}^{+0.22}$	${7.77}_{-0.68}^{+0.75}$	3.12 ± 0.13	0
fit 1	2.89	6.28	3.22	1.01
fit 2	${0.85}_{-0.54}^{+0.59}$	${17.16}_{-3.70}^{+3.49}$	${3.14}_{-0.15}^{+0.16}$	${1.23}_{-0.11}^{+0.09}$
fit 3	${0.59}_{-0.42}^{+0.61}$	${16.21}_{-3.34}^{+3.60}$	${3.16}_{-0.15}^{+0.16}$	${1.19}_{-0.10}^{+0.12}$
fit 4	${0.65}_{-0.45}^{+0.63}$	${16.35}_{-3.58}^{+3.68}$	${3.14}_{-0.15}^{+0.16}$	1.20 ± 0.11

.

Regardless of the choice of spectral fit for the 2012 Chandra observation [67], the values of $Q_{\mathrm{imp}}$ and $T_{\mathrm{b}}$ are comparable. The cooling models only differ from the prediction of $T_{\mathrm{core}}$ because this cooling parameter is constrained by the late-time observations, and as such depends on the inferred value of $k_{\mathrm{B}}{T}_{\mathrm{eff}}^{\infty }$ from the 2012 Chandra observation [67]. As shown by the dashed curves in Figure 6, Figure 7, Figure 8 and Figure 9, and consistent with the results of [30], cooling models with BCS superfluidity predict that the crust returned to thermal equilibrium with the core about 1000 days after the end of the outburst. Therefore, the 2012 Chandra observation at 56113 MJD cannot be explained if the effective surface temperature $T_{\mathrm{eff}}^{\infty }$ is inferred from spectral fits 2, 3, and 4 (see the dashed curves in Figure 7, Figure 8 and Figure 9). This observation can only be reproduced if no further cooling of the crust is assumed between the 2008 and 2012 Chandra observations, corresponding to fit 1 (see the dashed curve in Figure 6). Our results are consistent with those of [35,36,37,40,68].

3.2. With Gapless Superfluidity

Figure 10, Figure 11, Figure 12 and Figure 13 display the marginalized posterior probability distributions of the model parameters for MXB 1659-29 post-outburst I with superfluid neutrons in the gapless regime using the accreted crust model of [45,46]. The cooling curves are shown in Figure 6, Figure 7, Figure 8 and Figure 9, with the associated best-fit parameters displayed in Table 2.

When allowing for gapless superfluidity, we obtain an excellent fit to all cooling observations. In particular, the 2012 Chandra observation with $k_{\mathrm{B}}{T}_{\mathrm{eff}}^{\infty }$ inferred from spectral fits relying on further crust cooling scenario (corresponding to fits 2, 3, and 4) is now explained; the cooling curves are represented by the red solid lines in Figure 7, Figure 8 and Figure 9. For these cooling models, MXB 1659-29 was still cooling before the onset of outburst II in 2015 (about 5650 days after the end of outburst I); if this source had remained quiescent, the thermal equilibrium between the crust and the core would have been reached about 20,000 days after the end of outburst I. Such a delay in the thermal relaxation is consistent with the estimation provided by Equation (25) in [39]. The cooling model with the last data points provided by fit 1 (no further cooling between 2008 and 2012) deserves further discussion; its marginalized posterior probability distributions display a bimodal structure (see Figure 10), making irrelevant the use of the median to extract the optimal cooling parameters. The parameters that provide the best fit to the cooling data (corresponding to one of the modes) are as follows: $T_{\mathrm{core}}=2.89\times {10}^7$ K, $Q_{\mathrm{imp}}=6.28$, $T_{\mathrm{b}}=3.22\times {10}^8$ K, and ${\mathbb{V}}_n/{\mathbb{V}}_{Ln}=1.01$. The resulting cooling curve is shown in Figure 6. The cooling model with gapless superfluidity leads to an equally good fit as the one obtained assuming BCS superfluidity, with comparable associated cooling curves. Therefore, the physical interpretation of the cooling data is ambiguous in this case.

4. Conclusions

In this paper, we have investigated the cooling of MXB 1659-29 after the end of the outburst I considering different spectral fits of the 2012 Chandra observation reported in [67]. Cooling simulations have been performed within the thermodynamically consistent approach of [45,46], which takes the influence of neutron diffusion into account. Neutron superfluidity in the inner crust (characterized by the ¹S₀ pairing gap obtained from the quantum Monte Carlo computations in [73]) is either in the static BCS regime (i.e., ${\mathbb{V}}_n=0$) or the dynamical gapless regime (i.e., ${\mathbb{V}}_n\ge {\mathbb{V}}_{Ln}$). The uncertainties of the fitting parameters have been assessed using MCMC simulations.

When restricting the analysis to BCS superfluidity, observations can only be reproduced by adopting spectral fit 1 ($k_{\mathrm{B}}{T}_{\mathrm{eff}}^{\infty }=55.0\pm 3.0$ eV) for the 2012 Chandra observation. This model implies that the crust–core thermal equilibrium was restored. Nonetheless, this cooling scenario, tacitly assumed in [35,36,37,40,68], implies an increase in the density of the hydrogen column, the origin of which remains unclear. Scenarios relying on further crust cooling (spectral fits 2, 3, and 4 of the 2012 Chandra observations) cannot be explained within the traditional cooling model with neutron pairing gaps predicted by microscopic calculations, as shown by the dashed curves in Figure 7, Figure 8 and Figure 9.

On the other hand, gapless superfluidity provides an excellent fit to all cooling data. In particular, scenarios relying on further crust cooling are now well explained (see the red solid curves in Figure 7, Figure 8 and Figure 9). The post-outburst I cooling of MXB 1659-29 (assuming no further cooling between 2008 and 2012, i.e., fit 1) can also be accounted for within gapless superfluidity. However this cooling model leads to an equally good fit as the standard cooling model with superfluid neutrons in BCS regime. As shown in Figure 6, both cooling curves are very similar; therefore, the physical interpretation is ambiguous in this case.

A finite neutron superfluid velocity ${\mathbb{V}}_n$ in the crust of neutron stars can be sustained by the pinning of quantized neutron vortices [38]. Whether the superfluid velocity could exceed Landau’s velocity, as is required for the existence of gapless superfluidity, remains to be investigated. The maximum value of the neutron superflow is limited by the pinning force on individual vortices. Estimates of this pinning force vary significantly depending on the considered approach (see [48,49,81,82,83,84], Table 1 of [85], and references therein). Usually, this pinning force is computed between neutron vortices and nuclei. However, vortices are also expected to pin to proton fluxoids in the deepest layers of the crust [86] and in the core [87,88,89,90,91]. These additional pinning sites provide a higher pinning force [81] and are supported by the observation of glitches in the isolated Crab and Vela pulsars [92]. Estimating the associated maximum superflow beyond which neutron vortices unpin is challenging (see Section 3 of [39] for estimates of Landau’s velocities and pinning forces), as crustal inhomogeneities can increase the critical superflow [93] by an order of magnitude [94]. For these reasons, the possibility of gapless superfluidity in neutron star crusts is not implausible and should be considered in the interpretation of the cooling of qSXTs. The importance of superfluid hydrodynamics in neutron star cooling calls for further studies of the relevant vortex dynamics. The existence of neutron superflow in the inner crust of qSXTs could have other observational consequences, and as such could be further tested. At some point (most likely during the outburst), neutron vortices may unpin, causing an anti-glitch (or under certain circumstances a glitch; see [95]). However, this requires accurate timing of these sources. The very recent detection of an anti-glitch in the rotation-powered pulsar PSR B0540–69 [96], interpreted as a “Partial ‘evaporation’ of the superfluid component,” could be an independent signature of gapless superfluidity (see [97]). This calls for further studies. A similar suppression of superfluidity has also been suggested to explain the cooling of HETE J1900.1–2455 after its ten-year-long outburst [98,99]. Thus, this object appears to be a very promising source for testing gapless superfludity.