© ESO, 2017

1. Introduction

The properties of an accretion disc strongly depend on the parameters of the compact object and the binary system as a whole. At the same time, they also determine the observational appearance of an accreting object. As a consequence, some of the physical phenomena typical for one class of objects may not be observable in another. For instance, transitions between different ionisation states of an accretion disc are commonly considered to be responsible for bright outbursts in dwarf novae and soft X-ray transients (see, e.g. reviews by Cherepashchuk 2000; Lasota 2001; Warner 2003), but previously were not considered for X-ray pulsars (XRPs). The transition is associated with the so-called thermal-viscous instability that arises as a result of the partial ionisation of hydrogen in an accretion disc (Meyer & Meyer-Hofmeister 1984; Smak 1984; Mineshige et al. 1993; Cheng et al. 1992; Chen et al. 1997).

The ionisation state of the plasma in the disc determines the disc opacity and the equation of state, and, therefore, the viscosity (see, e.g. the review by Lasota 2001). In a cold disc that consists mainly of atomic hydrogen, the viscosity and thus the accretion rate are low, so that the accreted matter accumulates in the disc. In the hot state, the hydrogen is mainly ionised and the disc viscosity dramatically increases, which allows most of the accumulated matter to be rapidly accreted onto a compact object. The critical effective temperature defining the two states is the hydrogen ionisation temperature, which is ~6500 K (see Sect. 3.5 in Warner 2003). The transition between the two states is accompanied by a heating wave originating either from the inner or outer part of the accretion disc (Smak 1984). The accretion process becomes stable again when the temperature falls for some reason below the critical value across the entire disc, that is, in the case of a low mass-accretion rate (Lasota 1997), $\mathit{Ṁ}\mathit{<}\mathit{Ṁ}\mathrm{cold}\mathrm{\simeq }\mathrm{3.5}\mathrm{\times }{\mathrm{10}}^{\mathrm{15}}\hspace{0.17em}{\mathit{r}}_{\mathrm{10}}^{\mathrm{2.65}}\hspace{0.17em}{\mathit{M}}_{\mathrm{1.4}}^{-0.88}\mathrm{g}\hspace{0.17em}{\mathrm{s}}^{-1}\mathit{,}$(1)where r₁₀ is the inner disc radius¹ and M_1.4 is the neutron star mass in units of 1.4 M_⊙.

From another perspective, at very low mass accretion rates onto a magnetized neutron star, the so-called “propeller effect” may occur as an abrupt luminosity drop below some critical value (Illarionov & Sunyaev 1975). This rapid cessation of accretion is caused by a centrifugal barrier produced by the rotating magnetosphere if it moves faster than the local Keplerian velocity. In other words, when the magnetospheric radius ${\mathit{R}}_{\mathrm{m}}\mathrm{\simeq }\mathrm{2.5}\mathrm{\times }{\mathrm{10}}^{\mathrm{8}}\hspace{0.17em}\mathit{k}\hspace{0.17em}{\mathit{M}}_{\mathrm{1.4}}^{\mathrm{1}\mathit{/}\mathrm{7}}\hspace{0.17em}{\mathit{R}}_{\mathrm{6}}^{\mathrm{10}\mathit{/}\mathrm{7}}\hspace{0.17em}{\mathit{B}}_{\mathrm{12}}^{\mathrm{4}\mathit{/}\mathrm{7}}\hspace{0.17em}{\mathit{L}}_{\mathrm{37}}^{\mathrm{-}\mathrm{2}\mathit{/}\mathrm{7}}\hspace{0.17em}\mathrm{cm}$(2)(the neutron star magnetic momentum is taken to be μ = BR³/ 2) is larger than the corotation radius, ${\mathit{R}}_{\mathrm{c}}\mathrm{=}{\left(\frac{\mathit{GM}{\mathit{P}}^{\mathrm{2}}}{\mathrm{4}{\mathit{\pi }}^{\mathrm{2}}}\right)}^{\mathrm{1}\mathit{/}\mathrm{3}}\mathrm{\simeq }\mathrm{1.68}\mathrm{\times }{\mathrm{10}}^{\mathrm{8}}\hspace{0.17em}{\mathit{M}}_{\mathrm{1.4}}^{\mathrm{1}\mathit{/}\mathrm{3}}\hspace{0.17em}{\mathit{P}}^{\mathrm{2}\mathit{/}\mathrm{3}}\mathrm{cm}\mathit{.}$(3)The limiting luminosity for the onset of the propeller can be estimated by equating the corotation and magnetospheric radii (see, e.g. Tsygankov et al. 2016a): ${\mathit{L}}_{\mathrm{prop}}\mathrm{\simeq }\frac{\mathit{GM}\mathit{Ṁ}\mathrm{prop}}{\mathit{R}}\mathrm{\simeq }\mathrm{4}\mathrm{\times }{\mathrm{10}}^{\mathrm{37}}{\mathit{k}}^{\mathrm{7}\mathit{/}\mathrm{2}}{\mathit{B}}_{\mathrm{12}}^{\mathrm{2}}{\mathit{P}}^{\mathrm{-}\mathrm{7}\mathit{/}\mathrm{3}}{\mathit{M}}_{\mathrm{1.4}}^{\mathrm{-}\mathrm{2}\mathit{/}\mathrm{3}}{\mathit{R}}_{\mathrm{6}}^{\mathrm{5}}\hspace{0.17em}{\mathrm{ergs}}^{-1}\mathit{,}$(4)where P is the neutron star rotational period in seconds and Ṁ is the mass-accretion rate. Factor k accounts for the details of interaction of the accretion flow with the magnetosphere, relating its size to the Alfvén radius, that is, k = R_m/R_A. In the case of disc accretion, it is usually assumed to be k = 0.5 (Ghosh & Lamb 1978).

In the case of rapidly rotating accreting XRPs (using 4U 0115+63 and V 0332+53 as case studies), Tsygankov et al. (2016a) showed that an observation of the transition of the accretion disc to the cold state is impossible because of the onset of the propeller regime at much higher mass accretion rates than the rate required for the accretion disc to have a temperature below 6500 K at the inner radius. At the same time, the overall shape and energetics of the outburst were shown to correspond to expectations from the disc instability model.

XRPs with spin periods that are sufficiently long to reduce the centrifugal barrier are required to reach a sufficiently low mass-accretion rate to switch the accretion disc to the cold state before the transition to the propeller regime. A few XRPs with long spin periods exist, one of which, GRO J1008−57, shows very predictable Type I outbursts every periastron passage with long quiescence periods between the outbursts (Kühnel et al. 2013). This source was selected for the monitoring campaign that covered a full orbital cycle and aimed to study the source properties at a very low mass-accretion rate.

GRO J1008−57 was discovered as an XRP with a period of 93.587 ± 0.005 s by the BATSE instrument on board the Compton Gamma-Ray Observatory (CGRO) during the bright outburst in 1993 (Stollberg et al. 1993). Its transient nature was associated with the Be type of the optical companion (B1-B2 Ve star; Coe et al. 2007). GRO J1008−57 shows giant, Type II, and Type I (associated with the periastron passage) outbursts. The distance to the source was estimated to be 5.8 kpc (Riquelme et al. 2012). The orbital parameters of the binary system were determined using the data from several observatories: the orbital period P_orb = 249.46 ± 0.10 d, the projected semi-major axis a_x sin i = 530 ± 60 lt-s, the longitude of periastron ω = −26 ± 8 deg, and the eccentricity e = 0.68 ± 0.02 (Levine & Corbet 2006; Coe et al. 2007; Kuehnel et al. 2012). The presence of a cyclotron line at ~88 keV in the spectrum of the source was first suggested based on CGRO/OSSE data (Shrader et al. 1999) and was later confirmed with Suzaku at E_cyc = 75.5 keV (Yamamoto et al. 2013). This allows us to estimate a magnetic field of the neutron star of ~8 × 10¹² G.

In this work we analyse the data obtained with the Swift/XRT telescope during one full orbital cycle of Be/XRP GRO J1008−57 between the two consequent Type I outbursts in January and September 2016. The unique combination of high sensitivity and flexibility of scheduling for the Swift observatory allowed us to perform a detailed long-term observational campaign aimed at reaching accretion regimes that have never been investigated before. Knowledge of the magnetic field strength of the neutron star as well as of other parameters of the system allowed us to interpret the observed temporal behaviour in terms of the accretion disc instability and to construct the model of accretion from a cold disc in highly magnetized neutron stars.

2. Data analysis and results

2.1. Swift/XRT observations

The Swift observatory (Gehrels et al. 2004) provides the possibility of performing long-term observation campaigns of faint X-ray sources using the onboard focusing XRT telescope (Burrows et al. 2005). This work is based on XRT data collected between two consequent Type I outbursts in January and September 2016 (Kretschmar et al. 2016; Nakajima et al. 2016). Observations were performed every 3–4 days in photon-counting (PC) mode. The spectrum in each observation was extracted using the online tools² provided by the UK Swift Science Data Centre (Evans et al. 2009).

Spectra were fitted with the XSPEC package using the Cash statistic (Cash 1979) after they were grouped to have at least one count per bin. To avoid any problems caused by the calibration uncertainties at low energies³, we restricted our spectral analysis to the 0.7–10 keV band.

To estimate the bolometric correction factor, we used the results obtained by Kühnel et al. (2013), who demonstrated the dependence of the source spectrum on its luminosity. Using the spectral parameters in the faint state of GRO J1008−57, we obtained an estimate for the bolometric correction factor K_bol ~ 1.6. We note that the remaining uncertainty in the correction factor does not influence any of the conclusions. In the following analysis, we apply this correction to the Swift/XRT data and refer to the bolometric fluxes and luminosities, unless stated otherwise.

2.2. Timing analysis

The light curve of GRO J1008−57 obtained from the Swift/XRT data is shown in Fig. 1. Originally, the observations were requested in order to detect the transition of GRO J1008−57 to the propeller regime (see Sect. 3) at the limiting luminosity of L_prop = 5.6 × 10³³ erg s^-1 (shown in the upper panel of Fig. 1 with the horizontal dashed line). However, at some point in the declining phase of the outburst, the source unexpectedly stopped fading around MJD 57 440 at a luminosity of around 10³⁵ erg s^-1 (shown in the same figure with the dotted line), which is one and a half orders of magnitude higher than L_prop. During the next seven months, the source continued to fade, but at a much lower rate characterised by an e-folding time of τ ~ 145 days in contrast to τ ~ 5.5 days before the transition. Strictly speaking, the flux decline during the low-level state is better fitted with a power-law of index ~−0.7.

Eventually, the monitoring covered the whole orbital cycle, with the sharp brightening of the source on MJD 57 648 signifying the next Type I outburst during the subsequent periastron passage (Nakajima et al. 2016). The transition of the source to the propeller state has therefore not been observed.

Despite the short duration of individual Swift observations and the long spin period of the source, the pulsations are easily detectable during the outburst. Similarly to what has been described in earlier reports (Kühnel et al. 2013; Bellm et al. 2014), the pulse profile exhibits two peaks with a relatively high pulsed fraction of 40−50%, as is typical for most of XRPs (Lutovinov & Tsygankov 2009). However, no pulsations were detected after the observation 00031030034 (MJD 57 430.51), when the source entered the low-level stable accretion state. On the other hand, the number of photons per observation becomes too low at this stage (a few tens to a few hundreds), so that the non-detection of the pulsations might simply be due to the insufficient counting statistics. For all low-level observations, the 3σ upper limits for the amplitude of the pulsations calculated based on the H-test (de Jager et al. 1989) and following the approach described in Brazier (1994) indeed lie in range 40−95%, as illustrated in Fig. 1d. The non-detection of the pulsations in individual Swift observations at low fluxes is therefore expected, taking their sensitivity to pulsed flux into account.

Fig. 1 Panel a: Bolometric luminosity of GRO J1008−57 obtained by the Swift/XRT telescope. The luminosity is calculated from the unabsorbed flux assuming a distance d = 5.8 kpc and fixed NH = 2.0 × 1022 cm-2. The dashed red and dotted blue lines show the predicted luminosity Lprop for the transition to the propeller regime and the observed transition to the low-level stable accretion regime, respectively. Panels b and c represent variations of the photon index and absorption value with time, respectively. Black symbols correspond to the fixed NH, red triangles and blue open squares to the absorption as a free parameter. In panel d the estimated 3σ upper limits for the pulsed fraction as function of time are shown for the Swift/XRT data with the horizontal dashed line showing the typical pulsed fraction measured during the bright state. The pulsations are clearly detected in all observations, which are expected to be sufficiently sensitive (i.e. below this line), but not elsewhere.

On the other hand, a search for the pulsations in the combined light-curve is complicated by the sparse sampling of the light curve and the uncertainty in the orbital parameters of the system. We therefore conclude that the non-detection of the pulsations by Swift does not contradict a presence of pulsations given the available counting statistics and light-curve sampling.

Somewhat stronger limits on the amplitude of the pulsations in the quiescence can be obtained from the 5 ks long Chandra observation 14639 of the source carried out in May 2013 (MJD 56 440.72) between the Type I outbursts. The bolometric luminosity calculated in the same way as for Swift data is ~10³⁵ erg s^-1, which is comparable with the values measured by Swift during our observations in the low-level stable state. To search for the pulsations, we first extracted photons from a circle with radius 7.4′′ centred on the source position after applying the standard screening criteria described in the instrument documentation. This yields 3215 source photons, which is comparable to the yield for individual Swift/XRT observations during the outburst and only a factor of two less than the total number of photons detected by Swift in the low-level state. On the other hand, the Chandra observation is not affected by the uncertainty on the orbital parameters. To search for the pulsations, we first corrected the photon arrival times for the effects of orbital motion in solar and binary systems using the ephemeris reported by Kühnel et al. (2013) and then used the same approach as for the Swift data. No significant pulsations around the source pulse frequency could be detected with the 3σ upper limit of ~20% for periods in the range 80−100 s. We note, however, that Chandra data are susceptible to pile-up, which potentially reduces the sensitivity to pulsed flux. The pile-up fraction is lower for pixels in the wings of the point-spread function and higher in its core, meaning that it depends on the count-rate in a given pixel. Therefore we repeated the simulation above for each pixel within a circle with radius of 10 pixels centred on the source. All events detected within a single frame (3.2 s) were considered as one event and noted to estimate the total pile-up fraction, which is around 15%. The upper limit on the pulsed fraction was then estimated using a concatenated event list from all pixels as described above, and it indeed was significantly higher at ~35%. While slightly lower than the typical pulsed fraction observed during the outburst, this by no means excludes pulsations at low fluxes, especially considering the fact that the pulsed fraction is known to be variable and that we ignored the background for this estimate. We therefore conclude that additional observations, preferably with XMM-Newton, are required in order to study the pulsations in the low-level stable accretion regime.

2.3. Spectral variability

All spectra from individual observations can be fitted with a simple absorbed power-law model. To illustrate the stability of the spectral shape over the luminosity and time, in Fig. 2 we present three spectra of GRO J1008−57 obtained in the brightest state of the January 2016 outburst (ObsId 00031030028; red circles), around the transition luminosity (ObsId 00031030035; blue squares), and deep inside the stable low-level state (ObsIds 00031030066-85; magenta crosses) with luminosities 6.9 × 10³⁶, 3.1 × 10³⁵, and 5.7 × 10³⁴ erg s^-1, respectively. No significant differences are immediately suggested by the data.

Fig. 2 Spectra of GRO J1008−57 obtained by Swift/XRT in different luminosity states: the brightest state of the January 2016 outburst (ObsId 00031030028; red circles), the transiting luminosity (ObsId 00031030035; blue squares), and deep inside the stable low-level state (ObsIds 00031030066-85; magenta crosses).

A more detailed analysis, however, reveals a minor change in the absorption and photon index values in the data collected in the bright (outburst) and low-level states. To obtain sufficient count statistics in the low-level state (between MJD 57 440 and 57 630), we binned the available data into three wide bins (see Fig. 1 and Table A.1).

As a result, we found that during the outburst (before MJD 57 440 and after MJD 57 648) the mean value of N_H is lower than ~1.8 × 10²² cm^-2 , which is just slightly higher than the Galactic value in this direction (N_H = 1.4 × 10²² cm^-2; Kalberla et al. 2005). On the other hand, after the transition its value increases up to N_H = (2.0 ± 0.1) × 10²² cm^-2. The difference is most prominent in the parameters of the averaged spectra. The variability of the spectral parameters during the whole orbital cycle of GRO J1008−57 is shown in Fig. 1 for single and averaged observations. The middle and bottom panels illustrate variations in photon index and absorption column with time, respectively. Black symbols correspond to the absorption column fixed at N_H = 2.0 × 10²² cm^-2, while red triangles and blue open squares show the fits with free absorption.

The observational log as well as the best-fit spectral parameters for the single and averaged observations are presented in Table A.1 for the free and fixed absorption values.

3. Discussion

Accreting magnetized neutron stars (XRPs, accreting millisecond pulsars, and accreting magnetars) exhibit a complex behaviour that is determined by the interaction of their magnetospheres with the accreting matter. Particularly, observational evidence for the propeller effect was found in a number of sources with magnetic fields from ~10⁸ G (for the accreting millisecond pulsar SAX J1808.4–3658; Campana et al. 2008) to ~10¹⁴ G (for the first pulsating ULX M82 X-2; Tsygankov et al. 2016b). The XRPs typically have magnetic fields somewhere in between these values, and some of them also exhibit a transition to the propeller regime (Stella et al. 1986; Cui 1997; Campana et al. 2001; Tsygankov et al. 2016a; Lutovinov et al. 2017).

Our observational campaign was originally aimed at detecting the transition of GRO J1008−57 to the propeller regime. In the case of GRO J1008−57, Eq. (4) gives L_prop = 5.6 × 10³³ erg s^-1. However, instead of a sharp drop in flux, GRO J1008−57 entered a stable accretion regime at a luminosity of about 10³⁵ erg s^-1 (see Fig. 1), about one and a half orders of magnitude higher than L_prop.

We note that unlike V 0332+53 and 4U 0115+63, where an obvious spectral softening was observed after the transition to the propeller regime (Tsygankov et al. 2016a; Wijnands & Degenaar 2016), no such changes have been observed in GRO J1008−57 after the transition to the stable accretion state. This strongly suggests that the emission mechanism remains the same, that is to say, the source continues to accrete even though the data quality was insufficient for detecting pulsations.

3.1. Accretion disc stability

As was shown by Lasota (1997), stable accretion onto the compact object is possible when the mass accretion rate is high enough to keep the entire accretion disc hot and fully ionised (temperature >6500 K) or when the mass accretion rate is so low that the temperature is below the critical value even at the inner radius. At intermediate levels of the mass accretion rate, the appearance of the region with a temperature of 6500 K will cause a rapidly propagating heating or cooling front through the disc, leading to the fast variability of the source flux. In this section we place the XRPs into the context of the disc instability model and make predictions for pulsars with different properties (see also the discussion in Tsygankov et al. 2016a).

We consider conditions under which an XRP may accrete matter from the recombined (cold) disc. Particularly, two criteria have to be fulfilled: (1) the mass accretion rate has to be high enough to allow the matter to penetrate the centrifugal barrier, and at the same time, (2) the mass accretion rate has to be sufficiently low to allow the disc to remain sufficiently cold ( <6500 K) at every radius.

The first criterion was discussed above and is realized when the magnetospheric radius R_m is smaller than the corotation radius R_c: ${\mathit{R}}_{\mathrm{m}}\mathrm{\le }{\mathit{R}}_{\mathrm{c}}\mathit{.}$(5)Condition (5) is fulfilled when the accretion luminosity is higher than the threshold luminosity presented above (see Eq. (4)): L ≥ L_prop.

The second criterion is satisfied when the mass accretion rate is below the value given by Eq. (1) (Ṁ <Ṁ_cold; Lasota 1997). Substituting the magnetospheric radius instead of r into Eq. (1), we obtain another condition for the stable accretion from a cold disc: $\mathit{L}\mathit{<}{\mathit{L}}_{\mathrm{cold}}\mathrm{=}\mathrm{9}\mathrm{\times }{\mathrm{10}}^{\mathrm{33}}\hspace{0.17em}{\mathit{k}}^{\mathrm{1.5}}\hspace{0.17em}{\mathit{M}}_{\mathrm{1.4}}^{\mathrm{0.28}}\hspace{0.17em}{\mathit{R}}_{\mathrm{6}}^{\mathrm{1.57}}\hspace{0.17em}{\mathit{B}}_{\mathrm{12}}^{\mathrm{0.86}}\mathrm{erg}\hspace{0.17em}{\mathrm{s}}^{-1}\mathit{.}$(6)Below this level, the temperature in the accretion disc is lower than 6500 K at R >R_m.

If the threshold luminosity for the propeller regime is higher than the luminosity that corresponds to the transition to the cold disc (L_prop >L_cold), then the decrease in mass accretion rate during the outburst decay will cause the transition to the propeller state. In the opposite case (L_prop <L_cold), the accretion disc will switch to the cold state with low viscosity, allowing stable accretion even at a very low rate.

Interestingly, the final state of the source after an outburst is determined by two fundamental parameters of the neutron star: the magnetic field, and the spin period. Equating the expressions for luminosities L_cold and L_prop, we can derive the critical value of the spin period as a function of the neutron star magnetic field: ${\mathit{P}}^{\mathrm{\ast }}\mathrm{=}\mathrm{36.6}\hspace{0.17em}{\mathit{k}}^{\mathrm{6}\mathit{/}\mathrm{7}}\hspace{0.17em}{\mathit{B}}_{\mathrm{12}}^{\mathrm{0.49}}\hspace{0.17em}{\mathit{M}}_{\mathrm{1.4}}^{-0.17}\hspace{0.17em}{\mathit{R}}_{\mathrm{6}}^{\mathrm{1.22}}\mathrm{s}\mathit{.}$(7)If the spin period P <P^∗, a pulsar will end up in the propeller regime. Otherwise, the source will start to accrete stably from the cold disc.

3.2. More accurate condition for accretion from a cold disc

The condition given by Eq. (6) is based on the assumption that the disc temperature is highest at the inner radius given by R_m. However, an effective temperature distribution over the radial coordinate depends on the exact boundary condition and viscous stress at the magnetospheric radius. Condition (6) can be derived more accurately. There is an uncertainty in the location, where the stress disappears. This uncertainty is described by the parameter β: β = 1 corresponds to a disappearance of the stress at the magnetospheric radius R_m, while β = 0 corresponds to the case when the stress disappears at r ≪ R_m. Then the distribution of effective temperature T_eff over the radial coordinate is given by (Frank et al. 2002) ${\mathit{\sigma }}_{\mathrm{SB}}{\mathit{T}}_{\mathrm{eff}}^{\mathrm{4}}\mathrm{=}\frac{\mathrm{3}}{\mathrm{8}\mathit{\pi }}\frac{\mathit{L}}{{\mathit{r}}^{\mathrm{3}}}\hspace{0.17em}\mathit{R}\hspace{0.17em}\left[\mathrm{1}\mathrm{-}\mathit{\beta }{\left(\frac{{\mathit{R}}_{\mathrm{m}}}{\mathit{r}}\right)}^{\mathrm{1}\mathit{/}\mathrm{2}}\right]\mathit{,}$(8)where σ_SB is the Stefan-Boltzmann constant. The maximum effective temperature, ${\mathit{\sigma }}_{\mathrm{SB}}{\mathit{T}}_{\mathrm{eff}\mathit{,}\max }^{\mathrm{4}}\mathrm{=}\mathit{A}\hspace{0.17em}\frac{\mathrm{3}}{\mathrm{8}\mathit{\pi }}\frac{\mathit{L}}{{\mathit{R}}_{\mathrm{m}}^{\mathrm{3}}}\hspace{0.17em}\mathit{R,}$(9)is achieved at the radius ${\mathit{r}}_{\max }\mathrm{=}\{\begin{array}{c}\\ \frac{\mathrm{49}}{\mathrm{36}}\hspace{0.17em}{\mathit{\beta }}^{\mathrm{2}}\hspace{0.17em}{\mathit{R}}_{\mathrm{m}}\mathit{,}& \mathrm{if}\mathit{\beta }\mathrm{\ge }\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\mathit{,}\\ {\mathit{R}}_{\mathrm{m}}\mathit{,}& \mathrm{if}\mathit{\beta }\mathit{<}\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\mathit{,}\end{array}$(10)where $\mathit{A}\mathrm{=}\{\begin{array}{c}\\ \mathrm{0.057}\hspace{0.17em}{\mathit{\beta }}^{-6}\mathit{,}& \mathrm{if}\mathit{\beta }\mathrm{\ge }\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\mathit{,}\\ \mathrm{1}\mathrm{-}\mathit{\beta ,}& \mathrm{if}\mathit{\beta }\mathit{<}\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\mathrm{·}\end{array}$(11)As a result, condition (6) can be rewritten in more accurate way. In this approach the maximum effective temperature in the disc is lower than 6500 K throughout the disc when the accretion luminosity is $\mathit{L}\mathrm{\le }{\mathit{L}}_{\mathrm{cold}}^{\mathrm{\left(}\mathrm{2}\mathrm{\right)}}\mathrm{\simeq }\mathrm{7}\mathrm{\times }{\mathrm{10}}^{\mathrm{33}}{\mathit{A}}^{\mathrm{-}\mathrm{7}\mathit{/}\mathrm{13}}\hspace{0.17em}{\mathit{k}}^{\mathrm{21}\mathit{/}\mathrm{13}}\hspace{0.17em}{\mathit{M}}_{\mathrm{1.4}}^{\mathrm{3}\mathit{/}\mathrm{13}}\hspace{0.17em}{\mathit{R}}_{\mathrm{6}}^{\mathrm{23}\mathit{/}\mathrm{13}}{\mathit{B}}_{\mathrm{12}}^{\mathrm{12}\mathit{/}\mathrm{13}}\hspace{0.17em}{\mathit{T}}_{\mathrm{6500}}^{\mathrm{28}\mathit{/}\mathrm{13}}\mathrm{erg}\hspace{0.17em}{\mathrm{s}}^{-1}\mathit{,}$(12)where T₆₅₀₀ = T_eff/ 6500 K. This estimate is more accurate than Eq. (6) as it accounts for interaction of the disc with the magnetosphere. In the case when the stress disappears at R_m, condition (12) gives L_cold higher by a factor of a few than Eq. (6).

As can be seen from Eq. (12), our prediction of the transition to the accretion from the cold disc (${\mathit{L}}_{\mathrm{cold}}^{\mathrm{\left(}\mathrm{2}\mathrm{\right)}}\mathrm{~}\mathrm{7}\mathrm{\times }{\mathrm{10}}^{\mathrm{34}}$ erg s^-1) coincides with the observed value within the factor of 2–3. Taking into account existing systematic uncertainties (the distance to the system, the physics of the coupling between the disc and the star, the width of the coupling region, and the uncertainty in the transiting temperature), we consider this match as support of our physical picture.

Another source of uncertainty can be associated with the irradiation of the accretion disc by the central object. Some theoretical models have indeed explained the long duration of the soft X-ray transients by irradiation of the outer regions of the accretion disc, which keeps them in hot ionised state for a longer time (see, e.g. King & Ritter 1998). There are also models of the soft X-ray transient bursts that do not need external irradiation for explanation of accretion discs dynamics (see, e.g. Lipunova 2015, and references therein), which might be more appropriate for our case. The observed outburst duration seems to be compatible with the cooling-wave propagation timescale. Indeed, the accretion disc is expected to extend to 4 × 10¹⁰ cm close to the peak of the outburst (assuming an effective temperature T_eff = 6500 K at this radius). The cooling-wave velocity V_cw is a few times lower than the heating wave velocity V_hw (Cannizzo et al. 1988), which can be estimated as V_hw ≈ αV_s, where V_s ≈ 10⁶ cm s^-1 is the sound speed (Meyer 1984; Cannizzo 1993). The propagation time of the cooling wave from the radius 4 × 10¹⁰ cm inward to R_m is therefore expected to be about 10–20 days (assuming α = 0.1), which is comparable with observed duration.

Using the updated Eq. (12) for the critical accretion luminosity ${\mathit{L}}_{\mathrm{cold}}^{\mathrm{\left(}\mathrm{2}\mathrm{\right)}}$, we can derive an updated equation for the critical value of the spin period P^∗, which, as described above, determines the pulsar behaviour at very low mass-accretion rate. Now instead of Eq. (7) we have ${\mathit{P}}^{\mathrm{\ast }}\mathrm{=}\mathrm{40.7}\hspace{0.17em}{\mathit{k}}^{\mathrm{21}\mathit{/}\mathrm{23}}\hspace{0.17em}{\mathit{B}}_{\mathrm{12}}^{\mathrm{6}\mathit{/}\mathrm{13}}\hspace{0.17em}{\mathit{M}}_{\mathrm{1.4}}^{\mathrm{-}\mathrm{5}\mathit{/}\mathrm{13}}\hspace{0.17em}{\mathit{R}}_{\mathrm{6}}^{\mathrm{18}\mathit{/}\mathrm{13}}\hspace{0.17em}{\mathit{A}}^{\mathrm{3}\mathit{/}\mathrm{13}}\hspace{0.17em}{\mathit{T}}_{\mathrm{6500}}^{\mathrm{-}\mathrm{12}\mathit{/}\mathrm{13}}\mathrm{s}\mathit{.}$(13)This equation can be used to predict the behaviour of any XRP (with a known magnetic field) expected in the case of transient activity. This is especially important for XRPs in Be binary systems (BeXRPs), which are known to be transient sources (see, e.g. Reig 2011) with magnetic fields measured from the cyclotron lines in their spectra (Walter et al. 2015). For illustration, we show some known BeXRPs as well as the accreting millisecond pulsar SAX J1808.4−3658, the intermediate pulsar GRO J1744−28, and the accreting magnetar M82 X-2 on the B−P plane in Fig. 3. All sources in this plane are divided with the prediction from Eq. (13) into two groups: (i) those entering the propeller regime at a low mass-accretion rate (below the line), and (ii) sources where stable accretion from the cold disc continues at any accretion rate (above the line). Persistent low-luminous BeXRPs (Reig & Roche 1999) are shown in green. Clearly, GRO J1008−57 resides in the area corresponding to the sources with accretion from the cold disc. The majority of sources residing below the P^∗(B) line have previously been shown to exhibit transitions to the propeller regime (for a review see Tsygankov et al. 2016a). It is important to note that peak luminosities in BeXRPs outbursts are ranging from 10³⁷ to 10³⁹ erg s^-1, exceeding all possible values of L_cold and L_prop. During the outburst decline, such sources therefore inevitably end up in one of the above-mentioned states.

Fig. 3 Collection of some known BeXRPs (shown in black), as well as the accreting millisecond pulsar SAX J1808.4−3658, the intermediate pulsar GRO J1744−28, and the accreting magnetar M82 X-2 (all three shown in blue) on the B−P plane. The solid and dashed lines correspond to the prediction of P∗(B) from Eq. (13) for β = 1 and two different values of parameter k: 0.5 and 0.7, respectively. This line separates sources where the propeller regime is possible from sources with stable accretion from the cold disc. Persistent low-luminous BeXRPs are shown in green. GRO J1008−57 is shown in red and resides in the area corresponding to the sources with accretion from the cold disc.

4. Conclusion

We analysed the Swift/XRT observations of GRO J1008−57 obtained between two subsequent Type I outbursts in January and September 2016. The original idea was to detect the transition of the source to the propeller regime, which is accompanied by an abrupt decrease in source flux and a softening of its spectrum. However, during the declining phase of the outburst, the source unexpectedly stopped its fading and entered a stable accretion state that was characterised by an accretion rate of the order of ~10¹⁴−10¹⁵ g s^-1 and a hard spectrum. We associate this state with accretion from the cold (low-ionised) disc with a temperature below ~6500 K.

We proposed a model of accretion from the cold disc in the systems harbouring neutron stars with strong magnetic fields (i.e. XRPs). The basic idea of the model is that in slowly rotating neutron stars the centrifugal barrier caused by the rotating magnetosphere is greatly suppressed in comparison to the fast rotating neutron stars, leading to a much lower threshold luminosity for the transition to the propeller regime. This allows such sources to reach mass-accretion rates that are so low that the temperature throughout the accretion disc becomes lower than the hydrogen recombination limit of ~6500 K.

When this mass accretion rate is reached, the fast fading of the source intensity should stop and further accretion with a low rate in the stable accretion regime from the cold disc with very low viscosity is expected to continue. This behaviour was observed in the pulsar GRO J1008−57 between two consequent Type I outbursts in January and September 2016.

Our model has strong predictive power. Particularly, the transition to the accretion regime from a cold disc is expected to be observed in all XRPs with a certain combination of pulse period and magnetic field strength. Other manifistations of the cold disc accretion associated with a change in disc structure could be anticipated. A change of the inner disc radius can be expected to affect the spin evolution of the pulsar, aperiodic variability properties, pulse profiles, and the energy spectrum of the source. A detailed calculation of the accretion disc structure in this case is ongoing and will be published elsewhere.

Acknowledgments

We are grateful to Ilia Potravnov for a number of useful comments and to the Swift team for the execution of our ToO request. This work was supported by the Russian Science Foundation grant 14-12-01287 (S.S.T., A.A.M., A.A.L., V.S.), the Academy of Finland grant 268740, and the National Science Foundation grant PHY-1125915 (J.P.). V.D. thanks the Deutsches Zentrums for Luft- und Raumfahrt (DLR) and the DFG for financial support (grant DLR 50 OR 0702). We also acknowledge the support of COST Action MP1304.

References

Appendix A: Additional table

All Tables

All Figures

Fig. 1 Panel a: Bolometric luminosity of GRO J1008−57 obtained by the Swift/XRT telescope. The luminosity is calculated from the unabsorbed flux assuming a distance d = 5.8 kpc and fixed NH = 2.0 × 1022 cm-2. The dashed red and dotted blue lines show the predicted luminosity Lprop for the transition to the propeller regime and the observed transition to the low-level stable accretion regime, respectively. Panels b and c represent variations of the photon index and absorption value with time, respectively. Black symbols correspond to the fixed NH, red triangles and blue open squares to the absorption as a free parameter. In panel d the estimated 3σ upper limits for the pulsed fraction as function of time are shown for the Swift/XRT data with the horizontal dashed line showing the typical pulsed fraction measured during the bright state. The pulsations are clearly detected in all observations, which are expected to be sufficiently sensitive (i.e. below this line), but not elsewhere.

In the text

	Fig. 2 Spectra of GRO J1008−57 obtained by Swift/XRT in different luminosity states: the brightest state of the January 2016 outburst (ObsId 00031030028; red circles), the transiting luminosity (ObsId 00031030035; blue squares), and deep inside the stable low-level state (ObsIds 00031030066-85; magenta crosses).
In the text

Fig. 3 Collection of some known BeXRPs (shown in black), as well as the accreting millisecond pulsar SAX J1808.4−3658, the intermediate pulsar GRO J1744−28, and the accreting magnetar M82 X-2 (all three shown in blue) on the B−P plane. The solid and dashed lines correspond to the prediction of P∗(B) from Eq. (13) for β = 1 and two different values of parameter k: 0.5 and 0.7, respectively. This line separates sources where the propeller regime is possible from sources with stable accretion from the cold disc. Persistent low-luminous BeXRPs are shown in green. GRO J1008−57 is shown in red and resides in the area corresponding to the sources with accretion from the cold disc.

In the text