Bright mini-outburst ends the 12-year long activity of the black hole candidate Swift J1753.5-0127

TL;DR

This study monitors the final stages of a 12-year outburst of the black hole candidate Swift J1753.5-0127, revealing a mini-outburst and reflare, and suggests a link between disk temperature and outburst decay.

Contribution

It introduces a new classification method for rebrightening events and links mini-outbursts to short orbital periods and disk temperature conditions.

Findings

Mini-outburst peaked at flux consistent with prior decay extrapolation.

Optical color indicates outer disk temperature of ~11,000 K during rapid fade.

Mini-outbursts occur in short orbital period systems (<7h).

Abstract

We present optical, UV and X-ray monitoring of the short orbital period black hole X-ray binary candidate Swift J1753.5-0127, focusing on the final stages of its 12$-$year long outburst that started in 2005. From September 2016 onward, the source started to fade and within three months, the optical flux almost reached the quiescent level. Soon after that, using a new proposed rebrightening classification method we recorded a mini-outburst and a reflare in the optical light curves, peaking in February (V$\rm\sim$17.0) and May (V$\rm\sim$17.9) 2017, respectively. Remarkably, the mini-outburst has a peak flux consistent with the extrapolation of the slow decay before the fading phase preceding it. The following reflare was fainter and shorter. We found from optical colors that the temperature of the outer disk was $\sim 11$,000 K when the source started to fade rapidly. According to the disk instability model, this is close to the critical temperature when a cooling wave is expected to form in the disk, shutting down the outburst. The optical color could be a useful tool to predict decay rates in some X-ray transients. We notice that all X-ray binaries that show mini-outbursts following a main outburst are short orbital period systems ($<$ 7 h). In analogy with another class of short period binaries showing similar mini-outbursts, the cataclysmic variables of the RZ LMi type, we suggest mini-outbursts could occur if there is a hot inner disk at the end of the outburst decay.

Tables

Table 1. Table 1 Log of LCOGT observations from June 2016.

MJD	magnitude($i\prime$)	MJD	magnitude($i\prime$)	MJD	magnitude($R$)	MJD	magnitude($V$)
57576.408966	16.664$\pm$0.041	57852.653244	17.206$\pm$0.007	57558.574655	16.858$\pm$0.040	57558.571983	16.821$\pm$0.075
57582.337375	16.619$\pm$0.033	57852.667005	17.147$\pm$0.008	57576.406211	16.849$\pm$0.007	57576.403534	16.871$\pm$0.012
57592.379523	16.698$\pm$0.029	57852.673729	17.136$\pm$0.013	57582.334629	16.792$\pm$0.008	57582.33196	16.816$\pm$0.014
57601.423749	16.680$\pm$0.024	57852.678244	17.135$\pm$0.008	57592.376769	16.853$\pm$0.012	57592.374098	16.870$\pm$0.019
57622.373905	16.659$\pm$0.033	57852.686403	17.139$\pm$0.009	57601.420983	16.868$\pm$0.008	57601.418325	16.914$\pm$0.012
57650.438382	17.196$\pm$0.030	57852.691101	17.118$\pm$0.013	57622.371154	16.889$\pm$0.010	57622.368481	16.936$\pm$0.019
57657.303600	17.858$\pm$0.031	57852.698035	17.115$\pm$0.037	57650.435627	17.427$\pm$0.012	57650.432968	17.554$\pm$0.031
57681.395282	18.441$\pm$0.024	57852.709628	17.079$\pm$0.035	57657.300838	18.029$\pm$0.027	57657.298173	18.209$\pm$0.068
57693.199752	19.344$\pm$0.024	57852.717024	17.157$\pm$0.114	57681.392516	18.845$\pm$0.036	57681.389858	18.956$\pm$0.069
57698.204232	19.786$\pm$0.022	57852.72825	17.114$\pm$0.161	57693.196992	19.603$\pm$0.117	57698.19881	20.450$\pm$0.296
57700.039837	19.747$\pm$0.026	57852.729595	17.115$\pm$0.346	57698.201471	20.371$\pm$0.189	57783.663806	18.169$\pm$0.064
57700.194786	19.967$\pm$0.020	57852.734499	17.108$\pm$0.300	57783.666481	18.047$\pm$0.033	57797.623031	17.068$\pm$0.046
57783.669241	17.744$\pm$0.020	57852.74075	17.089$\pm$0.047	57797.62571	16.953$\pm$0.018	57798.621437	16.965$\pm$0.038
57797.628466	16.779$\pm$0.019	57852.746995	17.102$\pm$0.015	57798.622952	16.982$\pm$0.022	57799.386816	17.010$\pm$0.031
57798.619832	16.700$\pm$0.027	57852.753815	17.132$\pm$0.016	57799.38975	16.963$\pm$0.018	57799.775908	16.967$\pm$0.066
57799.383835	16.728$\pm$0.025	57852.762352	17.184$\pm$0.018	57799.778842	16.956$\pm$0.041	57800.773383	17.024$\pm$0.116
57799.772894	16.627$\pm$0.020	57852.765903	17.157$\pm$0.035	57800.774915	16.935$\pm$0.041	57801.384364	17.042$\pm$0.032
57800.771778	16.695$\pm$0.019	57852.773000	17.100$\pm$0.033	57801.387291	16.943$\pm$0.020	57801.769249	16.965$\pm$0.169
57801.381370	16.706$\pm$0.020	57852.778253	17.116$\pm$0.023	57801.770752	16.855$\pm$0.051	57802.768061	17.028$\pm$0.068
57801.767629	16.695$\pm$0.021	57853.033500	17.161$\pm$0.041	57801.773653	16.942$\pm$0.032	57804.766226	17.033$\pm$0.085
57802.765060	16.657$\pm$0.075	57853.475840	17.202$\pm$0.051	57802.770988	17.047$\pm$0.054	57804.772667	17.028$\pm$0.082
57807.466797	16.782$\pm$0.062	57856.258205	17.288$\pm$0.017	57804.768896	16.947$\pm$0.038	57807.394323	17.051$\pm$0.019
57808.111048	16.793$\pm$0.031	57856.050972	17.269$\pm$0.015	57804.775336	16.951$\pm$0.042	57807.463757	17.057$\pm$0.019
57809.108328	16.746$\pm$0.043	57857.626176	17.452$\pm$0.017	57807.46984	16.987$\pm$0.013	57808.108029	17.075$\pm$0.018
57810.105714	16.798$\pm$0.035	57861.993962	17.869$\pm$0.020	57808.114082	17.031$\pm$0.012	57809.105297	17.078$\pm$0.020
57811.102868	16.813$\pm$0.038	57862.094179	18.079$\pm$0.017	57809.111372	16.997$\pm$0.013	57810.102676	17.064$\pm$0.029
57814.101241	16.837$\pm$0.039	57865.000261	18.517$\pm$0.017	57810.108736	16.969$\pm$0.015	57811.099824	17.108$\pm$0.021
57816.086183	16.810$\pm$0.092	57866.986331	19.081$\pm$0.014	57811.10591	17.022$\pm$0.014	57814.104309	17.080$\pm$0.019
57818.080712	16.779$\pm$0.053	57868.000318	18.976$\pm$0.018	57814.107262	17.065$\pm$0.013	57816.089192	17.082$\pm$0.017
57818.570966	16.812$\pm$0.041	57868.04205	18.920$\pm$0.008	57816.09213	17.035$\pm$0.011	57818.083761	17.070$\pm$0.020
57820.075244	16.743$\pm$0.041	57868.050433	18.908$\pm$0.015	57818.08672	17.014$\pm$0.013	57818.565527	17.039$\pm$0.016
57823.067088	16.808$\pm$0.033	57870.405355	18.377$\pm$0.020	57818.568201	16.995$\pm$0.009	57820.078283	17.066$\pm$0.018
57824.064272	16.816$\pm$0.035	57870.609638	18.283$\pm$0.039	57820.081237	17.018$\pm$0.013	57823.070125	17.118$\pm$0.034
57826.310092	16.773$\pm$0.048	57870.625248	18.276$\pm$0.020	57823.07308	17.051$\pm$0.017	57824.067304	17.109$\pm$0.057
57828.053927	16.818$\pm$0.036	57871.426129	18.011$\pm$0.023	57824.070279	17.025$\pm$0.030	57826.31414	17.119$\pm$0.035
57830.062119	16.954$\pm$0.020	57871.752167	17.778$\pm$0.049	57826.317063	17.042$\pm$0.020	57828.056969	17.115$\pm$0.034
57835.098306	16.847$\pm$0.015	57873.325745	17.273$\pm$0.014	57828.059935	17.047$\pm$0.020	57830.065173	17.182$\pm$0.065
57846.035396	16.882$\pm$0.028	57874.361536	17.533$\pm$0.014	57830.068147	17.208$\pm$0.038	57835.101317	17.223$\pm$0.021
57846.045207	16.895$\pm$0.028	57877.257184	17.567$\pm$0.014	57835.104243	17.105$\pm$0.012	57846.038429	17.259$\pm$0.018
57847.032727	16.888$\pm$0.021	57877.615258	17.565$\pm$0.014	57846.041383	17.162$\pm$0.011	57846.048233	17.272$\pm$0.017
57850.244833	17.021$\pm$0.034	57878.212070	17.516$\pm$0.017	57847.038727	17.163$\pm$0.011	57847.035765	17.284$\pm$0.018
57850.248495	17.056$\pm$0.037	57878.587685	17.641$\pm$0.017	57851.281824	17.340$\pm$0.016	57851.2789	17.453$\pm$0.026
57850.257411	17.055$\pm$0.052	57879.584894	17.718$\pm$0.017	57853.039507	17.412$\pm$0.019	57853.036542	17.520$\pm$0.035
57850.263663	17.034$\pm$0.018	57879.572086	17.572$\pm$0.016	57856.264113	17.695$\pm$0.052	57856.261181	17.870$\pm$0.090
57850.270962	17.030$\pm$0.052	57880.582335	17.897$\pm$0.016	57857.623409	17.676$\pm$0.054	57857.620736	18.014$\pm$0.142
57850.277968	17.021$\pm$0.057	57881.579571	17.892$\pm$0.015	57862.100176	18.264$\pm$0.028	57862.097209	18.294$\pm$0.038
57850.284849	17.034$\pm$0.039	57881.585786	17.958$\pm$0.015	57865.006267	18.865$\pm$0.032	57865.0033	18.993$\pm$0.044
57850.292388	16.979$\pm$0.118	57882.580111	17.904$\pm$0.015	57866.992334	19.371$\pm$0.061	57866.989368	20.290$\pm$0.199
57850.343168	16.995$\pm$0.212	57885.679478	18.657$\pm$0.026	57868.006302	19.521$\pm$0.051	57868.003333	19.549$\pm$0.067
57850.347353	17.039$\pm$0.138	57889.557676	18.396$\pm$0.025	57870.615676	18.813$\pm$0.034	57868.046237	19.721$\pm$0.060
57850.354417	17.060$\pm$0.023	57893.644986	18.154$\pm$0.013	57874.365153	17.730$\pm$0.021	57868.054629	19.588$\pm$0.053
57850.361243	17.046$\pm$0.070	57898.038157	19.034$\pm$0.013	57877.618386	17.769$\pm$0.021	57870.612631	18.900$\pm$0.048
57850.366324	17.065$\pm$0.048	57898.702179	18.929$\pm$0.013	57879.575198	17.885$\pm$0.019	57874.363361	17.899$\pm$0.033
57850.370319	17.089$\pm$0.042	57899.770325	18.705$\pm$0.013	–	–	57877.616871	18.121$\pm$0.038
57850.382741	17.007$\pm$0.286	57908.607852	20.909$\pm$0.022	–	–	57879.573685	18.180$\pm$0.033
57850.384097	17.018$\pm$0.337	57930.011298	21.352$\pm$0.013	–	–	57881.580338	18.441$\pm$0.107
57851.275926	17.065$\pm$0.278	57932.261395	20.916$\pm$0.018	–	–	57889.55952	18.230$\pm$0.259
57852.628351	17.189$\pm$0.195	57932.394386	21.069$\pm$0.058	–	–	57893.639086	18.656$\pm$0.044
57852.628407	17.191$\pm$0.220	57934.480869	20.791$\pm$0.075	–	–	57898.041195	19.803$\pm$0.085
57852.637699	17.152$\pm$0.197	57934.841549	20.210$\pm$0.037	–	–	57898.704951	19.368$\pm$0.083
57852.640745	17.216$\pm$0.140	–	–	–	–	57899.764824	19.234$\pm$0.084
57852.646999	17.299$\pm$1.000	–	–	–	–	57932.398345	21.198$\pm$0.216

Equations

$$T_{\mathrm{irr}}={\left(\mathcal{C}\frac{\dot{M}{c}^2}{4\pi {\sigma }_{\mathrm{SB}}{R}^2}\right)}^{1/4}\approx 18,800{\mathcal{C}}_{-3}^{1/4}{{\dot{M}}_{16}}^{1/4}{R}_{10}^{-1/2}\mathrm{K},$$

$$R_D(\max )=2.1\times 10^{10}{m}^{1/3}{P}_{\mathrm{orb}}^{2/3}\mathrm{cm},$$

$${\dot{M}}_{\mathrm{crit}}\approx 9.5\times 10^{14}{\mathcal{C}}_{-3}^{-0.36}{R}_{D,10}^{2.39}{m}^{-0.64}\mathrm{g}{\mathrm{s}}^{-1},$$

$$t_{\mathrm{dec}}\simeq \frac{R^2}{3\nu },$$

Full text

1. Introduction

Black hole X-ray transients (BHXTs) are binary systems consisting of a stellar-mass black hole (BH) primary accreting matter from a non-collapsed donor secondary. The vast majority spend most of their time in quiescence, where the X-ray luminosity is low. Episodically, during outburst, the X-ray flux increases by orders of magnitude, approaching in some cases the Eddington luminosity limit ($L_{\rm Edd}$). BHXTs exhibit a number of different spectral states during outbursts (e.g. low-hard, high-soft; Belloni & Motta, 2016), where the accretion properties change.

BHXT outbursts are thought to be triggered by a thermal-viscous instability in the accretion disk. In the disk instability model (DIM, for a review see Lasota, 2001), as matter accumulates in the disk during quiescence, the surface density increases until a threshold is exceeded at a certain radius in the disk. Hydrogen becomes ionized, the disk heats up quickly and the source goes into outburst.

The DIM can broadly explain the quiescent to outburst cycle of BHXTs. However, some observed properties cannot be well reproduced by it. Most BHXTs directly fade into quiescence with an exponential decay after the outburst peak, whereas some BHXTs (e.g. GRO J0422$+$32, GRS 1009$-$45, MAXI J1659–152, GRS 1739–278; Chen et al., 1997; Homan et al., 2013; Yan & Yu, 2017), show several rebrightenings after the outburst peak. Some of these rebrightenings have been defined as mini-outbursts by Chen et al. (1997) because they happen soon after the flux reaches close to the quiescence level, and they are of smaller amplitude and shorter duration compared to the normal outburst (see e.g. Dubus et al., 2001, hereafter D01). A sudden reactivation soon after the end of an outburst decay has been observed also in neutron star (NS) systems (e.g. Hartman et al., 2011; Patruno et al., 2016). Mini-outbursts are of much greater amplitude than the more common “reflares” that typically occur during the fade of the main outburst before reaching quiescence (Lasota, 2001). Reflares are theoretically expected to arise from a sequence of heating and cooling front reflections in the disk according to the DIM, but could alternatively be caused by:

X-rays heating the companion, increasing the mass accretion rate (outburst ‘echos’; see Lasota, 2001; Dubus et al., 2001; Kalemci et al., 2014, and references therein), 2. 2.

synchrotron emission from the reactivation of the jet in the hard state decay (e.g. Kalemci et al., 2013), 3. 3.

in the case of neutron star accretors, the activation/deactivation of the propeller effect reducing/increasing the mass accretion rate due to the rapidly rotating neutron star magnetosphere (Hartman et al., 2011; Patruno et al., 2016).

Even rarer are ‘multi-peak’ outbursts, where a second peak (or several peaks) following the main outburst reach a similar flux level to the first peak and may last longer (e.g. the BH systems GRO J1655–40; XTE J1118+480 and the neutron star system IGR J00291+5934; Chen et al., 1997; Chaty et al., 2003; Lewis et al., 2010). The properties of mini-outbursts are not well explained by the DIM, but require specific conditions of the accretion flow such as a hot inner disk residing near the end of the first outburst. This has been adopted to explain how the DIM can reproduce the mini-outbursts of some dwarf novae (DN; e.g. Hameury et al., 2000), but to date has not successfully been applied to mini-outbursts of BHXTs.

Swift J1753.5–0127 (hereafter J1753) was discovered by the Swift Burst Alert Telescope (BAT) on 2005 June 30 (Palmer et al., 2005) and showed a typical fast rise, exponential-decay (FRED) light curve. However, after decaying from the outburst peak, instead of returning to quiescence, J1753 remained active, with BAT fluxes varying between 0.001 and 0.03 count/s (15$-$50 keV). The X-ray spectral and timing properties during outburst indicate that the source is a BH X-ray binary that remained in the low-hard state (e.g. Cadolle Bel et al., 2007; Zhang et al., 2007) for the vast majority of the time (Shaw et al., 2016a).

The orbital period of J1753 is constrained to be 3.24 hours (Zurita et al., 2008), with an alternative claim of 2.85 hours proposed by Neustroev et al. (2014) which has been challenged by Shaw et al. (2016a). The source distance is poorly constrained, $\sim$2.5$-$8 kpc, and the inclination $i$ is constrained by model-dependent methods to be $\sim 55^{\circ}$ (e.g. Zurita et al., 2008; Froning et al., 2014; Plotkin et al., 2017; Gandhi et al., 2018). Time-resolved optical spectroscopy suggests that the primary mass is greater than $7.4\mathrm{M}_{\odot}$ (Shaw et al., 2016a). During the outburst the radio flux of J1753 (Cadolle Bel et al., 2007; Soleri et al., 2010), which is consistent with optically thick synchrotron emission likely originating in a compact jet (e.g. Tomsick et al., 2015), is typically lower than that of BHXTs following the standard track in the radio vs. X-ray diagram (Gallo et al., 2012; Rushton et al., 2016), making J1753 one of the radio-faint BHXTs. However, recent observations show that the source was close to the standard track during the rise and decay of the rebrightenings discussed below (Plotkin et al., 2017).

After being active for $\sim$11 years, J1753 became faint at X-ray, UV, optical, and radio wavelengths, almost reaching the previous measured optical quiescence level (V$\sim 21$ mag, Cadolle Bel et al., 2007) in November 2016 (Russell et al., 2016; Shaw et al., 2016c; Al Qasim et al., 2016; Plotkin et al., 2016). Unexpectedly, within $\sim 3$ months J1753 was bright again at all wavelengths (e.g. Al Qasim et al., 2017; Zhang et al., 2017a; Bright et al., 2017).

Here, we present seven years of multi-band optical monitoring of J1753 with the Faulkes and Las Cumbres Observatory (LCO) telescopes, complemented with archival optical data, and UV and soft X-ray coverage provided by Swift. In particular, we concentrate on the latest stage of the outburst (2016 June$-$2017 June), when the source faded and exhibited rebrightening episodes.

2. Rebrightenings classification

Previously, some of the rebrightening phenomena (e.g., reflares, glitches, mini-outbursts) have been classified qualitatively in Chen et al. (1997). Since then, more rebrightening phenomena in X-ray transients have been discovered and labelled (e.g., multi-peak outbursts, secondary outbursts). However, a systematic classification scheme has not been developed for the rebrightenings, which has caused several inconsistencies in the literature (e.g. V404 Cyg Muñoz-Darias et al., 2017)

In order to classify the rebrightening phenomena quantitatively, we introduce here a new observation-based classification scheme, which first asks whether or not the source has reached quiescence preceding each rebrightening episode. We consider a source to reach quiescence if either (a) the optical magnitude is within 0.5 mag of the quiescent level (or the X-ray flux is within a factor of 2 of the X-ray quiescent level), or (b) if the extrapolation of the outburst decay rate reaches the quiescent level before the rebrightening is detected. Then we define $F_{i}/F_{out}$ as the flux ratio between each rebrightening peak flux and the outburst peak flux, $\Delta t_{\rm i}$ as the time separating the start of the quiescent period from the start of each rebrightening, and $t_{\rm out}$ as the duration of the main outburst.

In practice, for a rebrightening event (following Fig. 1):

If the source flux does not reach quiescence before the rebrightening, it can be classified as:

1. (a)

A glitch, if $F_{i}/F_{out}<0.7$ and the light curve slope in the decay of the rebrightening is similar to the light curve slope during the main outburst decay. 2. (b)

A reflare, if $F_{i}/F_{out}<0.7$, but the slopes of the light curve of the outburst decay and that following the rebrightening are different. 3. (c)

A multi-peak outburst, if $F_{i}/F_{out}\geq 0.7$.

Since multiple rebrightening events can occur, the above can be used to classify each subsequent rebrightening until the flux reaches quiescence. 2. 2.

If the flux reaches quiescence before a rebrightening, it can be classified as:

1. (a)

A mini-outburst, if $\Delta t_{\rm i}/t_{\rm out}\leq 1$ and $F_{i}/F_{out}<0.7$. 2. (b)

A new outburst, if $\Delta t_{\rm i}/t_{\rm out}>1$ or $\Delta t_{\rm i}/t_{\rm out}\leq 1$ and $F_{i}/F_{out}\geq 0.7$.

In the case of a mini-outburst: until the flux reaches quiescence, any following rebrightenings have to be compared with the mini-outburst properties since they are part of the mini-outburst; once the flux reaches quiescence again, the following rebrightening has to be compared with the main outburst.

This method provides a unique classification for most of the rebrightenings observed in BH and NS transients. However, some caveats have to be taken into account: (1) The classification depends on the energy band (e.g. for the same source, simultaneous optical and X-ray light curves could lead to different classifications). (2) If the source is recurrent (e.g. GX 339–4, for which $\Delta t_{\rm i}/t_{\rm out}\leq 1$ is typically true for a large fraction of its outbursts), all rebrightenings should be categorized as new outbursts. (3) If there are observational gaps in the light curves with duration comparable or longer than the outburst length only a potential classification can be given. (4) We chose $F_{i}/F_{out}=0.7$ to separate strong and weak rebrightenings, and $\Delta t_{\rm i}/t_{\rm out}=1$ to separate mini-outbursts from new outbursts, so that known rebrightenings fall under the same classification as the definitions in Chen et al. (1997). These values can be further tuned in the future, when more data will become available.

3. Observations and data reduction

3.1. Optical photometry

We observed J1753 with the 2-m robotic Faulkes Telescopes North  (located at Haleakala on Maui, USA) and South (at Siding Spring, Australia) (e.g. Lewis, 2018), and the 1-m Las Cumbres Observatory (LCO) telescopes (Brown et al., 2013). These observations are part of an ongoing monitoring campaign of $\sim 40$ low-mass X-ray binaries (Lewis et al., 2008). Data were taken in the $V$, $R$, and $i\arcmin$-band filters from 2010 April, and were reduced using the LCO automatic pipeline. Photometry was performed using PHOT in IRAF. Photometric calibration was achieved using stars with magnitude errors of $<0.05$ in the Pan-STARRS1 and APASS catalogues (Magnier et al., 2013; Henden et al., 2015). The 2010–2013 January data were published in Shaw et al. (2013) using flux calibration from stars listed in Zurita et al. (2008). We also use archival $I$ and $V$-band data of J1753 published in Hynes et al. (2009) and Soleri et al. (2010). The observation logs are provided in Table 1

3.2. Swift observations

The Swift X-Ray Telescope (XRT; Burrows et al., 2005) data were reduced using the Swift tools within heasoft v. 6.23 (Blackburn, 1995). All observations were reprocessed using the tool xrtpipeline. The source light curves and spectra were extracted in the 0.3$-$10.0 keV band using a 20-arcsec circular extraction region centered on the source position. Background data were extracted from an annular region with an inner radius of 30 arcsec and an outer radius of 60 arcsec. In some photon counting (PC) mode observations (with average count rates higher than 0.5 counts/s), about 2$-$4 pixels were removed from the center of the extraction region of the source to account for the pile-up. The 15–50 keV Burst Alert Telescope (BAT; Barthelmy et al., 2005) daily light curve was also downloaded111https://swift.gsfc.nasa.gov/results/transients/ and a 5-day binning was applied to it.

The Ultraviolet/Optical Telescope (UVOT; Roming et al., 2005) on board Swift operated in imaging mode during all observations. In each UVOT observation, the source was observed with at least one of the six available filters: $uvw2$, $uvm2$, $uvw1$, $u$, $b$, and $v$. The UVOT data were analyzed following Poole et al. (2008). We extracted the source photons from a circular region with a radius of 4 arcsec. Background data were extracted from an annular region with an inner radius of 15 arcsec and an outer radius of 25 arcsec. We used the task uvotsource to determine the optical/UV flux.

4. Results

4.1. First rapid flux decay

We show the optical and X-ray (15$-$50 keV) light curves of J1753 in Fig. 2. The beginning of the X-ray light curve shows the decay of a FRED type outburst (see also Figure 1 in Zhang et al., 2007). This is followed by an unusually long period of low-level activity ($\sim$11 years), with several lower intensity peaks. After a short transition to a low-luminosity soft state in 2015 March–April (MJD $\sim$57100; Shaw et al., 2016b), the source flux went again below the BAT detection limit by the end of September 2016 (MJD $\sim$57660), when J1753 faded toward quiescence.

After the initial outburst peak in 2005, the optical light curve remained remarkably steady, around $V\sim 16.6$; $i\arcmin\sim 16.4$ mag from 2010 (MJD $\sim$55200) to 2014 (MJD $\sim$57000). The optical flux then gradually faded by 0.6 mag in both $V$ and $i\arcmin$-bands, during the 2015 transition to the low-luminosity soft state, before subsequently recovering back in the hard state. From 2016 September (MJD $\sim$57630) onward, the optical flux started a rapid decay (hereafter, the “first decay”) and within $\sim 3$ months (by MJD 57700) the optical flux had faded by a factor of $\sim 30$, becoming consistent with the quiescent flux upper limit (V$\sim 21$ mag; Cadolle Bel et al., 2007; Neustroev et al., 2016). Assuming an exponential decay of the optical flux, we measure from MJD $\sim 57630$ to MJD $\sim 57700$ an average decay rate of $\sim$0.051(3), 0.059(2), and 0.052(1) mag/day in $V$, $R$, and $i\arcmin$-bands, respectively. All quoted errors in the text are at $1\sigma$ c.l. if not otherwise specified.

We show in Fig. 3 a zoom-in of the multi-wavelength optical light curve from 2016 May onward to highlight the rebrightening episodes. Only three observations were taken by Swift during 2016; before the first decay, and on November 6 and 7 near the end of the first decay (MJD 57699). The source could not be detected with XRT and UVOT in November ( or with radio facilities; Plotkin et al., 2016, 2017), further implying that the source was close to or in quiescence for the first time since its discovery (Shaw et al., 2016c). Then, up to the end of January 2017, the source was Sun constrained and not visible from the ground.

4.2. Bright mini-outburst

When J1753 was again visible from the ground, on January 30, 2017 (MJD 57783), it became clear that it had unexpectedly brightened at optical wavelengths (Al Qasim et al., 2017, see Fig. 3). Subsequent radio (Bright et al., 2017), UV, and X-ray observations confirmed the flux increase in all bands (Kong, 2017). According to our new classification method, this rebrightening after the main outburst is a mini-outburst (i.e. the flux reached quiescence, $\Delta t_{\rm 1}/t_{\rm out}\ll 1$, and $F_{1}/F_{out}\sim 0.35$ in the V-band). The total duration of the first peak of the mini-outburst is $\sim 90$ days (MJD 57780–57870).

We find that the optical, UV and X-ray light curves are well correlated from MJD 57800 onward. In the X-ray–UV bands the mini-outburst seems to peak around MJD 57807 ( dot-dashed vertical line in Fig. 3), which is about seven days after the optical flux peak (dashed vertical line around MJD 57800). However, there are no UV data taken at the time of the optical peak and the UV and X-ray coverage is sparse compared to that in the optical bands.

Interestingly, we find that the optical flux peaks at a value consistent with the extrapolation of the slow fading before the first decay, suggesting the source somehow retains information about the previous flux decay (this was also the case for the first mini-outburst of GRO J0422+32 in 1993; e.g. Callanan et al., 1995). After the peak, J1753 remained bright for approximately 50 days, slowly fading. During the slow fade, the optical flux first decreased up to MJD $\sim$57845 at a rate of 0.0037(5), 0.0039(6) and 0.0034(4) mag/day in $V$, $R$ and $i\arcmin$-bands, respectively (assuming an exponential decay). After MJD $\sim$57845, the optical flux started to decrease rapidly (assuming an exponential decay) at a rate of 0.158(8), 0.112(4), and 0.098(4) mag/day in $V$, $R$, and $i\arcmin$-bands (with a corresponding decay in UV and X-ray fluxes as well). This is about $\sim 2-3$ times faster than the decay rate of the first decay in 2016.

After reaching a minimum flux corresponding to $V\sim 20.3$ mag on MJD 57866, the optical flux started to rise again. We recorded a new rebrightening episode peaking at $V=$ 17.9 mag on MJD 57874 (Bernardini et al., 2017). Since the source did not reach quiescence before this second rebrightening, its properties have to be compared with that of the mini-outburst of which it is part of. In particular, $F_{2}/F_{out}\sim 0.4$, and the slope after the rebrightening peak is different from the slope of the mini-outburst decay,. Consequently, we classify the second rebrightening as a $\sim$45 days long reflare (of the mini-outburst). There is at least another reflare shortly after the first (MJD 57889).

After the reflares the source went to quiescence again (likely between MJD 57900 and 57908). We combined our quiescent data into deep $V$ and $i\arcmin$-band images, and calculated an average quiescent magnitude of $V=$ 22.17 $\pm$ 0.25; $i\arcmin$ $=$ 21.00 $\pm$ 0.14 during June 2 to July 5, 2017 (see also Zhang et al., 2017b).

4.3. Color evolution

Throughout the fading and rebrightening periods there were large variations in the optical $V$ to $i\arcmin$ color (Fig. 3, bottom panel). In Fig. 4 we show the color–magnitude diagram (CMD), using 2-day bins of the data. The symbols correspond to different stages of the evolution of J1753. The color is bluer when brighter, and the general trend of the whole data set is well described by a simple model (Maitra & Bailyn, 2008; Russell et al., 2011) of a single temperature, constant area blackbody heating and cooling. At the lower temperatures, the optical emission originates in the Rayleigh-Jeans tail of the blackbody, whereas at higher temperatures it is coming from near the curved peak of the blackbody; this causes the colour changes. We adopt the same method as Russell et al. (2011) to apply this model to the data of J1753. The model assumes an optical extinction of E(B-V)=$0.45\pm 0.09$ (Froning et al., 2014), the Cardelli et al. (1989) extinction law and the an orbital period of $P_{orb}$ = 3 hr. The source consistently follows the model during the fading and brightening periods of the mini-outburst and the reflares, implying that the optical emission is dominated by a characteristic disk radius with a temperature of $\sim 15,000$ K during the period from 2010 to 2015, cooling down to $\sim 4,000$ K when the source is near quiescence (although the companion star may dominate at these low fluxes). This region has to be either the thermally emitting outer regions of the disk, or the irradiated outer disk (e.g. Hynes, 2005). We cannot distinguish this based on the emission area, since this depends on the uncertain distance of the source. However, in Shaw et al. (2019), sophisticated broadband fitting is applied to data from the rebrightening episodes of J1753, and a fully irradiated disk is favoured from their modeling.

According to the irradiated DIM (IDIM), as the irradiated accretion disk gradually cools during the initial decay after outburst peak, a cooling wave develops in the outer disk that is at a critical photospheric temperature, shutting down the outburst rapidly (King & Ritter 1998, D01). This temperature corresponds to the minimum column density for the upper branch of the ‘S-curve’, which for an irradiated disk is $T_{\rm crit}\sim 11,000$ K (see Equation A.2 in Lasota et al., 2008). We find from Fig. 4 that the temperature of the emitting region is $\sim 11,000-12,000$ K just when the source starts to fade rapidly. This occurs twice – once at the start of the first decay in 2016 (MJD 57622–57650) and again near the end of the mini-outburst in 2017 (MJD 57848–57852). These are shown in the CMD in Fig. 4 as thick blue and red lines, respectively. On both occasions the source crosses $T_{\rm crit}$, immediately fading rapidly, as expected in the IDIM as the cooling wave propagates through the disk. A similar process may have occurred in GRO J0422+32, in which there is a similar bluer-when-brighter behavior during the main outburst and the following two rebrightening events, that according to our classification method are two mini-outbursts (see Fig. 4 of Chevalier & Ilovaisky, 1995).

This can explain why the source did not fade rapidly before 2016 – because the outer disk temperature remained above $T_{\rm crit}$, so the entire disk was likely to be above the critical temperature. While the disk remained in this quasi-steady state above $T_{\rm crit}$, the mass transfer rate from the companion star must have been approximately equal to the mass accretion rate onto the compact object, since the viscous timescale is on the order of a month; much shorter than the 11-year ‘standstill’ period. In addition, there was no slow fade during the second mini-outburst in May–June 2017, because the disk remained below $T_{\rm crit}$, only reaching $\sim 9,000$ K, so it was likely that cooling fronts were still active, rapidly ending the outburst.

We note that there appear to be excursions away (redder) from the blackbody model near the peaks of the two mini-outbursts in Fig. 4. This could be due to a changing area of the blackbody, or a separate transient component redder than the disk that contributed to the optical flux at the time (e.g. synchrotron emission from the jet; Russell et al., 2011).

5. Discussion

The characteristics of the outburst of J1753 are peculiar in many ways. Three aspects of the outburst evolution are particularly interesting and not easily explained under the IDIM.

• •

Firstly, the 11-year steady ‘standstill’ period following the initial FRED is atypical, and possibly unique. This requires the mass accretion rate onto the black hole to be comparable to the mass transfer rate from the companion star. We find that the standstill can be due to the outer disk not cooling sufficiently, $T_{\rm crit}$ not being reached, preventing a cooling front from forming. Why it took 11 years for the source to reach $T_{\rm crit}$ is unknown, although the 2015 low-luminosity soft state provides a clue as it precedes the rapid fade. J1753 is a short orbital period system so its irradiated disk is relatively small, so it is essentially missing the cool, outer regions of a larger disk. This may have played a role in keeping the whole disk hotter than the critical temperature.

• •

Secondly, rather than having low-amplitude reflares during the outburst fade, the source first reached close to quiescence and then suddenly brightened by $>3$ magnitudes in optical and by $\sim 3$ or more orders of magnitude in X-ray during the mini-outburst. The amplitude of the rise into the mini-outburst is greater than expected from reflares, which have optical amplitudes of $\sim 1$–2 mag at most (D01).

• •

Thirdly, the recurrence time between the end of the outburst and the start of the mini-outburst is very short; only a few months or less. This is insufficient time for the disk to be evaporated and filled in again before the mini-outburst. A further clue regarding the timing of the mini-outburst was suggested by Plotkin et al. (2017), see also Shaw et al. (2013). They identified a long-term X-ray modulation during the standstill, with a period of $\sim 400$ days and pointed out that the mini-outburst might be an extension of this long-term modulation (see their Fig. 3).

Mini-outbursts are rare events whereas reflares are commonly reported in the optical and X-ray light curves of BHXTs. Similar mini-outbursts have been observed from the BHXTs GRO J0422$+$32 (R-band, Shrader et al., 1994; Chen et al., 1997), MAXI J1659–152 (X-rays, Homan et al., 2013), GRS 1739–278 (X-rays, Yan & Yu, 2017; Corral-Santana et al., 2018), A0620–00 (optical, Charles, 1998) and possibly GRS 1009$-$45 (V-band, Chen et al., 1997).

The above systems all have short orbital periods, in the range $P_{\it orb}\sim$2.5–7 h except GRS 1739–278, for which $P_{\it orb}$ is unknown, and we here suggest it may have a short orbital period. In fact, mini-outbursts have been detected from three out of the five BHXTs with periods $\leq 5.1$ h (see table 2 in Shahbaz et al., 2013). The other two are XTE J1118$+$480 (Chaty et al., 2003), which in 2000 had a multi-peak outburst according to our classification, and Swift J1357.2–0933, which did not have a mini-outburst following its 2011 or 2016 outbursts (but coverage was poor in the months after the 2011 outburst; see Russell et al., 2018, and references therein). The optical, X-ray, and radio (see also Plotkin et al., 2017; Shaw et al., 2019) monitoring of the mini-outburst and following reflares of J1753 represent the most complete multi-wavelength coverage of such an event for a BHXT to date.

The optical light curves of the BHXT mini-outbursts are morphologically very similar to those of another class of short orbital period (few hours) binaries, the dwarf novae (DN) of the RZ LMi-type (Hameury et al., 2000, and references therein) which exhibit short ‘superoutburst’ duty cycles of as low as $\sim 20$ days (e.g. Osaki, 1995). Dwarf novae are cataclysmic variables, i.e. close binary systems that host accreting white dwarf primaries, and show much shorter outburst durations and recurrence times than BH or NS X-ray transients (see Schreiber et al., 2003; Britt et al., 2015; Zhang et al., 2017c). Intriguingly, the first mini-outburst following a normal outburst of a BHXT or DN seems to usually peak at a flux level consistent with the extrapolation of the decay curve that precedes it. Mini-outbursts may be closely related to a critical temperature of the accretion disk corresponding to that flux. For J1753 we have demonstrated that this is indeed the case; when the rapid fade begins, the outer disk temperature is cooling and crossing the critical photospheric temperature needed for a cooling wave to develop, shutting down the outburst.

The DIM is able to reproduce the outburst light curves of DN mini-outbursts under the condition that the inner disk remains hot at the end of an outburst. In this case, when a cooling front develops in the disk the density/temperature immediately below the front is high enough to start a new heating wave. The inward moving cooling front is ‘reflected’ and an outgoing heating front starts a new (mini)outburst (see e.g. D01).

Perhaps the same mechanism could cause mini-outbursts in BHXTs if, like in DN, there is a hot inner disk near the end of the outburst. There were no soft X-ray observations during the first fade of J1753 in 2016, so the inner disk radius is unknown at this time. However, in 2015 the source was observed in a unusual faint soft state, during which Shaw et al. (2016b) measured a disk inner radius of R${}_{\rm in}\sim 12$–$28R_{g}$, at a temperature of $kT_{\rm in}=0.25$ keV and an X-ray luminosity of $\sim 6\times 10^{-3}L_{\rm Edd}$. J1753 is therefore certainly capable of maintaining a hot inner disk at low luminosities. Presumably, during the first fade of J1753 in 2016 the source was in the low-hard state, which is usually characterized by a truncated disk. However, in the low-hard state of J1753 before 2016, evidence for the presence of a faint inner disk that resides at $<20R_{g}$ has been provided by iron line fitting, a soft excess in the continuum and variability spectra (Wilkinson & Uttley, 2009; Kalamkar et al., 2013; Tomsick et al., 2015; Kajava et al., 2016). This is unusual for a BHXT in the low-hard state, and this could be an additional reason for J1753 to have a mini-outburst while many other (longer period) systems do not.

We propose that BHXT mini-outbursts are actually fairly common in short-period ($<7$ h) BHXTs, and may be due to the presence of a hot inner disk at the end of the outburst decay, and arise from a sequence of heating and cooling front reflections in the accretion disk. Testing this would require a deep soft X-ray observation during an outburst fade, to confirm the presence of a hot inner disk just before a mini-outburst. We speculate that in BHXTs with longer orbital periods, their disks are larger and therefore the outermost regions of their disks are cooler than in short period BHXTs. The inward traveling cooling front may therefore take longer to reach the hot inner regions of the disk in long period systems compared to short period systems, by which time the inner disk may be truncated. Truncation typically occurs over several days–weeks during an outburst decay, which is a similar order of magnitude to the timescale for a heating/cooling front to move in/out (see e.g. Bernardini et al., 2016). In the short period systems, the temperature at the outer radius of their smaller disks will be higher than the temperature at the outer radius of the larger accretion disks of the longer period systems (for the same accretion rate). The lack of the cool outer regions in short period XTs may cause the heating/cooling fronts to remain hot enough to initiate a new (mini)outburst, or the inner disk truncation radius may be somehow regulated by the small size of the disk and/or the hot flow at smaller radii. In addition, the hard-state nature of most of the outburst of J1753 (and most other short period BHXTs) provides a clue about the state of the disk, i.e. that the irradiating source is hard and possibly extended (the hot flow / Comptonizing region is likely to be physically thicker than the inner disk), making it able to illuminate and heat the outer disk more readily than the soft photons from the inner disk. It is worth noting that long outbursts lasting several years or decades have been reported in some long period XTs, e.g. GRS 1915+105 ($>20$ years; $P_{\it orb}=33.5$ days), GRO J1655–40 (16 months; $P_{\it orb}=2.6$ days), KS 1731–260 (12.5 years; $P_{\it orb}\sim 10$ hours) (Sobczak et al., 1999; Wijnands et al., 2001; Deegan, Combet, & Wynn, 2009; Zurita et al., 2010), although V404 Cyg had two very short outbursts in 2015–2016 (Muñoz-Darias et al., 2017). However, the neutron star system EXO 0748–676 had an outburst lasting 24 years, and it has a 3.8 hr period (e.g. Hynes & Jones, 2009).

5.1. Standstill

Since the beginning of the J1753 outburst is so well explained by the IDIM, it is worth investigating what makes its end different from what the standard version of the model predicts. During the long standstill period, the outer disk temperature is observed to be $\sim 15,000$ K. This is most likely an irradiation temperature on the disk surface222$F_{\rm irr}/F_{\rm vis}\sim{\mathcal{C}}R/R_{\rm in}$, so for $R/R_{\rm in}\gtrsim 10^{3}$, the irradiation flux is always larger than the flux produced by viscosity.. Therefore

[TABLE]

where $\dot{M}=\dot{M}_{16}10^{16}\rm\,g\,s^{-1}$. The mean flux during standstill is $F_{X}^{s}\approx 10^{-9}\,\rm erg\,cm^{-2}s^{-1}$ corresponding to an accretion rate $\dot{M}_{s}\approx 1.3\times 10^{16}\rm g\,s^{-1}$. For such an accretion rate, a temperature of $\sim 15,000$ K, corresponds to $R\approx 1.8\times 10^{10}\,{\mathcal{C}}^{1/2}_{-3}$cm.

On the other hand, since the standstill corresponds to a steady stable disk, one must have $\dot{M}_{s}>\dot{M}_{\rm crit}^{+}(R_{D}^{s})$, where $R_{D}^{s}$ is the disk outer radius during standstill. The disk radius shrinks during the outburst decay, and a sudden enhancement of the mass-transfer rate might reduce its value. For $\dot{M}_{s}=1.3\times 10^{16}\rm g\,s^{-1}$, the disk during standstill is stable when $R_{D}^{s}<4.8\times 10^{10}{\mathcal{C}}^{0.15}_{-3}$cm, consistent with the value deduced from the irradiation temperature. A fluctuation lowering the mass transfer rate from the companion star could end the standstill and trigger the decay of the outburst. Note that a detailed analysis of the standstill period based on broadband SED modelling is presented in Shaw et al. (2019).

5.2. The outburst of J1753 according to the IDIM

Here, we investigate how the initial decay from maximum of the J1753 outburst can be described by the Irradiated DIM (IDIM). We assume a distance $d=3$ kpc, a black-hole mass in solar units, $m=7$, and an orbital period $P_{\rm orb}=3$ hr. Using for the maximum disk radius the formula (see e.g. Lasota et al., 2008)

[TABLE]

where $P_{\rm h}$ the orbital period in hours, one obtains for J1753 $R_{D}({\rm max})=8.3\times 10^{10}$cm.

According to the IDIM, the maximum accretion rate (at the luminosity peak) is that of a (quasi)steady disk accreting at constant rate of $\sim 3\dot{M}_{\rm crit}^{+}(R_{D})$, where

[TABLE]

is the value of the minimum critical accretion for a hot, irradiated disk at its outer radius $R_{D}=R_{D,10}10^{10}$cm (Lasota et al., 2008), and ${\mathcal{C}}=10^{-3}{\mathcal{C}}_{-3}$ is a constant characterizing the outer-disk irradiation by a central point-like source. We ignore the very weak dependence on the viscosity parameter $\alpha$.

Assuming ${\mathcal{C}}_{-3}=1$ (Lasota et al. 2008; see however Tetarenko et al. 2018), one obtains for the accretion rate at maximum of the J1753 outburst $\dot{M}_{\rm max}\approx 1.3\times 10^{17}\rm g\,s^{-1}$. For an accretion efficiency $\eta=0.1$ this corresponds to a peak luminosity $L_{\rm max}\approx 1.2\times 10^{37}\,\rm erg\,s^{-1}$. The observed X-ray flux at the peak of J1753 outburst was $F_{X}^{p}\approx 10^{-8}\,\rm erg\,cm^{-2}s^{-1}$ which at the assumed distance corresponds to $L_{\rm max}\approx 1.2\times 10^{37}\,\rm erg\,s^{-1}$, which is in excellent agreement with the IDIM. Obviously this excellence is a coincidence in view of the uncertainties in the values of the distance (Gandhi et al., 2018, which could well be further than the assumed 3 kpc; the GAIA parallax distance is unreliable;), ${\mathcal{C}}$ and $\eta$. Nevertheless, it shows that the IDIM can be successfully fit to describe the initial peak of the outburst of J1753, at least until the standstill.

This is confirmed by estimating the outburst decay time which according to the IDIM is

[TABLE]

where $\nu=\alpha c_{s}^{2}/\Omega$ is the Shakura–Sunyaev kinematic viscosity coefficient, $c_{s}\propto T_{c}^{1/2}$ the sound speed, $T_{c}$ the disk midplane temperature, and $\Omega=(GM/R^{3})^{1/2}$. Taking the critical midplane temperature $T_{\rm c}^{+}\approx 14,500\,\rm K$ corresponding to $\dot{M}_{\rm crit}^{+}$ (Lasota et al., 2008) and the parameters assumed for J1753, one obtains for the decay timescale $t_{\rm dec}\approx 93\,\alpha_{0.2}^{-1}\,\rm days$, where $\alpha_{0.2}=\alpha/0.2$ (i.e. we adopt $\alpha=0.2$). This is again in excellent agreement with observations of the first decay lasting $\sim 3$ months.

5.3. Optical fade during the 2015 soft state

As mentioned in Section 3.1, the optical emission appears to gradually decrease by $\sim 0.6$ mag in 2015 around MJD 57100, which is when the source made a transition to the faint soft state (Shaw et al., 2016b). The optical flux then recovers, brightening back in the hard state, before starting the slow decay (Fig. 2). A fade of the optical/infrared flux is common in the soft state in BHXTs, and is usually attributed to the disappearance of the synchrotron jet component (e.g. Kalemci et al., 2013). However, this optical fade is usually more prominent in the infrared, and less at optical wavelengths, with a large change in the color in the soft state. Such a color change is not apparent in J1753; in Fig. 4 we find that the soft state data (green crosses) lie on the blackbody line, as do the hard state data just before and after the soft state, with a very similar color. The optical fade in the soft state is therefore unlikely to be due to a synchrotron component disappearing in this state. An excess has been detected in J1753 in the hard state, but only at longer wavelengths (Froning et al., 2014; Rahoui et al., 2015), not at optical wavelengths.

The soft X-ray flux increases in the soft state, as the inner disk is close to the BH and the inner disk temperature increases. Concurrently the hard X-rays decrease, as the power law component fades in the soft state (this is visible in the Swift BAT light curve; Fig.2). If the optical emission originates in the outer regions of the disk, the optical flux should remain approximately constant over the transition to/from the soft state (e.g. Russell et al., 2011). If the irradiated outer accretion disk produces the optical emission instead, the optical will be positively correlated with the X-ray emission. The optical flux of J1753 does not react to the increase of soft X-rays, which implies it would be reprocessed hard X-rays. Since the X-ray power law in the hard state likely resides up to at least $\sim 100$ keV (e.g. Tomsick et al., 2015; Kajava et al., 2016), and may even extend beyond 600 keV (Cadolle Bel et al., 2007), the bolometric luminosity is likely to be higher in the hard state than in the soft state. Moreover, as well as being more energetic, harder X-rays can penetrate further into the disk atmosphere, and so heat the disk more efficiently. In addition, if the X-ray power law is elevated above the disk, such as in the lamppost model for the Comptonized region, these hard photons will more readily illuminate the outer disk, compared to the inner regions of the disk itself. To summarize, the optical fade in the soft state is either caused by a decrease of intrinsic emission from the outer disk (for some unknown reason), or a decrease in the irradiation bump (if this is reprocessed hard X-rays).

6. Conclusions

In this paper we have presented new optical, UV and X-ray observations of the short orbital period BHXT Swift J1753.5–0127, focusing on the final stages of its 12$-$year long outburst that started in 2005.

(i) We recorded a bright mini-outburst with a reflare at the end of the main outburst decay. We found from the optical colors that the temperature of the outer disk was $\sim 11$,000 K when the source started to fade rapidly on both occasions. The mini-outburst had a peak flux consistent with the extrapolation of the previous decay before the first fade, which is similar to some mini-outbursts in other BHXTs.

(ii) According to the IDIM, the $\sim 11$,000 K is consistent with being the critical temperature (for an irradiated disk) when a cooling wave forms, accelerating the fade of the outburst, as is observed. The optical color could be a useful tool to predict decay rates in some X-ray transients.

(iii) Based on available observations, we suggest the X-ray binaries that have known mini-outbursts following an outburst are all short orbital period systems ($<7$ h). Their smaller accretion disks may be a requirement for mini-outbursts to occur. Another requirement may be the existence of a hot inner disk at the end of the outburst, as was previously used to explain the mini-outbursts from RZ LMi type cataclysmic variables.

(iv) In this paper we also introduced for the first time a new method to classify rebrightening events quantitatively in NS/BH-XRBs.

Acknowledgments

The Faulkes Telescopes are maintained and operated by the Las Cumbres Observatory (LCO). The X-ray data are obtained from the High Energy Astrophysics Science Archive Research Center (HEASARC), provided by NASA’s Goddard Space Flight Center and NASA’s Astrophysics Data System Bibliographic Services. We thank the Swift Science Operations Team for performing the observations. The research reported in this publication was supported by Mohammed Bin Rashid Space Centre (MBRSC), Dubai, UAE, under Grant ID number 201701.SS.NYUAD. GZ acknowledges funding support from the CAS Pioneer Hundred Talent Program Y7CZ181001. FB is founded by the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement no. 664931. AWS is supported by an NSERC Discovery Grant and a Discovery Accelerator Supplement. PAC is grateful to the Leverhulme Trust for the award of a Leverhulme Emeritus Fellowship. RMP acknowledges support from Curtin University through the Peter Curran Memorial Fellowship. LCO data from this work are being used by FL and RD in their educational project, “Black Holes In My School”. PK (teacher) and DP (pupil) represent one of the pilot schools in this project. JPL was supported by the Polish NCN grant 2015/19/B/ST9/01099 and the French National Space Center CNES. JCAMJ is the recipient of an Australian Research Council Future Fellowship (FT140101082).

7. appendix