X-RAY OUTBURSTS OF LOW-MASS X-RAY BINARY TRANSIENTS OBSERVED IN THE RXTE ERA

Among 187 LMXBs in the catalog of Liu et al. (2007), 103 sources are identified as transients, 95 of which are monitored by the RXTE/ASM. We also found 11 LMXBTs in the ASM data archive that are not included in the LMXB catalog of Liu et al. (2007). Then there are a total of 106 LMXBTs with RXTE/ASM light curves, but four of them (GRS 1741.9–2853, AX J1745.6–2901, 1 A 1742–289, CXOGC J174540.0–290031) are in the Galactic center region, which cannot be resolved by the RXTE/ASM. So we excluded these four sources in our analysis. There are another three sources (GRS 1915+105, KS 1731–260, and Cir X-1) that have outbursts of very long duration during the RXTE era. The rise or decay phases of the outbursts of these three sources are not available during the RXTE era, so we do not take these three sources as classic transient sources on the observational timescale of the RXTE/ASM. IGR J17091–3624 is also excluded because of its very poor data coverage.

Thus a total of 98 LMXBTs with available RXTE/ASM data are included in our analysis. Their long-term X-ray behavior during the RXTE era roughly satisfied the criteria for XN in Tanaka & Shibazaki (1996). The one-dwell light curve of each source retrieved from the public ASM products database,¹ which covers the period from early 1996 to late 2011, was uniformly rebinned with one day duration bins in order to improve the sensitivity (∼10 mCrab for the daily average data), and the effects of the Type I and Type II X-ray bursts in NS LMXBTs are insignificant in the one-day binned data. The X-ray flux of each source is reported in Crab units ($1\;{\rm Crab}=75\;\;{\rm ct}\;{{{\rm s}}^{-1}}$, estimated from the light curve of the Crab Nebula observed by the ASM). We only selected as our samples outbursts with X-ray peak fluxes larger than 0.1 Crab over a significance of 3σ, so our samples only include bright outbursts. We describe how to select the outburst peak flux in Section 2.3 in detail.

The object 4U 1608–522 sometimes stayed in a low flux level for as long as about 80 days after an outburst, then showed a secondary outburst, such as the 2004 and 2005 outbursts (during MJD 53072–53202 and MJD 53424–53691, respectively). We took the primary and secondary outbursts as two independent samples in our analysis. The 2007–2009 outburst of 4U 1608–522 (see, e.g., Linares et al. 2009, or Figure 1 during MJD 54258–55006) is constituted of two primary outbursts and a series of small outbursts. These series of outbursts contributed three outburst samples in our analysis.

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Finally, we ended up with a total of 110 outbursts in 36 LMXBTs, which includes 22 BH systems and 14 NS systems, respectively. More than half of the LMXBTs in our sample only have one bright outburst during the RXTE era (see Table 1). MXB 1730–33 has the most amount of bright outbursts, up to 23 in the 16 yr. System parameters collected from the literature for these LMXBTs are shown in Table 1. Figure 1 shows long-term X-ray light curves of five NS and BH LMXBTs with frequent outbursts during 1996–2011. We measured the peak X-ray luminosity, rate of change of luminosity on a daily timescale, e-folding rise or decay timescale, outburst duration, and total radiated energy for each of the outbursts. All of the results are collected in Table 8. We plot one outburst from each LMXBT in Figure 2 as examples.

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Table 1.  System Parameters of LMXBTs in Our Sample

Source	D(${\rm Kpc}$)	${{M}_{{\rm X}}}\;({{M}_{\odot }})$ a	${{P}_{{\rm orb}}}({\rm hr})$	${{N}_{{\rm outburst}}}$	References
SWIFT J1539.2–6227	⋯	⋯	⋯	1	Krimm et al. (2011)
4U 1543–47	7.5 ± 1	9.4 ± 1	26.95	1	Orosz (2003), Orosz et al. (1998)
XTE J1550–564	5.3 ± 2.3	10.56 ± 1.0	37.25	2	Orosz et al. (2002)
4U 1630–47	10 ± 5	⋯	⋯	8	Callanan et al. (2000)
XTE J1650–500	2.6 ± 0.7	5 ± 2.3	7.63	1	Homan et al. (2006), Orosz et al. (2004)
XTE J1652–453	⋯	⋯	⋯	1	Markwardt et al. (2009)
GRO J1655–40	3.2 ± 0.2	7.02 ± 0.22	62.88	2	Orosz & Bailyn (1997), Hjellming & Rupen (1995)
MAXI J1659–152	7.45 ± 2.15	5.8 ± 2.2	2.41	1	Kuulkers et al. (2013), Yamaoka et al. (2012)
GX 339–4	5.75 ± 0.8	12.3 ± 1.4	42.14	5	Shaposhnikov & Titarchuk (2009)
XTE J1720–318	6.5 ± 3.5	⋯	⋯	1	Chaty & Bessolaz (2006)
GRS 1739–278	7.25 ± 1.25	⋯	⋯	1	Greiner et al. (1996)
H1743–322	9.1 ± 1.5	13.3 ± 3.2	⋯	6	Shaposhnikov & Titarchuk (2009)
XTE J1748–288	>8 = 10 ± 2	⋯	⋯	1	Miller et al. (2001)
SLX 1746–331	⋯	⋯	⋯	3	Skinner et al. (1990)
XTE J1752–223	3.5 ± 0.4	9.6 ± 0.9	⋯	1	Shaposhnikov et al. (2010)
SWIFT J1753.5–0127	6 ± 2	⋯	3.24	1	Cadolle Bel et al. (2007), Zurita et al. (2008)
XTE J1755–324	⋯	⋯	⋯	1	Revnivtsev et al. (1998)
XTE J1817–330	3 ± 2	⋯	⋯	1	Sala et al. (2007)
XTE J1818–245	3.55 ± 0.75	⋯	⋯	1	Cadolle Bel et al. (2009)
SWIFT J1842.5–1124	⋯	⋯	⋯	1	Krimm et al. (2008)
XTE J1859+226	4.2 ± 0.5	7.7 ± 1.3	6.58	1	Shaposhnikov & Titarchuk (2009), Corral-Santana et al. (2011)
XTE J2012+381	⋯	⋯	⋯	1	Campana et al. (2002)
4U1608–522	5.8 ± 2	⋯	12.89	15	Güver et al. (2010)
XTE J1701–462	8.8 ± 1.3	⋯	⋯	1	Lin et al. (2009)
RX J1709.5–2639	10 ± 2	⋯	⋯	3	Jonker et al. (2004)
XTE J1723–376	<13 = 10 ± 3	⋯	⋯	1	Galloway et al. (2008)
MXB 1730–33	8.8 ± 3	⋯	⋯	23	Kuulkers et al. (2003)
XTE J1739–285	<10.6 = 8 ± 2	⋯	⋯	1	Kaaret et al. (2007)
GRO J1744–28	8.5 ± 0.5	⋯	284.02	2	Nishiuchi et al. (1999), Finger et al. (1996)
IGR J17473–2721	6.4 ± 0.96	⋯	⋯	2	Chen et al. (2010)
EXO 1745–248	8.7 ± 3	⋯	⋯	3	Cohn et al. (2002), Kuulkers et al. (2003)
1 A 1744–361	<9 = 6 ± 3	⋯	1.62	1	Bhattacharyya et al. (2006)
4U 1745–203	8.47 ± 0.4	⋯	⋯	2	Ortolani et al. (1994)
SAX J1750.8–2900	6.79 ± 0.14	⋯	⋯	2	Kaaret et al. (2002), Galloway et al. (2008)
2 S 1803–245	<7.3 = 5 ± 2	⋯	9	1	Cornelisse et al. (2007)
Aql X-1	5 ± 1	⋯	18.95	11	Rutledge et al. (2001), Welsh et al. (2000)

Note.

^aWe assumed the NS mass to be 1.4 ± 0.1 ${{M}_{\odot }}$.

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We identified 110 bright LMXBT outbursts in the observations of RXTE/ASM (including 68 NS LMXBT outbursts and 42 BH LMXBT outbursts). The numbers of outbursts in every year is shown in Figure 3. Because of the detector problems of RXTE/ASM since late 2010, the sky coverage of RXTE/ASM was highly reduced (A. Levine 2012, private communication), so only one outburst (the 2011 outburst of 4U 1608–52) can be identified in the 2011 data. For the period of 15 years from early 1996 to late 2010, an annual average of 7.3 bright outburst events of LMXBTs were detected by RXTE/ASM, including 4.5 NS LMXBT outbursts and 2.8 BH LMXBT outbursts. This average number of outbursts per year is roughly three times larger than the estimation in Chen et al. (1997), which may be due to the improvement of the sky coverage of the RXTE/ASM. Our outburst sample consists of bright outbursts with a peak flux larger than 0.1 Crab. Therefore, the outburst rate of the LMXBTs we report here is independent of RXTE/ASM sensitivity.

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We chose the maximal flux value in the daily light curve of an outburst as the peak flux. We excluded two kinds of data points when we identified the peak flux of an outburst, which are unlikely to be real detections and may be from instrumental or systematic errors. The first kind is that the flux is larger than the fluxes on two adjacent days by more than a factor of two (e.g., maximum fluxes of the outbursts of RX J1709.5–2639 and MAXI J1659–152 in Figure 2). The second kind is that the error of the flux is larger than the two adjacent days by more than a factor of two (e.g., maximum flux of an outburst of GX 339–4 in Figure 2). In addition, the ASM data of two outbursts have large data gaps near the outburst peaks (the 1996 outburst of 4U 1608–522 and the 1996 outburst of GRO J1744–28). In such cases, the peak flux we measured may be the lower limit of the true value.

In order to convert the observed photon count rate into flux, we assumed that the X-ray spectrum of each LMXBT is the same as the Crab Nebula in the 2–12 keV energy band. For simplicity, we used a conversion of 1 Crab as $2.12\times {{10}^{-8}}$ ergs cm⁻² s⁻¹ (2–12 keV), which is estimated according to the spectrum of the Crab Nebula in the 2–10 keV energy band (Kirsch et al. 2005). The column densities of the sources in our sample are in the range of $0.15-8\times {{10}^{22}}$ cm⁻², and most of them (29 out of 36) are below $2\times {{10}^{22}}$ cm⁻². We used WebPIMMS to estimate the flux of the spectra of typical soft and hard states of LMXBTs at a column density of $2\times {{10}^{22}}$ cm⁻² with an input RXTE/ASM count rate of the Crab Nebula (∼75 c s⁻¹), and we found that the differences from the flux of the Crab Nebula are about 10% and 25% in the 2–12 keV energy band for the soft state and the hard state, respectively. So for most sources, the uncertainty of the flux we estimated according to the Crab Nebula spectra is less than 25%. However, for the large column density cases (such as 4U 1630–472, ∼8 $\times {{10}^{22}}$ cm⁻²), we may underestimate the flux by 90% in the 2–12 keV energy band. We collected the values of distances and masses of our selected LMXBTs from the literature in order to estimate the X-ray luminosity (the NS mass is assumed to be 1.4 $\pm 0.1\;{{M}_{\odot }}$). Then the peak X-ray luminosity L_peak of the sources (14 NS LMXBTs and 9 BH LMXBTs) with known distances and masses are scaled in units of L_Edd, where L_Edd is taken as $1.3\times {{10}^{38}}\;(M/{{M}_{\odot }})\;{\rm ergs}\;{{{\rm s}}^{-1}}$. These estimations could bring large uncertainties due to the differences in spectral shapes, hydrogen column density, inclination angle, and radiation efficiency between those of the Crab Nebula and those of the sources in our sample, in addition to the distances and the masses (in't Zand et al. 2007). Notice that the X-ray flux is measured in 2–12 keV. The bolometric X-ray luminosity could be three times larger than the luminosity in 2–12 keV, including the uncertainties caused by the diverse energy spectra and column densities (in't Zand et al. 2007).

Among the 110 outbursts we selected, the brightest outburst during the RXTE era is the 1998 outburst of XTE J1550–564, which reached up to 6.5 Crab. The faintest outburst we selected is the 2004 outburst of 4U 1608–522, the peak X-ray flux of which is about 0.12 Crab. The most luminous outburst during the RXTE era is the 2002 outburst of 4U 1543–475 for the sources with known distances, which reached $5.4\times {{10}^{38}}\;{\rm ergs}\;{{{\rm s}}^{-1}}$ in 2–12 keV, and the bolometric X-ray luminosity could be $1.6\times {{10}^{39}}\;{\rm ergs}\;{{{\rm s}}^{-1}}$. The minimum peak X-ray luminosity of our selected outbursts is $9.6\times {{10}^{36}}\;{\rm ergs}\;{{{\rm s}}^{-1}}$ from the 2001 outburst of XTE J1650–500. The peak X-ray luminosity of the 1996 outburst of GRO J1744–28 is about 1.3 L_Edd in 2–12 keV, which is the highest Eddington luminosity in our sample. But RXTE/ASM only covered the decay phase of this outburst, so the peak X-ray luminosity we measured could be the lower limit of the true value. The bolometric X-ray luminosity is about three times larger than that in 2–12 keV (e.g., in't Zand et al. 2007), so the actual peak X-ray luminosity of the 1996 outburst of GRO J1744–28 is at least 3.9 L_Edd. The minimum peak X-ray in our sample is about 0.01 L_Edd in 2–12 keV, which is from the 2010 outburst of XTE J1752–223.

Figure 4 shows the distributions of the peak luminosities and fluxes of all of the selected outbursts, including the BH LMXBTs, the NS LMXBTs, and all of the LMXBTs, respectively. The statistical results of the average peak X-ray luminosity in a logarithmic scale are shown in Table 2. The distribution of the peak X-ray fluxes in Crab units has a cutoff at the low flux end due to the selection criteria of 0.1 Crab, and the distribution above 0.1 Crab is roughly consistent with a power-law form. From the left panel of Figure 4, we can see that the outbursts with peak flux larger than 1.5 Crab all belong to BH LMXBTs. The distribution of peak luminosities in units of ${\rm ergs}\;{{{\rm s}}^{-1}}$ and L_Edd both are roughly consistent with a Gaussian-like distribution in a logarithmic scale. The average peak X-ray luminosity (in units of ergs s⁻¹) of BH LMXBTs is nearly two times larger than that of NS LMXBTs; we can see those from both the middle panel of Figure 4 and Table 2. The average peak X-ray luminosity (in units of L_Edd) of the NS LMXBTs is four times larger than that of the BH LMXBTs (see the right panel of Figure 4 and Table 2). The average BH mass with a mass determination in our sample is about 9 ${{M}_{\odot }}$, and the NS mass is assumed as 1.4 ${{M}_{\odot }}$. By considering the difference in average peak X-ray luminosity in units of ${\rm ergs}\;{{{\rm s}}^{-1}}$ between BH and NS LMXBTs, we obtain a peak X-ray luminosity in units of L_Edd of NS LMXBTs that is about four times larger than that of BH LMXBTs, which is consistent with our measurement.

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Table 2.  Statistical Results of the Average Peak X-ray Luminosity

	All LMXBTs	NS LMXBTs	BH LMXBTs
$\lt {\rm log} ({{F}_{{\rm peak}}})\gt$	−0.440 ± 0.348	−0.531 ± 0.266	−0.292 ± 0.413
$\lt {{F}_{{\rm peak}}}\gt ({\rm Crab},2-12\;{\rm keV})$	$0.363_{-0.200}^{+0.447}$	$0.294_{-0.135}^{+0.249}$	$0.510_{-0.313}^{+0.812}$
$\lt {\rm log} ({{L}_{{\rm peak}}})\gt$	37.668 ± 0.331	37.579 ± 0.267	37.846 ± 0.375
$\lt {{L}_{{\rm peak}}}\gt ({{10}^{38}}\;{\rm ergs}\;{{{\rm s}}^{-1}},2-12\;{\rm keV})$	$0.466_{-0.248}^{+0.531}$	$0.380_{-0.174}^{+0.322}$	$0.701_{-0.406}^{+0.964}$
$\lt {\rm log} ({{L}_{{\rm peak}}}/{{L}_{{\rm Edd}}})\gt$	−0.817 ± 0.398	−0.681 ± 0.267	−1.281 ± 0.423
$\lt {{L}_{{\rm peak}}}/{{L}_{{\rm Edd}}}\gt (2-12\;{\rm keV})$	$0.152_{-0.091}^{+0.228}$	$0.209_{-0.096}^{+0.177}$	$0.052_{-0.033}^{+0.086}$

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The e-folding time is calculated by

$$\begin{eqnarray}&&\tau =\frac{{\Delta }t}{{\rm ln} A}\end{eqnarray} \tag{ 1 }$$

where ${\Delta }t$ is the time needed for the X-ray flux to increase by a factor of A. Note that the actual rise phase may not be well described by a single exponential form. We selected two data points in the light curve to measure the e-folding time for simplicity. Then τ corresponds to the e-folding time between the two data points in the light curve; that is, for an interval of τ, the X-ray flux increases by a factor of $e\simeq 2.71828$. The rise timescale defined in this way is independent of the instrumental sensitivity.

In practice, from the outburst peak backward in time to the beginning of an outburst, we selected a data point marked as ${{F}_{90\%}}$, which has the closest flux to 90% of the outburst peak flux F_peak in the range between the F_peak and the first two adjacent data points below 80% of the F_peak, so the ${{F}_{90\%}}$ we selected could actually be in the range of $80-100\%\;{{F}_{{\rm peak}}}$ in the real data. Then we selected a data point named ${{F}_{50\%}}$, which is the closest to 50% of F_peak in the range between the first data point below 66% of F_peak and the first two adjacent data points below 33% of F_peak, so the ${{F}_{50\%}}$ could actually be in the range of $33-66\%\;{{F}_{{\rm peak}}}$. We need to select two F_50% values if the outburst has a multipeak profile (e.g., the outburst of GRO J1655–40 in Figure 2). A data point named ${{F}_{10\%}}$ with a flux closest to $10\%$ of F_peak was selected in the range between the first data point below 20% of F_peak and the first two adjacent data points below 1% of F_peak, so the ${{F}_{10\%}}$ could actually be in the range of $1-20\%\;{{F}_{{\rm peak}}}$. During the procedures indicated above, we excluded the data points in the daily light curves of unusual fluctuation for which a certain flux is larger or smaller than the fluxes of the two adjacent days by more than a factor of two (e.g., the data point on MJD 53589 of XTE J1818–245 and the data point on MJD 52859 of H1743–322 in Figure 2). Then the e-folding rise timescales ${{\tau }_{{\rm rise},10-90\%}}$, ${{\tau }_{{\rm rise},10-50\%}}$, and ${{\tau }_{{\rm rise},50-90\%}}$ are calculated following Equation (1).

In some outbursts, ${{F}_{10\%}}$ or ${{F}_{50\%}}$ cannot be identified because of data gaps (e.g., in Figure 2, the data gaps during MJD 50100–50130 in the outburst of GRS 1739−278 and the data gaps during MJD 52600–52640 in the outburst of XTE J1720–318). There are other cases for which we could not identify the data points ${{F}_{10\%}}$ or ${{F}_{50\%}}$ because the X-ray flux increased too quickly on the daily timescale (e.g., the outburst of MXB 1730–333 in Figure 2).

To reverse the order of the data during the decay phase, we used the same method as in the rise phase to select the ${{F}_{90\%}}$, ${{F}_{50\%}}$, and ${{F}_{10\%}}$. Then we obtained the e-folding decay timescales for the different decay episodes corresponding to ${{F}_{10\%}}$–${{F}_{90\%}}$, ${{F}_{10\%}}$–${{F}_{50\%}}$, and ${{F}_{50\%}}$–${{F}_{90\%}}$, respectively. Figure 5 shows the distributions of the e-folding rise and decay timescales, including the BH LMXBTs, the NS LMXBTs, and all of the LMXBTs, respectively. Table 3 lists the statistical results of the average e-folding rise and decay timescales in a logarithmic scale. Generally speaking, the e-folding decay timescale (∼15 days) is about three times larger than the e-folding rise timescale (∼5 days; see ${{\tau }_{{\rm rise},10-90\%}}$ and ${{\tau }_{{\rm decay},10-90\%}}$ in Table 3). From Figure 5 and Table 3, we can see that the ${{\tau }_{{\rm rise},10-50\%}}$ is about two times smaller than ${{\tau }_{{\rm rise},50-90\%}}$ (both for BH and NS LMXBTs). Both the ${{\tau }_{{\rm rise},10-90\%}}$ and ${{\tau }_{{\rm rise},50-90\%}}$ of BH LMXBTs are two times larger than those of NS LMXBTs, but the ${{\tau }_{{\rm rise},10-50\%}}$ of BH and NS LMXBTs are similar. All three e-folding decay timescales of BH LMXBTs are larger than those of NS LMXBTs by a factor of about two. The e-folding decay timescales are similar between the decay episodes corresponding to ${{F}_{10\%}}$–${{F}_{50\%}}$ and ${{F}_{50\%}}$–${{F}_{90\%}}$ (both for BH and NS LMXBTs), which suggests that the decay phase (from ${{F}_{90\%}}$ to ${{F}_{10\%}}$) of most outbursts is roughly of exponential form.

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Table 3.  Statistical Results of Average e-Folding Rise and Decay Timescales

	All LMXBTs	NS LMXBTs	BH LMXBTs
$\lt {\rm log} ({{\tau }_{{\rm rise},10-90\%}})\gt$	0.726 ± 0.477	0.610 ± 0.432	0.913 ± 0.493
$\lt {{\tau }_{{\rm rise},10-90\%}}\gt ({\rm days})$	$5.326_{-3.552}^{+10.663}$	$4.072_{-2.564}^{+6.926}$	$8.185_{-5.555}^{+17.288}$
$\lt {\rm log} ({{\tau }_{{\rm rise},10-50\%}})\gt$	0.565 ± 0.516	0.509 ± 0.523	0.639 ± 0.504
$\lt {{\tau }_{{\rm rise},10-50\%}}\gt ({\rm days})$	$3.672_{-2.553}^{+8.377}$	$3.230_{-2.261}^{+7.533}$	$4.356_{-2.992}^{+9.552}$
$\lt {\rm log} ({{\tau }_{{\rm rise},50-90\%}})\gt$	0.847 ± 0.509	0.726 ± 0.415	1.007 ± 0.578
$\lt {{\tau }_{{\rm rise},50-90\%}}\gt ({\rm days})$	$7.038_{-4.857}^{+15.677}$	$5.323_{-3.275}^{+8.509}$	$10.166_{-7.483}^{+28.349}$
$\lt {\rm log} ({{\tau }_{{\rm decay},10-90\%}})\gt$	1.176 ± 0.392	1.032 ± 0.348	1.410 ± 0.347
$\lt {{\tau }_{{\rm decay},10-90\%}}\gt ({\rm days})$	$15.013_{-8.927}^{+22.022}$	$10.774_{-5.944}^{+13.260}$	$25.689_{-14.128}^{+31.393}$
$\lt {\rm log} ({{\tau }_{{\rm decay},10-50\%}})\gt$	1.086 ± 0.414	0.925 ± 0.380	1.339 ± 0.332
$\lt {{\tau }_{{\rm decay},10-50\%}}\gt ({\rm days})$	$12.194_{-7.490}^{+19.415}$	$8.418_{-4.908}^{+11.773}$	$21.829_{-11.675}^{+25.098}$
$\lt {\rm log} ({{\tau }_{{\rm decay},50-90\%}})\gt$	1.186 ± 0.490	1.050 ± 0.462	1.398 ± 0.461
$\lt {{\tau }_{{\rm decay},50-90\%}}\gt ({\rm days})$	$15.330_{-10.373}^{+32.075}$	$11.221_{-7.348}^{+21.291}$	$25.032_{-16.381}^{+47.397}$

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The e-folding rise or decay timescale we measured might include the plateau phase or secondary flare for the outbursts with plateau or multipeak profiles. Some outbursts have a plateau or secondary flare during the decay (such as the outbursts of XTE J1701–462 and XTE J2012+381 in Figure 2). It is quite interesting to study the decay before and after the plateau or secondary flare; this may reveal the mechanism of the plateau or the secondary flare. In the 1997 outburst of Aql X-1 and the 1998 outburst of XTE J2012+381, the ${{F}_{90\%}}$–${{F}_{50\%}}$ and ${{F}_{50\%}}$–${{F}_{10\%}}$ during the decay covered the phases before and after a short plateau or a secondary flare. The ${{\tau }_{{\rm decay},10-50\%}}$ and ${{\tau }_{{\rm decay},50-90\%}}$ were about six and 14 days for the outburst of Aql X-1, which means that the flux decreases faster after the plateau than before. And ${{\tau }_{{\rm decay},10-50\%}}$ and ${{\tau }_{{\rm decay},50-90\%}}$ were about 18 and 19 days for the outburst of XTE J2012+381, which means that the flux decreases exponentially at the same rate before and after the secondary flare. Chen et al. (1997) has also mentioned that the source undergoes an exponential decay or sharp cutoff after a plateau phase. Because a detailed study of the outburst profiles is beyond the scope of this work, we cannot draw a conclusion about this issue at the moment.

We also measured the rate of change of flux according to $\dot{F}={\Delta }F/{\Delta }t$ for the different rise or decay episodes corresponding to ${{F}_{10\%}}$–${{F}_{50\%}}$, ${{F}_{50\%}}$–${{F}_{90\%}}$, and ${{F}_{10\%}}$–${{F}_{90\%}}$, respectively. For the sources with known distances and masses, we scaled the $\dot{L}$ in units of L_Edd. Figure 6 shows the distributions of $\dot{L}$ of different rise and decay episodes in a logarithmic scale, and Table 4 shows the statistical results of the average $\dot{L}$ in a logarithmic scale. From Figure 6 and Table 4, we can see that the ${{\dot{L}}_{{\rm rise},50-90\%}}$ is about two times larger than ${{\dot{L}}_{{\rm rise},10-50\%}}$, for both the BH and NS LMXBTs, and ${{\dot{L}}_{{\rm decay},50-90\%}}$ is also about two times larger than ${{\dot{L}}_{{\rm decay},10-50\%}}$. These suggest that the X-ray luminosity changes much more dramatically when it is near the peak X-ray luminosity.

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Table 4.  Statistical Results of the Average $\dot{L}$

	All LMXBTs	NS LMXBTs	BH LMXBTs
$\lt {\rm log} ({{\dot{L}}_{{\rm rise},10-90\%}})\gt$	−1.994 ± 0.692	−1.756 ± 0.457	−2.797 ± 0.755
$\lt {{\dot{L}}_{{\rm rise},10-90\%}}\gt ({{L}_{{\rm Edd}}}/{\rm day})$	$0.010_{-0.008}^{+0.040}$	$0.018_{-0.011}^{+0.033}$	$0.002_{-0.001}^{+0.007}$
$\lt {\rm log} ({{\dot{L}}_{{\rm rise},10-50\%}})\gt$	−2.108 ± 0.742	−1.848 ± 0.553	−2.820 ± 0.736
$\lt {{\dot{L}}_{{\rm rise},10-50\%}}\gt ({{L}_{{\rm Edd}}}/{\rm day})$	$0.008_{-0.006}^{+0.035}$	$0.014_{-0.010}^{+0.037}$	$0.002_{-0.001}^{+0.007}$
$\lt {\rm log} ({{\dot{L}}_{{\rm rise},50-90\%}})\gt$	−1.819 ± 0.740	−1.584 ± 0.468	−2.453 ± 0.954
$\lt {{\dot{L}}_{{\rm rise},50-90\%}}\gt ({{L}_{{\rm Edd}}}/{\rm day})$	$0.015_{-0.012}^{+0.068}$	$0.026_{-0.017}^{+0.051}$	$0.004_{-0.003}^{+0.028}$
$\lt {\rm log} ({{\dot{L}}_{{\rm decay},10-90\%}})\gt$	−2.379 ± 0.595	−2.148 ± 0.313	−3.164 ± 0.660
$\lt {{\dot{L}}_{{\rm decay},10-90\%}}\gt ({{L}_{{\rm Edd}}}/{\rm day})$	$0.004_{-0.003}^{+0.012}$	$0.007_{-0.004}^{+0.008}$	$0.001_{-0.001}^{+0.002}$
$\lt {\rm log} ({{\dot{L}}_{{\rm decay},10-50\%}})\gt$	−2.458 ± 0.594	−2.212 ± 0.316	−3.269 ± 0.575
$\lt {{\dot{L}}_{{\rm decay},10-50\%}}\gt ({{L}_{{\rm Edd}}}/{\rm day})$	$0.003_{-0.003}^{+0.010}$	$0.006_{-0.003}^{+0.007}$	$0.001_{-0.0004}^{+0.001}$
$\lt {\rm log} ({{\dot{L}}_{{\rm decay},50-90\%}})\gt$	−2.102 ± 0.707	−1.887 ± 0.466	−2.813 ± 0.894
$\lt {{\dot{L}}_{{\rm decay},50-90\%}}\gt ({{L}_{{\rm Edd}}}/{\rm day})$	$0.008_{-0.006}^{+0.032}$	$0.013_{-0.009}^{+0.025}$	$0.002_{-0.001}^{+0.011}$

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We showed the correlation between ${{\dot{L}}_{{\rm rise}}}$ and L_peak scaled in L_Edd in Figure 7. The Spearman correlation coefficients of these three correlations are 0.70, 0.67, and 0.71 at a significance of $7.28\sigma$, $6.31\sigma$, and $6.87\sigma$ for the different rise episodes corresponding to ${{F}_{10\%}}$–${{F}_{90\%}}$, ${{F}_{10\%}}$–${{F}_{50\%}}$, and ${{F}_{50\%}}$–${{F}_{90\%}}$, respectively. These results demonstrate that there is a significant positive correlation between ${{\dot{L}}_{{\rm rise}}}$ and L_peak (see Figure 7), which confirms the previous results in Yu & Yan (2009). We used a linear function in a logarithmic scale (${\rm log} {{\dot{L}}_{{\rm rise}}}=A+B\times {\rm log} {{L}_{{\rm peak}}}$) to fit the three correlations with a Bayesian approach following Kelly (2007). Then we got ${\rm log} {{\dot{L}}_{{\rm rise},10-90\%}}=(-0.83\pm 0.14)$ $\;+\;(1.48\pm 0.15)\times {\rm log} {{L}_{{\rm peak}}}$, ${\rm log} {{\dot{L}}_{{\rm rise},10-50\%}}=(-0.93\pm 0.15)$ $\;+\;(1.47\pm 0.16)\times {\rm log} {{L}_{{\rm peak}}}$, and ${\rm log} {{\dot{L}}_{{\rm rise},50-90\%}}=(-0.59\pm 0.17)$ $\;+\;(1.48\pm 0.17)\times {\rm log} {{L}_{{\rm peak}}}$, respectively. The intrinsic scatters of these correlations are 0.36 ± 0.05, 0.37 ± 0.06, and 0.43 ± 0.06 dex, respectively. The best-fit results are also plotted in Figure 7. We found that the L_peak is also positively correlated with ${{\dot{L}}_{{\rm decay}}}$ in units of L_Edd (see Figure 7). The Spearman correlation coefficients are 0.59, 0.56, and 0.60 at a significance of $5.84\sigma$, $5.39\sigma$, and $5.97\sigma$ for the different decay episodes corresponding to ${{F}_{10\%}}$–${{F}_{90\%}}$, ${{F}_{10\%}}$–${{F}_{50\%}}$, and ${{F}_{50\%}}$–${{F}_{90\%}}$, respectively, which shows that the positive correlations between L_peak and ${{\dot{L}}_{{\rm decay}}}$ are also very significant. We then used the same function (${\rm log} {{\dot{L}}_{{\rm decay}}}=A+B\times {\rm log} {{L}_{{\rm peak}}}$) to fit the three correlations with a Bayesian approach in Kelly (2007), and we got ${\rm log} {{\dot{L}}_{{\rm decay},10-90\%}}=(-1.36\pm 0.14)$ $+\;(1.32\pm 0.15)\times {\rm log} {{L}_{{\rm peak}}}$, ${\rm log} {{\dot{L}}_{{\rm decay},10-50\%}}=(-1.44\pm 0.15)\;+\;(1.29\pm 0.16)\;\times \;{\rm log} {{L}_{{\rm peak}}}$, and ${\rm log} {{\dot{L}}_{{\rm decay},50-90\%}}\;=\;(-0.98\pm 0.16)+(1.43\pm 0.17)\times \;{\rm log} {{L}_{{\rm peak}}},$ respectively. The intrinsic scatters of the three correlations are 0.34 ± 0.04, 0.36 ± 0.05, and 0.40 ± 0.05 dex, respectively. The best-fit results are also plotted in Figure 7. In Figure 7, it seems that these positive correlations also hold in the individual source GX 339–4. In addition, we also found a positive correlation between the e-folding rise timescale and the orbital period (see Figure 8). The Spearman correlation coefficients are 0.51, 0.63, and 0.29 at a significance of $3.15\sigma$, $3.99\sigma$, and $1.45\sigma$ for ${{\tau }_{10-90\%}}$, ${{\tau }_{10-50\%}}$, and ${{\tau }_{50-90\%}}$, respectively, but no correlation between the e-folding decay timescale and the orbital period was found.

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For outbursts with ${{F}_{10\%}}$ identified both in the rise and the decay phase, we estimated the duration of an outburst as the time interval between these two F_10% values. In this way, our estimation of outburst duration is independent of the instrument sensitivity. Figure 9 shows the distributions of the outburst durations for the available outbursts, and Table 5 shows the statistical results of the average duration in a logarithmic scale. Based on our definition, the average duration of outbursts in LMXBTs is about 54 days. From Figures 9 and 5, we can see that the average duration of BH LMXBTs (∼88 days) is larger than that of NS LMXBTs (∼39 days) by a factor of about two.

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Table 5.  Statistical Results of the Average Outburst Duration

	All LMXBTs	NS LMXBTs	BH LMXBTs
$\lt {\rm log} ({\rm Duration})\gt$	1.729 ± 0.355	1.595 ± 0.307	1.945 ± 0.321
$\lt {\rm Duration}\gt$ (days)	$53.641_{-29.968}^{+67.904}$	$39.316_{-19.934}^{+40.436}$	$88.183_{-46.095}^{+96.579}$

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We estimated the duty cycle for each LMXBT using the outburst duration and the number of outbursts we measured. This parameter is very important in understanding the luminosity function and evolution of LMXBs (such as Belczynski et al. 2004; Fragos et al. 2008, 2009), which is defined as the fraction of the lifetime that a transient source is in the outburst phase. Because of the highly reduced sky coverage of RXTE/ASM since late 2010, we only used the data before 2011 to estimate the duty cycle. We summed all of the outburst durations during the period 1996–2010 and divided by the total observation time (∼5472 days) to estimate the duty cycle for each LMXBT. For the outbursts, we could not measure the outburst duration because of the lack of identification of ${{F}_{10\%}}$ in the rise or decay phases, so we used the ${{F}_{50\%}}$ or F_90% (if lacking ${{F}_{50\%}}$) instead.

Figure 10 shows the distribution of the duty cycles we estimated for all 36 LMXBTs. The average of duty cycles in a logarithmic scale is $0.025_{-0.016}^{+0.045}$, $0.035_{-0.019}^{+0.044}$, and $0.020_{-0.014}^{+0.042}$ for all LMXBTs, NSs, and BHs, respectively. However, many LMXBTs in our sample only have one outburst during the entire RXTE era, so the duty cycle we estimated is actually the upper limit of the true value. The evolution history of the LMXBTs is much longer than the 15 yr long RXTE/ASM observation, so the behavior during the RXTE era may not be typical in the long evolution history. We do at least get an order-of-magnitude estimate of the duty cycles, which is helpful in the theoretical modeling of the luminosity function and the evolution of LMXBTs.

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Some sources in our sample were also investigated by Chen et al. (1997), including two NS LMXBTs, Aql X-1 and 4U 1608–522, and three BH LMXBTs, 4U 1543–475, 4U 1630–472, and GRO J1655–40. We have combined the results in Chen et al. (1997) to improve the estimation of the duty cycles of these sources. We used the average outburst duration in the RXTE era for outbursts with no duration measurements in Chen et al. (1997). The duty cycles of Aql X-1 and 4U 1608–522 from 1970 to 2011 are about 0.08 and 0.09, which are the same as that in the RXTE era. The duty cycle of 4U 1543–475 from 1971 to 2011 is about 0.05, which is one order of magnitude larger than that in the RXTE era. The duty cycles of GRO J1655–40 from 1994 to 2011 and 4U 1630–472 from 1971 to 2011 are about 0.17 and 0.21, which are larger than the 0.12 and 0.16 estimated for the outbursts only in the RXTE era.

Grindlay et al. (2014) has discovered historical optical outbursts in four BH LMXBTs. The data of XTE J1118+480 are released in the DASCH (Digital Access to a Sky Century @ Harvard) project. We identified 10 optical outbursts during the period from about 1911 to 1989 from the DASCH data of XTE J1118+480. There is an X-ray outburst of XTE J1118+480 during the RXTE era, which is not included in our sample because the peak flux was less than 0.1 Crab. If we assume that all of the historical optical outbursts were associated with X-ray outbursts and that the outburst durations of those optical outbursts are similar to that of the X-ray outbursts in the RXTE era, the duty cycle of XTE J1118+480 from 1911 to 2011 is about 0.05, which is a little larger than the estimation (0.03) from only the data of the RXTE era.

We integrated the X-ray flux over an outburst from ${{F}_{10\%}}$ in the rise phase to ${{F}_{10\%}}$ in the decay phase to estimate the total X-ray fluence for each available outburst. The X-ray energy spectra of LMXBTs usually varies dramatically during an outburst. In order to estimate the X-ray flux more accurately, we extracted the ASM light curves in three energy bands and scaled the X-ray flux in Crab units with conversion factors of 1 Crab = 27, 23, 25 c s⁻¹ for 2–3, 3–5, and 5–12 keV, respectively. Then we converted the photon fluxes of three energy bands into fluxes by assuming a Crab Nebual-like X-ray spectrum, and we summed the fluxes in the three energy bands to obtain the X-ray flux in the 2–12 keV band. Then we measured the total X-ray fluence by summing the 2–12 keV X-ray flux from ${{F}_{10\%}}$ in the rise phase to ${{F}_{10\%}}$ in the decay phase. The data gaps in the daily light curve due to sparse coverage were filled with values inferred by linear interpolation. For sources with known distances, we obtained the total energy E radiated during the outburst. We also measured the total radiated energy during the rise and decay phases in this way.

Figure 11 shows the distributions of the total radiated energy E in the 2–12 keV band. Table 6 shows the statistical results of the average total radiated energy in a logarithmic scale. As can be seen from Table 6 and Figure 11, the average total energy of BH LMXBTs in 2–12 keV ($\sim 2.47\times {{10}^{44}}$ ergs) is about five times larger than that of NS LMXBTs ($\sim 0.55\times {{10}^{44}}$ ergs). The average total energy is about $0.91\times {{10}^{44}}$ ergs. Assuming a bolometric X-ray flux correction factor of three and a constant radiative efficiency of 0.1, this corresponds to a total mass of $1.52\times {{10}^{-9}}{{M}_{\odot }}$ on average being accreted during an outburst. The total radiated energy during the decay phase is about two times larger than that during the rise phase for both BH and NS LMXBTs, which suggests that the central compact object will accrete more mass during the decay phase (see Table 6). There is a cutoff of the distribution of E for BH LMXBTs at the lower E end, but it does not show in the NS LMXBT distribution (see left panel of Figure 11), which may suggest that a minimum disk mass is required to trigger an outburst for BH LMXBTs.

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Table 6.  Statistical Results of the Average Total Radiated Energy

	All LMXBTs	NS LMXBTs	BH LMXBTs
$\lt {\rm log} ({{E}_{{\rm total}}})\gt$	43.958 ± 0.565	43.741 ± 0.468	44.392 ± 0.491
$\lt {{E}_{{\rm total}}}\gt ({{10}^{44}}\;{\rm ergs},2-12\;{\rm keV})$	$0.908_{-0.660}^{+2.425}$	$0.551_{-0.363}^{+1.065}$	$2.468_{-1.671}^{+5.180}$
$\lt {\rm log} ({{E}_{{\rm rise}}})\gt$	43.469 ± 0.594	43.253 ± 0.476	43.901 ± 0.577
$\lt {{E}_{{\rm rise}}}\gt ({{10}^{44}}\;{\rm ergs},2-12\;{\rm keV})$	$0.295_{-0.220}^{+0.862}$	$0.179_{-0.119}^{+0.357}$	$0.796_{-0.585}^{+2.206}$
$\lt {\rm log} ({{E}_{{\rm decay}}})\gt$	43.795 ± 0.580	43.597 ± 0.520	44.191 ± 0.489
$\lt {{E}_{{\rm decay}}}\gt ({{10}^{44}}\;{\rm ergs},2-12\;{\rm keV})$	$0.624_{-0.460}^{+1.749}$	$0.396_{-0.276}^{+0.913}$	$1.554_{-1.050}^{+3.236}$

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We further plotted the correlation between the peak X-ray luminosity and total radiated energy (see Figure 12). The Spearman correlation coefficients are 0.79, 0.71, and 0.80 at a significance of $9.51\sigma$, $8.00\sigma$, and $10.02\sigma$ for the E, E_rise, and E_decay, respectively, which demonstrate that there is a strong positive correlation between the peak X-ray luminosity and the total radiated energy. We used a linear function in a logarithmic scale (${\rm log} E=A+B\times {\rm log} {{L}_{{\rm peak}}}$) to fit the three correlations with a Bayesian approach in Kelly (2007). Then we got ${\rm log} E=(-13.26\pm 6.53)+(1.52\pm 0.17)\times {\rm log} {{L}_{{\rm peak}}}$, ${\rm log} {{E}_{{\rm rise}}}=(-11.18\pm 7.31)+(1.45\pm 0.19)\times {\rm log} {{L}_{{\rm peak}}}$, and ${\rm log} {{E}_{{\rm decay}}}=(-15.76\pm 6.40)+(1.58\pm 0.17)\times {\rm log} {{L}_{{\rm peak}}}$, respectively. The intrinsic scatters of these correlations are 0.29 ± 0.04, 0.35 ± 0.05, and 0.30 ± 0.04 dex, respectively. The best-fit results are also plotted in Figure 12. These correlations seem to also hold in an individual source MXB 1730-33 (see Figure 12). We also found that the total radiated energy is positively correlated with the e-folding rise and decay timescale (see Figure 13). For the rise timescale, the Spearman correlation coefficients are 0.57, 0.42, and 0.49 at a significance of $5.97\sigma$, $3.78\sigma$, and $4.55\sigma$ for ${{\tau }_{{\rm rise},10-90\%}}$, ${{\tau }_{{\rm rise},10-50\%}}$, and ${{\tau }_{{\rm rise},50-90\%}}$, respectively. For the decay timescale, the Spearman correlation coefficients are 0.73, 0.66, and 0.63 at a significance of $8.43\sigma$, $7.12\sigma$, and $6.69\sigma$ for ${{\tau }_{{\rm decay},10-90\%}}$, ${{\tau }_{{\rm decay},10-50\%}}$, and ${{\tau }_{{\rm decay},50-90\%}}$, respectively. So these positive correlations are all very significant. We used a linear function in a logarithmic scale (${\rm log} \tau =A+B\times {\rm log} E$) to fit all six correlations with a Bayesian approach in Kelly (2007). Then we got ${\rm log} {{\tau }_{{\rm rise},10-90\%}}=(-23.31\pm 3.76)+(0.55\pm 0.09)\times {\rm log} E$, ${\rm log} {{\tau }_{{\rm rise},10-50\%}}=(-18.08\pm 4.75)+(0.42\pm 0.11)\times {\rm log} E$, ${\rm log} {{\tau }_{{\rm rise},50-90\%}}=(-22.08\pm 4.41)+(0.52\pm 0.10)\times {\rm log} E$, ${\rm log} {{\tau }_{{\rm decay},10-90\%}}=(-26.49\pm 2.49)+(0.63\pm 0.06)\times {\rm log} E$, ${\rm log} {{\tau }_{{\rm decay},10-50\%}}=(-26.19\pm 3.02)+(0.62\pm 0.07)\times {\rm log} E$, and ${\rm log} {{\tau }_{{\rm decay},50-90\%}}=(-26.00\pm 3.73)+(0.62\pm 0.08)\;\times {\rm log} E$, respectively. The intrinsic scatters of all of the above six correlations are 0.39 ± 0.03, 0.47 ± 0.04, 0.44 ± 0.04, 0.22 ± 0.02, 0.27 ± 0.03, and 0.37 ± 0.03 dex, respectively. The best-fit results are also plotted in Figure 13. The correlation between ${{\tau }_{{\rm rise},10-90\%}}$ and E also holds in an individual source GX 339–4, and the correlation between ${{\tau }_{{\rm decay},10-90\%}}$ and E holds in an individual source H1743–322 (see the upper panels of Figure 13). We also found a marginal correlation between the total radiated energy E and the orbital period P_orb among different sources (see Figure 15). The Spearman correlation coefficient is 0.49 at a significance of $3.03\sigma$. This correlation seems to only hold at an orbital period of less than about 100 hr because of the large deviation from this correlation of GRO J1744–28 with orbital period 284 hr. We need more data to test whether the correlation is valid at orbital periods larger than 100 hr.

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For the sources with unknown distances, we estimated the total fluence for each available outburst. If these outbursts also follow the correlations between E and ${{\tau }_{{\rm rise}}}$ or ${{\tau }_{{\rm decay}}}$ shown in Figure 13, we can infer the distances for these sources according to the correlations. We used a linear model in a logarithmic scale (${\rm log} E=A+B\times {\rm log} \tau$) to fit the correlations with a Bayesian approach in Kelly (2007). The best-fit parameters are shown in Table 7. There are five sources with unknown distances in our sample (see Table 1). We identified three outbursts in SLX 1746–331 and one outburst in four other sources (SWIFT J1539.2–6227, SWIFT J1842.5–1124, XTE J1755–324, and XTE J2012+381). The e-folding rise or decay timescales of these five sources can be seen in Table 8. We calculated six values for the distance at most for each outburst according to the fitted results of different e-folding rise or decay timescales in Table 7 by assuming that those outbursts follow the correlations. We used the average value and standard deviations of the distances from the different correlations as our estimation. The distances for these five sources are 10.81 ± 3.52, 9.32 ± 1.53, 12.87 ± 2.32, 11.09 ± 4.52, and 7.49 ± 2.16 kpc for SLX 1746–331, SWIFT J1539.2–6227, SWIFT J1842.5–1124, XTE J1755–324, and XTE J2012+381, respectively.

Table 7.  Parameters of the Best-fit Results of ${\rm log} E=A+B\times {\rm log} \tau$

	A	B
${{\tau }_{{\rm rise},10-90\%}}$ vs. E	43.46 ± 0.10	0.68 ± 0.11
${{\tau }_{{\rm rise},10-50\%}}$ vs. E	43.72 ± 0.10	0.49 ± 0.13
${{\tau }_{{\rm rise},50-90\%}}$ vs. E	43.48 ± 0.12	0.60 ± 0.12
${{\tau }_{{\rm decay},10-90\%}}$ vs. E	42.65 ± 0.12	1.14 ± 0.10
${{\tau }_{{\rm decay},10-50\%}}$ vs. E	42.92 ± 0.12	1.00 ± 0.11
${{\tau }_{{\rm decay},50-90\%}}$ vs. E	43.09 ± 0.13	0.75 ± 0.10

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Table 8.  Parameters of LMXBT Outbursts

Source	tpeak	Fpeak	${{\tau }_{{\rm rise},10-90\%}}$	${{\tau }_{{\rm rise},10-50\%}}$	${{\tau }_{{\rm rise},50-90\%}}$	${{\tau }_{{\rm decay},10-90\%}}$	${{\tau }_{{\rm decay},10-50\%}}$	${{\tau }_{{\rm decay},50-90\%}}$	Outburst Duration	E
	(MJD)	(Crab)	(days)	(days)	(days)	(days)	(days)	(days)	(days)	(1044 ergs)
Swift J1539.2–6227	54815	0.28 ± 0.02	7.12 ± 2.15	1.77 ± 0.85	8.28 ± 1.16	12.04 ± 5.11	13.88 ± 7.70	6.11 ± 1.90	47.00	⋯
4U 1543–475	52445	3.79 ± 0.02	0.79 ± 0.01	0.34 ± 0.01	2.37 ± 0.02	8.82 ± 0.11	8.73 ± 0.15	9.08 ± 0.23	27.00	6.11 ± 1.63
XTE J1550–564	51075	6.52 ± 0.04	4.36 ± 0.02	6.12 ± 0.04	1.21 ± 0.01	19.17 ± 0.13	28.19 ± 0.27	2.65 ± 0.03	51.00	5.32 ± 4.62
	51663	0.95 ± 0.01	9.07 ± 0.17	11.95 ± 0.40	3.09 ± 0.15	8.66 ± 0.48	5.68 ± 0.51	18.20 ± 3.20	41.00	0.89 ± 0.77
4U 1630–472	50247	0.34 ± 0.02	46.20 ± 4.40	3.21 ± 0.50	115.97 ± 12.51	22.59 ± 1.38	14.72 ± 1.03	44.64 ± 6.64	145.00	7.00 ± 7.00
	50864	0.37 ± 0.01	5.38 ± 0.22	2.33 ± 0.12	15.49 ± 0.75	16.15 ± 0.67	13.97 ± 1.07	20.78 ± 2.21	51.00	2.13 ± 2.13
	51359	0.25 ± 0.02	24.02 ± 4.05	13.20 ± 2.78	64.58 ± 12.23	11.30 ± 0.98	7.68 ± 1.67	18.83 ± 7.60	78.00	2.25 ± 2.25
	51884	0.56 ± 0.05	5.09 ± 0.60	1.20 ± 0.17	27.20 ± 6.49	18.68 ± 1.13	17.08 ± 1.84	21.26 ± 3.17	72.00	4.62 ± 4.62
	52627	0.84 ± 0.04	38.57 ± 1.33	2.65 ± 0.11	8.26 ± 0.77	19.43 ± 2.27	17.40 ± 2.80	24.53 ± 3.77	140.00	11.39 ± 11.39
	53747	0.31 ± 0.02	19.36 ± 6.50	9.62 ± 4.06	55.40 ± 18.98	44.94 ± 5.99	42.18 ± 8.66	52.50 ± 14.70	146.00	5.52 ± 5.52
	54479	0.26 ± 0.02	7.53 ± 5.45	5.21 ± 4.59	18.43 ± 6.31	61.45 ± 9.13	29.75 ± 5.97	151.02 ± 15.08	167.00	6.55 ± 6.55
	55224	0.32 ± 0.02	10.13 ± 2.35	2.66 ± 0.88	29.91 ± 7.42	11.27 ± 1.24	5.54 ± 0.98	23.31 ± 4.25	53.00	2.76 ± 2.76
XTE J1650–500	52172	0.56 ± 0.00	7.21 ± 1.11	1.37 ± 0.30	20.99 ± 0.91	29.12 ± 3.39	21.22 ± 3.42	50.56 ± 3.60	81.00	0.35 ± 0.19
XTE J1652–453	55016	0.18 ± 0.01	3.28 ± 1.73	1.90 ± 1.35	7.21 ± 2.68	23.72 ± 8.37	24.22 ± 12.04	22.45 ± 6.67	61.00	⋯
GRO J1655–40	50284	3.21 ± 0.02	37.00 ± 0.95	14.54 ± 0.53	91.69 ± 1.24	172.34 ± 1.47	49.93 ± 0.66	168.22 ± 4.05	461.00	16.18 ± 2.02
	53508	4.31 ± 0.03	22.87 ± 0.32	4.11 ± 0.08	5.72 ± 0.08	51.84 ± 0.41	62.40 ± 0.65	20.41 ± 0.43	171.00	4.90 ± 0.61
MAXI J1659–152	55476	0.63 ± 0.04	⋯	⋯	1.40 ± 0.17	7.24 ± 2.21	9.57 ± 4.13	1.48 ± 0.23	⋯	⋯
GX 339–4	50867	0.33 ± 0.02	46.65 ± 5.34	34.42 ± 5.99	78.01 ± 14.54	114.78 ± 34.43	27.48 ± 11.09	372.52 ± 61.78	432.00	6.22 ± 1.73
	52482	0.88 ± 0.01	49.03 ± 1.59	20.66 ± 0.92	63.21 ± 2.39	78.13 ± 3.03	77.87 ± 4.18	78.83 ± 3.40	317.00	10.41 ± 2.90
	53310	0.53 ± 0.02	39.25 ± 5.50	27.72 ± 5.25	72.10 ± 2.53	64.08 ± 6.79	60.87 ± 9.31	73.27 ± 12.25	273.00	4.54 ± 1.26
	54149	1.03 ± 0.02	14.19 ± 0.82	16.45 ± 1.29	8.15 ± 0.42	25.04 ± 0.79	18.24 ± 0.90	42.91 ± 2.77	95.00	2.73 ± 0.76
	55317	0.66 ± 0.02	21.33 ± 1.59	18.11 ± 1.94	30.77 ± 4.08	139.03 ± 36.04	139.03 ± 46.65	139.03 ± 15.91	382.00	5.57 ± 1.55
XTE J1720–318	52652	0.44 ± 0.01	21.30 ± 13.28	⋯	⋯	24.88 ± 3.98	23.17 ± 5.77	30.10 ± 12.81	113.00	0.98 ± 1.06
GRS 1739–278	50159	0.84 ± 0.02	⋯	⋯	3.64 ± 0.20	84.36 ± 4.59	97.32 ± 7.25	48.70 ± 3.95	⋯	⋯
H 1743–322	52752	1.39 ± 0.01	9.11 ± 0.19	10.32 ± 0.33	6.20 ± 0.20	76.14 ± 18.18	72.85 ± 23.87	21.74 ± 1.77	179.00	15.67 ± 5.17
	53234	0.33 ± 0.02	19.97 ± 4.52	7.54 ± 2.31	55.15 ± 10.46	17.23 ± 1.19	13.36 ± 1.39	26.99 ± 2.93	96.00	3.12 ± 1.03
	53602	0.20 ± 0.01	3.75 ± 0.86	2.45 ± 0.73	8.05 ± 1.94	11.64 ± 7.99	10.49 ± 9.73	14.93 ± 1.69	42.00	0.70 ± 0.23
	54466	0.33 ± 0.04	9.43 ± 3.71	14.05 ± 8.92	1.36 ± 0.44	13.51 ± 2.24	13.87 ± 2.94	12.21 ± 4.02	53.00	1.36 ± 0.45
	54988	0.25 ± 0.01	5.24 ± 1.28	6.22 ± 2.20	2.93 ± 0.71	12.98 ± 6.12	14.73 ± 9.99	8.65 ± 3.29	41.00	0.72 ± 0.24
	55425	0.22 ± 0.01	3.20 ± 13.69	2.86 ± 14.18	5.35 ± 1.84	7.94 ± 7.07	8.83 ± 11.10	5.77 ± 2.80	37.00	0.64 ± 0.21
XTE J1748–288	50970	0.57 ± 0.01	1.35 ± 0.25	1.18 ± 0.25	2.99 ± 0.24	14.45 ± 1.63	9.87 ± 1.43	30.73 ± 1.70	37.00	2.32 ± 0.93
SLX 1746–331	52740	0.37 ± 0.01	2.78 ± 0.21	2.28 ± 0.28	3.55 ± 0.20	25.09 ± 4.09	15.02 ± 3.51	49.32 ± 5.09	60.00	⋯
	54383	0.14 ± 0.01	1.72 ± 1.10	1.64 ± 1.16	2.43 ± 0.76	10.13 ± 3.04	8.26 ± 3.42	15.36 ± 4.55	30.00	⋯
	54475	0.16 ± 0.02	13.36 ± 7.14	8.25 ± 6.00	29.94 ± 20.38	33.91 ± 6.49	28.17 ± 8.73	45.12 ± 12.87	107.00	⋯
XTE J1752–223	55219	0.52 ± 0.02	37.67 ± 6.50	29.99 ± 6.32	58.74 ± 22.33	33.66 ± 4.49	32.01 ± 5.81	38.69 ± 8.78	161.00	1.07 ± 0.25
Swift J1753.5–0127	53558	0.21 ± 0.01	3.79 ± 0.44	0.93 ± 0.17	9.74 ± 1.41	28.47 ± 2.95	28.84 ± 4.25	27.48 ± 3.34	69.00	0.51 ± 0.34
XTE J1755–324	50656	0.18 ± 0.00	3.78 ± 1.02	3.89 ± 1.39	3.44 ± 0.41	38.65 ± 5.52	46.78 ± 9.13	14.95 ± 2.19	102.00	⋯
XTE J1817–330	53767	1.94 ± 0.02	2.45 ± 0.06	0.82 ± 0.03	7.20 ± 0.26	24.55 ± 0.41	27.59 ± 0.81	15.68 ± 0.95	57.00	0.81 ± 1.08
XTE J1818–245	53596	0.53 ± 0.01	0.89 ± 0.05	0.55 ± 0.04	2.22 ± 0.20	26.24 ± 2.14	28.54 ± 3.21	20.16 ± 1.64	55.00	0.25 ± 0.11
Swift J1842.5–1124	54737	0.15 ± 0.01	7.79 ± 2.18	8.62 ± 3.83	6.23 ± 1.52	30.43 ± 14.59	28.57 ± 19.32	34.98 ± 5.54	89.00	⋯
XTE J1859 + 226	51467	1.31 ± 0.01	3.30 ± 0.03	3.42 ± 0.05	3.02 ± 0.08	17.16 ± 0.38	21.42 ± 0.70	5.84 ± 0.20	50.00	1.02 ± 0.24
XTE J2012 + 381	50967	0.21 ± 0.00	2.65 ± 0.19	1.98 ± 0.26	3.44 ± 0.28	31.90 ± 5.26	18.19 ± 4.04	19.43 ± 1.62	84.00	⋯
4U 1608–522	50094	0.60 ± 0.01	⋯	⋯	4.14 ± 0.37	25.51 ± 1.08	4.63 ± 0.29	69.08 ± 2.87	⋯	⋯
	50850	0.95 ± 0.01	1.44 ± 0.07	1.18 ± 0.07	2.64 ± 0.14	21.26 ± 0.45	22.23 ± 0.67	18.74 ± 0.38	54.00	1.86 ± 1.28
	51896	0.42 ± 0.03	⋯	⋯	⋯	11.32 ± 1.39	12.82 ± 2.84	8.75 ± 1.71	⋯	⋯
	52220	0.65 ± 0.02	2.10 ± 0.52	0.99 ± 0.35	4.75 ± 0.46	4.21 ± 0.50	⋯	⋯	19.00	0.56 ± 0.39
	52482	0.79 ± 0.01	3.51 ± 0.17	2.24 ± 0.14	8.11 ± 0.27	21.62 ± 0.46	11.55 ± 0.50	51.63 ± 5.17	64.00	2.07 ± 1.42
	52711	0.34 ± 0.01	1.49 ± 0.09	1.19 ± 0.14	1.99 ± 0.24	6.29 ± 0.57	7.19 ± 0.74	2.52 ± 0.55	22.00	0.27 ± 0.19
	53084	0.26 ± 0.01	2.63 ± 1.31	1.49 ± 0.98	6.18 ± 1.31	2.74 ± 0.22	1.36 ± 0.15	5.61 ± 0.63	14.00	0.16 ± 0.11
	53192	0.12 ± 0.06	6.57 ± 2.12	8.70 ± 4.50	2.95 ± 0.36	3.02 ± 0.71	3.58 ± 1.42	2.18 ± 0.28	19.00	0.06 ± 0.04
	53494	0.67 ± 0.02	26.61 ± 5.76	2.56 ± 0.82	8.93 ± 1.86	19.75 ± 0.99	18.95 ± 1.42	21.51 ± 0.89	100.00	3.05 ± 2.11
	53654	0.27 ± 0.00	7.66 ± 1.35	9.30 ± 2.55	4.61 ± 0.33	12.24 ± 3.43	11.36 ± 4.45	14.49 ± 1.77	58.00	0.52 ± 0.36
	54273	0.57 ± 0.01	3.53 ± 0.26	⋯	⋯	5.50 ± 0.33	3.18 ± 0.25	14.06 ± 1.15	20.00	0.47 ± 0.32
	54402	0.65 ± 0.00	5.04 ± 0.29	3.04 ± 0.26	9.50 ± 0.42	16.12 ± 3.41	11.73 ± 3.23	30.64 ± 2.21	51.00	1.43 ± 0.99
	54986	0.16 ± 0.01	16.96 ± 3.71	12.69 ± 3.82	28.79 ± 4.97	6.06 ± 1.29	4.45 ± 1.36	10.34 ± 2.71	62.00	0.31 ± 0.22
	55265	0.57 ± 0.01	1.24 ± 0.08	0.58 ± 0.05	2.92 ± 0.20	3.67 ± 0.33	2.67 ± 0.32	6.88 ± 0.61	14.00	0.32 ± 0.22
	55669	0.71 ± 0.01	10.14 ± 2.93	10.83 ± 3.63	5.88 ± 2.45	31.59 ± 3.23	30.68 ± 4.59	1.71 ± 0.29	97.00	2.10 ± 1.45
XTE J1701–462	53767	0.91 ± 0.01	11.95 ± 2.26	7.86 ± 1.97	24.88 ± 1.55	220.68 ± 13.41	295.43 ± 25.99	39.86 ± 2.03	566.00	33.06 ± 9.77
RX J1709.5–2639	50456	0.22 ± 0.01	1.66 ± 0.87	⋯	⋯	31.97 ± 7.19	25.94 ± 6.13	52.75 ± 33.75	77.00	1.69 ± 0.68
	52306	0.18 ± 0.02	32.34 ± 24.38	43.52 ± 47.34	7.12 ± 1.91	11.53 ± 3.08	5.21 ± 2.00	29.27 ± 13.82	102.00	1.92 ± 0.77
	54431	0.26 ± 0.03	7.24 ± 1.98	6.83 ± 3.03	7.92 ± 1.74	27.06 ± 9.57	13.78 ± 7.40	55.06 ± 18.43	73.00	1.97 ± 0.79
XTE J1723–376	51189	0.16 ± 0.02	22.74 ± 11.25	32.81 ± 23.96	1.34 ± 0.30	45.19 ± 4.60	21.42 ± 5.54	12.18 ± 5.63	147.00	1.27 ± 0.76
MXB 1730–33	50188	0.26 ± 0.01	0.78 ± 0.15	0.55 ± 0.11	4.38 ± 0.89	8.40 ± 1.16	9.28 ± 2.01	6.74 ± 0.82	22.00	0.44 ± 0.30
	50387	0.30 ± 0.01	⋯	⋯	2.37 ± 0.33	11.81 ± 3.95	12.41 ± 5.49	9.97 ± 1.09	⋯	⋯
	50625	0.38 ± 0.01	2.33 ± 0.42	⋯	⋯	7.65 ± 0.79	9.46 ± 1.49	4.18 ± 0.28	24.00	0.47 ± 0.32
	50843	0.27 ± 0.01	5.71 ± 1.23	⋯	⋯	9.20 ± 0.75	10.75 ± 1.27	5.19 ± 0.70	32.00	0.50 ± 0.34
	51047	0.16 ± 0.02	1.49 ± 1.58	1.64 ± 2.30	1.02 ± 0.22	7.39 ± 1.13	10.32 ± 2.20	1.68 ± 0.84	20.00	0.35 ± 0.24
	51250	0.28 ± 0.01	3.42 ± 0.69	3.90 ± 1.00	1.37 ± 0.28	6.96 ± 0.55	8.89 ± 1.06	2.88 ± 0.63	30.00	0.35 ± 0.24
	51453	0.24 ± 0.02	1.99 ± 0.26	⋯	⋯	9.72 ± 3.02	9.18 ± 3.51	12.19 ± 3.83	30.00	0.42 ± 0.29
	51739	0.15 ± 0.01	3.03 ± 1.02	1.04 ± 0.74	5.80 ± 3.35	6.03 ± 1.16	5.58 ± 1.56	7.08 ± 1.11	19.00	0.21 ± 0.14
	51953	0.17 ± 0.01	0.85 ± 0.49	⋯	⋯	7.66 ± 1.09	8.94 ± 1.69	3.44 ± 0.44	22.00	0.31 ± 0.21
	52051	0.16 ± 0.01	1.32 ± 0.36	0.82 ± 0.28	3.51 ± 0.81	9.04 ± 1.66	8.94 ± 2.45	9.27 ± 1.70	24.00	0.29 ± 0.20
	52387	0.14 ± 0.01	2.93 ± 0.48	0.53 ± 0.11	11.97 ± 1.65	4.49 ± 3.21	4.36 ± 4.17	4.88 ± 0.79	21.00	0.26 ± 0.18
	53103	0.17 ± 0.01	2.35 ± 0.53	2.45 ± 0.66	1.89 ± 0.64	7.80 ± 3.46	7.73 ± 4.55	8.02 ± 1.46	26.00	0.33 ± 0.23
	53602	0.13 ± 0.02	4.84 ± 0.88	0.74 ± 0.19	2.76 ± 0.52	6.52 ± 0.89	6.15 ± 1.35	7.20 ± 1.19	22.00	0.29 ± 0.20
	53749	0.22 ± 0.02	10.70 ± 4.51	⋯	⋯	3.54 ± 0.77	3.80 ± 1.01	2.39 ± 0.90	31.00	0.46 ± 0.31
	53907	0.21 ± 0.02	0.80 ± 0.20	0.48 ± 0.14	2.46 ± 0.70	9.66 ± 2.35	11.94 ± 3.96	3.43 ± 0.72	23.00	0.31 ± 0.22
	54265	0.16 ± 0.01	1.97 ± 0.69	⋯	⋯	9.56 ± 3.06	7.96 ± 4.58	10.45 ± 2.76	28.00	0.35 ± 0.24
	54430	0.21 ± 0.01	1.35 ± 0.39	1.25 ± 0.45	1.78 ± 0.25	7.21 ± 4.29	9.74 ± 8.38	1.47 ± 0.29	20.00	0.24 ± 0.17
	54734	0.13 ± 0.02	4.39 ± 0.96	3.41 ± 1.06	6.82 ± 1.29	4.36 ± 1.71	3.88 ± 2.25	5.58 ± 2.76	21.00	0.22 ± 0.15
	54842	0.27 ± 0.01	1.60 ± 1.73	⋯	⋯	4.04 ± 1.58	4.14 ± 2.09	3.68 ± 0.95	14.00	0.30 ± 0.20
	54976	0.14 ± 0.01	0.89 ± 0.34	0.40 ± 0.20	2.30 ± 0.41	5.85 ± 4.09	7.73 ± 7.70	1.40 ± 0.27	17.00	0.16 ± 0.11
	55074	0.18 ± 0.02	1.15 ± 0.20	0.80 ± 0.20	2.06 ± 0.49	6.63 ± 1.70	4.11 ± 1.53	12.66 ± 2.88	18.00	0.23 ± 0.16
	55220	0.22 ± 0.02	18.67 ± 9.28	12.98 ± 9.61	33.29 ± 23.84	13.01 ± 3.89	6.77 ± 2.94	4.16 ± 1.25	76.00	0.77 ± 0.53
	55395	0.19 ± 0.01	0.89 ± 0.35	0.79 ± 0.58	1.02 ± 0.30	8.28 ± 4.43	3.69 ± 2.56	25.09 ± 13.76	22.00	0.26 ± 0.18
XTE J1739–285	51475	0.16 ± 0.01	0.98 ± 0.33	⋯	⋯	40.22 ± 10.22	42.80 ± 15.67	15.34 ± 6.12	86.00	0.55 ± 0.28
GRO J1744–28	50094	1.29 ± 0.02	⋯	⋯	⋯	42.18 ± 1.57	28.11 ± 1.52	72.99 ± 2.57	NaN	⋯
	50474	0.81 ± 0.02	19.78 ± 1.69	18.09 ± 2.20	24.50 ± 2.84	28.19 ± 1.42	20.40 ± 1.39	50.66 ± 4.07	102.00	6.50 ± 0.77
IGR J17473–2721	53347	0.15 ± 0.02	4.65 ± 2.97	5.32 ± 4.49	2.30 ± 1.59	16.70 ± 13.06	⋯	⋯	56.00	0.13 ± 0.05
	54642	0.42 ± 0.02	29.42 ± 3.41	41.67 ± 7.17	3.09 ± 0.47	14.19 ± 3.33	8.78 ± 2.61	33.40 ± 8.01	95.00	1.38 ± 0.41
EXO 1745–248	51776	0.53 ± 0.02	3.89 ± 0.28	3.85 ± 0.31	4.31 ± 1.00	18.13 ± 1.84	19.98 ± 3.14	13.99 ± 2.05	53.00	1.91 ± 1.32
	52459	0.22 ± 0.02	4.97 ± 1.02	4.77 ± 1.30	5.58 ± 1.50	5.74 ± 1.44	2.22 ± 0.76	15.78 ± 3.51	29.00	0.55 ± 0.38
	55486	0.61 ± 0.03	33.24 ± 4.98	12.94 ± 4.77	68.23 ± 33.77	40.55 ± 11.71	47.50 ± 21.89	28.54 ± 4.18	166.00	7.34 ± 5.06
1 A 1744–361	52955	0.15 ± 0.01	2.53 ± 2.45	0.63 ± 0.68	19.30 ± 4.95	29.58 ± 26.49	9.12 ± 10.94	91.63 ± 45.22	82.00	0.41 ± 0.41
4U 1745–203	52179	0.18 ± 0.01	6.67 ± 0.82	5.12 ± 1.08	9.58 ± 1.79	8.84 ± 2.19	3.11 ± 1.00	28.54 ± 9.56	46.00	0.72 ± 0.07
	53526	0.32 ± 0.02	8.41 ± 1.94	5.31 ± 1.71	16.11 ± 2.44	17.29 ± 3.24	20.28 ± 4.49	8.36 ± 3.55	54.00	1.25 ± 0.12
SAX J1750.8–2900	51971	0.14 ± 0.01	3.56 ± 1.16	3.98 ± 2.12	2.81 ± 0.86	14.79 ± 4.94	19.17 ± 9.37	4.61 ± 1.36	39.00	0.16 ± 0.02
	54588	0.17 ± 0.02	7.63 ± 1.70	9.90 ± 3.30	2.80 ± 0.68	20.80 ± 8.50	18.89 ± 10.93	25.38 ± 5.83	69.00	0.51 ± 0.03
2 S 1803–245	50933	0.75 ± 0.02	4.87 ± 0.14	3.21 ± 0.16	8.58 ± 0.74	16.24 ± 0.78	12.43 ± 0.82	26.00 ± 1.61	51.00	0.98 ± 0.79
Aql X-1	50481	0.38 ± 0.01	3.56 ± 0.44	2.39 ± 0.46	5.60 ± 0.33	11.97 ± 0.68	6.24 ± 0.50	26.67 ± 1.68	35.00	0.45 ± 0.18
	50683	0.24 ± 0.00	7.93 ± 2.15	5.34 ± 1.95	15.41 ± 1.57	13.12 ± 1.68	5.73 ± 1.05	14.04 ± 1.69	51.00	0.37 ± 0.15
	50901	0.47 ± 0.00	6.95 ± 0.34	2.90 ± 0.20	19.13 ± 1.75	11.27 ± 0.67	5.19 ± 0.40	32.98 ± 1.30	57.00	1.05 ± 0.42
	51319	0.39 ± 0.01	2.50 ± 0.17	⋯	⋯	7.67 ± 0.39	3.75 ± 0.30	16.10 ± 1.21	22.00	0.25 ± 0.10
	51833	0.68 ± 0.02	2.92 ± 0.05	3.10 ± 0.07	2.27 ± 0.09	11.97 ± 0.32	7.17 ± 0.26	25.24 ± 1.22	47.00	1.13 ± 0.45
	52335	0.21 ± 0.00	6.82 ± 1.16	3.37 ± 0.73	21.48 ± 6.31	9.04 ± 1.40	3.73 ± 0.81	23.22 ± 2.70	38.00	0.26 ± 0.10
	52707	0.67 ± 0.01	2.84 ± 0.14	2.79 ± 0.31	2.97 ± 0.52	9.90 ± 0.32	5.04 ± 0.21	26.07 ± 1.53	40.00	1.00 ± 0.40
	53167	0.28 ± 0.02	14.47 ± 1.28	22.20 ± 3.24	3.23 ± 0.27	2.92 ± 0.84	2.84 ± 1.02	3.27 ± 0.60	42.00	0.18 ± 0.07
	54365	0.26 ± 0.00	4.83 ± 0.41	3.53 ± 0.46	7.64 ± 0.75	3.94 ± 0.25	2.79 ± 0.26	6.66 ± 0.34	26.00	0.22 ± 0.09
	55160	0.30 ± 0.01	6.50 ± 1.26	5.18 ± 1.40	9.85 ± 0.98	2.92 ± 1.13	2.71 ± 1.36	3.63 ± 0.61	23.00	0.19 ± 0.08
	55454	0.40 ± 0.02	3.64 ± 0.52	⋯	⋯	11.90 ± 1.97	6.49 ± 1.36	32.86 ± 3.69	42.00	0.35 ± 0.14

Note.

^aThe date when the outburst reaches its peak flux in the ASM light curve.

Download table as:  ASCIITypeset images: 1 2 3

The intrinsic scatters of the six different E–τ correlations are quite large, from about 0.3 to 0.5 dex. An average intrinsic scatter of 0.4 dex implies that the 1σ uncertainties of the distances inferred according to those correlations can be up to a factor of 1.6. We further applied this method to sources with known distances and compared the inferred distances with the distances we collected from the literature (see Figure 14). The differences between these two quantities for most of the sources are less than a factor of 1.6, which supports the idea that the large scatter of Figure 14 is due to the intrinsic scatters in the E–τ correlations. A few sources such as 4U 1543–47 show a large discrepancy, which is caused by the large deviation from the E–τ correlations of this source.

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We have performed a systematic study of the LMXBT outburst properties in the 2–12 keV band using the data from RXTE/ASM. We defined parameters that are used to describe outburst properties and presented the statistical results of these parameters, including peak X-ray luminosity, rate of change of luminosity during the rise or decay phase on a daily timescale, e-folding rise or decay timescale, outburst duration, and total radiated energy. Readers need to keep in mind that these parameters depend on the energy band: all of our measurements are in the 2–12 keV band. The bolometric luminosity and total radiated energy could be three times larger than what we measured in 2–12 keV (e.g., in't Zand et al. 2007). Please note that our sample only includes the bright outbursts with a peak count rate larger than 0.1 Crab.

• 1.  
  The average peak X-ray luminosity of outbursts in our sample is about 4.7 × 10³⁷ ergs s⁻¹ (see L_peak in Table 2).
• 2.  
  The average rate of change of luminosity is about 0.01 L_Edd/day during the rise phase (see ${{\dot{L}}_{{\rm rise},10-90\%}}$ in Table 4) and 0.004 L_Edd/day during the decay phase (see ${{\dot{L}}_{{\rm decay},10-90\%}}$ in Table 4).
• 3.  
  The average e-folding rise timescale is about five days (see ${{\tau }_{{\rm rise},10-90\%}}$ in Table 3), and the average e-folding decay timescale is about 15 days (see ${{\tau }_{{\rm decay},10-90\%}}$ in Table 3).
• 4.  
  The average outburst duration is about 54 days (see Table 5).
• 5.  
  The average total radiated energy of all LMXBTs is about 0.91×10⁴⁴ ergs for the bright outbursts (see $\lt {{E}_{{\rm total}}}\gt$ in Table 6).

We have also found correlations between some outburst properties or between outburst properties and binary system parameters.

• 1.  
  A positive correlation between the rate of change of luminosity and the peak X-ray luminosity in units of L_Edd in both the rise and decay phases (see Figure 7).
• 2.  
  A positive correlation between the peak X-ray luminosity and the total radiated energy (see Figure 12).
• 3.  
  A positive correlation between the e-folding rise or decay timescale and the total radiated energy (see Figure 13).
• 4.  
  A weak positive correlation between the orbital period and the total radiated energy (see Figure 15).
• 5.  
  A weak positive correlation between the orbital period and the e-folding rise timescale (see Figure 8).

Once a new LMXBT is discovered, the most important thing is to determine the nature of the compact object. If there is no evidence of an NS from a Type I burst or pulsation detection or if the mass of the compact object cannot be measured, then people normally use the X-ray spectral and timing properties to infer the nature of the compact object.

Our statistical study of the outbursts of LMXBTs shows that the average values of most parameters have significant differences between BH LMXBTs and NS LMXBTs, except ${{\tau }_{{\rm rise},10-50\%}}$.

• 1.  
  The average peak count rate and luminosity of BH LMXBTs is about two times larger than that of NS LMXBTs.
• 2.  
  The average e-folding rise timescales (except ${{\tau }_{{\rm rise},10-50\%}}$) of BH LMXBTs are about two times larger than those of NS LMXBTs.
• 3.  
  The average e-folding decay timescales of BH LMXBTs are all more than two times larger than those of NS LMXBTs.
• 4.  
  The average outburst duration of BH LMXBTs is about two times larger than that of NS LMXBTs.
• 5.  
  The average total radiated energy of BH LMXBTs is about five times larger than that of NS LMXBTs.

Therefore, these parameters are very helpful in inferring the nature of the compact object in a statistical sense. When a new transient is discovered, we can measure the outburst properties (including peak X-ray flux, rise or decay timescale, outburst duration, and total radiated energy), as described in Section 2, and compare them with the distributions of all of the available parameters of BH LMXBTs and NS LMXBTs to jointly infer the nature of the central compact object.

Yu & Yan (2009) have suggested that the nonstationary accretion that is characterized by the rate of increase of the X-ray luminosity plays an important role in determining the luminosity of the hard-to-soft state transition and the peak X-ray luminosity of an outburst. In order to further investigate the role of nonstationary accretion, we studied the rate of change of luminosity in the rise and decay phases. The rate of change of luminosity in the rise phase is about two times larger than that in the decay phase (both for BH and NS LMXBTs). The correlation between the rate of change of luminosity and peak X-ray luminosity scaled in L_Edd exists both in the rise and decay phases (see Figure 7), which confirms the previous results and further supports the schematic picture of Figure 28 in Yu & Yan (2009).

The nonstationary accretion corresponding to transient outbursts is probably set up by an initial condition: disk mass (Yu et al. 2004, 2007; Wu et al. 2010a, 2010b). In our study, the total radiated energy E should correspond to the disk mass accreted during an outburst if a constant radiative efficiency can be approximately assumed. There are three outbursts followed by immediate secondary outbursts in our sample, including the 2002 and 2010 outbursts of 4U 1630–47 and the 1998 outburst of XTE J1550–564. Based on our definition, the E only includes the radiated energy during the primary outburst in these cases. But only three data points cannot erase the positive correlations related with E. Thus we concluded that the disk mass accreted during an outburst is correlated with peak X-ray luminosity, rise timescale, and decay timescale, which supports the idea that disk mass plays a major role in determining outburst properties inferred from observations (Yu et al. 2004, 2007; Wu et al. 2010a, 2010b). The peak X-ray luminosity is a function of disk mass, which is actually expected from the DIM (King & Ritter 1998; Shahbaz et al. 1998; Dubus et al. 2001; Lasota 2001), because the peak mass accretion rate ${{\dot{M}}_{{\rm peak}}}$ depends on the maximum radius over which the heating front can propagate.

We found a weak correlation between the total radiated energy E and the orbital period P_orb (see Figure 15). For a fixed mass ratio between the compact star and the companion star, the orbital period constrains the size of the Roche lobe and the maximum disk size as well. Only if a specific ratio of the Roche lobe was filled by the accretion disk, and a specific ratio of mass stored in the disk was accreted onto the compact star during an outburst, can we see such a positive correlation between E and P_orb. This correlation is also expected from the DIM (Lasota 2001) because the mass accreted during an inside-out outburst is mainly determined by the disk radius where the heating front can reach; the critical surface density change is very small between different systems. If the irradiation is strong enough to ionize the whole disk, the heat front can propagate to the outer radius, and most of the disk mass will be accreted by the compact object during the outburst (King & Ritter 1998; Shahbaz et al. 1998; Dubus et al. 2001). The average peak X-ray luminosity of our sample is about 4.7 × 10³⁷ ergs s⁻¹ (see Table 2), which is larger than the critical luminosity required to ionize the entire disk for a typical disk size given in Shahbaz et al. (1998), so we can see that the correlation between the total radiated energy E and the orbital period P_orb. GRO J1744–28 has the longest orbital period in our sample, which means the largest disk size. A possible reason for GRO J1744–28 showing a large deviation from the correlation (see Figure 15) is that the peak X-ray luminosity is not high enough to ionize the whole disk.

A statistical study of the hard-to-soft state transitions in Galactic bright X-ray binaries shows no indication of a luminosity saturation or cutoff in the correlation between the hard-to-soft transition luminosity and the peak X-ray luminosity of the outburst or flare up to more than 30% of L_Edd (Yu & Yan 2009). This indicates that more luminous hard states and outbursts than those observed in our Galaxy are allowed by physics. If the luminosity function of Galactic LMXBT outbursts can be estimated, we would see that one outburst of Galactic LMXBTs can reach an Eddington outburst in a few tens of years (Yu & Yan 2009). A super-Eddington outburst candidate is the 1999 outburst of V4641 Sgr (see Orosz et al. 2001; Revnivtsev et al. 2002). However, this source was classified as HMXB in the subsequent studies (Orosz et al. 2001; Liu et al. 2006). On the other hand, the 1999 outburst of V4641 Sgr shows properties that are largely different from the classic outburst of an LMXBT. The rise and decay phases of this outburst were extremely short: it reached 12 Crab within 8 hr from quiescence and returned to quiescence within 2 hr. We speculated that this outburst has a mechanism different from the classic outburst of an LMXBT.

Two transient ultraluminous X-ray sources (ULXs) in M31 were identified as LMXBs (Henze et al. 2012; Kaur et al. 2012; Middleton et al. 2012), which confirmed our previous prediction of the occurrence of transient ULXs in Yu & Yan (2009), because M31 is significantly larger than the Milky Way. Monitoring observations performed with Chandra, XMM-Newton, and Swift covered the entire outburst phase for XMMU J004243.61+412519 and the decay phase of CXOM31 J004253.1+411422. We collected the X-ray light curves of both transient ULXs from the literature (Kaur et al. 2012; Middleton et al. 2013) in order to compare the outburst properties with the Galactic LMXBTs. We followed the same methods in Section 2 to measure the peak X-ray luminosity, rate of change of luminosity, e-folding rise and decay timescales, outburst duration, and total radiated energy (see Table 9). The light curves of these two ULXs were obtained in 0.3–10 keV, which is different from the RXTE/ASM light curves of Galactic LMXBTs. Comparing the parameters of the outbursts of the two transient ULXs in M31 (see Table 9) with NS LMXBTs and BH LMXBTs, we tend to suggest that the nature of the central compact objects is like BHs. The masses of central compact objects in these two ULXs are both assumed to be 10 ${{M}_{\odot }}$.

Table 9.  Parameters of Outbursts of Transient ULXs in M31

Source	tpeak	Lpeak	${{\tau }_{{\rm rise},10-90\%}}$	${{\tau }_{{\rm rise},10-50\%}}$	${{\tau }_{{\rm rise},50-90\%}}$	${{\tau }_{{\rm decay},10-90\%}}$	${{\tau }_{{\rm decay},10-50\%}}$	${{\tau }_{{\rm decay},50-90\%}}$	Outburst Duration	E
	MJD	(1039 ergs s−1)	(days)	(days)	(days)	(days)	(days)	(days)	(days)	(1044 ergs)
XMMU J004243.61+412519	55957	1.26 ± 0.02	8.69 ± 0.48	⋯	⋯	75.57 ± 3.05	53.31 ± 3.91	105.80 ± 6.91	195.00	112.50 ± 0.43
CXOM31 J004253.1+411422	55182	>3.77 ± 0.04	⋯	⋯	⋯	33.05 ± 6.57	38.63 ± 9.81	12.84 ± 1.33	⋯	⋯

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The outburst parameters of the two M31 transient ULXs are also plotted in the correlations we found in the Galactic LMXBTs. As can be seen in Figures 7 and 12, the two transient ULXs in M31 follow the correlation between the rate of change of luminosity and the peak X-ray luminosity and the correlation between the peak X-ray luminosity and the total radiated energy. But as shown in Figure 13, the two transient ULXs seem to locate below the extrapolation of the correlations between the e-folding rise or decay timescale and the total radiated energy found in Galactic LMXBTs. Because of the large intrinsic scatters of the correlations, we need more data to verify whether the correlations have a saturation or break at the higher luminosity end. In the future, much more sensitive all-sky monitors will provide us with a large sample of light curves of transient ULXs in nearby galaxies, which will offer a significant advantage in studying the properties of super-Eddington outbursts because these sources have more populations and smaller absorption compared to Galactic sources and shorter timescales compared to super-Eddington active galactic nuclei. These two transient ULXs in M31 roughly follow the correlations found in the Galactic LMXBTs, and their outburst properties are also similar to the Galactic LMXBTs, which demonstrates that the sub-Eddington and super-Eddington outbursts are driven by the same mechanism.

In the original form of the DIM, the rise or decay time corresponds to the time duration over which the heating or cooling front propagates. The velocity of the heating front is of the order of $\alpha {{c}_{s}}$ (Meyer 1984), where c_s is the sound speed, so the rise time t_rise is $\sim R/\alpha {{c}_{s}}$ (Frank et al. 2002). If the rise time corresponds to the propagation time through the entire accretion disk, it will be ∼100 hr if assuming a disk size of $\sim {{10}^{11}}$ cm, which is comparable with the rise timescale we observed (${{\tau }_{{\rm rise},10-90\%}}\sim 5$ days; see Table 3). But the rise timescale we measured is in the X-ray band, so it may correspond to the heating front propagation time through the X-ray emission region, which is much smaller.

Dubus et al. (2001) considered a modified DIM, including the effects of irradiation and evaporation, and assumed that the inner region of a disk is replaced by an advection-dominated accretion flow at the beginning of an outburst. The inner radius of the truncated disk decreases until its minimum value is reached when ${{\dot{M}}_{{\rm in}}}={{\dot{M}}_{{\rm ev}}}({{R}_{{\rm min}}})$, where ${{\dot{M}}_{{\rm in}}}$ is the mass accretion rate and ${{\dot{M}}_{{\rm ev}}}$ is the evaporation rate. The rise time t_rise is the viscous time that the disk needs to reach R_min from the initial truncated radius R_tr. For the typical parameters of LMXBTs, the t_rise is of the order of several days (Dubus et al. 2001), which is comparable with the rise timescale we observed. But we did not measure the rise timescale by taking the quiescence as the beginning. In practice, the rise time from quiescence is difficult to estimate from the RXTE/ASM data because the sensitivity is not enough to determine the quiescent flux level.

Our statistical results show that the outburst decay of most outbursts is consistent with an exponential form down to ${{F}_{10\%}}$. It has been found that the irradiation of the accretion disk by the central X-rays can naturally explain the exponential decay (King & Ritter 1998). The decay timescale (∼20–40 days) given by King & Ritter (1998) is comparable to the average decay timescale we measured within error bars (6–37 days; see ${{\tau }_{{\rm decay},10-90\%}}$ in Table 3). King & Ritter (1998) also showed that the e-folding decay timescale is of the order of the viscous timescale of the outer radius and increases with the orbital period, but saturates at ${{\tau }_{{\rm decay}}}\;\sim$ 40 days for the binary systems with orbital periods longer than about a day. We have checked for this but did not find a positive correlation between orbital period and e-folding decay timescale for the data where ${{\tau }_{{\rm decay}}}$ is below 40 days. The broadband spectral evolution (including the optical, ultraviolet, and X-ray bands) shows that the outer disk radius decreases during the outburst decay (see, e.g., Hynes et al. 2002). So the ${{\tau }_{{\rm decay}}}$ as a proxy for the viscous timescale of the outer disk radius cannot trace the size of the Roche lobe. This might be the reason that we did not find a statistical positive correlation between ${{\tau }_{{\rm decay}}}$ and the orbital period.

However, we found a positive correlation between the e-folding rise timescale and the orbital period (see Figure 8). It is reasonable that the viscous time at the outer disk is positively correlated with the orbital period (Menou et al. 1999; Gilfanov & Arefiev 2005), so the rise timescale may relate to the viscous time of the outer disk. If the e-folding rise timescale we measured is a proxy for the viscous timescale at the initial truncated radius, the truncated radius must be related to the outer disk radius, which leads to the correlation between the e-folding rise timescale and the orbital period. How the truncated radius is related to the outer disk radius when the LMXBTs are in their quiescence deserves further studies.