The relation between outburst rate and orbital period in low-mass X-ray binary transients
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3, 2019Typeset using L A TEX twocolumn style in AASTeX61
THE RELATION BETWEEN OUTBURST RATE AND ORBITAL PERIOD IN LOW-MASS X-RAY BINARYTRANSIENTS J IE L IN ,
1, 2 Z HEN Y AN , Z HANWEN H AN , AND W ENFEI Y U Shanghai Astronomical Observatory and Key Laboratory for Research Galaxies and Cosmology, Chinese Academy of Sciences, 80 Nandan Road, Shanghai200030, China: [email protected] University of Chinese Academy of Sciences, 19 A Yuquan Road, Beijing 100049, China: [email protected] Yunnan Observatory, Academia Sinica, Kunming 650011, China
ABSTRACTWe have investigated the outburst properties of low-mass X-ray binary transients (LMXBTs) based on a comprehensive studyof the outbursts observed in the past few decades. The outburst rates were estimated based on the X-ray monitoring data from
Swift /BAT, RXTE/ASM and MAXI, and previous reports in the literature. We found that almost all LMXBTs with the orbitalperiod below ∼
12 hr showed only one outburst in these observations. There are systematic difference in the outburst ratebetween long-period ( P orb (cid:38)
12 hr) and short-period ( P orb (cid:46)
12 hr) systems. We infer that mass transfer rate is responsible forthe systematic difference, since the disk instability model (DIM) suggested that the mass transfer rate is a key factor affecting thequiescence time. The difference in outburst rate between long-period and short-period LMXBTs is probably due to the differentmass transfer mechanism at different evolutionary stages of the donors. Based on the evolutionary tracks of single stars, wederived the critical orbital period for X-ray binaries that harbor a subgiant donor in various metallicity. The critical orbital period( P orb , crit =12.4 hr) is consistent with the above orbital period boundary obtained from the statistics of outburst rates. Furthermore,we found a negative correlation between the outburst rate and the orbital period in the samples for which the luminosity class ofthe donor star is III/IV. The best-fitting power-law index for the black hole subsamples is roughly consistent with the theoreticalprediction for those systems with a donor star evolved off the main sequence. Keywords: accretion, accretion disks — black hole physics — X-rays: binaries a r X i v : . [ a s t r o - ph . H E ] J a n J IE L IN ET AL . INTRODUCTIONLow-mass X-ray binaries (LMXBs) are binary systems inwhich a donor star transfers mass onto a black hole (BH)or a neutron star (NS) through Roche lobe overflows. Thedonor star is usually a main-sequence star, a subgiant, or evena white dwarf (WD). Among LMXBs, low-mass X-ray bi-nary transients (LMXBTs) spend most of their time in X-rayquiescence (typically L ≤ erg s − ) and occasionally turninto outbursts, during which their X-ray luminosities can in-crease by many orders of magnitude (see Chen et al. 1997,hereafter CSL97).Similar to those of dwarf novae, the outburst behavior ofLMXBTs is usually believed to be the result of disk insta-bility (Lasota 2001; Osaki 1995). In the disk instabilitymodel (DIM), the trigger mechanism is the thermal instabil-ity within the accretion disk. The disk thermal instabilityoccurs when the surface density exceeds the maximum valueon the cold branch or the surface density is lower than theminimum value on the hot branch (Lasota 2001, 2016).When and where the instability is triggered are usually re-lated to the following two processes: mass accumulation atthe outer edge of the disk, and the viscous diffusion of thematter accumulated in the outer disk. Two correspondingtimescales matter. The mass-accumulation time is the timeduration for accumulating mass up to the critical surface den-sity of the cold branch at the outer disk. The viscous drifttime is the time for the matter in the disk to drift appreciablyin the radial direction and trigger the disk instability in the in-ner disk. Hence, theoretically, the duration for a transient tostay in quiescence is determined by the shorter one of the twotimescales (Smak 1984; Osaki 1995; Lasota 2001). Accord-ing to Osaki (1995), the mass-accumulation time depends onthe mass transfer rate and the outer disk radius, while the vis-cous drift time is independent of the mass transfer rate fromthe companion star and is weakly dependent on the disk outerradius. Therefore, the mass transfer rate and outer disk radiusare thought to be the key factors that determine whether thematter will tend to accumulate on the outer edge or propagateinward (Lasota 2001; Dubus et al. 2001). The mass transferrate is tightly related with the orbital period, mass, and theevolutionary stage of the donor (Webbink et al. 1983; King1988). On the other hand, the outer disk radius is also pos-itively correlated with the orbital period in LMXBs (Lasota2001; Lasota et al. 2008). Therefore, the quiescence timemight be related to the orbital periods, as well as the natureof donors.From the observational point of view, the quiescence timeof LMXBTs is difficult to measure owing to the limited sen-sitivity of X-ray monitors, and we cannot determine when atransient enters or leaves the quiescence state exactly. Mostrecent all-sky monitors offer good time coverage, but theirsensitivities are still far above the flux level of the quies- cence. On the other hand, the outburst rate, i.e., the numberof outbursts per year, which is the reciprocal of the averagerecurrence time, can be counted directly when a flux thresh-old is taken. Notice that the quiescence time is the time du-ration from the end of an outburst to the start of the nextone, which is shorter than the recurrence time, defined as thetime interval from the start of an outburst to the start of thenext outburst (Lasota 2001). For most LMXBTs now known,the outburst duty cycle is quite small (Yan & Yu 2015,here-after YY15;Tetarenko et al. 2016, hereafter TSHG16), so thequiescence time and the recurrence time in the current sam-ple are usually comparable. In the following, we present ourmeasurements of the outburst rates for all known LMXBTsin which the orbital period has been determined, aiming atobtaining the potential relation between the recurrence timeand the orbital period or the properties of the donor.1.1. Case A and Case B mass transfer
The most common mechanism driving mass transfer in abinary system is the expansion of the donor (see Pylyser &Savonije 1988). The radius of an intermediate-mass star (typ-ically 3 M (cid:12) ) grows in three different epochs: the phase ofcore-hydrogen burning (main-sequence stage), the phase be-fore ignition of core helium burning (subgiant branch and redgiant branch) and the phase after termination of core heliumburning (asymptotic giant branch). The mass transfer in thesethree phases is often referred to as case A, case B, and caseC mass transfer, respectively (see also Kolb 2010). Noticethat, for low-mass stars, their main-sequence lifetime couldbe longer than the age of the universe (or Hubble time), andthey never become (sub)giants if their initial masses are smallenough ( M , i (cid:46) . (cid:12) ). The radius of a donor star in caseC mass transfer is much larger than those in case B and caseA, which implies a much larger orbital separation and orbitalperiod. Such an orbital period exceeds the maximum orbitalperiod of known LMXBTs; therefore, case C mass transfer isbeyond the scope of our study.In case B mass transfer, the radius grows on thermal time t th of the donor star, which is approximately equal to theKelvin-Helmholtz time. In case A mass transfer, the radiusof the donor grows on its nuclear time t nuc , which is typicallya factor of 1000 times longer than the thermal time t th . There-fore, the case B mass transfer rate is much larger than the caseA mass transfer rate. Typically, the case B mass transfer ratecan reach 10 − M (cid:12) / yr, while the case A mass transfer rateis about several times 10 − M (cid:12) / yr (see de Kool et al. 1986;Kolb 2010). Since the size of (sub)giants is usually severaltimes or tens of times that of main-sequence stars, only thoseLMXBTs with a sufficiently large orbital period can harbora donor of a (sub)giant. In other words, case B mass transfercan be performed only if the LMXBTs have an orbital periodgreater than a critical orbital period . In this paper, we will HE RELATION BETWEEN OUTBURST RATE AND ORBITAL PERIOD IN LOW - MASS X- RAY BINARY TRANSIENTS DATAWe selected the LMXBT samples with confirmed measure-ments of the orbital periods based on the catalog of Knevittet al. (2014) and Tetarenko et al. (2016). We then searchedfor the X-ray outbursts of these samples mainly from thearchived data of the all-sky monitors, including RXTE/ASM,
Swift /BAT and MAXI, since they are complete for detectingX-ray outbursts of LMXBTs in the past two decades (YY15;TSHG16). We also searched for outbursting events in the lit-erature before the launch of RXTE. Due to limited sensitivityand some poor time coverage of the all-sky X-ray monitors,it is impossible to identify all the outbursts of LMXBTs inthe period we investigated. However, we can find all theoutbursts with the X-ray peak flux above a certain thresh-old from the all-sky monitoring data. Considering the typ-ical 1-day X-ray sensitivity (10–20 mcrab) of RXTE/ASM,
Swift /BAT and MAXI, we set the threshold to 50 mcrab ofour outburst sample, corresponding to about 10 erg s − at2–10 keV if a source distance of 8 kpc is taken.To ensure the completeness of sample, our samples includeall LMXBTs with known orbital periods except the followingsources. Swift
J1357.2–0933, XTE J0929–314, XTE J1710–281, IGR J17498–2921, XTE J1814–338, 4U 2129+47 (For-man & Jokipii 1978; Bozzo et al. 2007) and SWIFT J1749.4–2807 are excluded from our samples owing to their weak out-bursts (below 50 mcrab). AX J1745.6–2901 is also excludedbecause it is in the Galactic center and may be confused withanother X-ray source (e.g. CXOGC J174540.0-290027 andCXOGC J174540.0-290031). All our LMXBT samples arelisted in Table 1.We have examined the orbital periods of our samples. Asshown in Table 1, the measurements of the orbital periods arelimited to four common measurement methods: the Doppler-shifted absorption (or emission) lines, the periodic modula-tions in optical and IR band, the periodic dips in the X-rayemission, and the Doppler-delayed pulses. In our samples,MAXI J0556–332 has two candidate orbital periods (16.43and 9.754 hr). The orbital period of 4U 0042+32 may benot be reliable, since it is measured by periodic X-ray mod-ulation rather than the X-ray dips in the common cases. Forexample, Kaluzienski et al. (1980) also found an 8.2-hour pe-riodic X-ray modulation in Cen X-4, but its orbital period isbelieved to be 15.1 hr which was measured by the Doppler-shifted emission lines (Cowley et al. 1988) and the periodicmodulation in V band (Chevalier 1989). In order to investigate the evolutionary stage of the donorsin our samples, we collected the luminosity classes of thedonors from the literature. Typically, the luminosity classnotation “V” indicates that the star is at the main-sequencestage, and the notation “III” or “IV” indicates that the staris at (sub)giant stage. The luminosity class of the donorcould be jointly determined by the absolute magnitude andthe spectra (or by the width of spectral lines). Notice that theoptical and infrared emission from X-ray binaries might con-tain the contribution from the accretion disk or jet even dur-ing the quiescence, which means that luminosity class maybe not a very reliable indicator to classify the donors. Welisted all the known spectral classifications of the donors inTable 1. 2.1.
The number of outbursts
Firstly, we describe here how we selected the outburst sam-ples (during or after the RXTE era) from the X-ray moni-toring data in detail. We retrieved the orbital or dwell-by-dwell light curves from the public archives of RXTE/ASM,
Swift /BAT and MAXI (see, e.g. Levine et al. 1996; Mat-suoka et al. 2009; Krimm et al. 2013). Among these, thelight curves from RXTE/ASM in 2–12 keV correspond tothe period from MJD 50088 to ∼ MJD 55924, the lightcurves from
Swift /BAT in 15–50 keV correspond to the pe-riod from MJD 53414 to MJD 57900 and the light curvesfrom MAXI in 2–4 keV and 4–10 keV correspond to theperiod from MJD 55058 to MJD 57900. We excluded theMAXI data in the 10–20 keV energy band due to its rela-tively poor signal-to-noise ratio as compared to
Swift /BAT.All light curves were rebinned into daily averaged data, andall the X-ray intensity is converted into intensity in crabunit for the purpose of comparison among data from dif-ferent instruments. We used 1 Crab = 75 . − forRXTE/ASM, 1 Crab = 0 .
221 count s − cm − for Swift /BAT,1 Crab = 1 .
67 count s − cm − and 1 .
15 count s − cm − forMAXI at 2–4 keV and 4–10 keV energy bands, respectively.We then searched for any outbursts with the X-ray peakflux above 50 mcrab from the long-term light curves of dif-ferent instruments. In order to avoid from selecting fake out-bursts (e.g. the type I X-ray burst in NS XRBs), we appliedthe criterion by requiring that there should be at least threesuccessive data points with signal-to-noise ratio better than3 σ , i.e., the X-ray outbursts from our LMXBT samples alllasted more than 2 days. J IE L IN ET AL . Table 1 . Low-mass X-ray Binary Transients InformationSource Name Orbital Period a Method a Spectral Type b Discovery Number of Outbursts c Rate f (hr) (MJD) Era I d Era II e Era III Total (yr − )BH LMXBTsMAXI J1659–152 2.41 [1] D - 55464 [2] - 1/1 (1) 0/0 1/1 ≤ Swift
J1753.5–0127 3.2 [3 , A+M - 53551 [5] - 1/1 (1) 0/0 1/1 ≤ [6 , A+M K5-7V [6 , [8] - 1/2 0/0 1/2 ≤ [9] A+M M1(+/-1)V [9 , [11] [11] ≤ [12 , A+M K5-7V [13] [14] - 1/1 (1) 0/0 1/1 ≤ [15 , A+M K6-M0V [16] [17] [17] ≤ [18] M K4V [18] [19] - 1/1 (1) 0/0 1/1 ≤ [20] M K3-4V [21 , [23] [23] ≤ [24] A K5(+/-2)V [24] [25] [26 , ≤ [27] D - 56026 [28] - 0/0 1/1 1/1 ≤ [29] A+M K0-5V [29 , [31] [31] ≤ [32] A K5-7V [32 , [34] [34 , ≤ [36] M M0-5V [37 , [38] [38 , , [41] A+M A2V [42 , [43] [43 , , [46 , A K3III [46 , [48] - 5/5 (2) 0/0 5/5 0.234GX 339–4 42.1 [49] A F8-G2III [30] ∼ [50 , [50 , [52 , A G0-5III [53] [54] [54] [55 , A+M F6III [55 , [58] [58] [59] A K0IV [60] [61] [61] [62]
M K1-5III? [63] [64] [64] ≤ [65] P - 52691 [66] - 1 1 2 0.093XTE J1751–305 0.71 [67]
P - 52367 [68] - 1 0 1 ≤ Swift
J1756.9–2508 0.91 [69]
P - 54258 [70] - 1 0 1 ≤ [71] P - 53535 [72] - 1 0 1 ≤ [73] D - 42837 [74] >=2 [74 , ∗ [76] ∗∗ [77] P - 50338 [78] - 5 2 7 0.327GS 1826–238 2.09 [79 , , M - 47412 [81] [81] ≤ [82] P - 53341 [83] - 0 1 1 ≤ [84] P - 55086 [85] - 1 0 1 ≤ [86 , D+M G5-9V? [88] [89] [89] ≤ [90 , A+M K5-7V [91 , [93] - 1 0 1 ≤ [94] D K2(or later)V? [95] [96] [96] [97] D - 48143 [98] >=1 [98]
12 0 12 ∗∗ [99] M ForG? [99] [100]
10 (10) [100 , , , ,
13 (15) 5 28 0.594Cen X-4 15.1 [105 , A+M K3-7V [106 , , [109] [109 , , HE RELATION BETWEEN OUTBURST RATE AND ORBITAL PERIOD IN LOW - MASS X- RAY BINARY TRANSIENTS Table 1 – ContinuedSource Name Orbital Period a Method a Spectral Type b Discovery Number of Outbursts c Rate f (hr) (MJD) Era I d Era II e Era III Total (yr − )MAXI J0556–332 16.43 or 9.75 [112] A - 55572 [113] - 1 0 1 ≤ [114 , M K7V [116] ∼ [117]
19 (13) [117 ,
19 (11) 5 43 0.8974U 0042+32 278.4 [119] D ∗∗∗ G? [120] [121] [121] ≤ [122] P (GorK)III [123] [124] [124] Notes. a The methods used for the orbital period measurements: D (periodic dips), M (periodic modulations), A (Doppler-shifted absorption oremission lines), and P (Doppler-delayed pulses). b The spectral types here depend on the spectroscopy and photometry from optical or infrared observations. c We list the number of outbursts in era I (before 1996), era II (from early to late 2011, the RXTE era), and era III (after 2011). For the outburststhat span multiple periods, we attribute them to the period in which they began to outburst. For the BH LMXBTs, we also list the numbersbased on the outburst list of TSHG16 and use “/” to divide our and their results. d The numbers in the parentheses represent the numbers of the outbursts (above 50 mcrab) recorded by CSL97 for the outbursts before theRXTE era. e The numbers in the parentheses show the numbers of the outbursts that are counted by YY15 for the outbursts in the RXTE era. f We set the upper limit on the outburst rate for those LMXBTs in which only one outburst has been detected above the threshold during theentire history of X-ray observations. ∗ An extra outburst of 1A 1744–361 is based on the report of the MAXI team when it is not in the monitoring catalog of MAXI (Bahramianet al. 2013). ∗∗ The total numbers of outbursts for 1A 1744–361 and GRS 1747–312 are counted since the RXTE era (early 1996). ∗∗∗
Theorbital period of 4U 0042+32 was measured by periodic X-ray modulation rather than the X-ray dips in the common cases.
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Notice that some outbursts that started before the launchof RXTE operated were also detected by RXTE/ASM (e.g.the outburst of GRO J1744-28 in late 1995). We took theseoutbursts as the outbursts before the RXTE era. For those quasi-persistent sources that showed an outburst lasting morethan 10 years, such as GRS 1915+105, GS 1826–238, HETEJ1900.1–2455 (Degenaar et al. 2017) and EXO 0748–676 J IE L IN ET AL .(Wolff et al. 2008), we took them as one outburst from thesesources.1A 1744–361 and GRS 1747–312 were discovered be-fore the RXTE era. The peak fluxes of their outbursts wereslightly higher than 50 mcrab in the soft X-ray band and wereextremely weak in the hard X-ray band. In order to estimatethe outburst rate of these two sources accurately, we onlycounted their outbursts during or after the RXTE era and cal-culated the outburst rates in the corresponding period. Ad-ditionally, 1A 1744–361 is not in the monitorin-37 in NGC6441 (Bahramian et al. 2013). We counted this outburst too.The numbers of outbursts in our samples are listed in Ta-ble 1. For comparison, we listed the number of outburstsbefore the RXTE era based on Table 7 of CSL97 and thenumber of outbursts during the RXTE era from Table 1 ofYY15 . Notice that we only took the outbursts above 50mcrab from CSL97. We also listed the number of outburstsfor BH LMXBTs based on Table 14 of TSHG16 behind ourresults.There are some differences between our results and thosein YY15 because we also used
Swift /BAT data to search forthe outbursts and applied a threshold of 50 mcrab instead of100 mcrab. Because YY15 had included several secondaryoutbursts of 4U 1608–522, their number of outbursts for 4U1608–522 is more than our result. Furthermore, we excludedtwo outbursts from GRO J0422+32 in 1993 that are shown inTSHG16, because these two outbursts could only be detectedin the optical band rather than the X-ray band (Callanan et al.1995; Shrader et al. 1997). The flux of the long-durationoutburst in EXO 0748–676 was just 38 mcrab from the resultof CSL97 but the flux of this outburst increased to above 50mcrab in the RXTE era, so we included this outburst in oursamples. 2.2.
The outburst rates
The outburst rate was calculated by R out = N out / T , where N out is the number of outbursts since the LMXBT was dis-covered with X-ray observations and T is the time intervalin years since its discovery until MJD 57900. For thoseLMXBTs discovered during the operations of RXTE/ASM, Swift /BAT or MAXI, T is set to the interval from the datewhen RXTE/ASM started operation until MJD 57900, due tosimilar sensitivities of these missions. In general, for thoseLMXBTs in which only one outburst has been detected abovethe threshold during the entire history of X-ray observations,we can only set the upper limit on the outburst rate.Notice that some outbursts from these LMXBTs may havebeen missed owing to poor sensitivity and coverage in earlyX-ray missions. RESULTSWe plotted the outburst rate as a function of the orbital pe-riod for BH and NS LMXBTs in Figure 1. The outburst rates of those sources with good measurements are in the rangefrom less than 0.03 yr − to near once a year. It is surpris-ing that the outburst rate vs. orbital period relation shows asystematic difference in the systems with the orbital periodsbelow and above ∼
12 hr. An obvious feature is that almost allsamples with the orbital periods below ∼
12 hr showed onlyone outburst during the X-ray observational history. Thosetransients with frequent outbursts, e.g. Aql X-1 and 4U 1608-52, tend to have an orbital period above 12 hr. There is a verylarge dispersion (or a jump) in the relation between the out-burst rate and the orbital period at around ∼
12 hr. Further-more, for the samples with a luminosity class of III/IV, theoutburst rates show a broad trend to decrease with the orbitalperiod.3.1.
Different outburst rates in the LMXBTs below andabove ∼
12 hr
The outburst rate could be associated with the orbital pe-riod. The difference in the outburst rate below and above ∼ τ correlation coefficient from survival analysis toinvestigate the association between the outburst rate and thelevel of the orbital period.As shown in Figure 1, we have calculated the τ and the p -value for all possible boundaries by using the function cenken in CRAN package NADA within the R statistical software(Helsel 2005; Akritas et al. 1995). We have employed the tiecorrection to keep τ in the range [-1, 1] when a large num-ber of tied pairs arise in our data. For overall samples, thepeak ( τ = 0 .
53) appears at 10.4–12.4 hr with p = 1 . × − ( ∼ . σ ), which means that the outburst rate is associatedwith the level of the orbital period. On the other hand, wealso calculated Kendall’s τ for BH and NS subsamples in-dividually. The Kendall’s τ peaks at 12.5–14.7 hr ( τ = 0 . p = 5 . × − ( ∼ . σ ) for BH subsamples and 9.8–12.4hr ( τ = 0 .
43) with p = 0 .
061 ( ∼ . σ ) for NS subsamples.Hence, the outburst rate and the orbital period are stronglydependent on each other in the BH systems. The associationis even stronger than that in the overall samples.In addition, we applied Fisher’s exact test to examine againthe significance of the association between the level of theoutburst rate (frequent or infrequent) and the level of orbitalperiod (long or short). The boundaries of the orbital periodwere set to the values that take the Kendall’s τ to peak valuesabove. In order to put all upper limits into the class of “in-frequent outburst”, the boundary of the outburst rate should HE RELATION BETWEEN OUTBURST RATE AND ORBITAL PERIOD IN LOW - MASS X- RAY BINARY TRANSIENTS O u t b u r s t r a t e ( y r − ) Class VClass III/IVClass unknownBHNSMAXI J0556-332 K e n d a ll ' s τ AllBHNS L o g p AllBHNS A v e r a g e r e c u rr e n c e t i m e ( y r ) Figure 1. Top panel:
The outburst rate as a function of orbital period for BH (circles) and NS (squares) LMXBTs. Those LMXBTs fromwhich only one outburst has been detected in historical X-ray observations are labeled with downward-pointing arrows. NS X-ray binary MAXIJ0556–332, of which the orbital period is uncertain, is marked as red open squares at 9.75 and 16.43 hr. Colors represent the luminosity classof the donor: V (blue), III/IV (red), and unknown (black). The horizontal dot-dashed line indicates the outburst rate of 0.05 yr − . The verticalorange dashed line indicates the derived lower limit of the orbital period for a BH X-ray binary to harbor a donor of a subgiant, as discussedin Section 4.2. The vertical blue dashed lines indicate the derived lower limit of orbital period for a BH X-ray binary to harbor a 0.1 M (cid:12) main-sequence donor star and the upper limit to harbor a 3 M (cid:12) main-sequence star, respectively. The best-fitting power-law models to the BHsubsamples with a donor of luminosity class III/IV are represented by the red solid/dotted lines (excluding/including GRS 1915+105). Middlepanel and bottom panel:
The Kendall’s τ and corresponding p -value for various boundaries of the orbital period. The black, blue, and redsolid lines represent the results from all, BH and NS samples, respectively. The gray and blue areas represent the maximum τ and minimum p for all and BH samples, respectively. J IE L IN ET AL .be set to a value larger than 0.047 yr − . And the boundaryshould divide the samples into two subsamples that are asequal as possible. Hence, we set the boundary to 0.05 yr − .Based on the selection of these boundaries, the Fisher’s ex-act test gives p = 8 . × − ( ∼ . σ ) for overall samples, p = 1 . × − ( ∼ . σ ) for BH subsamples, and p = 0 . τ correlation coefficient above.3.1.1. Possible selection effect
Theory predicts a positive correlation between the orbitalperiod and the outburst peak luminosity in LMXBTs (King& Ritter 1998; Shahbaz et al. 1998), which is also confirmedby the X-ray observations (Wu et al. 2010). The LMXBTswith a shorter orbital period tend to have a lower outburstpeak luminosity, so quite a number outbursts from the short-period subsamples may be missed owing to the flux limit (50mcrab) that we set to form our samples. We need to address apotential selection effect and further investigate whether thesystematic difference between long-period and short-periodLMXBTs is affected by the flux criteria.The flux criterion, 50 mcrab, corresponds to about 2 × erg s − in 3–200 keV energy band for sources with a typi-cal distance of 8 kpc by assuming that their X-ray spectraare roughly similar to those of the Crab. Wu et al. (2010)have fitted the relation between the orbital period and thepeak luminosity in a large sample with a straight line and gotlog L peak / L Edd = ( − . ± . + (0 . ± .
08) log P orb . Wecan infer that the outbursts in a BH system (if M BH = 10 M (cid:12) )with an orbital period less than ∼ M NS = 1 . (cid:12) ) with an orbital period lessthan ∼
20 hr could be missed by our selection. Because theorbital periods of BH subsamples, listed in Table 1, are alllonger than 2 hr, the outburst rates measured in BH systemsshould not be affected by the flux criterion applied. There-fore, the association between outburst rate and orbital periodin the BH subsamples is reliable.On the other hand, the orbital periods of most NS subsam-ples, as listed in Table 1, are shorter than 20 hr, which meansthat the selection effect in NS LMXBTs cannot be ruled out.The 20 hour limit would correspond to 0.9 hr if the sourcedistance is changed to 3 kpc, so some nearby sources (e.g.Cen X-4) will not be affected. In short, we cannot concludethat there is a relation between the outburst rate and the or-bital period in NS LMXBTs.3.2.
The correlation between the outburst rate and theorbital period in the systems with a luminosity classIII/IV donor star
As shown in Figure 1, for the samples with a luminosityclass of III/IV, the outburst rate shows a broad trend to de-crease with the orbital period. We applied a Spearman’s rank correlation coefficient totest the correlation between the outburst rate and the or-bital period in the subsamples with a luminosity class ofIII/IV, and yielded a coefficient of − .
89 at a significanceof 99.4%(2.8 σ ), which indicates that the outburst rate andthe orbital period are anticorrelated in these LMXBTs. Wewere going to further investigate the correlations for the NSand BH, respectively. However, since only one NS samplewas identified as harboring a luminosity class III/IV donorstar, we then tested the correlation in the black hole subsam-ples with a luminosity class III/IV donor star and yielded thesame coefficient of − .
89 at a slightly lower significance of98.2% (2.4 σ ). And we are not sure that this negative cor-relation is applicable in the LMXBTs with orbital periodshorter than 30 hr, since our subsamples with a luminosityclass III/IV donor star all have an orbital period longer than30 hr. We will explore the possible explanation of this nega-tive correlation in Section 4.4. DISCUSSIONAs introduced in Section 1.1, the mass transfer rates arevery different between case A and case B mass transfers. InSection 4.2, we will discuss the critical orbital period for caseB mass transfer based on the evolutionary tracks of singlestars. The critical orbital period is consistent with the bound-ary orbital period, which separates the samples into very dif-ferent levels of outburst rates. Hence, the systematic differ-ence of outburst rates below and above ∼
12 hr can be asso-ciated with different evolutionary stages of the donors. Frompopular accretion theories, e.g. the DIM, the mass transferrate affects the quiescence time ( ≈ recurrence time). We dis-cuss the relation between the quiescence time and the masstransfer rate first and then discuss the orbital period rangecorresponding to the case A and case B mass transfers in thefollowing subsection.4.1. About the quiescence time
In theoretical models (e.g. the standard DIM), quiescencetime is usually mentioned rather than the recurrence time ofoutbursts. The recurrence time t rec can be expressed as thesum of the quiescence time and the duration of an outburst,namely, t rec = t qui + t out = t qui / (1 − τ ), where τ represents theduty cycle. TSHG16 shows that the average duty cycle oftransient BHBs is only 7.3%. This means that approximatelythe recurrence time can be taken as the quiescence time inBH systems.As mentioned in the introduction section, the quiescencetime can be estimated by the shorter one of the mass-accumulation time and the viscous drift time in the standardDIM (Osaki 1995, 1996). However, the standard DIM ischallenged by the long quiescence time observed, which canexceed 20–30 years in some short-period LMXBTs (typi-cally P orb <
12 hr), because the viscous drift time is shorter
HE RELATION BETWEEN OUTBURST RATE AND ORBITAL PERIOD IN LOW - MASS X- RAY BINARY TRANSIENTS α is given by the “standard value” (see Menou et al. 2000).Hameury et al. (1997) suggested that the disk would betruncated at a sufficiently large radius, which will increasethe quiescence time, because the viscous drift time is onlyvalid for r out / r in (cid:29) t qui ≈ (cid:15) M D , max ˙ M T − ˙ M in , (1)where M D , max is the maximum disk mass in the quiescence, (cid:15) represents the fraction of mass loss of the accretion diskduring outburst, ˙ M T is the mass transfer rate from the donor,and ˙ M in is the accretion rate at the inner edge of the disk. Inthe standard DIM, ˙ M in (cid:28) ˙ M T holds during the quiescence.The maximum disk mass can be estimated by assumingthat the accretion disk before an outburst is filled up to thecritical surface density. The integration of the critical surfacedensity to the disk area yields M D , max ∝ r . M − . (Hameuryet al. 1998; Menou et al. 1999b; Lasota 2001; Knevitt et al.2014). On the other hand, Paczynski (1977) have derivedthe maximum size of an accretion disk in a close binary sys-tem by assuming vanishingly small pressure in the disk. Asa result, the maximum size of the disk is a function of massratio and orbital period of the binary system. According toLasota et al. (2008), the outer disk radius is positively re-lated to the orbital period and the mass of the compact object,namely, r out ∝ P / M / . Hence, we obtained the relation be-tween the maximum disk mass and the orbital period, i.e. M D , max ∝ P . M / . Since the maximum disk mass increaseswith the orbital period at a given M , lower recurrence timein long-period LMXBTs ( P orb (cid:38)
12 hr) cannot be caused bythe term M D , max unless the viscosity parameter α is muchlower than the “standard value” in the short-period LMXBTs(Menou et al. 2000).Instead of adjusting the parameters (e.g. α and (cid:15) ) of thetheoretical models, a more intuitive explanation of our re-sults is that the mass transfer rate ˙ M T is systematically differ-ent between long-period and short-period LMXBTs owing todifferent evolutionary stages.4.2. The critical orbital period for case B mass transfer
As introduced in Section 1.1, case B mass transfer can beperformed only if the LMXBTs have an orbital period greaterthan a critical orbital period P orb , crit . We can infer the criticalorbital period based on the evolutionary tracks of single starsand the assumption that the radius of a single star just equals /M fl R / R fl Red giantSubgiantMain-sequence
BeyondHubble Time! BeyondHubble Time! BeyondHubble Time!
Figure 2.
Mass-radius relations for main-sequence stars (blue area),subgiants (orange area) and giants (red area) in solar metallicity(Z=0.019). The blue, orange, and red solid lines represent themass-radius relations of the ZAMS, the TAMS and the beginningof the red giant branch, respectively. The purple solid line indicatesthe mass-radius relation in a 14.1 billion years ( ≈ Hubble time)isochrone. The area to the left of this line exceeds the Hubble time.We used a cyan circle to highlight the intersection of the isochroneand the TAMS and also a brown circle to highlight the inflectionof the ZAMS. The dashed lines represent the mass-radius relationsof a Roche lobe filling donor star to the critical orbital period if M = 10 M (cid:12) (blue) or M = 1 . (cid:12) (red). to the Roche lobe radius. As shown in Figure 2, blue, orange,and red areas represent the mass-radius relation (MRR) ofmain-sequence stars, subgiants, and red giants in solar metal-licity (Z=0.019), respectively. These evolutionary tracks aretaken from the Podova Stellar Evolution Database (Girardiet al. 2000). The purple solid line is the MRR in a 14.1billion yr ( ≈ Hubble time) solar-metallicity isochrone takenfrom Girardi et al. (2000). Obviously, the age of a star cannotexceed Hubble time. Hence, we can obtain the smallest ra-dius and the lowest mass of a subgiant from the intersectionof the isochrone and terminal-age main sequence (TAMS, theorange solid line). Obviously, the LMXBT system with thecritical orbital period must just harbor such a subgiant. Forthe solar metallicity Z=0.019, the smallest radius is 1.2 R (cid:12) and the lowest mass is 0.9 M (cid:12) . On the other hand, the MRRof a Roche lobe filling donor star can be determined whenthe orbital period and the mass of the compact object areboth known. For an X-ray binary for which the orbital periodequals the critical orbital period, its MRR must pass throughthe intersection of the isochrone and the TAMS. Therefore,we obtain P orb , crit =12.9 hr for M = 10 M (cid:12) (blue dashed line)and P orb , crit =12.6 hr for M = 1 . (cid:12) (red dashed line). Ob-0 J IE L IN ET AL .viously, the critical orbital period is insensitive to the massof the compact object. Notice that we did not take into ac-count the cases of mass loss by stellar winds or mass transfer.These effects can lead to lower mass of the (sub)giants.We also took into account the cases under a wider rangeof metallicity, from Z=0.0004 to 0.03. We obtained that thelower limits of the critical orbital period are 12.7 hr for typ-ical BH systems ( M = 10 M (cid:12) ) and 12.4 hr for typical NSsystems ( M = 1 . (cid:12) ). This means that we cannot findany LMXBT system with the orbital period below 12.4 hrin which the donor performs case B mass transfer . There-fore, the mass transfer rates in the systems with periods be-low 12.4 hr are very low when they all only perform case Amass transfer, unless their orbital periods are short enoughfor significant angular momentum loss by gravitational waveradiation. From Eq. (1), a lower mass transfer rate will leadto a longer quiescence time (or recurrence time). Although M D , max is also dependent on the orbital period, M D , max ∝ P . cannot offset the gap of several hundred (or even thousand)times between case A and case B mass transfer rates unlessthe orbital period is much shorter than those case B binaries(typically P orb < ∼ ∼
12 hr.Back to our results, for the BH systems, P orb , crit = 12 . τ to a maximum value in Section 3.1.All BH subsamples below 12.7 hr show a very low outburstrate (<0 .
05 yr − ) and almost all BH subsamples above 12.7hr show a relatively higher outburst rate ( >0 .
06 yr − at least),except for a sample (GRS 1915+105) with a very large orbitalperiod.On the other hand, the critical orbital period should be ap-plicable to the NS LMXBTs in theory. However, we did notfind a significant difference in outburst rates for NS subsam-ples. In fact, the behaviors of some NS LMXBTs are not wellunderstood. For example, the orbital period of GRS 1747-312 is slightly lower than the critical orbital period, whichmeans that the donor of GRS 1747-312 is more likely to be amain-sequence star, but its outburst is very frequent.4.3. The wide period range for case A mass transfer
When the orbital period is above P orb , crit , the system canharbor a main-sequence star and a (sub)giant. From our sam-ples, the maximum mass of the known main-sequence donorsmay be ≈ . − . (cid:12) (Orosz et al. 1998; Orosz 2003;Tetarenko et al. 2016). Therefore, we assumed that the max-imum mass of main-sequence donors in LMXBTs is around3 M (cid:12) . To harbor a 3 M (cid:12) main-sequence star with Z=0.0004–0.03, the upper limit of the orbital period can reach 48.4 hr for M = 10 M (cid:12) or 41.6 hr for M = 1 . (cid:12) . The candidate massrange of the main-sequence donor becomes wider when P orb is greater than the period P orb , infle ( ≈ P orb , crit when Z=0.019)corresponding to the inflection of the MRR of the zero-agemain sequence (ZAMS, blue solid line in Figure 2), becausethe slope of the ZAMS becomes lower when its mass exceedsaround 1.47 M (cid:12) , which could be caused by the transition ofnuclear energy from the proton–proton cycle to the carbon–nitrogen cycle (Demircan & Kahraman 1991). The P orb , infle can reduce to ∼ t nuc ∝ M − . , a massive main-sequence donor (e.g. 2 M (cid:12) )can also contribute slightly substantial (case A) mass trans-fer (about several times 10 − M (cid:12) / yr).Similarly, for an X-ray binary to harbor a very low massmain-sequence star (e.g. 0 . (cid:12) ), its orbital period must belonger than 1.0 hr if M = 10 M (cid:12) or 1.1 hr if M = 1 . (cid:12) ,which are derived from the MRR of the ZAMS (Tout et al.1996). We have also plotted these limits on the orbital pe-riod for BHs in Figure 1 in order to intuitively understand theassociation among outburst rate (or recurrence time), orbitalperiod, and evolutionary stage. From ∼
12 to ∼
48 hr, thereare both case A binaries and case B binaries. This means thatthe mass transfer rates (and also the outburst rate) in this pe-riod range will show a very large dispersion with orbital pe-riod, which can be seen in our results as shown in Figure 1.The period range, 12–48 hr, is numerically consistent withthe “bifurcation” orbital period (Pylyser & Savonije 1988;King et al. 1996; Menou et al. 1999a), which separates theconverging binary systems ( P orb decreases with time) and thediverging binary systems ( P orb increases with time).Notice that, besides the main-sequence and (sub)giantdonor, LMXBTs also can harbor a donor of a brown dwarfor a WD. For example, derived from the evolutionary modelsof Chabrier et al. (2000), the upper limit of the period forbrown dwarfs is around 4 hr.4.4. The outburst rate decreases with the orbital period inthe system with a donor of a (sub)giant
As introduced in Section 3.2, we have found a negativecorrelation between the outburst rate and the orbital period inthe subsamples with a luminosity class of III/IV donor star.We then discuss this correlation under the framework of theDIM. From Eq. (1), since the ˙ M in is negligible in quiescence,the outburst rate (or recurrence time) is mainly determinedby disk mass M D , max and mass transfer rate ˙ M T .The luminosity classes III/IV and long orbital periods( P orb ≥
30 hr) of these subsamples indicate that their donorstars are very likely to be subgiant or giant stars. King (1988)has derived a relation between the mass transfer rate andthe orbital period for those systems with a donor evolvedoff main sequence, ˙ M T ∝ P . M . . And we have obtained M D , max ∝ P . M / based on the DIM in Section 4.1. Then,we obtain that t rec ≈ t quiec ∝ (cid:15) M / M − . P . . Hence, in aLMXBT with a donor of a (sub)giant, the recurrence time in- HE RELATION BETWEEN OUTBURST RATE AND ORBITAL PERIOD IN LOW - MASS X- RAY BINARY TRANSIENTS (cid:15) is constant, at a given M and M , the outburst rate decreases with orbital period asa power-law form R out ∝ P − . .Because the theoretical outburst rate (or recurrence time)is dependent on the mass of the compact object and thereis only one NS sample in the correlation, we therefore useda power-law model to fit the correlation in the BH subsam-ples, except GRS 1915+105, the outburst rate of which isonly an upper limit. By applying the Cash statistic (Cash1979), we obtained the relation between the outburst rate andthe orbital period as R out = 1 . + . − . × ( P orb /
12 h) − . + . − . yr − (see Figure 1). The uncertainties correspond to ∆ C = 1 inthe Cash statistic. For a comparison, we also fitted the re-lation; GRS 1915+105 was included, and its outburst ratewas taken as a measured value. Then, we obtained R out =1 . + . − . × ( P orb /
12 h) − . + . − . yr − , which has a slightly flat-ter slope.The best-fitting power-law index is roughly consistent withthe above estimation from the DIM, no matter whether GRS1915+105 is included or not. In conclusion, the negative cor-relation in these BH subsamples supports the idea that theDIM plus the Case B mass transfer would cause a longerrecurrence time in the system with a longer orbital period.However, the spread of outburst rates around the best-fittingrelation is broad. The large scatter of this negative correlationindicates that other parameters (such as M and M ) besidesthe orbital period affect the outburst rate (or recurrence time)as well.In our samples, two other BH LMXBTs (GRS 1716–249and 4U 1543–47) are worth to be mentioning. Their donorsare luminosity classes V, but their orbital periods are in themixture region for a BH LMXBT with a main sequence or a(sub)giant donor star (see the orange and blue dashed verticallines in Figure 1). We calculated the Spearman correlationcoefficient as − .
33 with p = 0 .
38 (insignificant), in the sam-ples consisting of GRS 1716–249, 4U 1543–47 and all sub-samples of which the luminosity classes are III/IV. The testindicates that these two sources do not follow the negativecorrelation that we found in luminosity class III/IV subsam-ples. The possible explanation is that these two sources maynot harbor (sub)giant donors, or they do not follow the masstransfer rates derived by King (1988).On the other hand, the essential mechanism to trigger out-bursts is the same between NS LMXBTs and BH LMXBTsin the DIM. The correlation also should be available for theNS LMXBTs of which the donor is a (sub)giant. In Fig-ure 1, some NS samples (e.g. Aql X-1, 4U 1608–52 and GRS1747–312) are in line with the best-fitting model for the BHsubsamples. Therefore, we suggest that these systems couldall harbor a subgiant donor. Notice that although the spectraltype of the donor in Aql X-1 is identified as K7V (Chevalieret al. 1999), but this will lead to an estimate of the distance of 2.5 kpc, which is much closer than the distance estimatedby type I X-ray bursts (Galloway et al. 2008; Rutledge et al.2001), and it is also difficult to explain how a K7V donor fillsa Roche lobe of a 19-hour orbit.4.5.
The orbital period distribution of low-mass X-raybinaries
The relation between the recurrence time and the orbitalperiod plays an important role in regulating the orbital perioddistribution of low-mass X-ray binaries, which is essential torestrict the evolution models of X-ray binaries.Since the detection probabilities of LMXBTs are stronglydependent on the orbital periods, the observed distributionof LMXBs suffers significant selection effects. The detec-tion probabilities can be estimated based on the peak lumi-nosity, the outburst duration, and the recurrence time in therelation with the orbital period (Knevitt et al. 2014; Arur &Maccarone 2018). As the key factor to build the orbital pe-riod distribution of LMXBs, however, the relation betweenthe recurrence time and the orbital period is usually not welldetermined. For example, from Knevitt et al. (2014), they de-rived that the recurrence time increases continuously with theorbital period when the orbital period is above 2 hr, which isinconsistent with the observed results. Therefore, we suggestthat the period distribution of LMXBs should be estimatedbased on the actual relation. SUMMARYWe have investigated the outburst rate in relation to the or-bital period for the current LMXBT samples of which theorbital period has been measured. By applying the Kendall’s τ and the Fisher’s exact test, we found that the outburst ratesare systematically different between long-period ( P orb (cid:38) P orb (cid:46)
12 hr) subsamples. By analyzingBH and NS subsamples separately, we found that the system-atic difference is still significant for the BH subsamples butnot very significant for NS subsamples. We take into accountthe selection effect due to the positive correlation betweenthe orbital period and outburst peak luminosity in LMXBTs.Then, we rule out the selection effect on BH systems, but wecannot rule out the effect on NS systems.We infer that the mass transfer rates are responsible forthe systematic difference of outburst rates since the DIMsuggested that the mass transfer rate is a key factor affect-ing the quiescence time of LMXBTs and the mass transferrates could be very different in long-period and short-periodLMXBTs owing to different evolutionary stages of donors.From the evolutionary tracks of stars, we derived that thecritical orbital period to harbor a (sub)giant donor or performcase B mass transfer is 12.7 hr for BH LMXBTs and 12.4hr for NS LMXBTs. The critical orbital period is consistentwith the boundary of the orbital period, which separates the2 J IE L IN ET AL .samples into very different levels of outburst rates. This canexplain why there is a systematic difference of outburst ratesbetween the systems with orbital periods below and above ∼
12 hr.We also determined a wide orbital period range, 1.1 to ∼ ∼
12 to ∼
48 hr will show a very largedispersion, which can be verified in our results.Furthermore, we found a negative correlation between theoutburst rate and the orbital period in the subsamples forwhich the luminosity class is III/IV. The high luminosityclasses and long orbital periods of these samples mean thattheir donor stars are very likely to be subgiants or giants.For those LMXBTs with a donor star evolved off the mainsequence, the relation between the outburst rate (or recur-rence time) and the orbital period follows a power-law formunder the framework of the DIM, and the power-law indexis roughly consistent with our best-fitting results for the BHsubsamples.On the other hand, the essential mechanism that triggersoutbursts is the same between NS and BH LMXBTs in thecurrent theory of the DIM. We cannot investigate the correla-tion for NS LMXBTs, since there is only one NS LMXBT (GRO J1744–28) for which the luminosity class is III/IV.And some NS LMXBTs (e.g. 4U 1608–52 and GRS 1747–312 ) seem in line with the negative correlation, although theluminosity classes of their donors are not credibly identifiedyet. We suggest that a similar negative correlation shouldexist in these NS LMXBTs harboring a (sub)giant donor.We would like to thank RXTE and
Swift
Guest ObserverFacilities at NASA Goddard Space Flight Center for provid-ing RXTE/PCA and
Swift /XRT products and
Swift /BAT andRXTE/ASM transient monitoring results. This research hasmade use of MAXI data provided by RIKEN, JAXA, andthe MAXI team. W.Y. would like to acknowledge the sup-port by the National Program on Key Research and Devel-opment Project (Grant no. 2016YFA0400804), by the Na-tional Natural Science Foundation of China under grant No.U1838203 and 11333005 and by the FAST fellowship, whichis supported by Special Funding for Advanced Users, bud-geted and administrated by Center for Astronomical Mega-Science, Chinese Academy of Sciences (CAMS). Z.H. wouldlike to acknowledge the support by National Natural Sci-ence Foundation of China under grant no. 11521303 andno. 11733008. Z.Y. would like to acknowledge the supportby the National Natural Science Foundation of China undergrant no. 11773055.REFERENCES
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