The advection-dominated accretion flow for the anti-correlation between the X-ray photon index and the X-ray luminosity in neutron star low-mass X-ray binaries
aa r X i v : . [ a s t r o - ph . H E ] J un MNRAS , 1–12 (2020) Preprint 11 June 2020 Compiled using MNRAS L A TEX style file v3.0
The advection-dominated accretion flow for the anti-correlationbetween the X-ray photon index and the X-ray luminosity inneutron star low-mass X-ray binaries
Erlin Qiao , ⋆ and B.F. Liu , Key Laboratory of Space Astronomy and Technology, National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, China School of Astronomy and Space Sciences, University of Chinese Academy of Sciences, 19A Yuquan Road, Beijing 100049, China
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
Observationally, an anti-correlation between the X-ray photon index Γ (obtained by fitting theX-ray spectrum between 0.5 and 10 keV with a single power law) and the X-ray luminosity L . − , i.e., a softening of the X-ray spectrum with decreasing L . − , is found in neu-tron star low-mass X-ray binaries (NS-LMXBs) in the range of L . − ∼ − erg s − .In this paper, we explain the observed anti-correlation between Γ and L . − within theframework of the self-similar solution of the advection-dominated accretion flow (ADAF)around a weakly magnetized NS. The ADAF model intrinsically predicts an anti-correlationbetween Γ and L . − . In the ADAF model, there is a key parameter, f th , which describesthe fraction of the ADAF energy released at the surface of the NS as thermal emission tobe scattered in the ADAF. We test the e ff ect of f th on the anti-correlation between Γ and L . − . It is found that the value of f th can significantly a ff ect the anti-correlation between Γ and L . − . Specifically, the anti-correlation between Γ and L . − becomes flatterwith decreasing f th as taking f th = . , . , . , . .
003 and 0 respectively. By com-paring with a sample of non-pulsating NS-LMXBs with well measured Γ and L . − , wefind that indeed only a small value of 0 . . f th . . Γ and L . − . Finally, we argue that the small value of f th . . ffi ciency ofNSs with an ADAF accretion may not be as high as ǫ ∼ ˙ MGMR ∗ / ˙ Mc ∼ . Key words: accretion, accretion discs – stars: neutron – black hole physics – X-rays: binaries
Neutron star low-mass X-ray binaries (NS-LMXBs) are binariesin which a NS accretes matter from its low-mass companion star( . M ⊙ ) via the Roche lobe. Most NS-LMXBs are X-ray tran-sients, spending most of their time in the quiescent state (oftentaken as the X-ray luminosity . erg s − , and the X-ray lu-minosity here referring to the luminosity between 0.5 and 10 keV)for years to decades. NS-LMXBs are generally discovered as theygo into outburst, during which the X-ray luminosity of some NS-LMXBs can generally increase by several orders of magnitudereaching up to ∼ − erg s − . The outburst lasts fromweeks to months, and then the sources decay into quiescent state(e.g. Campana et al. 1998; Wijnands et al. 2006, for summary). TheX-ray spectrum of NS-LMXBs with the X-ray luminosity above ∼ erg s − is generally well studied (e.g Lin et al. 2007, fordiscussions). While for NS-LMXBs in the low-luminosity regime ⋆ E-mail: [email protected] as the X-ray luminosity below ∼ erg s − , due to the lim-itation of the sensitivity of the X-ray instruments in orbit cur-rently, the X-ray spectrum is relatively less well understood (e.g.Degenaar et al. 2013; Armas Padilla et al. 2013b; D’Angelo et al.2015; Chakrabarty et al. 2014; Wijnands et al. 2015). In this pa-per, we focus on the X-ray spectrum of NS-LMXBs between thequiescent state and the outburst in the X-ray luminosity range of ∼ − erg s − .Generally, the X-ray spectrum between 0.5 and 10 keV ofNS-LMXBs in the X-ray luminosity range of L . − ∼ − erg s − can be satisfactorily described by a single power law,although the spectral fitting can be improved if an additional ther-mal soft X-ray component is added for the sources with su ffi -ciently good data available, especially in the range of L . − ∼ − erg s − (e.g. Armas Padilla et al. 2013b; Degenaar et al.2013). Wijnands et al. (2015) compiled a sample of NS-LMXBsfrom literatures with well measured X-ray spectra between 0.5 and10 keV in the range of L . − ∼ − erg s − . All thesources in the sample are non-pulsating systems, which means that c (cid:13) Erlin Qiao and B.F. Liu the e ff ect of the magnetic field on the X-ray spectrum is very lit-tle. The X-ray spectra of all the sources in the sample are once re-ported in the literatures to be fitted with a single power-law model,i.e., N ( E ) ∝ E − Γ (with Γ being the X-ray photon index between 0.5and 10 keV ). Then it is found that systematically there is a stronganti-correlation between the X-ray photon index Γ and the X-rayluminosity L . − for the sources in the sample as a whole in therange of L . − ∼ − erg s − .Such an anti-correlation between the X-ray photon index Γ and the X-ray luminosity L X 1 has also been widely reportedfor black hole (BH) X-ray transients in their hard state (e.g.Wu & Gu 2008; Plotkin et al. 2013; Homan et al. 2013; Liu et al.2019). In Wijnands et al. (2015), the authors further compared the Γ - L . − anti-correlation between NS-LMXBs and BH X-raytransients, from which it is found that NS-LMXBs have a sig-nificant softer X-ray spectrum for a fixed X-ray luminosity thanthat of BH X-ray transients. Additionally, the separation of the X-ray photon index Γ between NS-LMXBs and BH X-ray transientsis more obvious for the X-ray luminosity below ∼ erg s − than that of above ∼ erg s − (equivalent to that the slopeof the Γ - L . − anti-correlation in NS-LMXBs is steeper thanthat of in BH X-ray transients). The anti-correlation between Γ and L X (or L X / L Edd ) in BHs has been well explained in theframework of the advection-dominated accretion flow (ADAF)(e.g. Qiao & Liu 2010, 2013; Qiao et al. 2013; Yang et al. 2015).ADAF is a kind of geometrically thick, optically thin, hot accretionflow (Ichimaru 1977; Rees et al. 1982; Narayan & Yi 1994, 1995;Abramowicz et al. 1995; Manmoto et al. 1997; Yuan & Narayan2014, for review), which is di ff erent from the geometrically thin,optically thick, cool accretion disc (Shakura & Sunyaev 1973). Forthe ADAF around a BH, the electron temperature T e of the ADAFincreases very slightly with decreasing ˙ m ( ˙ m = ˙ M / ˙ M Edd , with˙ M Edd = L Edd / . c = . × M / M ⊙ g s − ), while the Comptonscattering optical depth τ es decreases with decreasing ˙ m by an equalmultiple. So the Compton y -parameter (defined as y = kT e m e c τ es , with τ es <
1) decreases with decreasing ˙ m , leading to a softer X-rayspectrum with decreasing ˙ m . Meanwhile, in the ADAF solution L X (or L X / L Edd ) decreases with decreasing ˙ m , so a softer X-ray spec-trum is predicted with decreasing L X (or L X / L Edd ), i.e., an anti-correlation between Γ and L X (or L X / L Edd ) is predicted.The physical mechanism for the anti-correlation between Γ and L . − in NS-LMXBs as proposed in Wijnands et al. (2015)is not very clear. The softer X-ray spectrum and the steeper slopeof the Γ − L . − anti-correlation in NS-LMXBs compared withthat of in BH X-ray transients are speculated to be very probablyresulted by the e ff ect of the hard surface of the NS. In this paper,we focus on the anti-correlation between Γ and L . − in NS-LMXBs, and explain it within the framework of the self-similar so-lution of the ADAF around a weakly magnetized NS (Qiao & Liu2018, 2020). In Qiao & Liu (2020), the authors updated the cal-culation of Qiao & Liu (2018) with the e ff ect of NS spin consid- Please note that in di ff erent literatures, Γ and L X are sometimes measuredin di ff erent energy bands, such as Islam & Zdziarski (2018) for discussions. L X is often replaced by the Eddington scaled X-ray luminosity L X / L Edd (with L Edd being the Eddington luminosity, and L Edd = . × M / M ⊙ erg s − ), and then the correlation between Γ and L X / L Edd can be extended up to supermassive BHs, as that of the anti-correlationbetween Γ and L X / L Edd established in low-luminosity active galacticnuclei (AGNs) (e.g. Gu & Cao 2009; Younes et al. 2011; Veledina et al.2011; Emmanoulopoulos et al. 2012; Hern´andez-Garc´ıa et al. 2013, 2014;Jang et al. 2014). ered. One of the key points in Qiao & Liu (2018, 2020) is thatthe radiative feedback between the hard surface of the NS and theADAF is considered, as suggested by several other authors previ-ously (e.g. Narayan & Yi 1995). Roughly speaking, the feedback(thermal emission) from the surface of the NS will cool the ADAFitself, which makes the electron temperature of the ADAF aroundNSs is lower than that of around BHs, consequently predicting asofter X-ray spectrum (in the band greater than ∼ f th , of the ADAF energy(including the internal energy, the radial kinetic energy and the ro-tational energy) transferred onto the surface of the NS is assumedto be thermalized at the surface of the NS as thermal emission tobe scattered in the ADAF itself. The authors self-consistently cal-culate the structure and the corresponding emergent spectrum ofthe ADAF by considering the radiative feedback between the NSand the ADAF. Physically, the value of f th is uncertain, and as willshow in this paper, the value of f th can significantly a ff ect the struc-ture and the corresponding emergent spectrum of the ADAF, con-sequently a ff ecting the Γ − L . − anti-correlation.In this paper, we test the value of f th on the X-ray spectrumand the slope of the Γ − L . − anti-correlation. It is found thatfor a fixed X-ray luminosity, a softer X-ray spectrum, i.e., a larger Γ , is predicted for taking a larger value of f th . Meanwhile, the Γ − L . − anti-correlation becomes flatter with decreasing f th as taking f th = . , . , . , . .
003 and 0 respectively. Bycomparing with a sample of non-pulsating NS-LMXBs with wellmeasured Γ (obtained by fitting the observed X-ray spectrum be-tween 0.5 and 10 keV with a single power law) in the range of L . − ∼ − erg s − , we find that indeed only a smallvalue of 0 . . f th . . Γ and L . − . The data in the sam-ple are searched from literatures, which are summarized at Table1 of Wijnands et al. (2015), Appendix of Parikh et al. (2017) andBeri et al. (2019) respectively. Theoretically, in our model, thereare two components, i.e., a thermal component and a power-lawcomponent, both of which can contribute a fraction of the X-rayluminosity between 0.5 and 10 keV respectively. We conclude thatin the range of L . − ∼ − erg s − , the observed soft-ening of the X-ray spectrum with decreasing L . − is due tothe increase of the fractional contribution of the thermal soft X-raycomponent, which is supported by both the theoretical and obser-vational results that the fractional contribution of the power-lawcomponent η ( η ≡ L power law0 . − / L . − ) decreases with decreasing L . − . While in the range of L . − ∼ − erg s − , ourexplanation for the observed softening of the X-ray spectrum withdecreasing L . − is a little uncertain, which is probably due toa complex relation between the thermal soft X-ray component andthe power-law component, or probably dominantly due to the soft-ening of the power-law component itself. Finally, we argue that thederived value of f th . . ffi ciency of NSs with an ADAFaccretion may not be as high as ǫ ∼ ˙ MGMR ∗ / ˙ Mc ∼ . MNRAS000
003 and 0 respectively. Bycomparing with a sample of non-pulsating NS-LMXBs with wellmeasured Γ (obtained by fitting the observed X-ray spectrum be-tween 0.5 and 10 keV with a single power law) in the range of L . − ∼ − erg s − , we find that indeed only a smallvalue of 0 . . f th . . Γ and L . − . The data in the sam-ple are searched from literatures, which are summarized at Table1 of Wijnands et al. (2015), Appendix of Parikh et al. (2017) andBeri et al. (2019) respectively. Theoretically, in our model, thereare two components, i.e., a thermal component and a power-lawcomponent, both of which can contribute a fraction of the X-rayluminosity between 0.5 and 10 keV respectively. We conclude thatin the range of L . − ∼ − erg s − , the observed soft-ening of the X-ray spectrum with decreasing L . − is due tothe increase of the fractional contribution of the thermal soft X-raycomponent, which is supported by both the theoretical and obser-vational results that the fractional contribution of the power-lawcomponent η ( η ≡ L power law0 . − / L . − ) decreases with decreasing L . − . While in the range of L . − ∼ − erg s − , ourexplanation for the observed softening of the X-ray spectrum withdecreasing L . − is a little uncertain, which is probably due toa complex relation between the thermal soft X-ray component andthe power-law component, or probably dominantly due to the soft-ening of the power-law component itself. Finally, we argue that thederived value of f th . . ffi ciency of NSs with an ADAFaccretion may not be as high as ǫ ∼ ˙ MGMR ∗ / ˙ Mc ∼ . MNRAS000 , 1–12 (2020) nti-correlation between Γ and L . − We considered the ADAF accretion around a weakly magne-tized NS within the framework of the self-similar solution asin Qiao & Liu (2018), which was further updated in Qiao & Liu(2020) with the e ff ect of NS spin included. As in Qiao & Liu(2020), the structure of the ADAF can be calculated by specifyingthe NS mass m ( m = M / M ⊙ ), the NS radius R ∗ , rotational frequencyof the NS ν NS (describing NS spin), the mass accretion rate ˙ m , theviscosity parameter α and the magnetic parameter β (with magneticpressure p m = B / π = (1 − β ) p tot , p tot = p gas + p m ) for describ-ing the micro physics of ADAF, and f th describing the fraction ofthe total energy of the ADAF, L ∗ , (including the internal energy,the radial kinetic energy and the rotational energy) transferred ontothe surface of the NS to be thermalized as thermal emission to bescattered in the ADAF itself. One can refer to Qiao & Liu (2020)for the detailed expression of L ∗ . The e ff ective temperature of thethermal emission at the surface of the NS can be expressed as, T ∗ = (cid:16) L ∗ f th π R ∗ σ (cid:17) / . (1)Here it is assumed that the radiation from the surface of the NS isisotropic. σ is the StefanBoltzmann constant. In this paper, we setthe NS mass m = .
4, and the NS radius R ∗ =
10 km. Here wewould like to mention that it is suggested that the NS radii R ∗ are ∼ −
12 km for m ∼ . − . R ∗ is very little for taking m = .
4, we expect that our main conclusions in this paper willnot be strongly influenced for simply taking m = . R ∗ = ν NS can a ff ect the rotational energy of the ADAF transferred onto thesurface of the NS, which is proportional to the di ff erence betweenthe rotational frequency of the NS and the rotational frequency ofthe ADAF at its inner boundary (equation 2 of Qiao & Liu 2020).Since the angular velocity of the ADAF is sub-Keplerian, intrinsi-cally the rotational energy of the ADAF transferred onto the sur-face of the NS is very small. Meanwhile, as shown in Section 4.1of (Qiao & Liu 2020), the e ff ect of ν NS on the structure of theADAF can nearly be neglected as for taking ν NS = , , ν NS = α is uncertain. In general,it is suggested that the value of the viscosity parameter is α . L . − predicted by the ADAF model is related with α [roughly with 0.1 α , see Narayan & Yi (1995) for analysis]. Inorder to safely cover the upper limit of the X-ray luminosity of L . − ∼ erg s − that we focus on in this paper. We simplytake the suggested maximum value of the viscosity parameter, i.e., α = β = .
95 as sug-gested that the magnetic field in the ADAF solution is very weakby magnetohydrodynamic simulations (Yuan & Narayan 2014, forreview). So we have two parameters left, i.e., f th and ˙ m . Finally,we calculate the emergent spectrum of the model with the methodof multiscattering of soft photons in the hot gas as in Qiao & Liu(2018, 2020). ff ect of f th In the panel (1) of Fig. 1, we plot the emergent spectra for ˙ m = . × − , 5 . × − , 2 . × − and 8 . × − respectively with f th = .
1. Specifically, from the emergent spectra, it is foundthat the X-ray photon index Γ (obtained by fitting the X-ray spec-trum between 0.5 and 10 keV with a single power law) increasesfrom 1.74 to 3.56 with L . − decreasing from 1 . × to2 . × erg s − (please note that throughout the paper L . − is calculated by integrating the theoretical emergent spectrum ofthe ADAF model). One can refer to Table 1 for the detailed nu-merical results of Γ . One can also refer to the black symbol ‘ + ’in Fig. 2 for Γ versus L . − for clarity. It is clear that thereis an anti-correlation between Γ and L . − . However, intrinsi-cally, from the emergent spectra of our model, there are two compo-nents, i.e., a thermal soft X-ray component and a power-law com-ponent, both of which can contribute to the X-ray luminosity be-tween 0.5 and 10 keV. It can be seen that the e ff ective temperature T ∗ of the thermal emission at the surface of the NS decreases from0 .
62 to 0 .
36 keV with L . − decreasing from 1 . × to2 . × erg s − . One can refer to Table 1 for the detailed nu-merical results of T ∗ . Meanwhile, we show that the fractional con-tribution of the power-law component η ( η ≡ L power law0 . − / L . − )decreases from 25 .
0% to to 2 .
10% with L . − decreasing from1 . × to 2 . × erg s − . One can refer to Table 1 for thedetailed numerical results of η , and the symbol ‘ + ’ in Fig. 3 for η versus L . − for clarity. The decrease of η with decreasing L . − can be understood as follows. Actually, both the thermalsoft X-ray luminosity and power-law luminosity decrease with de-creasing ˙ m , however, the decrease of the power-law luminosity isquicker than that of the thermal soft X-ray luminosity, resulting in adecrease of η with decreasing ˙ m . Since L . − decreases with de-creasing ˙ m , η decreases with decreasing L . − . We further showthat the photon index of the intrinsic power-law component Γ in (ob-tained by analyzing the X-ray spectrum of the model between 0.5and 10 keV with a thermal soft X-ray component added to thepower-law component) increases from 2.38 to 3.25 with L . − decreasing from 1 . × to 2 . × erg s − . One can referto Table 1 for the detailed numerical results of Γ in , and the bluesymbol ‘ + ’ in Fig. 2 for Γ in versus L . − for clarity.Similar calculations for the emergent spectra for di ff erent ˙ m with f th = . , . , . , .
003 and 0 are presented in the panel(2), (3), (4), (5) and (6) of Fig. 1 respectively. One can refer to Table1 for the detailed numerical results of Γ , T ∗ , η , Γ in and L . − fordi ff erent ˙ m with f th = . , . , . , .
003 and 0 respectively .One can also refer to Fig. 2 for Γ (black symbol) and Γ in (blue sym-bol) versus L . − , and Fig. 3 for η versus L . − respectivelyfor clarity.As we can see from Fig. 2, it is very clear that there is an anti-correlation between Γ and L . − for f th = . , . , . , . .
003 and 0 respectively. Meanwhile, the slope of the Γ − L . − anti-correlation becomes flatter with decreasing f th . From Fig. 2,we can see that Γ in increases with decreasing L . − for dif-ferent f th , meaning that the intrinsic power-law component soft-ens with decreasing the X-ray luminosity for di ff erent f th . Mean-while, it can be seen that the separation between Γ and Γ in becomes Since systematically the radiative e ffi ciency of the ADAF will increasewith increasing f th , which leads to the ADAF more easily to be collapsed,consequently resulting in the upper limit of ˙ m decrease with increasing f th .In this paper, we roughly take ˙ m = . × − for f th = .
1, ˙ m = . × − for f th = .
03, ˙ m = . × − for f th = .
01, ˙ m = . × − for f th = . m = . × − for f th = .
003 respectively as the upper limits of ˙ m .We take ˙ m = . × − for f th = ∼ erg s − matching the upperX-ray luminosity that we focus on in this paper.MNRAS , 1–12 (2020) Erlin Qiao and B.F. Liu more and more clear with decreasing L . − for di ff erent f th , sug-gesting that the softening of Γ with decreasing L . − is due tothe increase of the fraction contribution of the thermal soft X-raycomponent for di ff erent f th , which is confirmed by the trend of η versus L . − as that η decreases with decreasing L . − for f th = . , . , . , .
005 and 0 .
003 respectively as can be seen inFig. 3. Γ and L . − —a constraint to f th As has been shown in Section 3.1, we test the e ff ect of f th on the X-ray spectra and the anti-correlation between Γ and L . − withinthe framework of the self-similar solution of the ADAF around aweakly magnetized NS. It is found that, theoretically, the e ff ect of f th on the Γ − L . − anti-correlation is significant. In the fol-lowing, we compare the theoretical results for the Γ − L . − anti-correlation with observations, further constraining the value of f th . We collect a sample composed of fifteen non-pulsating NS-LMXBs from literatures with well measured X-ray spectra between0.5 and 10 keV in the range of L . − ∼ − erg s − .Since only the non-pulsating NS-LMXBs are collected, we expectthat the e ff ects of the magnetic field on the X-ray spectra are verylittle. Eleven sources in the sample, i.e., AX J1754.2-2754, 1RXSJ171824.2-402934, 1RXH J173523.7-354013, XTE J1709-267,IGR J17062-6143, 1RXS J170854.4-321857, SAX J1753.5-2349,Swift J174805.3-24463, Aql X-1, IGR J17494-3030 and XTEJ1719-291 are from Wijnands et al. (2015). Three sources, i.e.,1RXS J180408.9-342058, EXO 1745-248 and IGR J17361-4441are from Parikh et al. (2017). MAXI J1957 +
032 is from Beri et al.(2019). Twelve sources in the sample (except for IGR J17494-3030,XTE J1719-291 and IGR J17361-4441 in the sample) are con-firmed as NS binaries since the Type I X-ray bursts have been ob-served. The nature of IGR J17494-3030, XTE J1719-291 and IGRJ17361-4441 are not confirmed yet, however the three sources arevery likely to be NSs. One can refer to Armas Padilla et al. (2013c)for IGR J17494-3030, Armas Padilla et al. (2011) for XTE J1719-291, and Wijnands et al. (2015) for IGR J17361-4441 for discus-sions in detail.The X-ray spectra between 0.5 and 10 keV of all the fifteensources in the sample are once reported in the literatures to befitted with a single power law, i.e., N ( E ) ∝ E − Γ (with N ( E ) be-ing the photon number at a given energy E and Γ being the X-ray photon index between 0.5 and 10 keV). In general, the spec-tral fitting of the Swift
X-ray data with a single power law cantypically result in an accepted fit. On the other hand, the X-rayspectra between 0.5 and 10 keV of seven sources in the sam-ple, i.e., AX J1754.2-2754, 1RXS J171824.2-402934 and 1RXHJ173523.7-354013 (Armas Padilla et al. 2013b), XTE J1709-267(Degenaar et al. 2013), Swift J174805.3-24463 (Bahramian et al.2014), IGR J17494-3030 (Armas Padilla et al. 2013c), and XTEJ1719-291 (Armas Padilla et al. 2011) are once reported to be fit-ted with both the single power-law model and the two-componentmodel, i.e., a thermal soft X-ray component at lower energies plusa power-law component at higher energies. Specifically, it is oftenfound that if the high quality
XMM − Newton
X-ray data can beavailable, the spectral fitting results can be significantly improvedwith a thermal soft X-ray component added to the power-law com-ponent, especially in the range of L . − ∼ − erg s − . Asfor the relation of Γ versus L . − we focus on in this paper, weselect the data for the X-ray spectra fitted with a single power law as in Wijnands et al. (2015) and Parikh et al. (2017). The power-lawX-ray photon index Γ for all the data of the sources in the samplehas an error less than 0.5 (which results in that several publishedsources with the measurements of Γ are not included in this pa-per). One can refer to the blue, dark green, dark yellow, dark cyan,dark blue and cyan blue symbols in Fig. 4 for the observationaldata of Γ versus L . − for details. It is clear that there is ananti-correlation between Γ and L . − (except for IGR J17361-4441). Γ increases from ∼ . ∼ L . − decreasing from ∼ erg s − to ∼ erg s − . The best-fitting linear regressionwith the least square method for the observational data between Γ and L . − gives, Γ = − . × log L . − + . , (2)where the errors of the data are not considered. One can refer tothe black solid line in Fig. 4 for clarity. One should also note thatin fitting the data between Γ and L . − , we do not add IGRJ17361-4441, which may reflect a population of NSs with distinctvery hard X-ray spectra, and will be discussed separately in Section4.2. We plot the theoretical results of Γ versus L . − in Fig. 4for comparisons. One can refer to the red points of ‘ + ’, ‘ ∗ ’, ‘ ⋄ ’, ‘ △ ’,‘ (cid:3) ’ and the ‘red filled-circle’ for f th = .
1, 0 .
03, 0 .
01, 0 .
005 0 . Γ versus L . − can be covered by the theoretical curves for taking f th as 0 . . f th . .
1. Specifically, from Fig. 4, it can be seenthat in the range of L . − ∼ − erg s − , there are threecrossing points between the theoretical curves of Γ versus L . − and equation 2, corresponding to the crossing point between equa-tion 2 and the theoretical curve from right to left with f = .
1, 0 . .
01 respectively. That is to say in the range of L . − ∼ − erg s − , f th is roughly in the range of 0 . . f th . . L . − ∼ − erg s − , although thereis only one crossing point between equation 2 and the theoreticalcurve corresponding to f th = . f th as 0 . . f th . . f th constrained by the anti-correlation between Γ and L . − in the range of L . − ∼ − erg s − , we further investigate the observational data of T ∗ versus L . − and η versus L . − for comparison. Specifi-cally, we plot the observational data of T ∗ versus L . − in panel(1) of Fig. 5, and η versus L . − in panel (2) of Fig. 5 respec-tively for the seven sources in the sample, i.e., AX J1754.2-2754,1RXS J171824.2-402934, 1RXH J173523.7-354013, XTE J1709-267, Swift J174805.3-24463, IGR J17494-3030 and XTE J1719-291, the high quality X-ray spectra of which are once reported to befitted with a two-component model, i.e., a thermal soft X-ray com-ponent added to the power-law component. The theoretical resultsof the ADAF model for T ∗ versus L . − and η versus L . − are also plotted in panel (1) and panel (2) of Fig. 5 respectively ascomparison. Specifically, the red points of ‘ + ’, ‘ ∗ ’, ‘ ⋄ ’, ‘ △ ’ and ‘ (cid:3) ’are the theoretical results for f th = .
1, 0 .
03, 0 .
01, 0 .
005 and 0 . L . − ∼ − erg s − , despite the scatter, the value of f th constrained byboth the diagram of T ∗ versus L . − and η versus L . − areroughly consistent with that of constrained by the anti-correlationbetween Γ and L . − , i.e., 0 . . f th . . f th = . η decreases from ∼
60% to ∼
20% for L . − decreasing from ∼ erg s − to ∼ erg s − , and for f th = . η decreases from ∼ ∼
15% for L . − decreasing from ∼ erg s − to ∼ MNRAS , 1–12 (2020) nti-correlation between Γ and L . − Figure 1.
Panel (1): Emergent spectra of the ADAF around a weakly magnetized NS for f th = .
1. Panel (2): Emergent spectra of the ADAF around a weaklymagnetized NS for f th = .
03. Panel (3): Emergent spectra of the ADAF around a weakly magnetized NS for f th = .
01. Panel (4): Emergent spectra of theADAF around a weakly magnetized NS for f th = . f th = . f th =
0. In all the calculations, we take m = . R ∗ =
10 km, α = β = . erg s − . That is to say, in general, η decreases with decreas-ing L . − for f th = .
003 and 0 .
005 respectively in the range of L . − ∼ − erg s − . Further, combining the result in Sec-tion 3.1 that Γ in is smaller than Γ , and Γ in as a function of L . − is significantly deviated from Γ as a function of L . − , we con-clude that the softening of the X-ray spectrum is due to the increaseof the fractional contribution of the thermal soft X-ray componentin the range of L . − ∼ − erg s − .While in the range of L . − ∼ − erg s − , the case is a little complicated. In the diagram of T ∗ versus L . − , theobserved T ∗ is slightly higher than that of the model predictions,for which we think it is very possible that we do not consider thethermal emission by the crust cooling of the NS itself in the modelcalculations. And further due to the change of T ∗ is always in a verynarrow range, we think T ∗ is not good tracer for constraining thevalue of f th . In the diagram of η versus L . − , as we can see frompanel (2) of Fig. 5, in the range of L . − ∼ − erg s − ,the observed value of η is generally greater than 50%, meaning that MNRAS , 1–12 (2020)
Erlin Qiao and B.F. Liu
Table 1.
Radiative features of the ADAF around a weakly magnetized NS for di ff erent ˙ m with f th = . , . , . , . , .
003 and 0 respectively. Γ is theX-ray photon index between 0.5 and 10 keV obtained by fitting the X-ray spectrum with a single power law. T ∗ is the e ff ective temperature at the surface ofthe NS. η ≡ L power law0 . − / L . − is the fractional contribution of the power-law luminosity. Γ in is the photon index of the intrinsic power-law componentobtained by analyzing the X-ray spectrum between 0.5 and 10 keV with a thermal soft X-ray component added to the power-law component. L . − is theluminosity between 0.5 and 10 keV. m = . R ∗ =
10 km, α = β = .
95 and ν NS = f th ˙ m Γ T ∗ (keV) η (%) Γ in L . − (erg s − )0.1 7 . × − . × . × − . × . × − . × . × − . × . × − . × . × − . × . × − . × . × − . × . × − . × . × − . × . × − . × . × − . × . × − . × . × − . × . × − . × . × − . × . × − . × . × − . × . × − . × . × − . × . × − . × . × − . × . × − . × . × − . × . × − . × . × − . × . × − . × . × − . × . × − . × . × − . × . × − . × . × − . × in this luminosity range the X-ray spectrum is dominant by a powerlaw. This requires a smaller value of f th . . . . f th . . Γ and L . − . Here we would like to men-tion that, based on our sample, only very few observational pointsfall in this luminosity range in the diagram of η versus L . − .We expect that more high quality XMM − Newton
X-ray data ofthe sources in our sample (even other sources) can be available toconfirm such a discrepancy (or not) in the future. We would liketo address further, if the constrained value of f th is uncertain, theexplanation for the softening of the X-ray spectrum is uncertain ac-cordingly in the range of L . − ∼ − erg s − . One canrefer to Section 4.1 for the detailed discussions.As a comparison, the observational data of Γ versus L . − for several BH X-ray transients are plotted in Fig. 4. One can referto the gray points in Fig. 4. The data are summarized at Table 2 ofWijnands et al. (2015), including the BH sample of Plotkin et al.(2013), as well as other six BH X-ray transients, i.e., Swift J1357.2-0933 (Armas Padilla et al. 2013a), Swift J1753.5-0217,Swift J1753.5-0217 and GRO J1655-40 (Reis et al. 2010), MAXIJ1659-152 (Jonker et al. 2012) and H1743-322 (Jonker et al. 2010).The error of Γ for the BH sample is also less than 0.5 as that ofthe NS sample in this paper. The BH sample covers a wide X-ray luminosity range from a few times 10 erg s − down to a fewtimes 10 erg s − . In general, there is an anti-correlation between Γ and L . − in the range of L . − ∼ − erg s − . Γ increases from ∼ . ∼ L . − decreasing from ∼ erg s − to ∼ erg s − . Below the X-ray luminosity of L . − ∼ erg s − , the X-ray spectra gradually level o ff at anaveraged value of < Γ > ≈ .
1. In the X-ray luminosity range of L . − ∼ − erg s − that we focus on in this paper, it canbe seen that, generally the X-ray spectra for the BH X-ray transientsare harder than that of the NS-LMXBs. Especially, in the range of L . − ∼ − erg s − , the separation between the BHX-ray transients and the NS-LMXBs is very clear, i.e., the X-rayspectra of the BH X-ray transients are much harder than that of the MNRAS , 1–12 (2020) nti-correlation between Γ and L . − Figure 2.
X-ray photon index Γ (black symbol, obtained by fitting the X-ray spectrum of the model between 0.5 and 10 keV with a single powerlaw) as a function of the X-ray luminosity L . − , and photon index ofthe intrinsic power-law component Γ in (blue symbol, obtained by analyzingthe X-ray spectrum of the model between 0.5 and 10 keV with a thermalsoft X-ray component added to the power-law component) as a function of L . − . For f th =
0, the value of Γ is same with Γ in , see the red filled-circle for Γ (or Γ in ) as a function of L . − . Figure 3.
Fractional contribution of the power-law luminosity η as a func-tion of the X-ray luminosity L . − . NS-LMXBs. While in the range of L . − ∼ − erg s − ,there is a significant overlap of the data points, and the separationbetween the BH X-ray transients and the NS-LMXBs is not as clearas that of in the range of L . − ∼ − erg s − .We plot the theoretical result of Γ versus L . − based onthe self-similar solution of the ADAF around a stellar-mass BH(Qiao & Liu 2018). One can refer to the black filled-circle for Γ versus L . − in Fig. 4. The corresponding emergent spectra canbe seen in Fig. 6. In the calculation, we take BH mass m = R in = R S (with R S being theSchwarzschild radius, and R S = . × m cm), viscosity param-eter α =
1, and the magnetic parameter β = .
95 as that we takefor NS case. As we can see that, the theoretical result of Γ versus L . − predicted by the ADAF around a BH can roughly matchthe observational data for Γ versus L . − . We notice that the theoretical results of Γ versus L . − for BH case is very closeto that of the NS case for taking f th =
0. It is easy to understandthat, if taking f th =
0, it means that there is no radiative feedbackbetween the surface of the NS and the ADAF, which is mathemat-ically equivalent to the BH case that all the energy of the ADAFcrossing the event horizon of the BH will be ‘vanished’. The verylittle di ff erence of the theoretical results between the BH case andthe NS case for taking f th = ff ects of the dif-ferent mass between BHs and NSs. Γ and L . − In this paper, we explain the observed anti-correlation between theX-ray photon index Γ and the X-ray luminosity L . − baseda sample of non-pulsating NS-LMXBs in the range of L . − ∼ − erg s − within the framework of the self-similar solutionof the ADAF around a weakly magnetized NS. We conclude that inthe range of L . − ∼ − erg s − , the softening of the X-ray spectrum is due to the increase of the fractional contribution ofthe thermal soft X-ray component, while in the range of L . − ∼ − erg s − , our explanation for the softening of the X-rayspectrum is uncertain.As we can see from Section 3.2, in the range of L . − ∼ − erg s − , by comparing with the observational data ofthe anti-correlation between Γ and L . − (i.e., equation 2), it issuggested that f th is roughly in the range of 0 . . f th . .
1. Inthis case, the softening of the X-ray spectra with L . − can beexplained with a complex pattern, which is as follows. Specifically,we can see from Fig. 4 that there are three crossing points betweenthe theoretical curves and equation 2 in the range of L . − ∼ − erg s − . From right to left, the X-ray luminositiesof the three crossing points are L . − ∼ . × erg s − ,4 . × erg s − and 1 . × erg s − respectively, and the theoret-ical value of η of the three crossing points are η ∼
20% for f th = . ∼
25% for f th = .
03, and ∼
40% for f th = .
01 respectively as canbe seen from Fig. 3. The value of η increases for the three crossingpoints with decreasing L . − , which means that the fractionalcontribution of the thermal soft X-ray component decreases withdecreasing L . − . Here we would like to address that the e ff ectof the thermal component on the value of Γ does not only depend onthe relative strength of the thermal component, but also depends onthe temperature of the thermal component. If the temperature of thethermal component T ∗ is above ∼ . T max being at ≈ . T ∗ in L ν versus ν ), an increase of the strength of thethermal component will make the spectrum harder, i.e., the value of Γ decreased. This is because if T ∗ is above ∼ . T ∗ is below ∼ . Γ increased, sincein this case more energy of the thermal emission will contribute tothe soft X-rays, which intrinsically makes the spectrum softer. Soalthough the fractional contribution of the thermal soft X-ray com-ponent decreases with decreasing L . − , it is possible that theX-ray spectrum softens if the temperature of the thermal compo-nent T ∗ decreases from a value above ∼ . ∼ . T ∗ decreaseswith decreasing L . − as can be easily interpolated from Table MNRAS , 1–12 (2020)
Erlin Qiao and B.F. Liu
Figure 4.
X-ray photon index Γ as a function of the X-ray luminosity L . − . The blue, dark green, dark yellow, dark cyan, dark blue and cyan bluesymbols are the observational data for the fifteen non-pulsating NSs. The black solid line refers to the best-fitting linear regression of the observational dataof the NSs (except for IGR J17361-4441). The red points of ‘ + ’, ‘ ∗ ’, ‘ ⋄ ’, ‘ △ ’, ‘ (cid:3) ’ and the ‘red filled-circle’ are the theoretical results of the ADAF around aweakly magnetized NS for f th = .
1, 0 .
03, 0 .
01, 0 .
005 0 .
003 and 0 respectively, and in the calculation m = . R ∗ =
10 km, α = β = .
95 are takenrespectively. All the gray points are the observational data for BHs. The black filled-circle points are the theoretical results of the ADAF around a BH, and inthe calculation m = R in = R S , α = β = .
95 are taken respectively. f th = .
1, 0 .
03 and 0 .
01 respectively. Meanwhile, for the threecrossing points, since the value of η is relatively small, the evolu-tion of Γ is dominated by the evolution of the thermal soft X-raycomponent. So based on our model, it is suggested that the evolu-tion of Γ is governed by a complex relation between the thermal softX-ray component and the power-law component, and the increaseof Γ (softening of the X-ray spectrum) with decreasing L . − isdominantly due to the decrease of the temperature of the thermalcomponent from a value above ∼ . ∼ . L . − ∼ − erg s − , the X-ray spectra of the sourcesin our sample are dominant by a power-law component. Someother observations may also support such a scenario. For exam-ple, Weng et al. (2015) studied the X-ray spectral evolution of theNS X-ray transient XTE J1810-189 in 2008 outburst decay, show-ing that the X-ray spectrum of the RXTE / PCA data (between 3-25keV) is dominant by a power-law component ( η & × erg s − to 6 × erg s − ,and the X-ray spectra soften with decreasing the X-ray luminosityfrom 4 × erg s − to 6 × erg s − . Further, in Weng et al.(2015), the authors confirmed that the softening of the X-ray spec-trum (between 3-25 keV) with decreasing the X-ray luminosityfrom 4 × erg s − to 6 × erg s − is due to the softeningof the intrinsic power-law component itself rather than the increaseof the thermal soft X-ray component. However, here we would liketo mention that since RXTE / PCA data do not cover the spectrumbelow 3 keV, it is not very clear how the spectra evolve if the softX-ray data between 0.5 and 3 keV are considered.Finally, we would like to mention that if it is universal that inthe range of L . − ∼ − erg s − the X-ray spectrumbetween 0.5 and 10 keV is dominant by a power-law component,we suggest that a small value of f th , i.e., 0 . . f th . . L . − ∼ − erg s − as in the rangeof L . − ∼ − erg s − for explaining the observation. Forexample, if we take f th = . MNRAS000
01 respectively. Meanwhile, for the threecrossing points, since the value of η is relatively small, the evolu-tion of Γ is dominated by the evolution of the thermal soft X-raycomponent. So based on our model, it is suggested that the evolu-tion of Γ is governed by a complex relation between the thermal softX-ray component and the power-law component, and the increaseof Γ (softening of the X-ray spectrum) with decreasing L . − isdominantly due to the decrease of the temperature of the thermalcomponent from a value above ∼ . ∼ . L . − ∼ − erg s − , the X-ray spectra of the sourcesin our sample are dominant by a power-law component. Someother observations may also support such a scenario. For exam-ple, Weng et al. (2015) studied the X-ray spectral evolution of theNS X-ray transient XTE J1810-189 in 2008 outburst decay, show-ing that the X-ray spectrum of the RXTE / PCA data (between 3-25keV) is dominant by a power-law component ( η & × erg s − to 6 × erg s − ,and the X-ray spectra soften with decreasing the X-ray luminosityfrom 4 × erg s − to 6 × erg s − . Further, in Weng et al.(2015), the authors confirmed that the softening of the X-ray spec-trum (between 3-25 keV) with decreasing the X-ray luminosityfrom 4 × erg s − to 6 × erg s − is due to the softeningof the intrinsic power-law component itself rather than the increaseof the thermal soft X-ray component. However, here we would liketo mention that since RXTE / PCA data do not cover the spectrumbelow 3 keV, it is not very clear how the spectra evolve if the softX-ray data between 0.5 and 3 keV are considered.Finally, we would like to mention that if it is universal that inthe range of L . − ∼ − erg s − the X-ray spectrumbetween 0.5 and 10 keV is dominant by a power-law component,we suggest that a small value of f th , i.e., 0 . . f th . . L . − ∼ − erg s − as in the rangeof L . − ∼ − erg s − for explaining the observation. Forexample, if we take f th = . MNRAS000 , 1–12 (2020) nti-correlation between Γ and L . − Figure 5.
Panel (1): E ff ective temperature at the surface of the NS T ∗ as a function of the X-ray luminosity L . − . The blue, dark green, dark yellow anddark cyan symbols are the observational data. The red points of ‘ + ’, ‘ ∗ ’, ‘ ⋄ ’, ‘ △ ’ and ‘ (cid:3) ’ are the theoretical results of the ADAF around a weakly magnetizedNS for f th = .
1, 0 .
03, 0 .
01, 0 .
005 and 0 .
003 respectively. Panel (2): Fractional contribution of the power-law luminosity η as a function of the X-ray luminosity L . − . The blue, dark green, dark yellow and dark cyan symbols are the observational data. The red points of ‘ + ’, ‘ ∗ ’, ‘ ⋄ ’, ‘ △ ’ and ‘ (cid:3) ’ are the theoreticalresults of the ADAF around a weakly magnetized NS for f th = .
1, 0 .
03, 0 .
01, 0 .
005 and 0 .
003 respectively.
Figure 6.
Emergent spectra of the ADAF around a BH. In the calculations,we take m = R in = R S , α = β = . and 10 keV is nearly completely dominant by a power-law compo-nent in the range of L . − ∼ − erg s − . In this case, thesoftening of the X-ray spectrum is dominantly due to the softeningof the power-law component itself. One can refer to the panel (5) inFig. 1 for clarity or Table 1 for the detailed numerical results. How-ever, as can be seen in Fig. 4, the theoretical relation of Γ versus L . − for f th = .
003 is a little deviated from the observations,i.e., the theoretical value of Γ is systemically below equation 2 inthe range of L . − ∼ − erg s − . We suggest that the dif-ference between the observational data and the model predictionsfor f th = .
003 may be related with both the observations (i.e., moreprecise observational data are necessary to confirmed the relationbetween Γ and L . − in this luminosity range in the future) andthe model. Especially, in this paper, we calculate the structure andcorresponding emergent spectrum of the ADAF within the frame-work of the self-similar solution, and do not consider the outflowof the ADAF. We can expect that at least if the outflow (wind) ofthe ADAF is considered, the spectrum will systematically become softer, consequently better matching the observations in this lumi-nosity range. Here we would like to mention that recent numericalsimulations of the hot accretion flow (i.e., ADAF) have shown thatoutflow is indeed existed around NSs (Bu et al. 2020), as that ofaround BHs (Yuan et al. 2012a,b). In general, numerical simula-tions of the ADAF show that the inflow mass rate ˙ m ( r ) ∝ r s , wherethe index s is roughly in the range of 0.5-1. That is to say if theoutflow is considered, the distribution of ˙ m in radial direction willbe changed, especially ˙ m in the inner region will decrease signif-icantly. The emission from the inner region of the ADAF mainlycontributes to hard X-rays. A decrease of ˙ m in the inner region willmake the emission in the hard X-ray band decreased, consequentlymaking the X-ray spectrum become softer. L . − ∼ − erg s − Observationally, an anti-correlation between Γ and L . − inthe range of L . − ∼ − erg s − in NS-LMXBs hasbeen proposed in (Wijnands et al. 2015), which is further confirmedin Parikh et al. (2017) and in the present paper by adding somemore sources in the sample. Meanwhile it is suggested that it isvery possible that such an anti-correlation between Γ and L . − is ‘universal’ for most NS-LMXBs in the range of L . − ∼ − erg s − (Wijnands et al. 2015; Parikh et al. 2017). How-ever, we should note that some very hard state source in the rangeof L . − ∼ − erg s − , such as IGR J17361-4441 indeeddose not observe such a so-called ‘universal’ anti-correlation. Ac-tually, a class of NSs with very hard X-ray spectra have been identi-fied (Parikh et al. 2017). In Parikh et al. (2017), the authors studiedthe X-ray spectra of six NS X-ray binaries, i.e., 1RXS J180408.9-342058, EXO 1745-248, IGR J18245-2452, SAX J1748.9-2021,IGR J17361-4441 and SAX J1808.4-3658, it is found that four outof the six sources, i.e., 1RXS J180408.9-342058, EXO 1745-248,IGR J18245-2452 and IGR J17361-4441 show very hard X-rayspectra with Γ ∼ N H is usedin the spectral fitting. IGR J18245-2452 is an accreting millisecondX-ray pulsar (AMXP), so the very hard X-ray spectrum is likely tobe related with the stronger magnetic field. The two sources, 1RXS MNRAS , 1–12 (2020) Erlin Qiao and B.F. Liu
J180408.9-342058 and EXO 1745-248 show very hard X-ray spec-tra in the range of L . − ∼ − erg s − .For IGR J17361-4441, as we can see from Fig. 4 (dark blue‘ ⋄ ’), in the range of L . − ∼ − erg s − , the relationof Γ versus L . − significantly deviates from the so-called ‘uni-versal’ anti-correlation. IGR J17361-4441 shows very hard X-rayspectra with Γ ∼ Γ with L . − , and the physical origin of the very hard X-ray spectraobserved in the range of L . − ∼ − erg s − for IGRJ17361-4441 are unclear, and needed to studied in detail in the fu-ture. Here we would like to mention that although the nature of IGRJ17361-4441 as a NS-LMXB can not be discarded (Wijnands et al.2015, for discussions), alternatively, IGR J17361-4441 is once ex-plained as a tidal disruption event of a planet sized body by a whitedwarf (Del Santo et al. 2014). Γ and L . − can beextended below ∼ erg s − and above ∼ erg s − ? In this paper, we focus on the anti-correlation between the X-rayphoton index Γ and the X-ray luminosity L . − in NS-LMXBsin the range of L . − ∼ − erg s − . Observation-ally, as has been discussed in Wijnands et al. (2015) that althoughthere is scatter for some individual sources, in general, it has beendemonstrated that there is a universal anti-correlation between Γ and L . − in the range of L . − ∼ − erg s − . How-ever, whether the anti-correlation between Γ and L . − can beextended below ∼ erg s − and above ∼ erg s − are uncer-tain. When the X-ray luminosity is below ∼ erg s − , NS-LMXBs are often regarded as in the quiescent state, during whichthe X-ray spectra are very complex and diverse. The X-ray spec-tra in the quiescent state can be, (1) completely dominated by athermal soft X-ray component, or (2) completely dominated bya power-law component, or (3) described by the two-componentmodel, i.e., a thermal soft X-ray component plus a power-lawcomponent (Wijnands et al. 2017, for review). We notice a re-cent paper by Sonbas et al. (2018), in which the authors compileda sample composed of twelve non-pulsating NS-LMXBs (twelvedata points) in the range of L . − ∼ − erg s − .Six data points in the sample fall in the range of L . − ∼ − erg s − , and the X-ray spectra can be satisfactorilydescribed by a single power law. Finally, it is found that there isan anti-correlation between the X-ray photon index Γ and the X-ray luminosity L . − , and the slope of the anti-correlation is − . ± .
63, which is obviously steeper than the slope of − . L . − ∼ − erg s − . Inthe range of L . − ∼ − erg s − , the X-ray spec-tra of the sources in the sample of Sonbas et al. (2018) are verysoft. Γ increases from ∼ .
87 to ∼ . L . − decreasingfrom ∼ erg s − to ∼ erg s − . Although in Sonbas et al.(2018), the authors proposed a steeper anti-correlation between Γ and L . − for L . − . erg s − , we think that the re-lation of Γ versus L . − for L . − . erg s − is stilluncertain, which strongly depends on the sample selections (i.e.,which kinds of X-ray spectra selected). The physical origin of theX-ray spectra for L . − . erg s − is also uncertain, whichcould be dominated by crust cooling (e.g. Brown et al. 1998), mag-netospheric emission (e.g. Stella et al. 1994; Campana et al. 2002;Burderi et al. 2003), or low-level accretion (e.g., ADAF) onto NSs(e.g. Zampieri et al. 1995; Qiao & Liu 2020). In our opinion, it isvery necessary to fit the high quality X-ray spectra of NS-LMXBs for L . − . erg s − in detail to distinguish the di ff er-ent physical mechanisms, which however exceeds the scope of thepresent paper. In a word, we think that the relation of Γ versus L . − for L . − below ∼ erg s − is still needed to bestudied in detail in the future.When the X-ray luminosity is above ∼ erg s − , e.g. inthe range of L . − ∼ − erg s − , in general, the X-ray photon index Γ is ∼ . − L . − ∼ − erg s − .Specifically, by fitting the Swift
X-ray spectra of these three sourcesbetween 0.5 and 10 keV with a single power law, the authors foundthat the X-ray photon index of these three sources is very low, i.e., Γ ∼ . −
1, in the range of L . − ∼ − erg s − . The iden-tification of the unusually hard X-ray spectra makes the relation be-tween Γ and L . − in the range of L . − ∼ − erg s − very complicated. So if the anti-correlation correlation proposedin Wijnands et al. (2015) (also in this paper) can be extended tothe range of L . − ∼ − erg s − , the scatter is verylarge. As discussed in Parikh et al. (2017), the identified unusuallyhard X-ray spectra may represent a new distinct spectral state. Ifsuch a very hard X-ray spectrum can be explained with the typicalComptonization model, it requires a higher electron temperature ora higher Compton scattering optical depth compared with the typi-cal hard state of NS-LMXBs in this luminosity range. The physicalorigin for the higher electron temperature or the higher Comptonscattering optical depth for producing the X-ray photon index of Γ ∼ . − f th and the radiative e ffi ciency of weaklymagnetized NSs with an ADAF accretion In our model for the ADAF accretion around a weakly magnetizedNS, there is a key parameter, f th , describing the fraction of theADAF energy released at the surface of the NS as thermal emis-sion to be scattered in the ADAF. The value of f th is very impor-tant for determining the feedback between the NS and the ADAF,consequently a ff ecting the radiative e ffi ciency of the ADAF accre-tion around a NS. However, physically, due to our relatively poorknowledge on the interaction between the NS and the accretionflow under the extreme gravitational field of NS, the value of f th isuncertain, which is probable to be related with the state of matter,the magnetic field, as well as the thermodynamics of the accretionflow at the surface of the NS (Wijnands et al. 2015, for discussions,and the references therein).As we have mentioned previously, in our model of the ADAFaccretion, basically, there are two components for the X-ray spec-trum between 0.5 and 10 keV, i.e., a thermal soft X-ray componentand a power-law component. In a companion paper Qiao & Liu(2020), as the zeroth order approximation for testing the modelpredictions, we theoretically investigate the correlation between thefractional contribution of the power-law component η and the X-rayluminosity L . − for a sample of NS-LMXBs probably domi-nated by low-level accretion onto NSs in a wider X-ray luminosityrange from L . − ∼ erg s − to ∼ erg s − . It is foundthat a small value of f th , i.e., f th . . η and L . − . In this paper, we furthertest the model predictions by explaining the observed Γ − L . − anti-correlation based on a sample of non-pulsating NS-LMXBs in MNRAS , 1–12 (2020) nti-correlation between Γ and L . − the range of L . − ∼ − erg s − . We conclude that f th is between 0.003 and 0.1, which further confirms the previous con-clusion in Qiao & Liu (2020), i.e., f th . .
1. The small value of f th . . ffi ciency of weakly magnetized NSswith an ADAF accretion is not as high as the generally proposedvalue of ǫ ∼ ˙ MGMR ∗ / ˙ Mc ∼ .
2. We would like to mention thatthere is a very interesting paper D’Angelo et al. (2015), in whichthe authors investigated the radiative e ffi ciency of a NS X-ray tran-sient Cen X-4 by fitting its broad band spectrum at a luminosity of L X ∼ erg s − . In general, our ADAF model predicts a simi-lar radiative e ffi ciency as that of D’Angelo et al. (2015), one canrefer to Section 4.3 in Qiao & Liu (2020) for detailed discussions.As discussed in Section 4.3 of Qiao & Liu (2020), it is very pos-sible that the remaining fraction, i.e., 1- f th , of the ADAF energytransferred onto the surface of the NS could be converted to therotational energy of the NS. In this work, we explain the observed anti-correlation between theX-ray photon index Γ (obtained by fitting the X-ray spectrum be-tween 0.5 and 10 keV with a single power law) and the X-rayluminosity L . − in NS-LMXBs in the range of L . − ∼ − erg s − within the framework of the self-similar so-lution of the ADAF around a weakly magnetized NS. We showthat the ADAF model intrinsically can predict an anti-correlationbetween Γ and L . − . We test the e ff ect of a key parame-ter, f th , describing the fraction of the ADAF energy released atthe surface of the NS as thermal emission to be scattered inthe ADAF, on the anti-correlation between Γ and L . − . Wefound that the value of f th can significantly a ff ect the slope of the Γ − L . − anti-correlation. Specifically, the anti-correlation be-tween Γ and L . − becomes flatter with decreasing f th as taking f th = . , . , . , . .
003 and 0 respectively. By comparingwith a sample of non-pulsating NS-LMXBs with well measured Γ and L . − , it is found that the value of 0 . . f th . . Γ − L . − anti-correlation. Fi-nally, we argue that the small value of f th . . ffi ciency of NSs with an ADAF accretion may not be as high as ǫ ∼ ˙ MGMR ∗ / ˙ Mc ∼ . f th , of the ADAF energy transferredonto the surface of the NS, we suggest that one of the promisingpossibilities is that such energy could be converted to the rotationalenergy of the NS, the test of which is still needed in the future. ACKNOWLEDGMENTS
This work is supported by the National Natural Science Founda-tion of China (Grants 11773037 and 11673026), the gravitationalwave pilot B (Grants No. XDB23040100), the Strategic PioneerProgram on Space Science, Chinese Academy of Sciences (GrantNo. XDA15052100) and the National Program on Key Researchand Development Project (Grant No. 2016YFA0400804).
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