The radiative efficiency of neutron stars at low-level accretion
aa r X i v : . [ a s t r o - ph . H E ] J a n MNRAS , 1–9 (2020) Preprint 27 January 2021 Compiled using MNRAS L A TEX style file v3.0
The radiative e ffi ciency of neutron stars at low-level accretion Erlin Qiao , ⋆ and B.F. Liu , Key Laboratory of Space Astronomy and Technology, National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100101, 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
When neutrons star low-mass X-ray binaries (NS-LMXBs) are in the low-level accretionregime (i.e., L X . erg s − ), the accretion flow in the inner region around the NS isexpected to be existed in the form of the hot accretion flow, e.g., the advection-dominatedaccretion flow (ADAF) as that in black hole X-ray binaries. Following our previous studies inQiao & Liu 2020a and 2020b on the ADAF accretion around NSs, in this paper, we investigatethe radiative e ffi ciency of NSs with an ADAF accretion in detail, showing that the radiativee ffi ciency of NSs with an ADAF accretion is much lower than that of ǫ ∼ ˙ MGMR ∗ / ˙ Mc ∼ . L X (e.g., be-tween 0.5 and 10 keV), ˙ M calculated by ˙ M = L X R ∗ GM is lower than the real ˙ M calculatedwithin the framework of the ADAF accretion. The real ˙ M can be more than two orders ofmagnitude higher than that of calculated by ˙ M = L X R ∗ GM with appropriate model parame-ters. Finally, we discuss that if applicable, the model of ADAF accretion around a NS canbe applied to explain the observed millisecond X-ray pulsation in some NS-LMXBs (such asPSR J1023 + erg s − , since at this X-ray luminosity the calculated ˙ M with the model ofADAF accretion can be high enough to drive a fraction of the matter in the accretion flow tobe channelled onto the surface of the NS forming the X-ray pulsation. Key words: accretion, accretion discs – stars: neutron – black hole physics – X-rays: binaries
Currently, there are two types of accretion flow around compact ob-jects [black holes (BH) and neutron stars (NS)], i.e., the geometri-cally thin, optically thick, cold accretion disc (Shakura & Sunyaev1973), and the geometrically thick, optically thin, hot accretionflow, such as the advection-dominated accretion flow (ADAF)(Yuan & Narayan 2014, for review). The cold accretion disc with ahigher mass accretion rate is widely used to explain the optical / UVemission in luminous active galactic nuclei (AGNs), and the X-rayemission of X-ray binaries at the high / soft state (e.g. Mitsuda et al.1984; Makishima et al. 1986). While the ADAF with a lower massaccretion rate is often used to explain the dominant emission inlow-luminosity AGNs, as well as the low / hard and the quiescentstate of X-ray binaries (Done et al. 2007, for review).In general, the cold accretion disc is a kind of radiatively ef-ficient accretion flow in both BH case and NS case. In the approx-imation of the Newtonian mechanics, for a non-rotating BH, theradiative e ffi ciency of the accretion disc is ǫ =
12 ˙
MGM R S / ˙ Mc ∼ . G being the gravitational constant, ˙ M being the mass accre-tion rate in units of g s − , c being the speed of light, and R S being the ⋆ E-mail: [email protected]
Schwarzschild radius with R S = GM / c ≈ . × M / M ⊙ cm),i.e., half of the gravitational energy will be released out in the formof the electromagnetic radiation in the accretion disc. While for aNS, the radiative e ffi ciency of the accretion is ǫ = ˙ MGMR ∗ / ˙ Mc ∼ . M = . M ⊙ , and R ∗ = . ), which is obtained asthat half of the gravitational energy will be released out in the ac-cretion disc and the other half of the gravitational energy will bereleased out in a thin boundary layer between the accretion discand the surface of the NS in the form of the electromagnetic radia-tion (Gilfanov & Sunyaev 2014).In general, the ADAF solution is a kind of radiatively in-e ffi cient accretion flow in BH case. In the approximation of theNewtonian mechanics, for a non-rotating BH, the radiative e ffi -ciency of the ADAF is ǫ <
12 ˙
MGM R S / ˙ Mc ∼ . We take R ∗ = . ffi ciency of thegravitational energy release between a NS and a non-rotating BH. If the NSmass m = . R ∗ = . © Erlin Qiao and B.F. Liu out radiation. The fraction of the viscously dissipated energy storedin the gas of the ADAF as the internal energy is dependent on ˙ M ,and this fraction increases with decreasing ˙ M , which means thatthe radiative e ffi ciency of the ADAF around a BH decreases withdecreasing ˙ M . Specifically, when ˙ M is close the critical mass accre-tion rate ˙ M crit of the ADAF ( ˙ M crit ∼ α ˙ M Edd , with α being the vis-cosity parameter, and ˙ M Edd = L Edd / . c ≈ . × M / M ⊙ g s − being the Eddington scaled mass accretion rate, where L Edd is de-fined as L Edd = . × M / M ⊙ erg s − ), the value of ǫ is closeto 0.1 as that of the cold accretion disc around a BH (Xie & Yuan2012). However, when ˙ M is significantly less than ˙ M crit , the radia-tive e ffi ciency ǫ decreases dramatically with decreasing ˙ M , and thevalue of ǫ is much less than 0.1 (Xie & Yuan 2012, for discussions).In this paper, we focus on the radiative e ffi ciency of the ADAFsolution around a NS (strictly speaking, it is the radiative e ffi ciencyof NSs with an ADAF accretion, since a fraction of the ADAFenergy released at the surface of the NS can finally radiate outto be observed). The dynamics of the ADAF around a weaklymagnetized NS has been investigated by some authors previ-ously (Medvedev & Narayan 2001; Medvedev 2004; Narayan & Yi1995). In general, one of the most di ffi cult problems for the study ofthe ADAF around a NS is how to treat the dynamics and radiationof the boundary layer between the surface of the NS and the ADAF.The physics of the boundary layer can significantly a ff ect the globaldynamics and radiation of the ADAF (Medvedev & Narayan 2001;Medvedev 2004; D’Angelo et al. 2015). Recently, in a series of pa-pers, i.e., Qiao & Liu (2018), Qiao & Liu (2020a), and Qiao & Liu(2020b), we study the dynamics and the radiation of the ADAFaround a weakly magnetized NS in the framework of the self-similar solution of the ADAF by simplifying the physics of theboundary layer. Specifically, we introduce a parameter, f th , whichdescribes the fraction of the ADAF energy released at the surfaceof the NS as thermal emission to be scattered in the ADAF. Underthis assumption, i.e., considering the radiative feedback betweenthe surface of the NS and the ADAF, we self-consistently calculatethe structure and the corresponding emergent spectrum of the NSwith an ADAF accretion. The value of f th can a ff ect the radiativee ffi ciency of NSs with an ADAF accretion. Physically, the value of f th is uncertain. However, it has been shown that the value of f th canbe constrained in a relatively narrow range by comparing with theobserved X-ray spectra (typically between 0.5 and 10 keV) of neu-tron star low-mass X-ray binaries (NS-LMXBs), since the value of f th can a ff ect both the shape and the luminosity of the X-ray spectra(Qiao & Liu 2020a,b). The results in Qiao & Liu (2020a,b) jointlysuggest that the value of f th is certainly less than 0.1, and a smallervalue of f th ∼ .
01 is more preferred.In this paper, following the results of Qiao & Liu (2020a,b)for the constraints to the value of f th , we investigate the radia-tive e ffi ciency of NSs with an ADAF accretion for taking twotypical values of f th as that of f th = .
1, and f th = .
01 re-spectively. The radiative e ffi ciency is defined as ǫ bol = L bol / ˙ Mc (with L bol being the bolometric luminosity). Based on the emer-gent spectra of NSs with an ADAF accretion for the bolometricluminosity, we find that ǫ bol is nearly a constant with ˙ M for ei-ther taking f th = . f th = .
01. The value of ǫ bol for f th = . ǫ ∼ ˙ MGMR ∗ / ˙ Mc ∼ .
2, and ǫ bol for f th = .
01 is roughly two orders of magnitude lower than the valueof ǫ ∼ .
2. Then, we suggest that NSs with an ADAF accretion isradiatively ine ffi cient despite the existence of the hard surface. Fur-ther, we investigate the radiative e ffi ciency in some specific bands,e.g., ǫ . − and ǫ . − (defined as ǫ . − = L . − / ˙ Mc and ǫ . − = L . − / ˙ Mc ). ǫ . − is nearly same with ǫ . − for a fixed ˙ M with f th = . f th = .
01 respec-tively. ǫ . − (or ǫ . − ) is less than ǫ bol , so is certainly lessthan ǫ ∼ ˙ MGMR ∗ / ˙ Mc ∼ .
2. Meanwhile, ǫ . − (or ǫ . − )decreases very quickly with decreasing ˙ M . As a result, for a NS-LMXB, if we intend to use the observed X-ray luminosity (e.g.,between 0.5 and 10 keV) as the indicator for ˙ M , ˙ M calculated withthe formula of ˙ M = L X R ∗ GM is lower than that of calculated withour model of ADAF accretion around a NS. Obviously, given a X-ray luminosity, the di ff erence between the ˙ M calculated with ourmodel of ADAF accretion and the ˙ M calculated with the formulaof ˙ M = L X R ∗ GM depends on f th , with the di ff erence of the calculated˙ M increases with decreasing f th .Finally, in this paper, we argue that if applicable, the model ofADAF accretion around a NS can probably be used to explain theobserved millisecond X-ray pulsation in some NS-LMXBs (suchas PSR J1023 + erg s − , since at this X-ray luminosity the calculated ˙ M withthe model of ADAF accretion can be high enough, e.g., more thantwo orders of magnitude higher than that of calculated with theformula of ˙ M = L X R ∗ GM for taking f th = .
01, to drive a fraction ofthe matter in the accretion flow to be channelled onto the surfaceof the NS forming the X-ray pulsation. A brief summary on theADAF model around a NS and the constraints to the value of f th in Qiao & Liu (2020a,b) are introduced in Section 2. The resultsare shown in Section 3. The discussions are in Section 4 and theconclusions are in Section 5. The structure and the corresponding emergent spectra of the ADAFaround a NS are strictly investigated within the framework ofthe self-similar solution of the ADAF (Qiao & Liu 2018). InQiao & Liu (2020a,b), we update the code with the e ff ect of theNS spin considered compared with that of in Qiao & Liu (2018).In our model, there are seven parameters, i.e., the NS mass m ( m = M / M ⊙ ), NS radius R ∗ , NS spin frequency ν NS , mass ac-cretion rate ˙ m ( ˙ m = ˙ M / ˙ M Edd ), as well as the viscosity param-eter α , and the magnetic parameter β [with magnetic pressure p m = B / π = (1 − β ) p tot , p tot = p gas + p m ] for describing the mi-crophysics of the ADAF. The last parameter is, f th , describing thefraction of the ADAF energy released at the surface of the NS asthermal emission to be scattered in the ADAF to cool the ADAF it-self, which consequently controls the feedback between the ADAFand the NS. We always take m = .
4, and R ∗ in the range of 10-12.5km (Degenaar & Suleimanov 2018) [ R ∗ = . R ∗ =
10 km in Qiao & Liu (2020b)]. In general, ithas been proven that the e ff ect of the NS spin frequency ν NS onthe structure and the emergent spectra of the ADAF around a NS isvery little, and nearly can be neglected [see Figure 8 of Qiao & Liu(2020a) for taking ν NS = , , ,
700 Hz respectively]. So wefix ν NS = β = . L . − . erg s − ) can be described by a sin-gle power-law model, or a two-component model, i.e., a thermalsoft X-ray component plus a power-law component. In general,if the Swift
X-ray data are used, the spectral fitting with a sin-
MNRAS , 1–9 (2020) adiative e ffi ciency at low-level accretion gle power-law model can return an accepted fit. And if the high-quality XMM − Newton
X-ray data are used, the spectra fittingwith a two-component model can significantly improve the fit-ting results in some X-ray luminosity range , e.g., in the rangeof L . − ∼ − erg s − (e.g. Wijnands et al. 2015). InQiao & Liu (2020a), we test the e ff ect of α and f th on the X-rayspectra between 0.5 and 10 keV, and explain the fractional contribu-tion of the power-law component η ( η ≡ L power law0 . − / L . − ) (withthe spectra fitted with the two-component model) as a function ofthe L . − for a sample of non-pulsating NS-LMXBs in a widerange from L . − ∼ − erg s − . Observationally, thereis a positive correlation between η and L . − for L . − & a few times of 10 erg s − , and an anticorrelation between η and L . − for L . − . a few times of 10 erg s − . By comparingwith the observed correlation (both the positive correlation and theanticorrelation) between η and L . − , it is found that the e ff ectof α on the correlation between η and L . − is very little, andnearly can be neglected. Meanwhile, it is found that the correlationbetween η and L . − can be well matched by adjusting the valueof f th . The value of f th is constrained to be less than 0.1. Especially, f th = .
01 is more preferred [see Figure 7 of Qiao & Liu (2020a)].Further, in Qiao & Liu (2020b), based on the sample of non-pulsating NS-LMXBs in Wijnands et al. (2015), and adding somemore non-pulsating NS-LMXBs from Parikh et al. (2017) andBeri et al. (2019), we explain the anticorrelation between the X-ray photon index Γ (obtained by fitting the X-ray spectra between0.5 and 10 keV with a single power law) and L . − , i.e., thesoftening of the X-ray spectra with decreasing L . − , in therange of L . − ∼ − erg s − by adjusting the valueof f th . Moreover, it is shown that a fraction of the sources inQiao & Liu (2020b) are once reported to be fitted with the two-component model (with XMM − Newton
X-ray data), i.e., a ther-mal soft X-ray component plus a power-law component. Com-bining the explanations for the anticorrelation between the X-rayphoton index Γ and L . − with the X-ray spectra analyzedwith the single power-law model, and the positive correlation be-tween η and L . − with the two-component model for a frac-tion of the sources in the sample, we conclude that in the range of L . − ∼ − erg s − , the softening of the X-ray spectrais due to the increase of the thermal soft X-ray component, whilein the range of L . − ∼ − erg s − , the softening ofthe X-ray spectra is probably due to the evolution of the power-lawcomponent itself. As a summary, in the study above for explain-ing the anticorrelation between Γ and L . − , it has been shownthat the value of f th can be constrained to be less than 0.1, andis very probably to be much smaller values, i.e., ∼ . − . m = . R ∗ = . ν NS = α = . β = .
95, weinvestigate the radiative e ffi ciency of NSs with an ADAF accretionfor di ff erent ˙ m by taking two typical values of f th , i.e., f th = . f th = .
01 respectively.
We plot the emergent spectra of NSs with an ADAF accretion fordi ff erent ˙ m with f th = . f th = . L . − , the X-ray luminosity between0.5 and 100 keV L . − , and the bolometric luminosity L bol fordi ff erent ˙ m with f th = .
1. Based on the emergent spectra in panel(2) of Fig. 1, a similar calculation is done for L . − , L . − ,and L bol for di ff erent ˙ m with f th = .
01. In panel (1) of Fig. 2, weplot L . − , L . − , and L bol as a function of ˙ m for f = . f = .
01 respectively. Specifically, for f = .
1, it can be seenthat, all the three quantities L . − , L . − , and L bol decreasewith decreasing ˙ m . Meanwhile, it can be seen that, for f = . L . − as a function of ˙ m is nearly overlapped with L . − asa function of ˙ m . This is because all the X-ray spectra are very softfor di ff erent ˙ m with f th = .
1, the value of L . − and L . − is nearly same for a fixed ˙ m . It also easy to see that, for f = .
1, thevalue of L bol is always greater than L . − (or L . − ). Mean-while, the separation between L bol and L . − (or L . − ) be-comes larger and larger with decreasing ˙ m . In general, the trends of L . − , L . − , and L bol as a function of ˙ m for f = .
01 aresimilar to that of for f = . L . − , L . − , and L bol for f = .
01 are systematically lowerthan that of for f = . m respectively. We further plot the formula L = ˙ MGMR ∗ (note: ˙ M = ˙ m ˙ M Edd , M = mM ⊙ ) as a comparison, one can referto the dashed line in panel (1) of Fig. 2 for clarity. It can be seenthat all the three luminosities, i.e., L . − , L . − and L bol , arelower than the luminosity calculated with the formula of L = ˙ MGMR ∗ for a fixed ˙ M (or ˙ m ).We define three quantities for the radiative e ffi ciency in di ff er-ent bands, i.e., ǫ . − = L . − / ˙ Mc , (1) ǫ . − = L . − / ˙ Mc , (2) ǫ bol = L bol / ˙ Mc . (3)In panel (2) of Fig. 2, we plot ǫ . − , ǫ . − and ǫ bol as a func-tion of ˙ m with f = .
1, and f = .
01 respectively. Specifically, for f th = . ǫ bol is nearly a constant for di ff erent ˙ m with ǫ bol ∼ . ǫ . − as a function of ˙ m is nearly overlapped with ǫ . − asa function of ˙ m . For f th = . ǫ . − decreases from ∼ . ∼ . × − for ˙ m decreasing from 1 . × − to 1 . × − ,and ǫ . − decreases from ∼ .
016 to ∼ . × − for ˙ m de-creasing from 1 . × − to 1 . × − . In general, the trends of ǫ . − , ǫ . − and ǫ bol as a function of ˙ m for f th = .
01 aresimilar to that of for f th = .
1. For f th = . ǫ bol is also nearlya constant, decreasing slightly with decreasing ˙ m , i.e., ǫ bol decreas-ing from ∼ .
006 to ∼ .
002 for ˙ m decreasing from 1 . × − to 5 . × − . For f th = . ǫ . − as a function of ˙ m is alsonearly overlapped with ǫ . − as a function of ˙ m . Specifically, ǫ . − decreases from ∼ .
002 to ∼ . × − for ˙ m decreasingfrom 1 . × − to 5 . × − . ǫ . − decreases from ∼ .
004 to ∼ . × − for ˙ m decreasing from 1 . × − to 5 . × − . Also asa comparison, we plot the radiative e ffi ciency ǫ calculated with theformula of ǫ ∼ ˙ MGMR ∗ / ˙ Mc , see the dashed line in panel (2) of Fig.2. The value of ǫ is ∼ .
2, which is roughly one order of magnitudehigher than ǫ bol for f = . ǫ bol for f = . f th = . f th = .
01, the predicted luminosity, i.e., L . − , L . − and L bol from our model of ADAF accretion, are all lower thanthe luminosity predicted by the formula of L = ˙ MGMR ∗ for a fixed˙ M (or ˙ m ). This in turn means that, given a value of L . − , MNRAS000
01, the predicted luminosity, i.e., L . − , L . − and L bol from our model of ADAF accretion, are all lower thanthe luminosity predicted by the formula of L = ˙ MGMR ∗ for a fixed˙ M (or ˙ m ). This in turn means that, given a value of L . − , MNRAS000 , 1–9 (2020)
Erlin Qiao and B.F. Liu L . − or L bol , the obtained ˙ M (or ˙ m ) from our model ofADAF accretion is greater than that of calculated with the for-mula of ˙ M = L R ∗ GM ( L can be L . − , L . − or L bol ). Herewe just take two X-ray luminosities between 0.5 and 10 keV, i.e., L . − = . × erg s − and L . − = . × erg s − as examples for calculating ˙ M . One can refer to Fig. 3 for the illus-trations and Table 1 for the detailed numerical results. Specifically,for L . − = . × erg s − , the mass accretion rate ˙ M cal-culated with the formula of ˙ M = L . − R ∗ GM is 2 . × g s − ,which is ∼
27 times less than the mass accretion rate ˙ M . cal-culated with our model of ADAF accretion for taking f th = . ∼
192 times less than the mass accretion rate ˙ M . calcu-lated with our model of ADAF accretion for taking f th = .
01. For L . − = . × erg s − , ˙ M calculated with the formula of˙ M = L . − R ∗ GM is 3 . × g s − , which is ∼
23 times lessthan ˙ M . calculated with our model of ADAF accretion for tak-ing f th = .
1, and is ∼
183 times less than ˙ M . calculated withour model of ADAF accretion for taking f th = .
01. The relativelyhigher ˙ M calculated from our model of ADAF accretion has veryclear physical meanings, as will be discussed in Section 3.2. erg s − Recently, the millisecond X-ray pulsations have been observedin several NS X-ray sources, such as, the transitional millisec-ond pulsar (tMSP) PSR J1023 + erg s − . This challenges thetraditional accretion disc theory for the formation of the X-ray pul-sation at such a low X-ray luminosity, since in general at this lowX-ray luminosity (if the mass accretion rate calculated with formulaof ˙ M = L X R ∗ GM ), the corotation radius of the NS accreting system isless than the magnetospheric radius. In this case, the ‘propeller’ ef-fect may work, expelling (a fraction of the) matter in the accretionflow to be leaving away from the NS (Illarionov & Sunyaev 1975).In this paper, we suggest that, if applicable, our model of ADAF ac-cretion may be applied to explain the observed millisecond X-raypulsation at the X-ray luminosity of a few times of 10 erg s − . Thisis because at this X-ray luminosity, ˙ M calculated from our modelof ADAF accretion for taking an appropriate value of f th , such as f th = .
01, can be more than two orders of magnitude higher thanthat of calculated with the formula of ˙ M = L X R ∗ GM to make the mag-netospheric radius less than the corotation radius. In this case, asthe accretion flow moves inward, if the radius is less than the mag-netospheric radius, (a fraction of) the matter in the accretion flowwill be magnetically channelled onto the surface of the NS, leadingto the formation of the X-ray pulsation.For clarity, we list the expression of the corotation radius R c and the magnetospheric radius R m respectively as follows. Thecorotation radius R c is expressed as, R c = GM π ν / We take these two X-ray luminosities since the X-ray pulsations havebeen confirmed in some NS-LMXBs in these luminosities, as will be dis-cussed in Section 3.2. ≃ . M . M ⊙ / ν NS
500 Hz − / , (4)where M is the NS mass, and ν NS is the NS spin frequency. Themagnetospheric radius R m is expressed as (Spruit & Taam 1993;D’Angelo & Spruit 2010; D’Angelo et al. 2015), R m = ηµ Ω NS ˙ M / ≃
24 km η / B ∗ G / R ∗
10 km / × ν NS
500 Hz − / ˙ M g s − − / , (5)where µ = B ∗ R ∗ is magnetic dipole moment (with B ∗ beingthe magnetic field at the surface of the NS, R ∗ being the NSradius), Ω NS is the rotational angular velocity of the NS (with Ω NS = πν NS ), η M is the mass accretion rate in units of g s − . We investigate the re-lation between R c and R m for PSR J1023 + PSR J1023 + the millisecond X-ray pulsation of PSRJ1023 + L . − ∼ . × erg s − . The spin frequency of PSRJ1023 + ν NS =
592 Hz (Archibald et al. 2015). As we cansee from Table 1, at the X-ray luminosity of L . − ∼ . × erg s − , the mass accretion rate ˙ M calculated with the formulaof ˙ M = L . − R ∗ GM is 2 . × g s − . At this X-ray luminosity,the mass accretion rate ˙ M . calculated with our model of ADAFaccretion for f th = . . × g s − , and mass accretion rate˙ M . calculated with our model of ADAF accretion for f th = . . × g s − . If we assume M = . M ⊙ , R ∗ = . B ∗ = G and η = .
1, accord-ing to equation (4), R c ≈ . R m0 ≈ . R m0 . ≈ . R m0 . ≈ . R m0 , R m0 . and R m0 . being the magnetospheric radii calculated by tak-ing the mass accretion rate as ˙ M , ˙ M . and ˙ M . respectively). Itcan be seen that, if the mass accretion rate, i.e. ˙ M , is calculatedwith the formula of ˙ M = L . − R ∗ GM , R m0 > R c , theoretically,in this case the pulsation cannot be formed. If the mass accretionrate, i.e. ˙ M . , is calculated with our model of ADAF accretion for f th = . R m0 . > R c , theoretically, the pulsation also cannot beformed. While, if the mass accretion rate, i.e. ˙ M . , is calculatedwith our model of ADAF accretion for f th = . R m0 . < R c ,theoretically, the pulsation can be formed (Illarionov & Sunyaev1975). XSS J12270-4859 and IGR J17379-3747: the millisecond X-ray pulsation of both XSS J12270-4859 and IGR J17379-3747 arediscovered at the X-ray luminosity of L . − ∼ . × erg s − .The spin frequency of XSS J12270-4859 and IGR J17379-3747 are ν NS =
593 Hz and ν NS =
468 Hz respectively. As we can see fromTable 1, at the X-ray luminosity of L . − = . × erg s − ,the mass accretion rate ˙ M calculated with the formula of ˙ M = L . − R ∗ GM is 3 . × g s − . At this X-ray luminosity, themass accretion rate ˙ M . calculated with our model of ADAF ac-cretion for f th = . . × g s − , and the mass accre-tion rate ˙ M . calculated with our model of ADAF accretion for f th = .
01 is 6 . × g s − . Also, if we assume M = . M ⊙ , R ∗ = . B ∗ = G and η = . MNRAS , 1–9 (2020) adiative e ffi ciency at low-level accretion Figure 1.
Panel (1): emergent spectra of NSs with an ADAF accretion for di ff erent ˙ m with f th = .
1. Panel (2): emergent spectra of NSs with an ADAFaccretion for di ff erent ˙ m with f th = . Figure 2.
Panel (1): X-ray luminosity L . − , L . − and bolometric luminosity L bol as a function of ˙ m respectively. The symbols of black ‘ + ’, blue‘ + ’, and red ‘ + ’ refer to L . − , L . − , and L bol respectively from our model of ADAF accretion with f th = .
1. The symbols of black ‘ △ ’, blue ‘ △ ’,and red ‘ △ ’ refer to L . − , L . − , and L bol respectively from our model of ADAF accretion with f th = .
01. The dashed line refers to the luminositycalculated with the formula of L = ˙ MGMR ∗ . Panel (2): radiative e ffi ciency ǫ . − , ǫ . − and ǫ bol as a function of ˙ m . The symbols of black ‘ + ’, blue ‘ + ’,and red ‘ + ’ refer to ǫ . − , ǫ . − and ǫ bol respectively from our model of ADAF accretion with f th = .
1. The symbols of black ‘ △ ’, blue ‘ △ ’, and red‘ △ ’ refer to ǫ . − , ǫ . − and ǫ bol respectively from our model of ADAF accretion with f th = .
01. The dashed line refers to the radiative e ffi ciencycalculated with the formula of ǫ = ˙ MGMR ∗ / ˙ Mc . Table 1.
Mass accretion rate obtained for a given X-ray luminosity L . − based on the curves in Fig. 3. Specifically, ˙ m and ˙ M are obtained with theformula of ˙ M = L . − R ∗ GM . ˙ m and ˙ M are in units of ˙ M Edd and g s − respectively. ˙ m . and ˙ M . are obtained from our model results of NSs with anADAF accretion for f th = .
1. ˙ m . and ˙ M . are in units of ˙ M Edd and g s − respectively. ˙ m . and ˙ M . are obtained from our model results of NSs with anADAF accretion for f th = .
01. ˙ m . and ˙ M . are in units of ˙ M Edd and g s − respectively. L . − ( erg s − ) ˙ m ( ˙ M Edd ) ˙ M (g s − ) ˙ m . ( ˙ M Edd ) ˙ M . (g s − ) ˙ m . ( ˙ M Edd ) ˙ M . (g s − )3 . × . × − . × . × − . × . × − . × . × . × − . × . × − . × . × − . × + M ,is calculated with the formula of ˙ M = L . − R ∗ GM , R m0 > R c , andif the mass accretion rate, i.e. ˙ M . , is calculated with our modelof ADAF accretion for f th = . R m0 . > R c . In these two cases,theoretically, the pulsation cannot be formed. If the mass accretionrate, i.e. ˙ M . , is calculated with our model of ADAF accretion for f th = . R m0 . < R c . In this case, theoretically, the pulsation canbe formed (Illarionov & Sunyaev 1975). One can refer to Table 2for the detailed numerical results of R c , R m0 , R m0 . , and R m0 . forXSS J12270-4859 and IGR J17379-3747 respectively.Here, we would like to remind that we take a fixed value ofthe NS spin frequency, i.e., ν NS =
500 Hz, for plotting L . − asa function of ˙ m as in Fig. 3, and the corresponding calculations for MNRAS000
500 Hz, for plotting L . − asa function of ˙ m as in Fig. 3, and the corresponding calculations for MNRAS000 , 1–9 (2020)
Erlin Qiao and B.F. Liu
Figure 3.
X-ray luminosity L . − as a function of ˙ m . The symbols of‘ + ’ and ‘ △ ’ refer to L . − from our model of ADAF accretion with f th = . f th = .
01 respectively. The dashed line refers to the luminositycalculated with the formula of L = ˙ MGMR ∗ . The horizontal black-dotted linerefers to the X-ray luminosity of L . − = × erg s − , and thehorizontal blue-dotted line refers to the X-ray luminosity of L . − = × erg s − . The vertical black-dotted lines from left to right refer tothe value of ˙ m of the crossing points between L . − = × erg s − and the formula of L = ˙ MGMR ∗ , our model of ADAF accretion with f th = . f th = .
01 respectively. The verticalblue-dotted lines from left to right refer to the value of ˙ m of the crossingpoints between L . − = × erg s − and the formula of L = ˙ MGMR ∗ ,our model of ADAF accretion with f th = .
1, and our model of ADAFaccretion with f th = .
01 respectively. ˙ M . (or ˙ m . ) and ˙ M . (or ˙ m . ) in Table 1, which is a little di ff er-ent from the observed value of ν NS =
592 Hz for PSR J1023 + ν NS =
593 Hz for XSS J12270-4859, and ν NS =
468 Hz for IGRJ17379-3747. However, we would like to remind again that it hasbeen proven that the e ff ect of the NS spin frequency, e.g., for tak-ing ν NS = , ,
500 and 700 Hz, on the structure and the emergentspectra the ADAF around a NS is very little, and nearly can be ne-glected (Qiao & Liu 2020a). So fixing the spin frequency at 500 Hzis a good approximation for comparing with the observational dataof PSR J1023 + ff ect of the large-scale magnetic field, which will bediscussed in Section 4.1.Finally, we would like to mention that several other scenar-ios have been discussed for explaining the formation of the mil-lisecond X-ray pulsation at the X-ray luminosity of a few timesof 10 erg s − (e.g. Archibald et al. 2015; Papitto et al. 2015;Patruno et al. 2016; Bult et al. 2019), some of which are summa-rized as follows. (1) The coupling between the magnetic fieldlines and the highly conducing accretion flow maybe is weak,which can lead to di ff usion of the gas in the accretion flow inwardvia Rayleigh-Taylor instability (Kulkarni & Romanova 2008), ora large-scale compression of the magnetic field (Romanova et al.2005; Ustyugova et al. 2006; Zanni & Ferreira 2013). In thesecases, even though the magnetospheric radius is greater than thecorotation radius, it is possible that a fraction of the gas in theaccretion flow to overcome the centrifugal barrier of the mag-netic field to be accreted onto the surface of the NS forming theX-ray pulsation. (2) The interaction between the magnetosphereand the accretion disc is complex, and the ‘propeller’ e ff ect can reject the infalling matter in the accretion disc only if the mag-netic field at the magnetospheric radius rotates significantly fasterthan the accretion disc (Spruit & Taam 1993). If this is not thecase, instead, the magnetospheric radius of the accretion disc couldbe trapped near the corotation radius (Siuniaev & Shakura 1977;D’Angelo & Spruit 2010, 2012), which has been used to explainthe observed 1 Hz modulation in AMXP SAX J1808.4-3658 andNGC 6440 X-2 (Patruno et al. 2009; Patruno & D’Angelo 2013). ff ect of the large-scale magnetic field of ∼ G onthe radiative e ffi ciency of NSs with an ADAF accretion In this paper, we investigate the radiative e ffi ciency of weakly mag-netized NSs with an ADAF accretion for taking two typical valuesof f th = . f th = .
01 as suggested in Qiao & Liu (2020a)and Qiao & Liu (2020b). Then, we show that NSs with an ADAFaccretion is radiatively ine ffi cient, with which we further explainthe observed millisecond X-ray pulsations for PSR J1023 + erg s − . However, we should note that, in ourmodel of NSs with an ADAF accretion, we do not consider the ef-fect of the large-scale magnetic field on the emission of the ADAF,which probably will a ff ect the radiative e ffi ciency of the NSs withan ADAF accretion.In general, accreting millisecond X-ray pulsars (AMXPs)are believed to have a relatively weaker magnetic field of ∼ G (e.g. Wijnands & van der Klis 1998; Casella et al. 2008;Patruno & Watts 2012, for review), which is di ff erent from thestandard X-ray pulsars often with a stronger magnetic fieldof ∼ G (e.g. Coburn et al. 2002; Pottschmidt et al. 2005;Caballero & Wilms 2012; Revnivtsev & Mereghetti 2015, for re-view). Due to the relatively weaker magnetic field in AMXPs, it isoften suggested that the magnetic field in AMXPs does not signif-icantly a ff ect the X-ray spectra (e.g. Poutanen & Gierli´nski 2003),which seems to be supported by some observations by comparingthe X-ray spectra between the non-pulsating NS-LMXBs and theAMXPs. In general, it is found that there is no systematic di ff er-ence of the X-ray spectra between the non-pulsating NS-LMXBsand the AMXPs in the range of L . − ∼ − erg s − . Forexample, in Wijnands et al. (2015), the author compiled a samplecomposed of eleven non-pulsating NS-LMXBs, finding that sys-tematically there is an anticorrelation between the X-ray photonindex Γ (obtained by fitting the X-ray spectra between 0.5 and 10keV with a single power law) and the X-ray luminosity L . − in the range of L . − ∼ − erg s − . Further, the authorsadded three AMXPs, i.e., NGC 6440 X-2, IGR J00291 + Γ and L . − to comparewith the non-pulsating NS sample, showing that at a fixed X-rayluminosity, the X-ray spectra of the AMXPs appear to be slightlyharder than that of the non-pulsating NS-LMXBs. More accurately,the authors did 2D KS test to study whether the AMXP data areconsistent with the non-pulsating data. It is found that a 90 per centconfidence interval for the probability of 1 . × − − . × − thatthe AMXP data and the non-pulsating data have the same distribu-tion. However, given the fact that only three AMXPs are includedin this study, actually, the authors also reminded that they cannotdraw strong conclusions whether the presence of the magnetic fieldin AMXPs can alter the X-ray spectra (Wijnands et al. 2015). In afurther study of Parikh et al. (2017), the authors combined the data MNRAS , 1–9 (2020) adiative e ffi ciency at low-level accretion Table 2.
The corotation radius R c calculated with equation (4), as well as the magnetospheric radius R m0 , R m0 . and R m0 . calculated with equation (5), forPSR J1023 + R m0 , R m0 . and R m0 . are calculated respectively by taking the mass accretionrate as ˙ M , ˙ M . , and ˙ M . as listed in Table 1, i.e., ˙ M is obtained with the formula of ˙ M = L . − R ∗ GM , ˙ M . are obtained from our model results of NSswith an ADAF accretion for f th = .
1, and ˙ M . are obtained from our model results of NSs with an ADAF accretion for f th = .
01. In all the calculations,we take M = . M ⊙ , R ∗ = . B ∗ = G, and η = . + ν NS =
592 Hz) L . − ( erg s − ) R c (km) R m0 (km) R m0 . (km) R m0 . (km)3 . × . . . . ν NS =
593 Hz) L . − ( erg s − ) R c (km) R m0 (km) R m0 . (km) R m0 . (km)5 . × . . . . ν NS =
468 Hz) L . − ( erg s − ) R c (km) R m0 (km) R m0 . (km) R m0 . (km)5 . × . . . . in Wijnands et al. (2015) and some additional new data in the rangeof L . − ∼ − erg s − for the anticorrelation between theX-ray photon index Γ and the X-ray luminosity L . − , the au-thors showed that they did not find that the X-ray spectra of AMXPsare systematically harder than that of the non-pulsating sources astested in Wijnands et al. (2015), suggesting that the hardness of theX-ray spectra does not have strict connection with the presence ofthe dynamic e ff ect of the magnetic field.As for L . − . erg s − (generally defined as the qui-escent state), the X-ray spectra of non-pulsating NS-LMXBs arevery complex and diverse, which can be (1) completely dominatedby a thermal soft X-ray component, (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-law com-ponent (e.g. Wijnands et al. 2015, for discussions). For example,the X-ray spectra of the non-pulsating NS-LMXB Cen X-4 at theX-ray luminosity of L . − ∼ erg s − can be well fittedby the two-component model, i.e. a thermal soft X-ray componentplus a power-law component, revealing a harder X-ray photon in-dex of Γ ∼ − . Γ ∼ − L . − ∼ a fewtimes of 10 erg s − (Sonbas et al. 2018). For the three sources,i.e., PSR J1023 + L . − ∼ a few times of 10 erg s − , it is found thatthe X-ray spectra can be well fitted by a single power law with thephoton index Γ ∼ . + Γ ∼ . ∼ .
35 keV plus a thermal component of ∼ .
12 keV for IGRJ17379-3747, indicating a very soft X-ray spectrum (the Group 3data) (Bult et al. 2019).In summary, as discussed above we think that the e ff ect of themagnetic field of ∼ G in AMXPs on the emission of NSs withan ADAF accretion in the range of L . − ∼ − erg s − is very little, consequently the e ff ect of the magnetic field on theradiative e ffi ciency of NSs with an ADAF accretion is very lit-tle. As for at the X-ray luminosity of L . − ∼ a few times of10 erg s − , we think it is not very easy to say whether there is sig-nificant e ff ect of the magnetic field of ∼ G on the emission ofNSs with an ADAF accretion. Here, at least for PSR J1023 + f th (for decreasing the contribu-tion of the thermal soft X-ray component), i.e., f th approaching tozero (even smaller than 0.01 as taken in the present paper) [see Fig.7 in Qiao & Liu (2020a) for details]. So we think that our expla-nations for the observed millisecond X-ray pulsations at the X-rayluminosity of a few times of 10 erg s − with our model of ADAFaccretion by taking a small value of f th , i.e. f th = .
01 is a goodapproximation. Here, we would like to address that due to the ex-istence of the magnetic field of ∼ G, the boundary condition inthe region between the surface of the NS and the ADAF in AMXPsshould be di ff erent from that of in non-pulsating NSs, which how-ever has been incorporated into the e ff ect of the parameter f th ifwe only focus on this question from the viewpoint of emission.Finally, we also would like to address that a detailed study of thee ff ects of the large-scale magnetic field of ∼ G on the dynamicsand the emission of NSs with an ADAF accretion is still very nec-essary for the consistency between the model and the observationsfor AMXPs in the future, although the e ff ects of the magnetic fieldat the strength of ∼ G on the radiative e ffi ciency of NSs withan ADAF accretion maybe are not very obvious. ffi ciency ofNSs with an ADAF accretion in the future In our model of NSs with an ADAF accretion, there is a very impor-tant parameter, f th , which controls the feedback between the surfaceof the NS and the ADAF. The value of f th can a ff ect the radiativee ffi ciency of NSs with an ADAF accretion. As has been shown inQiao & Liu (2020a) and Qiao & Liu (2020b), the value of f th hasbeen constrained to be less than 0.1, and it seems that a smallervalue of f th , i.e., f th ∼ .
01 is more preferred. It is possible that theremaining fraction, i.e., 1- f th , of the ADAF energy transferred ontothe surface of the NS could be partially converted to the rotationalenergy of the NS, and could be partially absorbed by the NS andstored as the internal energy at the crust of the NS. The accretedmatter in the form of the ADAF (with relatively higher temperatureand lower density) and the carried energy itself may produce someadditionally observational features at the surface of the NS, whichcurrently however has not been well investigated, depending on theresulted changes of the temperature and the density of the matter inthe very thin layer at the surface of the NS (e.g. Galloway & Keek2021, for the related discussions). The study of the further e ff ectsof the accreted matter in the form of the ADAF at the surface of the MNRAS , 1–9 (2020)
Erlin Qiao and B.F. Liu
NS exceeds the research scope in the present paper, and definitelywill be carried out in the future.In Qiao & Liu (2020a) and Qiao & Liu (2020b), the constraintto the value of f th is based on some statistically observed correla-tions in non-pulsating NSs, such as the fractional contribution ofthe power-law component η as a function of L . − , as well asthe X-ray photon index Γ as a function of L . − . In order moreprecisely to constrain the value of f th , we expect that the detailedX-ray spectral fittings will be done for some typically single sourcein the future, such as the study for Cen X-4 (e.g. Chakrabarty et al.2014; D’Angelo et al. 2015).As discussed in Section 3.2, if our model of ADAF accretioncan be applied to explain the observed millisecond X-ray pulsa-tion at the X-ray luminosity of a few times of 10 erg s − for PSRJ1023 + f th , e.g., f th = .
01 is required. Based on some relatedresults from the model of ADAF accretion for taking f th = . ν NS . If we assume that the change of the NS spin is due to the ac-cretion, according to the conservation of angular momentum, wehave I ˙ Ω NS = ˙ M ( Ω ∗ − Ω NS ) R ∗ , (6)where I is the moment of inertia of the NS, Ω ∗ is the rotational an-gular velocity of the ADAF at R ∗ with Ω ∗ = πν ∗ (with ν ∗ beingthe angular frequency at R ∗ ), and Ω NS is the rotational angular ve-locity of the NS with Ω NS = πν NS . Rearranging equation (6), wecan express the change rate of the NS spin frequency as follows,˙ ν NS = ˙ M ( ν ∗ − ν NS ) R ∗ / I . (7)Given the value of ˙ M , ν ∗ , ν NS and I , we can calculate the change rateof the NS spin frequency ˙ ν NS . For example, for PSR J1023 + L . − ∼ × erg s − , the corresponding ˙ M is3 . × g s − based on our model of ADAF accretion for f th = .
01. With ˙ M = . × g s − , we recalculate the struc-ture of the ADAF for ν ∗ . The value of ν ∗ is 253 Hz. The momentof inertia I is ∼ . × g cm for taking the typical value of M = . M ⊙ and R ∗ = . ν NS of PSR J1023 + M , ν ∗ , ν NS and I into equation (7), we get ˙ ν NS ∼ − . × − Hz s − ,which is close to ( ∼ . ν NS ∼ − . × − Hz s − for PSR J1023 + ν ∗ from our model of ADAF accretion is less than ν NS ,which means that a negative torque will be exerted on the NS, con-sequently making the rotational energy of the NS transferred ontothe ADAF and the NS to be spin-down, rather than the ADAF en-ergy transferred onto the NS and the NS to be spin-up. Further,since a variable flat-spectrum of radio emission is revealed as PSRJ1023 + ν NS (Deller et al. 2015).A similar calculation for ˙ ν NS is done for XSS J12270-4859 andIGR J17379-3747 with the millisecond X-ray pulsations observedat the X-ray luminosity of ∼ × erg s − . At this X-ray luminos-ity, the mass accretion rate ˙ M is 6 . × g s − based on our modelof ADAF accretion for f th = .
01. With ˙ M = . × g s − , werecalculate the structure of the ADAF for ν ∗ . The value of ν ∗ is278 Hz. The spin frequency ν NS is 593 Hz for XSS J12270-4859,and is 468 Hz for IGR J17379-3747. The moment of inertia I is ∼ . × g cm for taking M = . M ⊙ and R ∗ = . M , ν ∗ , ν NS and I intoequation (7), we get ˙ ν NS ∼ − . × − Hz s − for XSS J12270-4859 and ˙ ν NS ∼ − . × − Hz s − for IGR J17379-3747. It isclear that the value of ˙ ν NS is negative (i.e., spin-down) for XSSJ12270-4859 and IGR J17379-3747 as for PSR J1023 + × erg s − are correct, thepredicted change rate of the NS spin frequency ˙ ν NS is at the level of ∼ − − Hz s − , which we expect can be tested by the observationsin the future. Further, if the change rate of the NS spin frequency˙ ν NS predicted by our model of ADAF accretion can be confirmedin the future, which actually in turn supports our idea in the presentpaper that NSs with an ADAF accretion is radiatively ine ffi cientdespite the existence of the hard surface. Finally, we would liketo address that the estimation of ˙ ν NS in this paper is based on ourmodel of ADAF accretion around a weakly magnetized NS, whichwill make the estimated value of ˙ ν NS uncertain as applied to theAMXP cases. So the consideration of the e ff ect of the magneticfield ( ∼ G) on the value of ˙ ν NS in AMXPs is still very neces-sary in the future, which however exceeds the scope in the presentpaper. Following the paper of Qiao & Liu (2020a) and Qiao & Liu(2020b) for the constraints to the value of f th controlling the feed-back between the surface of the NS and the ADAF, in this paper,we investigate the radiative e ffi ciency of NSs with an ADAF accre-tion within the framework of the self-similar solution of the ADAFby taking two typically suggested values of f th , i.e., f th = . f th = .
01 respectively. Then, we show that the radiative e ffi ciencyof NSs with an ADAF accretion is significantly lower than that of ǫ ∼ ˙ MGMR ∗ / ˙ Mc ∼ .
2. Specifically, the radiative e ffi ciency of ourmodel of NSs with an ADAF accretion for f th = . ǫ ∼ ˙ MGMR ∗ / ˙ Mc ∼ .
2, andthe radiative e ffi ciency of our model of NSs with an ADAF accre-tion for f th = .
01 is roughly two orders of magnitude lower thanthat of ǫ ∼ .
2. As a result, we propose that the lower radiative ef-ficiency of our model of ADAF accretion probably can be appliedto explain the observed millisecond X-ray pulsation in some NS-LMXBs (such as PSR J1023 + erg s − , since at this X-ray luminosity thereal ˙ M calculated with our model of ADAF accretion for taking anappropriate value of f th , such as f th = .
01, can be more than twoorders of magnitude higher than that of calculated with the formulaof ˙ M = L X R ∗ GM to ensure a fraction of the matter in the ADAF to bechannelled onto the surface of the NS forming the X-ray pulsation. ACKNOWLEDGMENTS
Erlin Qiao thanks the very useful discussions with Dr. Chichuan Jinfrom NAOC. This work is supported by the National Natural Sci-ence Foundation of China (Grants 11773037 and 11673026), thegravitational wave pilot B (Grant No. XDB23040100), the StrategicPioneer Program on Space Science, Chinese Academy of Sciences(Grant No. XDA15052100), and the National Program on Key Re-search and Development Project (Grant No. 2016YFA0400804).
MNRAS , 1–9 (2020) adiative e ffi ciency at low-level accretion DATA AVAILABILITY
The data underlying this article will be shared on reasonable re-quest to the corresponding author.
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