The advection-dominated accretion flow for the anticorrelation between the X-ray photon index and the X-ray luminosity in neutron star low-mass X-ray binaries

Abstract

<jats:title>ABSTRACT</jats:title> <jats:p>Observationally, an anticorrelation between the X-ray photon index Γ (obtained by fitting the X-ray spectrum between 0.5 and 10 keV with a single power law) and the X-ray luminosity L0.5-10 keV, i.e. a softening of the X-ray spectrum with decreasing L0.5-10 keV, is found in neutron star low-mass X-ray binaries (NS-LMXBs) in the range of $L_{\rm 0.5\!-\!10\,keV}\sim 10^{34}\!-\!10^{36}\ \rm erg\ s^{-1}$. In this paper, we explain the observed anticorrelation between Γ and L0.5–10 keV within the framework of the self-similar solution of the advection-dominated accretion flow (ADAF) around a weakly magnetized NS. The ADAF model intrinsically predicts an anticorrelation between Γ and L0.5–10 keV. In the ADAF model, there is a key parameter, fth, which describes the fraction of the ADAF energy released at the surface of the NS as thermal emission to be scattered in the ADAF. We test the effect of fth on the anticorrelation between Γ and L0.5–10 keV. It is found that the value of fth can significantly affect the anticorrelation between Γ and L0.5–10 keV. Specifically, the anticorrelation between Γ and L0.5–10 keV becomes flatter with decreasing fth as taking fth = 0.1, 0.03, 0.01, 0.005, 0.003, and 0, respectively. By comparing with a sample of non-pulsating NS-LMXBs with well measured Γ and L0.5–10 keV, we find that indeed only a small value of 0.003 ≲ fth ≲ 0.1 is needed to match the observed anticorrelation between Γ and L0.5–10 keV. Finally, we argue that the small value of fth ≲ 0.1 derived in this paper further confirms our previous conclusion that the radiative efficiency of NSs with an ADAF accretion may not be as high as $\epsilon \sim {\dot{M} GM\over R_{*}}/{\dot{M} c^2}\sim 0.2$.</jats:p>

The radiative efficiency of neutron stars at low-level accretion

“…Combining the explanations for the anticorrelation between the X-ray photon index Γ and L 0.5−10keV with the X-ray spectra analyzed with the single power-law model, and the positive correlation between η and L 0.5−10keV with the two-component model for a fraction of the sources in the sample, we conclude that in the range of L 0.5−10keV ∼ 10 34 − 10 35 erg s −1 , the softening of the X-ray spectra is due to the increase of the thermal soft X-ray component, while in the range of L 0.5−10keV ∼ 10 35 − 10 36 erg s −1 , the softening of the X-ray spectra is probably due to the evolution of the power-law component itself. As a summary, in the study above for explaining the anticorrelation between Γ and L 0.5−10keV , it has been shown that the value of f th can be constrained to be less than 0.1, and is very probably to be much smaller values, i.e., ∼ 0.003 − 0.005 (Qiao & Liu 2020b).…”

“…Further, in Qiao & Liu (2020b), based on the sample of nonpulsating NS-LMXBs in Wijnands et al (2015), and adding some more non-pulsating NS-LMXBs from Parikh et al (2017) and Beri et al (2019), we explain the anticorrelation between the Xray photon index Γ (obtained by fitting the X-ray spectra between 0.5 and 10 keV with a single power law) and L 0.5−10keV , i.e., the softening of the X-ray spectra with decreasing L 0.5−10keV , in the range of L 0.5−10keV ∼ 10 34 − 10 36 erg s −1 by adjusting the value of f th . Moreover, it is shown that a fraction of the sources in Qiao & Liu (2020b) are once reported to be fitted with the twocomponent model (with XMM − Newton X-ray data), i.e., a thermal soft X-ray component plus a power-law component.…”

“…Further, in Qiao & Liu (2020b), based on the sample of nonpulsating NS-LMXBs in Wijnands et al (2015), and adding some more non-pulsating NS-LMXBs from Parikh et al (2017) and Beri et al (2019), we explain the anticorrelation between the Xray photon index Γ (obtained by fitting the X-ray spectra between 0.5 and 10 keV with a single power law) and L 0.5−10keV , i.e., the softening of the X-ray spectra with decreasing L 0.5−10keV , in the range of L 0.5−10keV ∼ 10 34 − 10 36 erg s −1 by adjusting the value of f th . Moreover, it is shown that a fraction of the sources in Qiao & Liu (2020b) are once reported to be fitted with the twocomponent model (with XMM − Newton X-ray data), i.e., a thermal soft X-ray component plus a power-law component. Combining the explanations for the anticorrelation between the X-ray photon index Γ and L 0.5−10keV with the X-ray spectra analyzed with the single power-law model, and the positive correlation between η and L 0.5−10keV with the two-component model for a fraction of the sources in the sample, we conclude that in the range of L 0.5−10keV ∼ 10 34 − 10 35 erg s −1 , the softening of the X-ray spectra is due to the increase of the thermal soft X-ray component, while in the range of L 0.5−10keV ∼ 10 35 − 10 36 erg s −1 , the softening of the X-ray spectra is probably due to the evolution of the power-law component itself.…”

“…The last parameter is, f th , describing the fraction of the ADAF energy released at the surface of the NS as thermal emission to be scattered in the ADAF to cool the ADAF itself, which consequently controls the feedback between the ADAF and the NS. We always take m = 1.4, and R * in the range of 10-12.5 km (Degenaar & Suleimanov 2018) [R * = 12.5 km in Qiao & Liu (2020a), and R * = 10 km in Qiao & Liu (2020b)]. In general, it has been proven that the effect of the NS spin frequency ν NS on the structure and the emergent spectra of the ADAF around a NS is very little, and nearly can be neglected [see Figure 8 of Qiao & Liu (2020a) for taking ν NS = 0, 200, 500, 700 Hz respectively].…”

“…The physics of the boundary layer can significantly affect the global dynamics and radiation of the ADAF (Medvedev & Narayan 2001;Medvedev 2004;D'Angelo et al 2015). Recently, in a series of papers, i.e., Qiao & Liu (2018), Qiao & Liu (2020a), and Qiao & Liu (2020b), we study the dynamics and the radiation of the ADAF around a weakly magnetized NS in the framework of the selfsimilar solution of the ADAF by simplifying the physics of the boundary layer. Specifically, we introduce a parameter, f th , which describes the fraction of the ADAF energy released at the surface of the NS as thermal emission to be scattered in the ADAF.…”