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 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 $\epsilon \sim {\dot{M} GM\over R_{*}}/{\dot{M} c^2}\sim 0.2$.