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Observationally, an anti-correlation between the X-ray photon index $\Gamma$(obtained by fitting the X-ray spectrum between 0.5 and 10 keV with a singlepower law) and the X-ray luminosity $L_{\rm 0.5-10keV}$, is found in neutronstar low-mass X-ray binaries (NS-LMXBs) in the range of $L_{\rm 0.5-10keV}\sim10^{34}-10^{36}\ \rm erg\ s^{-1}$. In this paper, we explain the observedanti-correlation between $\Gamma$ and $L_{\rm 0.5-10keV}$ within the frameworkof the self-similar solution of the advection-dominated accretion flow (ADAF)around a weakly magnetized NS. The ADAF model intrinsically predicts ananti-correlation between $\Gamma$ and $L_{\rm 0.5-10keV}$. In the ADAF model,there is a key parameter, $f_{\rm th}$, which describes the fraction of theADAF energy released at the surface of the NS as thermal emission to bescattered in the ADAF. We test the effect of $f_{\rm th}$ on theanti-correlation between $\Gamma$ and $L_{\rm 0.5-10keV}$. It is found that thevalue of $f_{\rm th}$ can significantly affect the anti-correlation between$\Gamma$ and $L_{\rm 0.5-10keV}$. Specifically, the anti-correlation between$\Gamma$ and $L_{\rm 0.5-10keV}$ becomes flatter with decreasing $f_{\rm th}$as taking $f_{\rm th}=0.1, 0.03, 0.01, 0.005$, $0.003$ and $0$ respectively. Bycomparing with a sample of non-pulsating NS-LMXBs with well measured $\Gamma$and $L_{\rm 0.5-10keV}$, we find that indeed only a small value of$0.003\lesssim f_{\rm th}\lesssim 0.1$ is needed to match the observedanti-correlation between $\Gamma$ and $L_{\rm 0.5-10keV}$. Finally, we arguethat the small value of $f_{\rm th}\lesssim 0.1$ derived in this paper furtherconfirms our previous conclusion that the radiative efficiency of NSs with anADAF accretion may not be as high as $\epsilon \sim {\dot M GM\overR_{*}}/{\dot M c^2}\sim 0.2$.

monthly notices of the royal astronomical society

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