A thermal soft X-ray component is often detected in low-level accreting neutron stars (NSs), but is not detected in low-level accreting stellar-mass black holes (BHs). In this paper, we investigate the origin of such a thermal soft X-ray component in the framework of the selfsimilar solution of the advection-dominated accretion flow (ADAF) around NSs. It is assumed that a fraction, f th , of the energy transferred onto the surface of the NS is thermalized at the surface of the NS as the soft photons to be scattered in the ADAF. We self-consistently calculate the structure and the corresponding emergent spectrum of the ADAF by considering the radiative coupling between the soft photons from the surface of the NS and the ADAF itself. We show that the Compton y-parameter of the ADAF for NSs is systematically lower than that of BHs. Meanwhile, we find that the temperature of the thermal soft X-ray component in NSs decreases with decreasing mass accretion rate, which is qualitatively consistent with observations. We test the effect of f th on the structure, as well as the emergent spectrum of the ADAF. It is found that a change of f th can significantly change the temperature of the thermal soft X-ray component as well as the spectral slope in hard X-rays. Finally, it is suggested that the value of f th can be constrained by fitting the high-quality X-ray data, such as the XMM-Newton spectrum between 0.5-10 keV in the future work.

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