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
Qiao, Erlin; Liu, B. F.
August 2020, Monthly Notices of the Royal Astronomical Society, Volume 496, Issue 3, pp.2704-2714
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 L<SUB>0.5-10 keV</SUB>, I.e. a softening of the X-ray spectrum with decreasing L<SUB>0.5-10 keV</SUB>, 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 L<SUB>0.5-10 keV</SUB> 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 L<SUB>0.5-10 keV</SUB>. In the ADAF model, there is a key parameter, f<SUB>th</SUB>, 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 f<SUB>th</SUB> on the anticorrelation between Γ and L<SUB>0.5-10 keV</SUB>. It is found that the value of f<SUB>th</SUB> can significantly affect the anticorrelation between Γ and L<SUB>0.5-10 keV</SUB>. Specifically, the anticorrelation between Γ and L<SUB>0.5-10 keV</SUB> becomes flatter with decreasing f<SUB>th</SUB> as taking f<SUB>th</SUB> = 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 L<SUB>0.5-10 keV</SUB>, we find that indeed only a small value of 0.003 ≲ f<SUB>th</SUB> ≲ 0.1 is needed to match the observed anticorrelation between Γ and L<SUB>0.5-10 keV</SUB>. Finally, we argue that the small value of f<SUB>th</SUB> ≲ 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 $\epsilon \sim {\dot{M} GM\over R_{*}}/{\dot{M} c^2}\sim 0.2$ .
