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

TL;DR

This paper explains the anti-correlation between X-ray photon index and luminosity in neutron star low-mass X-ray binaries using the ADAF model, highlighting the role of thermal energy fraction at the neutron star surface.

Contribution

It demonstrates that the ADAF model can naturally produce the observed anti-correlation and identifies the thermal energy fraction parameter as key to matching observations.

Findings

The anti-correlation is sensitive to the thermal energy fraction parameter.

A small thermal energy fraction (0.003 to 0.1) matches observed data.

The radiative efficiency of neutron stars with ADAF may be lower than previously thought.

Abstract

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 single power law) and the X-ray luminosity $L_{\rm 0.5-10keV}$, is found in neutron star low-mass X-ray binaries (NS-LMXBs) in the range of $L_{\rm 0.5-10keV}\sim 10^{34}-10^{36}\ \rm erg\ s^{-1}$. In this paper, we explain the observed anti-correlation between $\Gamma$ and $L_{\rm 0.5-10keV}$ 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 anti-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 the ADAF energy released at the surface of the NS as thermal emission to be scattered in the ADAF. We test the effect of $f_{\rm th}$ on the anti-correlation between $\Gamma$ and $L_{\rm 0.5-10keV}$. It is found that the value 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. By comparing 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 observed anti-correlation between $\Gamma$ and $L_{\rm 0.5-10keV}$. Finally, we argue that the small value of $f_{\rm th}\lesssim 0.1$ 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 $\epsilon \sim {\dot M GM\over R_{*}}/{\dot M c^2}\sim 0.2$.