2023MNRAS.521.3237L

Observationally, two main spectral states, i.e. the low/hard state and the high/soft state, are identified in black hole X-ray binaries (BH-XRBs). Meanwhile, the transitions between the two states are often observed. In this paper, we re-investigate the transition luminosities in the framework of the self-similar solution of the advection-dominated accretion flow (ADAF). Specifically, we search for the critical mass accretion rate $\dot{m}_{\rm crit}$ of ADAF for different radii r, respectively. It is found that $\dot{m}_{\rm crit}$ decreases with decreasing r. By testing the effects of BH mass m, the magnetic parameter β and the viscosity parameter α, it is found that only α has significant effects on $\dot{m}_{\rm crit}\text{--}r$ relation. We define the minimum $\dot{m}_{\rm crit}$ (roughly at the innermost stable circular orbit) as the hard-to-soft transition rate $\dot{m}_{\rm tr:H\text{--} S}$, above which BH will gradually transit from the low/hard state to the high/soft state, and $\dot{m}_{\rm crit}$ at 30 Schwarzschild radii as the soft-to-hard transition rate $\dot{m}_{\rm tr:S\rightarrow H}$, below which BH will gradually transit from the high/soft state to the low/hard state. We derive fitting formulae of $\dot{m}_{\rm tr:H\rightarrow S}$ and $\dot{m}_{\rm tr:S\rightarrow H}$ as functions of α, respectively. By comparing with observations, it is found that the mean value of α are α ∼ 0.85 and α ∼ 0.33 for the hard-to-soft transition and the soft-to-hard transition, respectively, which indicates that two classes of α are needed for explaining the hysteresis effect during the state transition. Finally, we argue that such a constrained α may provide valuable clues for further exploring the accretion physics in BH-XRBs.