In a previous paper , we described new analytic formulae for optically-thick supercritical accretion flows ( Watarai 2006 , hereafter paper 1 ) .
Here we present analytic formulae for optically-thin one-temperature accretion flows including the advection-dominated regime , using the “ semi-iterative ” method described in paper 1 .
Our analytic formulae have two real solutions .
The first solution corresponds to the advection-dominated accretion flow ( ADAF ) , and the second solution corresponds to the radiation-dominated accretion flow described by Shapiro , Lightman , & Eardley ( the so-called SLE model ) .
Both solutions are given by a cubic equation for the advection parameter f , which is the ratio of the advection cooling rate Q _ { adv } to the viscous heating rate Q _ { vis } , i.e. , f = Q _ { adv } / Q _ { vis } .
Most previous studies assume that f is constant ( f \sim 1 for the ADAF ) .
However , it is clear that f should be a function of the physical parameters of the radiative-cooling dominated regime .
We found that the ratio f can be written as a function of the radius , mass accretion rate , and viscous parameter \alpha .
Using this formula , we can estimate the transition radius from the inner optically-thin ADAF to the outer optically-thick standard disk , which can be measured using observations of the quiescent state in black hole X-ray binaries .