New Analytical Formulae for Optically-Thin Accretion Flows

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 $Q_{\rm adv}$ to the viscous heating rate $Q_{\rm vis}$, i.e., $f=Q_{\rm adv}/Q_{\rm vis}$. 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.