Analytical Model of Disk Evaporation and State Transitions in Accreting Black Holes

A well-studied model of state transitions in black hole X-ray binaries postulates that gas evaporates from a thin accretion disk into a hot corona. We present a height-integrated version of this model which can be solved analytically and provides a physical understanding of different regimes. With radius r scaled in Schwarzschild units and coronal mass accretion rate ṁ_c in Eddington units, the results are independent of black hole mass. Thus state transitions should be similar in X-ray binaries and AGN. The analytical solution consists of two power-law segments separated at a break radius, r_b ~ 10³ (α / 0.3)^-2, where α is the viscosity parameter. Gas evaporates from the disk to the corona for r > r_b, but condenses back for r < r_b. At r ~ r_b, ṁ_c reaches its maximum, ṁ_c,max ≈ 0.03 (α / 0.3)³. If at r ≫ r_b the thin disk accretes at a rate less than ṁ_c,max, then the disk evaporates fully before reaching r_b, giving the hard state. If the thin disk has a larger accretion rate than ṁ_c,max, it does not evaporate fully at any radius, giving the thermal state. The basic height-integrated model has only viscous heating in the corona. We also present a generalized model which includes direct coronal heating by energy transport from the thin disk. Again, the analytical solutions are independent of black hole mass, and r_b remains the same. This model predicts strong coronal winds for r > r_b, and relativistic temperatures for r ≪ r_b.