of irradiation are considered: direct irradiation from the innermost hot disk and irradiation that might be reflected by a corona or disk atmosphere or chromosphere above the disk (Meyer, Meyer-Hofmeister 1994). The X-ray luminosity of the irradiation is given by $$\label{lx}
   L_{\rm X}(t)={\epsilon}{\dot M}_{\rm in}c^2,$$ where the efficiency $\epsilon$ is 0.057 and ${\dot M}_{\rm in}$ is the mass-accretion rate at the inner edge of the disk $R_{\rm in}$, taken to be 3$r_g$ as for a Schwarzschild black hole. We adopt a simple model for indirect irradiation by assuming that the luminosity of equation (\[lx\]) effectively arises from a point at the center of the disk ($R=0$). The indirect irradiation flux is then given by $$\label{fi}
    F_{\rm i}(R,t) = C_{\rm X} {{L_{\rm X}(t)}\over{4\pi R^2}},$$ where $C_{\rm X}$ is a constant. More sophisticated models incorporating radiative transfer would give $C_{\rm X}$ = $C_{\rm X}(R, t)$ (see discussion in Tuchman et al. 1990), but such time-dependent radiative transfer is too difficult at this time.

For direct irradiation, we follow the prescription of Fukue (1992; see also Kusaka et al. 1970). This model assumes that the irradiation from a hot, geometrically thin annulus near the inside of the disk can be approximated by that 