= Solution
A <passive disk> is heated predominantly by intercepted stellar radiation, rather than by its own <viscous dissipation>. A rough luminosity criterion, ignoring geometric factors, is
$$
\boxed{L_{\rm acc}\sim\frac{GM\dot M}{R_*}\ll L_*.}
$$
The intrinsic <accretion disk> luminosity is of order $GM\dot M/(2R_*)$, with additional accretion energy potentially released at the stellar surface. The local test is more precise: absorbed irradiation must dominate the viscous flux at the radius being considered. For an ordinary steady <Keplerian accretion disk>, sufficiently far outside its inner edge,
$$
F_{\rm vis}\simeq\frac{3GM\dot M}{8\pi R^3},\qquad
F_{\rm irr}\simeq\frac{L_*\alpha}{4\pi R^2},\qquad
\frac{F_{\rm vis}}{F_{\rm irr}}\simeq\frac{3GM\dot M}{2L_*R\alpha}\ll1,
$$
where $\alpha$ is the <irradiation grazing angle of a disk>. Reflection, very small incidence angles and shadowing can invalidate a simple total-luminosity comparison. The following calculation assumes an illuminated, absorbing surface.
At a point $(R,z_s)$ on the upper surface, the outward unit normal in the meridional plane is proportional to $(-z_s',1)$, and a light ray from the point source is directed along $(R,z_s)$. Its flux through the upper face is the incident inverse-square flux times minus their scalar product:
$$
F_d=\frac{L_*}{4\pi(R^2+z_s^2)}\frac{Rz_s'-z_s}{\sqrt{R^2+z_s^2}\sqrt{1+(z_s')^2}}.
$$
For a <thin disk> with small surface slope this becomes
$$
\boxed{F_d\simeq\frac{L_*}{4\pi R^2}\left(z_s'-\frac{z_s}{R}\right)
=\frac{L_*}{4\pi R}\frac{d}{dR}\left(\frac{z_s}{R}\right).}
$$
The tangent slope and ray slope differ by $\alpha\simeq z_s'-z_s/R$. Positive incidence requires <disk flaring>, $d(z_s/R)/dR>0$, and an unobstructed ray from the star. A negative value of the formula would mean that direct illumination of this face is absent, not that it receives a negative heating flux. The hierarchy $z_s\gg R_*$ makes the finite angular size of the star negligible here.
Let $a=\mathcal R/\mu$, so the <ideal gas> <equation of state> gives the <isothermal sound speed> $c_s^2=aT_c$. In the stellar vertical <gravitational potential>, <hydrostatic equilibrium> has vertical acceleration $-\Omega_K^2z$, where $\Omega_K^2=GM/R^3$. Integration of $c_s^2\partial_z\rho=-\rho\Omega_K^2z$ gives a Gaussian atmosphere with <disk scale height>
$$
H^2=\frac{aT_cR^3}{GM}.
$$
Consequently $q=z_s/R=fH/R$ satisfies $T_c=GMq^2/(af^2R)$. Substitute this into the stipulated <Stefan–Boltzmann law> $F_d=\sigma T_c^4$ and the irradiation flux:
$$
\boxed{\frac{dq}{dR}=C\frac{q^8}{R^3},\qquad
C=\frac{4\pi\sigma}{L_*f^8}\left(\frac{GM}{a}\right)^4.}
$$
Here $f$ specifies surface height in units of <disk scale height> and is unrelated to the wind-energy fraction in the preceding solution. This <separable differential equation> integrates exactly to
$$
q^{-7}=\frac{7C}{2R^2}+B=\frac{R_0^2}{R^2}+B,\qquad
\boxed{R_0^2=\frac{14\pi\sigma}{L_*f^8}\left(\frac{GM}{\mathcal R/\mu}\right)^4.}
$$
If the thin-disc solution is formally matched to $q(R_M)=1$, then $B=1-R_0^2/R_M^2$, giving
$$
q(R)=\left[1+R_0^2\left(\frac1{R^2}-\frac1{R_M^2}\right)\right]^{-1/7}
=\left(\frac R{R_0}\right)^{2/7}\left[1+\frac{R^2}{R_0^2}-\frac{R^2}{R_M^2}\right]^{-1/7}.
$$
It follows that the <flaring passive disk> asymptotic is
$$
\boxed{\frac{z_s}{R}\simeq\left(\frac R{R_0}\right)^{2/7},\qquad H\propto R^{9/7},\qquad T_c\propto R^{-3/7}.}
$$
This applies in the thin intermediate region $R\ll\min(R_0,R_M)$, in addition to the point-source and stellar-gravity assumptions. The corrections are of relative order $R^2/R_0^2+R^2/R_M^2$. In particular, the printed hierarchy $R\ll R_M$ must still be accompanied by the initial thin-disc condition $q\ll1$: if $R_M\gg R_0$, radii larger than $R_0$ cannot satisfy that approximation. The matching at $q=1$ is an extrapolation used to set an integration constant; the derivation is not a valid treatment of a thick surface at $R_M$ itself.
Identify the stellar surface <temperature> with its effective radiating <temperature>, so that <luminosity of a spherical blackbody> gives $L_*=4\pi R_*^2\sigma T_*^4$. The given stellar surface scale height has $H_*^2=aT_*R_*^3/(GM)$, hence $GM/(aT_*R_*)=(R_*/H_*)^2$. Substituting into $R_0$ yields
$$
\boxed{\frac{R_0}{R_*}=\sqrt{\frac72}\left(\frac{R_*}{fH_*}\right)^4\gg1.}
$$
Thus a small stellar surface scale height leaves a wide interval above $R_*$ and below $R_0$ in which the <thin disk> flaring law can apply. The point-source approximation imposes the stronger lower cutoff obtained from $z_s\gg R_*$:
$$
R\gg R_*^{7/9}R_0^{2/9}.
$$
This cutoff is still much smaller than $R_0$ when $R_0\gg R_*$. The full interval is therefore $R_*^{7/9}R_0^{2/9}\ll R\ll\min(R_0,R_M)$, whenever these bounds overlap. An assumed thickening radius must lie above the lower cutoff for all of the original approximations to hold simultaneously. The stellar luminosity-<temperature> relation is essential to the final comparison; independent arbitrary values of $L_*$ and $T_*$ would not imply it.
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