= Solution
For an unattenuated isotropic point source, the radiation flux is $L_*/(4\pi r^2)$. The <radiation pressure> force per particle divided by its mass $\bar m$ gives the outward acceleration
$$
a_{\rm rad}=\frac{\sigma L_*}{4\pi\bar m c\,r^2}.
$$
Thus the effective inward inverse-square coefficient is $GM-\sigma L_*/(4\pi\bar m c)$. Insert it into the preceding wind equation. The <radiative shift of an isothermal breeze maximum> is
$$
\boxed{r'_M=\frac{GM-\sigma L_*/(4\pi\bar m c)}{2a^2}
=\frac{GM\bar m-\sigma L_*/(4\pi c)}{2k_BT_0}.}
$$
Equivalently, with <opacity> $\kappa=\sigma/\bar m$ and an <Eddington luminosity> $L_{\rm Edd}=4\pi GMc/\kappa$, this is $r'_M=r_M(1-L_*/L_{\rm Edd})$. It is an exterior speed maximum only if the radius is positive and lies within the modeled wind region. For $L_*\geq L_{\rm Edd}$, the wind equation's right side is positive at every $r>0$, so an always-subsonic outflow decreases in speed and has no such interior maximum.
Because $\sigma$ is a cross section \b[per particle], its conversion to acceleration requires $\bar m$; that mass cannot be omitted dimensionally. The requested dependence on the four stated variables therefore assumes a fixed particle species. For the atomic hydrogen of part (e), set $\bar m\simeq m_H$ before molecule formation.
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