We are given a problem:

Consider a grey atmosphere in radiative equilibrium. If the total pressure ($P$) and energy density ($u$) are related by $$P=\frac{u}{\tau^2}$$ in a non-isotropic region, where $\tau $ is the optical depth, then obtain $I(0,\mu)/I(0,1)$ as a function of $\mu.$

Since it's said, that we are considering the grey atmosphere in radiative equilibrium. That means, $$\frac{dF}{d\tau}=0\rightarrow J=S$$ We also have $$\frac{dP}{d\tau}=\frac{F}{c}\rightarrow P=\frac{F}{c}\left(\tau+q\right)$$ But this can't be possible with what are given. Where I'm getting this wrong? Is it possible like $$q=0\ \ \&\ \ \frac{u}{\tau^2}=\frac{F}{c}\tau\rightarrow u=\frac{F}{c}\tau^3$$


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