= Hopf function of a grey atmosphere
{c}
{title2=$q(\tau)$}
The Hopf function encodes the departure of a plane-parallel <grey atmosphere> from a purely linear source profile. With physical bolometric <radiative flux> $F$ and standard <mean intensity> $J$, it is defined by $J=3F[\tau+q(\tau)]/(4\pi)$. In thermal <radiative equilibrium>, $T^4=3T_{\rm eff}^4[\tau+q(\tau)]/4$. The <Eddington closure approximation> with its approximate surface condition gives constant $q=2/3$; exact angular transfer generally gives a depth-dependent function.
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