Grey atmosphere

A grey atmosphere has frequency-independent opacity. In plane-parallel radiative equilibrium under the Eddington closure, its temperature obeys

Consider $I=\alpha [\beta +\tau +\mu +\gamma e^{-\delta \tau }/(1+ϵ\mu )]$ with real $|ϵ|<1$, $\alpha \gamma \ne 0$, and $\delta >0$. Its angular mean includes $A_0=\mathrm{atanh}ϵ/ϵ$. Substitution into $\mu I_{\tau }=I-J$ requires $1-A_0+(\delta -A_0ϵ)\mu =0$. Thus $A_0=1$ and $\delta =ϵ$. For real $ϵ$, $A_0=1$ only at zero, giving $\delta =0$, a contradiction. Independently, its flux contains a varying term proportional to $A_1=(1-A_0)/ϵ$, which is nonzero for nonzero real $ϵ$. Exact conservative transfer therefore needs a more general angular/depth structure than this one smooth exponential correction.

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, $T^4=3T_{\mathrm{eff}}^4[\tau +q(\tau )]/4$. The Eddington closure approximation with its approximate surface condition gives constant $q=2/3$; exact angular transfer generally gives a depth-dependent function.

For no incident radiation and outward intensity approximated as direction independent, the surface angular moments are $F=\pi I_0$ and $P_{\mathrm{rad}}(0)=2\pi I_0/(3c)$. Integrating the radiation momentum equation with the Eddington closure approximation yields the grey atmosphere law $T^4=3T_{\mathrm{eff}}^4(\tau +2/3)/4$. This is an approximate angular boundary condition, not isotropy across both hemispheres.

A plane-parallel atmosphere neglects curvature over its vertical thickness, so thermodynamic variables depend on one Cartesian height coordinate.

A plane-parallel atmosphere is geometrically accurate only if the relevant line-forming region is thin relative to the stellar radius. An isothermal estimate gives $H=k_{B}T/(\mu m_u{g}_{\mathrm{eff}})$. Radiative acceleration reduces the effective gravity and increases extension. A small photospheric $H/R$ does not justify a plane-parallel description of wind lines formed far above that photosphere.