Grey atmosphere

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

In a well-mixed convective column with fixed absorber mass and constant mass opacity, adding a perfectly transparent gas leaves total optical depth $D$ unchanged. The absolute effective emission pressure scales with total surface pressure, but their ratio remains $D$. In the thick-column emitting-level model, $T_s=T_e{D}^{R_a/c_{pa}}$; added gas warms the surface only if its specific-gas-constant to heat-capacity ratio exceeds that of the original gas. Pressure broadening would violate the fixed-opacity assumption and must be treated separately.

For a source linear in downward effective optical depth $t\in [0,D]$, two-stream approximation weighting $e^{-t}$ has normalized mean $t_{\mathrm{eff}}=1-D/(e^D-1)$. Constant mass opacity gives $p_{\mathrm{rad}}=p_s{t}_{\mathrm{eff}}/D$, tending to $p_s/2$ for small $D$ and $p_s/D$ for large $D$. The atmospheric flux is $(1-e^{-D})S(t_{\mathrm{eff}})$, separate from transmitted ground radiation. Exact angular transfer with literal vertical optical depth instead has weight $2E_2(t)$ and thick-limit mean $2/3$; the convention matters.

A grey atmosphere model with mass absorption coefficient proportional to pressure. Under hydrostatic balance, $\kappa ={\kappa }_{s}p/p_s$ gives downward optical depth $t={\kappa }_s{p}^2/(2g{p}_s)$. For $T=T_s(p/p_s{)}^a$, a non-scattering optically thick column emits hemispheric radiant flux $F\sim \sigma T_s^4{t}_s^{-2a}\Gamma (2a+1)/(a+1)$. The angular factor follows from integrating ray attenuation over the upward hemisphere; a vertical-ray approximation omits it. The extension of the emitting-depth integral to infinity requires fixed $a$ as $t_s\to \infty$.

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.