Opacity ($\kappa$)

Opacity is an extinction cross-section per unit mass. It controls the photon mean free path and therefore the radiative temperature gradient in an optically thick star.

The negative hydrogen ion supplies important continuum opacity in cool hydrogen-rich stellar atmospheres. A useful local approximate law is $\kappa ={\kappa }_0{\rho }^{1/2}{T}^9$; it is not valid over arbitrary temperatures and compositions. In an ideal gas power-law opacity radiative envelope matched to ${\nabla }_{\mathrm{phot}}=1/8$, it gives ${\nabla }_{\mathrm{rad}}=-1/3+(11/24)(T/T_{\mathrm{eff}}{)}^{9/2}$, rising rapidly toward convective instability.

Let $\mathcal{R}=k_B/({\mu }_m{m}_p)$ for constant mean molecular weight ${\mu }_m$ in proton-mass units, and let $\sigma$ be the Stefan–Boltzmann constant in the Stefan–Boltzmann law. The gas sound speed used here satisfies $c_s^2=\mathcal{R}T$, and $\Omega$ is the local orbital frequency. For the local opacity approximation $\kappa ={\kappa }_0{\rho }^{1/3}{T}^{10}$, a gas-supported optically thick alpha disk has $\rho \sim \Sigma /H$ and $\mathcal{R}T\sim {\Omega }^2{H}^2$. Integrated viscous heating is $F_{+}\sim \alpha \Sigma {\Omega }^3{H}^2$. Radiative diffusion gives $F_{-}\sim (\sigma /{\kappa }_0)T^{-6}{\Sigma }^{-4/3}{H}^{1/3}\sim (\sigma /{\kappa }_0){\mathcal{R}}^6{\Omega }^{-12}{\Sigma }^{-4/3}{H}^{-35/3}$. Equating the fluxes proves the thickness scaling. At fixed composition and $\alpha$, it gives $H/r\propto r^{53/82}{\Sigma }^{-7/41}$ and $c_s\propto r^{6/41}{\Sigma }^{-7/41}$ in Keplerian rotation. Radial powers therefore require a surface-density profile; they are not determined by opacity alone. The formula is local to the stated opacity regime and small disk aspect ratio.

The Kramers opacity law has $\kappa ={\kappa }_0\rho T^{-7/2}$ for bound-free and free-free absorption under its idealized ionized-gas assumptions.

Electron-scattering opacity is approximately independent of density and temperature in a fully ionized nonrelativistic plasma of fixed composition. For hydrogen mass fraction $X$, ${\kappa }_{\mathrm{es}}\simeq 0.2(1+X)\mathrm{c}{\mathrm{m}}^2{\mathrm{g}}^{-1}$.

Free-free opacity is absorption caused when a photon is absorbed during a Coulomb encounter between an electron and an ion. In the Kramers approximation its Rosseland mean scales approximately as ${\kappa }_{\mathrm{ff}}\propto \rho T^{-7/2}$.