Kramers' opacity law

Kramers' opacity law describes the opacity of a medium in terms of the ambient density and temperature, assuming that the opacity is dominated by bound-free absorption (the absorption of light during ionization of a bound electron) or free-free absorption (the absorption of light when scattering a free ion, inverse of bremsstrahlung). ^[1] It is often used to model radiative transfer, particularly in stellar atmospheres. ^[2] The relation is named after the Dutch physicist Hendrik Kramers, who first derived the form in 1923. ^[3]

The general functional form of the opacity law is ${\displaystyle {\bar {\kappa }}\propto \rho }$${\displaystyle T^{-7/2},}$ where

${\displaystyle {\bar {\kappa }}}$ is the resulting average opacity ((kg/m³)^-1/m),

${\displaystyle \rho }$ is the density and

${\displaystyle T}$ the temperature of the medium.

Often the overall opacity is inferred from observations, and this form of the relation describes how changes in the density or temperature (highly non-linear) will affect the opacity.

Calculation

The specific forms for bound-free and free-free absorption are:

• Bound-free$${\displaystyle {\bar {\kappa }}_{\text{bf}}=4.34\times 10^{25}{\frac {g_{\text{bf}}}{t}}\cdot Z(1+X)\cdot {\frac {\rho }{\rm {g/cm^{3}}}}\left({\frac {T}{\rm {K}}}\right)^{-7/2}{\rm {\,cm^{2}\,g^{-1}}},}$$
• Free-free $${\displaystyle {\bar {\kappa }}_{\text{ff}}=3.68\times 10^{22}g_{\text{ff}}\cdot (1-Z)(1+X)\cdot {\frac {\rho }{\mathrm {g/cm^{3}} }}\left({\frac {T}{\rm {K}}}\right)^{-7/2}\mathrm {cm^{2}\,g^{-1}} .}$$

By classical electron-scattering (Thomson) opacity depends on H-ion concentration alone: $${\displaystyle {\bar {\kappa }}_{\mathrm {es} }=0.2(1+X)\mathrm {\,cm^{2}\,g^{-1}} =0.02(1+X)\mathrm {\,m^{2}\,kg^{-1}} }$$ Compton scattering of electrons occurs at higher photon energy.

Here, ${\displaystyle g_{\text{bf}}}$ and ${\displaystyle g_{\text{ff}}}$ are the Gaunt factors of circa 1 (quantum-mechanical correction terms) associated with bound-free and free-free transitions respectively. The ${\displaystyle t}$ is an additional correction factor, typically having a value between 1 and 100. The opacity depends on the number density of electrons and ions in the medium, described by the fractional abundance (by mass):

• of elements heavier than helium, ${\displaystyle Z,}$
• and of hydrogen, ${\displaystyle X.}$ ^[3]

With only helium present (and classical behaviour) ${\displaystyle {\bar {\kappa }}_{\text{ff}}}$ is proportional to mass density and ${\displaystyle T^{-3.5},}$ valid also for ${\displaystyle {\bar {\kappa }}_{\text{bf}}}$ in lithium etc. medium.

References

1. Phillips (1999), p. 92.
2. Carroll (1996), p. 274–276.
3. Carroll (1996), p. 274.

Bibliography

• Carroll, Bradley; Ostlie, Dale (1996). Modern Astrophysics. Addison-Wesley.
• Phillips, A. C. (1999). The Physics of Stars. Wiley.