Engineering & Science
Thomson Electron Scattering Optical Depth (tau_es) & Photon Mean Free Path
Calculate electron-scattering optical depth tau_es = n_e * sigma_T * L and photon mean free path l_mfp = 1 / (n_e * sigma_T) using the exact classical Thomson cross section sigma_T = 6.6524587e-29 m^2.
Why photons take thousands of years to escape the Sun's core—and why the CMB polarized!
At photon energies well below the 511 keV electron rest energy (h*nu << m_e * c^2), free electrons scatter electromagnetic waves elastically with a wavelength-independent cross section derived by J. J. Thomson: sigma_T = (8 * PI() / 3) * (r_e ^ 2) = 6.6524587e-29 m^2 (0.6652 barns), where r_e = 2.81794e-15 m is the classical electron radius!
Across an ionized plasma column of free-electron density n_e (m^-3) and path length L (meters), the Thomson Scattering Optical Depth is tau_es = n_e * sigma_T * L (with transmitted unscattered fraction EXP(-tau_es)), and the photon Mean Free Path between collisions is l_mfp = 1 / (n_e * sigma_T)! In a dense stellar atmosphere / ICF corona (n_e = 1.0e20 m^-3, L = 5.0e8 m), tau_es = 3.3262 (optically thick > 1, mean free path 150,320.4 km), whereas a 10x more tenuous outer corona (n_e = 1.0e19 m^-3) becomes optically thin at tau_es = 0.3326!
| Ionized Plasma Column (L = 5.0e8 m) | Electron Density n_e (m^-3) | Thomson Optical Depth tau_es | Photon Mean Free Path (km) |
|---|---|---|---|
| Optically Thick Envelope | 1.0e20 m^-3 | 3.3262 (tau > 1: Diffusive!) | 150,320.4 km |
| Optically Thin Corona | 1.0e19 m^-3 | 0.3326 (tau < 1: Transparent) | 1,503,203.6 km |
| Invalid Density n_e <= 0 | Division by Zero Guard | Guarded Input Error | "Invalid Plasma Column" |
Thomson Electron-Scattering Optical Depth tau_es = n_e * sigma_T * L
Multiplies electron density B2 (m^-3), path length B3 (m), and CODATA Thomson cross section 6.6524587e-29 m^2.
=IF(OR(B2 <= 0, B3 <= 0), "Invalid Plasma Column", ROUND(B2 * 6.6524587e-29 * B3, 4))Thomson Photon Mean Free Path l_mfp = 1 / (n_e * sigma_T) in Kilometers
Computes meters 1 / (n_e * 6.6524587e-29) and divides by 1,000 m/km.
=IF(B2 <= 0, "Invalid Plasma Column", ROUND((1 / (B2 * 6.6524587e-29)) / 1000, 1))Try the example
Use the original values or change them to test your understanding.