# Eddington Luminosity or Eddington Limit

### Eddington Luminosity or Eddington Limit

The Eddington luminosity or Eddington limit, $L_{\rm Edd}$, is the luminosity for which the outward radiation force from a spherically-symmetric source emitter on matter exactly balances the inward gravitational force onto a centrally-located mass.

$L_{\rm Edd} = \frac{4\pi G M m_{p} c}{\sigma_{T}}$

or

$L_{\rm Edd} = 1.263 \times 10^{38} \left(\frac{M}{M_{\odot}}\right) \ \ \ {\rm erg s^{-1}}$

Here
$G = 6.674 \times 10^{-11} \rm \ m^{-3} \ kg^{-1} \ s^{-2}$;
$M$ is the central mass (e.g. a black hole);
$m_{p} = 1.67 \times 10^{-27} \ {\rm kg}$ is the proton mass;
$c = 2.998 \times 10^{8} \ {\rm m \ s^{-1}}$;
$\sigma_{T} = 6.65 \times 10^{-25} \ {\rm cm^{2}}$ is the Thomson cross section.
$M_{\odot} = 2 \times 10^{30} \ {\rm kg}$ is the mass of the Sun.

Assumptions:
-Electron mass is neglected compared to the proton mass.
-Equal number of protons and electrons is assumed.
-Spherical symmetry of the radiation source is critical in the derivation.
-No other forces aside from radiation pressure and gravitational free fall are
considered.