# Eddington Luminosity or Eddington Limit

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### 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.