The Eddington Limit

One of the representative astrophysical application of the Thomson scattering is the Eddington Limit, the possibly maximum luminosity of a star, which can keep electrons from being blown out of the sphere.

Figure 2: The Eddington Limit: Balancing of gravity and radiation pressure

As shown in FIG. 2, a free electron can maintain within the sphere when the radiation pressure is weaker than the gravitational force falling into the star center. The radiation (photon ) energy flux at distance $r$ from the center of the star is

$${dE\over dtdA} = {L\over 4\pi r^2},$$ (14)

where $L$, energy per unit second, is the luminosity of the star. For a photon, the momentum $p$ is related with energy $E$ by $p=E/c$, and the momentum flux is then given by

$${dp\over dtdA} = {L\over 4\pi cr^2}.$$ (15)

Then the momentum transfer rate, i.e. the radiation force to a free electron is given by

$${dp\over dt}=\sigma_T{dp\over dtdA} = \sigma_t{L\over 4\pi cr^2}.$$ (16)

This must be smaller than the gravitational force:

$$\sigma_t{L\over 4\pi cr^2} \leq {GMm_p\over r^2}$$

$$L \leq {4\pi G m_p c\over \sigma_T}M \equiv L_{edd}.$$ (17)

$L_{edd}$ is called the Eddington luminosity and calculated to be

$$L_{edd} = 1.2\times 10^{38} \left({M\over M_{\odot}}\right)\quad {\rm erg/sec},$$ (18)

where $M_{\odot}=2.0\times 10^{33} \mbox{g}$ is the solar mass.