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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
\begin{figure}\centering\includegraphics[width=.6\textwidth]{eddington.eps}
\par\end{figure}

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

\begin{displaymath}
{dE\over dtdA} = {L\over 4\pi r^2},
\end{displaymath} (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


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

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


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

This must be smaller than the gravitational force:
$\displaystyle \sigma_t{L\over 4\pi cr^2}$ $\textstyle \leq$ $\displaystyle {GMm_p\over r^2}$  
$\displaystyle L$ $\textstyle \leq$ $\displaystyle {4\pi G m_p c\over \sigma_T}M \equiv L_{edd}.$ (17)

$L_{edd}$ is called the Eddington luminosity and calculated to be
\begin{displaymath}
L_{edd} = 1.2\times 10^{38} \left({M\over M_{\odot}}\right)\quad {\rm erg/sec},
\end{displaymath} (18)

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


next up previous
Next: Photoelectric Absorption Up: Thomson Scattering Previous: Thomson Scattering
Shigeru Yoshida
2002-07-18