Exploring the Eddington Limit: Deriving the Formulation of Accretion Processes

[Aleolomorfo]

I am studyng accretion process on "Astrophysics in a nutshell" by Dan Maoz and I have some doubts about the derivation of the formula for the eddington limit. I understand what the edding limit is. The accretion rate cannot be arbitrarly large. The starting point is to consider an electron at a radius $r$ in an ionized gas that is taking part in an accretion flow towards some compact object. The accretion flow produces a luminosity per frequency interval $L_\nu$, and therefore the density of photons with energy $h\nu$ at $r$ is:
$$n_{ph}=\frac{L_\nu}{4\pi r^2 ch\nu}$$
I do not understand why the density of photons is written in this way. I see that it is dimensionally correct but I do not see the reason.
$\frac{L_\nu}{4\pi r^2}$ is the flux of photons with frequency $\nu$ but I do not understand why it is divided by $ch\nu$.

 

[phyzguy]

$L_{\nu}$ is not the flux of photons, it is the flux of energy. So you have to divide by the energy per photon to get the flux of photons.

 

[Ken G]

Yes, you first have to divide by the energy per photon, h*nu, to get it into a number flux (and I believe you mean flux per area), but then you also have to divide by the particle speed to get it into a density of particles per volume. You'd have to do the same thing with a flux of bullets. If you still don't see it, it might help to take the c up onto the left side of the equation, and think about what a number density times a speed is.

However, I would also point out that the easiest way to understand the Eddington limit is to think in terms of the momentum flux per area, not the photon flux and not the photon density. This is because to get the radiative force per gram, you simply take the momentum flux per area, and multiply by the cross section per gram. That's the simplest way to see what is going on.

 

[sophiecentaur]

therefore the density of photons with energy hνhνh\nu at rrr is:

Is there any special point in restating the Eddington limit in terms of photons?