What happens to the Gas Pressure when working out the Eddington Luminosity?

I am looking at how the Eddington Luminosity is derived and I do no understand why we only care about $P_{rad}$?

When working out $L_{Edd}$ you take the ratio between $dP_{rad}/dr$ and $dP_{hydrostatic}/dr$ which must be less or equal to than 1. If $P_{hydro} = P_{gas} + P_{rad}$ why don't we take $P_{gas}$ into account anywhere?

For reference:

$$L_{Edd} = \frac{4\pi c G M}{\kappa}$$

The Eddington luminosity is defined in this way - but I guess that is not the answer you are looking for!

However, if we ask what isotropic luminosity is required in order to balance the inward gravitational force on a lump of gas then yes we could consider gas pressure as well.

The correct formulation is that the pressure gradient due to gas plus radiation balances the density times local gravity.

The justification for neglecting gas pressure would depend on the application. For example, when using the mass limit to justify an approximate upper mass limit for stars, it is reasonably easy to show that the ratio of radiation pressure to gas pressure $\sim 0.1 (M/M_{\odot})^{2}$, where $M$ is the stellar mass in solar units. Thus for stars $>3M_{\odot}$, radiation pressure dominates, and because $L_{edd} \simeq 3\times10^{4} (M/M_{\odot})\ L_{\odot}$ and $L \simeq (M/M_{\odot})^{3.5}\ L_{\odot}$ for a main sequence star, then $L<L_{edd}$ means that $$(M/M_{\odot})^{3.5} < 3\times10^{4} (M/M_{\odot})$$ and hence $M < 61 M_{\odot}$. At this mass the radiation pressure will be nearly 400 times the gas pressure, justifying the use of a "gas pressure-free" Eddington limit.

I suspect, though I can't give chapter and verse, that for the intense radiation fields normally associated with the Eddington limit this will always be the case.

If we are talking about spherical accretion, then because the luminosity is constant at any radius, then the Eddington limit applies to material at any radius. But at greater radii from the central source, for a constant velocity inflow, the density will fall as $r^{-2}$, and the temperature will be lower at large radii. Thus the ratio of radiation to gas pressure will likely increase with radius and so at some point it will always be appropriate just to consider the radiation pressure.

What this implies is that reaching the Eddington limit could stop the accretion, but that even if it does not (e.g. clumpy accretion), then the increasing gas pressure at smaller radii could limit the accretion if the compressed gas is unable to cool.