I know there is a theoretical mass limit for stars around $300 M_\odot$. But I cannot find a way to calculate that number. I tried to use the "Eddington Limit", but I don't know if I'm on the right track. Any suggestions to calculate that limit?
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$\begingroup$ I don't think there is an absolute limit to the mass of a star. It also depends on metallicity. PopIII could have had even 1000 solar masses (e.g. see Hirano et al. 2014 10.1088/0004-637X/781/2/60). But sure, at solar metallicity, stars more massive than 200 or 300 solar masses would not be able to retain their envelope for long, due to the Eddington limit, as you suggest. I don't think this is easy to prove with pen and paper, though, but I'm ready to be proved wrong with amazing answers $\endgroup$– PrallaxCommented Jun 22, 2022 at 14:30
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A basic argument would go something like this:
If you assume that the opacity in the star's envelope is dominated by Thomson scattering from free electrons, then it is simple enough to show (e.g. see here), that for the gravitational force on the envelope to exceed the outwardly directed radiation pressure, then the stellar luminosity must obey $$ L < 1.2\times 10^{31} \left( \frac{M}{M_\odot}\right)\ {\rm Watts}\ , $$ the so-called Eddington luminosity limit.
From there, if you have a prescription for the luminosity of a high mass star as a function of its mass; for example $$L \sim 3.8\times 10^{26} \left(\frac{M}{M_\odot}\right)^{3}\ {\rm Watts}\, $$ is a commonly used approximation.
Just substitute this in for the left hand side of the first equation and obtain $$ \left(\frac{M}{M_\odot}\right)^{2} < 3.2\times 10^4 $$ $$ M < 178 M_\odot$$
The exact figure is much more uncertain than this because (i) the opacity in the envelope isn't just due to Thomson scattering and in particular is highly dependent on metallicity, (ii) the luminosity isn't just some simple power-law scaling with the stellar mass, (iii) it doesn't consider rotation or mass-loss and (iv) the above argument crudely assumes you can calculate an Eddington limit for the whole star, whereas it might be exceeded in some parts of the envelope and not others.
The only way you are going to calculate a more accurate number is by doing a full simulation of the evolution of a massive star - e.g. Sanyal et al. (2015).