<p>A basic argument would go something like this:</p>
<p>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 <a href="http://www-ppl.s.chiba-u.jp/lecture/radiation/node2.html" rel="nofollow noreferrer">here</a>), that for the gravitational force on the envelope to exceed the outwardly directed radiation pressure, then the stellar luminosity must obey
<span class="math-container">$$ L &lt; 1.2\times 10^{31} \left( \frac{M}{M_\odot}\right)\ {\rm Watts}\ , $$</span>
the so-called <a href="https://en.wikipedia.org/wiki/Eddington_luminosity" rel="nofollow noreferrer">Eddington luminosity limit</a>.</p>
<p>From there, if you have a prescription for the luminosity of a high mass star as a function of its mass; for example
<span class="math-container">$$L \sim 3.8\times 10^{26} \left(\frac{M}{M_\odot}\right)^{3}\ {\rm Watts}\, $$</span>
is a commonly used approximation.</p>
<p>Just substitute this in for the left hand side of the first equation and obtain
<span class="math-container">$$ \left(\frac{M}{M_\odot}\right)^{2} &lt; 3.2\times 10^4 $$</span>
<span class="math-container">$$ M &lt; 178 M_\odot$$</span></p>
<p>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.</p>
<p>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. <a href="https://www.aanda.org/articles/aa/full_html/2015/08/aa25945-15/aa25945-15.html" rel="nofollow noreferrer">Sanyal et al. (2015)</a>.</p>

CC BY-SA 4.0
2022-06-22T16:12:09.273Z
2022-06-22T16:39:54.960Z
2022-06-22T16:39:54.960Z
ProfRob
<p>I know there is a theoretical mass limit for stars around <span class="math-container">$300 M_\odot$</span>. But I cannot find a way to calculate that number. I tried to use the &quot;Eddington Limit&quot;, but I don't know if I'm on the right track. Any suggestions to calculate that limit?</p>

CC BY-SA 4.0
2022-06-22T14:30:04.963Z
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
Prallax
<star><mass><eddington-luminosity>
How can the upper limit for a star's mass be calculated?
Aegean
# How can the upper limit for a star's mass be calculated?

(Asked by: username_0 on 2022-06-22)

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?

**Tags:** star, mass, eddington-luminosity
> **Comments:**
> **username_1**: 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  

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## 1 answers

### Answer by username_2 on 2022-06-22. Score: 4 | Accepted answer

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](http://www-ppl.s.chiba-u.jp/lecture/radiation/node2.html)), 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](https://en.wikipedia.org/wiki/Eddington_luminosity).

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)](https://www.aanda.org/articles/aa/full_html/2015/08/aa25945-15/aa25945-15.html).


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