<p>I can offer mass-luminosity relations, rather than mass-radius or mass-temperature relations, which I don't know of a way to compute other than with simulations, which would take more time to set up than I have available.</p>
<p>If we assume that a star is an ideal gas and that energy escapes purely by radiative diffusion then given a particular law for the opacity, one can estimate mass-luminosity relations that include the mean molecular weight <span class="math-container">$\mu$</span>. You can see a lengthy, technical discussion of this in Chapter 7 of the (excellent) <a href="https://users-phys.au.dk/%7Ejcd/evolnotes/" rel="nofollow noreferrer">lecture notes on stellar evolution by Jørgen Christensen–Dalsgaard</a> but the main results are for either constant opacity (like pure electron scattering):
<span class="math-container">$$L\propto M^3\mu^4$$</span>
or the Kramers law of opacity (<span class="math-container">$\kappa\propto\rho T^{3.5}$</span>)
<span class="math-container">$$L\propto\frac{M^{5.5}\mu^{7.5}}{R^{0.5}}$$</span>
Given a composition containing fractional mass abundances <span class="math-container">$X$</span>, <span class="math-container">$Y$</span> and <span class="math-container">$Z$</span> for hydrogen, helium and metals (so <span class="math-container">$X+Y+Z=1$</span>), a totally ionised mixture has mean molecular weight
<span class="math-container">$$\mu=\frac{1}{2X+\frac{3}{4}Y+\frac{1}{2}Z}$$</span>
Some useful reference values are:</p>
<ul>
<li>a quasi-solar mixture <span class="math-container">$X=0.7$</span>, <span class="math-container">$Z=0.02$</span>, for which <span class="math-container">$\mu\approx0.62$</span>;</li>
<li>solar metallicity (<span class="math-container">$Z=0.02$</span>) but no hydrogen (<span class="math-container">$X=0$</span>), for which <span class="math-container">$\mu\approx1.34$</span>; and</li>
<li>pure metals <span class="math-container">$Z=1$</span>, for which <span class="math-container">$\mu=2$</span>.</li>
</ul>
<p><em>Edit</em>: I omitted that the first relation is usually applied to &quot;hot&quot; stars, broadly those main sequence stars more massive than about twice the Sun's mass. The latter is not too bad for main sequence stars around the Sun's mass. Neither reproduces observations exactly.</p>

CC BY-SA 4.0
2024-08-19T22:01:47.763Z
2024-08-23T14:15:23.213Z
2024-08-23T14:15:23.213Z
Warrick
<p>I'm building a star cluster for my space opera setting and I'd like to include some highly-metallic stars as anomalies and resources.</p>
<p><a href="https://en.wikipedia.org/wiki/Metallicity" rel="nofollow noreferrer">Wikipedia</a> defines stellar metallicity as the fraction of a star's composition that isn't hydrogen or helium (i.e. if a star is 98% percent hydrogen and helium and 2% heavier elements, it is 2% 'metallic').</p>
<p>Also according to Wikipedia, all else being equal, a more metallic star will be larger, cooler, and dimmer than a less metallic star. However, I can't find any information beyond this, even off of Wikipedia, save that it's a result of the star's interior being more &quot;opaque&quot; and hindering the transfer of energy from the core outwards.</p>
<p>I know that the luminosity and radius of a main-sequence star are related to its mass and can be estimated with the equations <span class="math-container">$L=(M_*/M_\odot)^{3.5}$</span> and <span class="math-container">$R=(M_*/M_\odot)^{0.8}$</span> and the surface temperature then estimated with the equation <span class="math-container">$T=\frac{\sqrt[4]{L}}{\sqrt{R}}$</span>.</p>
<p>Metallicity obviously changes these relations, but can the equations above be modified to account for it? If they can, how?</p>

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2024-07-27T18:59:12.220Z
Hopefully, the question is better. Bump?
TerranAmbassador
CC BY-SA 4.0
2024-07-27T23:45:22.610Z
Thanks for the bounty!
TerranAmbassador
CC BY-SA 4.0
2024-07-28T14:17:39.127Z
Those are the equations for low mass stars, but for metals to burn high masses are needed.  You are basically asking about Wolf-Rayet stars.
eshaya
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2024-07-28T14:23:34.453Z
I don't necessarily need the metals to burn, just be present.
TerranAmbassador
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2024-07-28T14:44:21.940Z
They need to burn to support the star.  Otherwise, you have planets and black holes.
eshaya
CC BY-SA 4.0
2024-07-28T15:23:02.230Z
No? Our sun is a metallic star, ~1.3%, and it's a low-mass star that only burns hydrogen.
TerranAmbassador
CC BY-SA 4.0
2024-07-28T15:40:04.170Z
@eshaya are you sure? Metalicity is certainly important in the atmosphere, but how much of the power generation in the center of our Sun is really coming from something other than hydrogen fusing to helium. While it seems that according to Wikipedia's[Solar core; Energy conversion](https://en.wikipedia.org/wiki/Solar_core#Energy_conversion), p-p is believed to be responstible for about 90% and CNO the other 10%, neither actually consume anything other than hydrogen. CNO borrows carbon but politely puts it back on the shelf in good condition when finished with it.
uhoh
CC BY-SA 4.0
2024-07-29T03:00:53.933Z
Perhaps I misunderstood the question.  I thought this was a science fiction opera and highly-metallic stars meant they were mostly metals.  You did not specify the metallicity range that you are interested in.
eshaya
CC BY-SA 4.0
2024-07-29T04:03:11.457Z
@eshaya ah, I see what you mean.
uhoh
CC BY-SA 4.0
2024-08-02T22:34:08.903Z
I would expect the luminosity to go down as metallically increases because of line blanketing (metals absorb radiation and re-emit it at a higher wavelength) and an increased opacity caused by the metals (causes the star to swell and therefore cool). I don't know if the net effect of a higher opacity would decrease the luminosity, though.
Astrovis
<star><metallicity>
How does metallicity change the mass-radius and mass-luminosity relations in main-sequence stars?
TerranAmbassador
# How does metallicity change the mass-radius and mass-luminosity relations in main-sequence stars?

(Asked by: username_0 on 2023-11-18)

I'm building a star cluster for my space opera setting and I'd like to include some highly-metallic stars as anomalies and resources.

[Wikipedia](https://en.wikipedia.org/wiki/Metallicity) defines stellar metallicity as the fraction of a star's composition that isn't hydrogen or helium (i.e. if a star is 98% percent hydrogen and helium and 2% heavier elements, it is 2% 'metallic').

Also according to Wikipedia, all else being equal, a more metallic star will be larger, cooler, and dimmer than a less metallic star. However, I can't find any information beyond this, even off of Wikipedia, save that it's a result of the star's interior being more "opaque" and hindering the transfer of energy from the core outwards.

I know that the luminosity and radius of a main-sequence star are related to its mass and can be estimated with the equations $L=(M\_\*/M\_\odot)^{3.5}$ and $R=(M\_\*/M\_\odot)^{0.8}$ and the surface temperature then estimated with the equation $T=\frac{\sqrt[4]{L}}{\sqrt{R}}$.

Metallicity obviously changes these relations, but can the equations above be modified to account for it? If they can, how?

**Tags:** star, metallicity
> **Comments:**
> **username_0**: Hopefully, the question is better. Bump?  
> **username_0**: Thanks for the bounty!  
> **username_1**: Those are the equations for low mass stars, but for metals to burn high masses are needed.  You are basically asking about Wolf-Rayet stars.  
> **username_0**: I don't necessarily need the metals to burn, just be present.  
> **username_1**: They need to burn to support the star.  Otherwise, you have planets and black holes.  
> **username_0**: No? Our sun is a metallic star, ~1.3%, and it's a low-mass star that only burns hydrogen.  
> **username_2**: @eshaya are you sure? Metalicity is certainly important in the atmosphere, but how much of the power generation in the center of our Sun is really coming from something other than hydrogen fusing to helium. While it seems that according to Wikipedia's[Solar core; Energy conversion](https://en.wikipedia.org/wiki/Solar_core#Energy_conversion), p-p is believed to be responstible for about 90% and CNO the other 10%, neither actually consume anything other than hydrogen. CNO borrows carbon but politely puts it back on the shelf in good condition when finished with it.  
> **username_1**: Perhaps I misunderstood the question.  I thought this was a science fiction opera and highly-metallic stars meant they were mostly metals.  You did not specify the metallicity range that you are interested in.  
> **username_2**: @eshaya ah, I see what you mean.  
> **username_3**: I would expect the luminosity to go down as metallically increases because of line blanketing (metals absorb radiation and re-emit it at a higher wavelength) and an increased opacity caused by the metals (causes the star to swell and therefore cool). I don't know if the net effect of a higher opacity would decrease the luminosity, though.  

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## 1 answers

### Answer by username_4 on 2024-08-19. Score: 3

I can offer mass-luminosity relations, rather than mass-radius or mass-temperature relations, which I don't know of a way to compute other than with simulations, which would take more time to set up than I have available.

If we assume that a star is an ideal gas and that energy escapes purely by radiative diffusion then given a particular law for the opacity, one can estimate mass-luminosity relations that include the mean molecular weight $\mu$. You can see a lengthy, technical discussion of this in Chapter 7 of the (excellent) [lecture notes on stellar evolution by Jørgen Christensen–Dalsgaard](https://users-phys.au.dk/%7Ejcd/evolnotes/) but the main results are for either constant opacity (like pure electron scattering):
$$L\propto M^3\mu^4$$
or the Kramers law of opacity ($\kappa\propto\rho T^{3.5}$)
$$L\propto\frac{M^{5.5}\mu^{7.5}}{R^{0.5}}$$
Given a composition containing fractional mass abundances $X$, $Y$ and $Z$ for hydrogen, helium and metals (so $X+Y+Z=1$), a totally ionised mixture has mean molecular weight
$$\mu=\frac{1}{2X+\frac{3}{4}Y+\frac{1}{2}Z}$$
Some useful reference values are:

* a quasi-solar mixture $X=0.7$, $Z=0.02$, for which $\mu\approx0.62$;
* solar metallicity ($Z=0.02$) but no hydrogen ($X=0$), for which $\mu\approx1.34$; and
* pure metals $Z=1$, for which $\mu=2$.

*Edit*: I omitted that the first relation is usually applied to "hot" stars, broadly those main sequence stars more massive than about twice the Sun's mass. The latter is not too bad for main sequence stars around the Sun's mass. Neither reproduces observations exactly.


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