<p>Yes, you are correct in so far as what the Chandrasekhar limit is for a white dwarf supported by ideal electron degeneracy pressure and using a Newtonian equation of hydrostatic equilibrium.</p>
<p>Neither of these latter assumptions is true for real (or hypothetical) white dwarfs of any composition.</p>
<p>First, there is a negative correction to the ideal gas pressure caused by Coulomb interactions between the electrons and ions in the gas. Second, at high densities, inverse beta decay will start to remove free electrons and build nuclei with larger <span class="math-container">$\mu_e$</span>. This will destabilise the white dwarf at the critical density where the electron Fermi energy is high enough to start the inverse beta decay process. Third, General Relativity is required for high-mass white dwarfs. The GR equation of hydrostatic equilibrium has pressure on the right hand side, so that increasing pressure ultimately becomes self-defeating and causes instability at a finite density</p>
<p>All the above effects <em>lower</em> the Chandrasekhar mass from the ideal case you quote. By how much depends on which process may trigger instability first, which in turn depends on <span class="math-container">$\mu_e$</span> and the inverse beta decay threshold for specific nuclei. For instance, it is thought to be about 1.38 solar masses for a carbon white dwarf, rather than the 1.44 solar masses that the ideal case would predict. I think the limit for a &quot;hydrogen white dwarf&quot; would be <em>much</em> lower than <span class="math-container">$5.7 M_\odot$</span> because the threshold energy for inverse beta decay for protons is much lower than for carbon nuclei, so would occur at much lower densities.</p>

CC BY-SA 4.0
2021-09-12T20:13:16.860Z
Which assumptions are you referring to?
zucculent
CC BY-SA 4.0
2021-09-12T19:22:53.083Z
2021-10-27T23:04:57.027Z
2021-10-27T23:04:57.027Z
ProfRob
<p>The &quot;vary wildly&quot; point is not supported by the accepted stellar evolution models that always end up with some carbon-12/oxygen-16 mixture at the degenerate core stage near the Chandrasekhar limit.</p>
<p>Yes, the Chandrasekhar limit will be different for white dwarfs with different composition.</p>
<p>You are right for mercury and hydrogen &quot;white dwarf&quot; (i.e. a degeneracy-pressure supported object).</p>
<p>On the other hand, a white dwarf made of hydrogen will be VERY hard and slow to construct without failing either into a main-sequence star (if you just pile up hydrogen without cooling it) or an Ia-very-hydrogen-rich supernova (if you DO cool it carefully but fail at some point when there is already some amount of degenerate phase).</p>
<p>And whatever means to construct such a beast you invent, it will fail the &quot;white dwarf&quot; definition because it has to be cold in order to be stable. It will be a <a href="https://en.wikipedia.org/wiki/Black_dwarf" rel="nofollow noreferrer">black dwarf</a>.</p>
<p>Or maybe an overweight rogue planet (since it has never been a main sequence star and is not a remnant of one)?</p>
<p>Mercury is easier - it won't fuse.</p>

CC BY-SA 4.0
2021-09-07T00:47:52.457Z
I miscommunicated some of my points. Obviously the Chandrasekhar limit is very consistent among real white dwarves. But if one was somehow made of hydrogen, the Chandrasekhar limit would be very different than it was made of, say, lead. Thanks for pointing these out. I've made some edits that I hope will make things clearer to future readers.
zucculent
CC BY-SA 4.0
2021-09-18T17:41:01.657Z
An old brown dwarf is (nearly) a white dwarf made of hydrogen.
ProfRob
CC BY-SA 4.0
2021-09-18T21:19:37.360Z
@ProfRob indeed it is. But it is way below the mass limit we discuss.
fraxinus
CC BY-SA 4.0
2021-09-06T23:20:37.510Z
2021-09-07T10:10:47.667Z
2021-09-07T10:10:47.667Z
fraxinus
<p>Disclaimer: I'm going to be using the term &quot;white dwarf&quot; to refer to any spherical celestial body made of electron degenerate matter. If I had a better term, I would use it.</p>
<p>The Chandrasekhar limit for white dwarfs can be approximated with the equation <span class="math-container">$M_{limit}=\frac{\omega_3^0 \sqrt{3\pi}}{2} \left(\frac{\hbar c}{G}\right)^\frac32 \frac{1}{\left(\mu_e m_H\right)^2}$</span>. The most relevant part of the equation here is <span class="math-container">$\frac{1}{\left(\mu_e m_H\right)^2}$</span>. <span class="math-container">$m_H$</span> is the mass of a hydrogen atom and <span class="math-container">$\mu_e$</span> is the average molecular weight per electron, and it depends on what the white dwarf is made of. In real white dwarfs, <span class="math-container">$\mu_e$</span> is around 2, since the elements in a white dwarf have mostly equal numbers of protons and neutrons, so the Chandrasekhar limit is around 1.4 <span class="math-container">$M_\odot$</span>.</p>
<p>But it seems like <span class="math-container">$\mu_e$</span>, and therefore <span class="math-container">$M_{limit}$</span>, could theoretically be very different if the white dwarf was made of different elements. If a white dwarf was (somehow) made of pure hydrogen, for example, <span class="math-container">$\mu_e$</span> would be basically 1, and <span class="math-container">$M_{limit}$</span> would evaluate to <span class="math-container">$5.7 M_\odot$</span>. If a white dwarf was made of mercury-204, which has the highest mass-to-electron ratio among stable nuclei, <span class="math-container">$\mu_e$</span> would be around 2.55, and the Chandrasekhar for such a white dwarf would be around <span class="math-container">$0.88 M_\odot$</span>. Therefore, it looks like the Chandrasekhar limit for any stable white dwarf would have to be between <span class="math-container">$0.88 M_\odot$</span> and <span class="math-container">$5.7 M_\odot$</span>. Do I have this right? Obviously we'd never have a white dwarf made of hydrogen or mercury.</p>

CC BY-SA 4.0
2021-10-28T03:54:09.777Z
I'm sure most know this, but if you get it rotating fast enough, you can push the mass quite a fair bit higher, like this rare occurrence:     https://www.universetoday.com/145215/two-white-dwarfs-merged-together-into-a-single-ultramassive-white-dwarf/#:~:text=Two%20White%20Dwarfs%20Merged%20Together%20Into%20a%20Single,named%20WDJ0551%2B4135%2C%20is%20about%20150%20light%20years%20away.
userLTK
<white-dwarf><degenerate-matter>
Hypothetical upper and lower bounds for Chandrasekhar limit based on composition?
zucculent
# Hypothetical upper and lower bounds for Chandrasekhar limit based on composition?

(Asked by: username_0 on 2021-09-05)

Disclaimer: I'm going to be using the term "white dwarf" to refer to any spherical celestial body made of electron degenerate matter. If I had a better term, I would use it.

The Chandrasekhar limit for white dwarfs can be approximated with the equation $M\_{limit}=\frac{\omega\_3^0 \sqrt{3\pi}}{2} \left(\frac{\hbar c}{G}\right)^\frac32 \frac{1}{\left(\mu\_e m\_H\right)^2}$. The most relevant part of the equation here is $\frac{1}{\left(\mu\_e m\_H\right)^2}$. $m\_H$ is the mass of a hydrogen atom and $\mu\_e$ is the average molecular weight per electron, and it depends on what the white dwarf is made of. In real white dwarfs, $\mu\_e$ is around 2, since the elements in a white dwarf have mostly equal numbers of protons and neutrons, so the Chandrasekhar limit is around 1.4 $M\_\odot$.

But it seems like $\mu\_e$, and therefore $M\_{limit}$, could theoretically be very different if the white dwarf was made of different elements. If a white dwarf was (somehow) made of pure hydrogen, for example, $\mu\_e$ would be basically 1, and $M\_{limit}$ would evaluate to $5.7 M\_\odot$. If a white dwarf was made of mercury-204, which has the highest mass-to-electron ratio among stable nuclei, $\mu\_e$ would be around 2.55, and the Chandrasekhar for such a white dwarf would be around $0.88 M\_\odot$. Therefore, it looks like the Chandrasekhar limit for any stable white dwarf would have to be between $0.88 M\_\odot$ and $5.7 M\_\odot$. Do I have this right? Obviously we'd never have a white dwarf made of hydrogen or mercury.

**Tags:** white-dwarf, degenerate-matter
> **Comments:**
> **username_1**: I'm sure most know this, but if you get it rotating fast enough, you can push the mass quite a fair bit higher, like this rare occurrence:     https://www.universetoday.com/145215/two-white-dwarfs-merged-together-into-a-single-ultramassive-white-dwarf/#:~:text=Two%20White%20Dwarfs%20Merged%20Together%20Into%20a%20Single,named%20WDJ0551%2B4135%2C%20is%20about%20150%20light%20years%20away.  

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## 2 answers

### Answer by username_2 on 2021-09-12. Score: 3

Yes, you are correct in so far as what the Chandrasekhar limit is for a white dwarf supported by ideal electron degeneracy pressure and using a Newtonian equation of hydrostatic equilibrium.

Neither of these latter assumptions is true for real (or hypothetical) white dwarfs of any composition.

First, there is a negative correction to the ideal gas pressure caused by Coulomb interactions between the electrons and ions in the gas. Second, at high densities, inverse beta decay will start to remove free electrons and build nuclei with larger $\mu\_e$. This will destabilise the white dwarf at the critical density where the electron Fermi energy is high enough to start the inverse beta decay process. Third, General Relativity is required for high-mass white dwarfs. The GR equation of hydrostatic equilibrium has pressure on the right hand side, so that increasing pressure ultimately becomes self-defeating and causes instability at a finite density

All the above effects *lower* the Chandrasekhar mass from the ideal case you quote. By how much depends on which process may trigger instability first, which in turn depends on $\mu\_e$ and the inverse beta decay threshold for specific nuclei. For instance, it is thought to be about 1.38 solar masses for a carbon white dwarf, rather than the 1.44 solar masses that the ideal case would predict. I think the limit for a "hydrogen white dwarf" would be *much* lower than $5.7 M\_\odot$ because the threshold energy for inverse beta decay for protons is much lower than for carbon nuclei, so would occur at much lower densities.

> **Comments:**
> **username_0**: Which assumptions are you referring to?  

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### Answer by username_3 on 2021-09-06. Score: 1

The "vary wildly" point is not supported by the accepted stellar evolution models that always end up with some carbon-12/oxygen-16 mixture at the degenerate core stage near the Chandrasekhar limit.

Yes, the Chandrasekhar limit will be different for white dwarfs with different composition.

You are right for mercury and hydrogen "white dwarf" (i.e. a degeneracy-pressure supported object).

On the other hand, a white dwarf made of hydrogen will be VERY hard and slow to construct without failing either into a main-sequence star (if you just pile up hydrogen without cooling it) or an Ia-very-hydrogen-rich supernova (if you DO cool it carefully but fail at some point when there is already some amount of degenerate phase).

And whatever means to construct such a beast you invent, it will fail the "white dwarf" definition because it has to be cold in order to be stable. It will be a [black dwarf](https://en.wikipedia.org/wiki/Black_dwarf).

Or maybe an overweight rogue planet (since it has never been a main sequence star and is not a remnant of one)?

Mercury is easier - it won't fuse.

> **Comments:**
> **username_0**: I miscommunicated some of my points. Obviously the Chandrasekhar limit is very consistent among real white dwarves. But if one was somehow made of hydrogen, the Chandrasekhar limit would be very different than it was made of, say, lead. Thanks for pointing these out. I've made some edits that I hope will make things clearer to future readers.  
> **username_2**: An old brown dwarf is (nearly) a white dwarf made of hydrogen.  
> **username_3**: @ProfRob indeed it is. But it is way below the mass limit we discuss.  

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