<p>The paper (<a href="https://arxiv.org/abs/2309.12644" rel="nofollow noreferrer">arXiv version</a>) actually finds a mass limit of <span class="math-container">$M_{\mathrm{TOV}}=2.25^{+0.08}_{-0.07}M_{\odot}$</span>; it is believed that above this limit, subject to some caveats, a neutron star would further collapse into a black hole. The range <span class="math-container">$M\sim2.5\mathrm{-}3M_{\odot}$</span> really refers to compact objects that have been detected in gravitational wave events; they're mentioned just to indicate that according to the analysis, they are in fact likely the lowest-mass black holes known. LIGO/Virgo gravitational wave data alone could not determine whether that class of objects are neutron stars or black holes.</p>
<p>&quot;Non-rotating&quot; really does mean non-rotating. The equation governing models, the <a href="https://en.wikipedia.org/wiki/Tolman%E2%80%93Oppenheimer%E2%80%93Volkoff_equation" rel="nofollow noreferrer">Tolman-Oppenheimer-Volkoff equation</a>, assumes the star is not rotating. Since we would expect all neutron stars to rotate to <em>some</em> degree, and since the paper is based on a selection of real neutron stars, it might appear the data isn't applicable! However -- and this is a point the paper and articles maybe don't emphasize enough -- most reasonable spin periods do not significantly change the mass limit of a neutron star. For all but the fastest-spinning neutron stars, the difference would not be more than a percent or so (<a href="https://arxiv.org/abs/1709.07889" rel="nofollow noreferrer">Alsing et al. 2018</a>), and it is only for the very shortest possible spin periods that we would see deviations of up to 20% (<a href="https://ui.adsabs.harvard.edu/abs/2016MNRAS.459..646B/abstract" rel="nofollow noreferrer">Breu &amp; Rezzolla 2016</a>). No known pulsars spin fast enough to change the limit by more than ~2%.</p>
<p>The authors therefore assume that most of the neutron stars in the sample can be modeled as slow rotators. Three (PSR J0952-060, PSR J2215+5135, and PSR J0740+6620) have short enough rotation periods that an adjustment must be made, which they do using relations derived from simulations. They therefore believe the results are valid.</p>
<p>For what it's worth, since the paper's computed uncertainties on the mass limit are ~4%, any variation due to rotation among the other neutron stars in the sample is a factor of a few less, and those uncertainties dominate. It certainly doesn't affect their conclusion regarding the type of objects in the <span class="math-container">$2.5\mathrm{-}3M_{\odot}$</span> range.</p>

CC BY-SA 4.0
2024-03-18T12:32:46.240Z
Just a note. The fastest rotating pulsar is already only at about 1/3 of  the Keplerian limit. So the formulae in Breu & Rezzolla suggests the TOV limit is ony increased by 2% at most.
ProfRob
CC BY-SA 4.0
2024-03-22T14:35:46.863Z
@ProfRob Thanks, my language in the second paragraph was not ideal; I should have made it clear that the last sentence was referring to models and not specific pulsars.
HDE 226868
CC BY-SA 4.0
2024-03-16T04:16:28.883Z
2024-03-22T14:37:12.830Z
2024-03-22T14:37:12.830Z
HDE 226868
<p>HDE226868's answer is fine and I won't repeat that. However, an answer to your headline question is that a non-rotating neutron star is effectively one where the rotation does not significantly alter the structure.</p>
<p>A criterion to decide if rotation is important is to compare the angular acceleration with the gravitational acceleration at the surface. Rotation is unimportant if
<span class="math-container">$$ r \omega^2 = \frac{4\pi^2 r}{P^2} \ll \frac{GM}{r^2}\ $$</span>
and thus
<span class="math-container">$$P^2 \gg \frac{4\pi^2 r^3}{GM}\ , $$</span>
where <span class="math-container">$r$</span> and <span class="math-container">$M$</span> are the neutron star radius and mass, and <span class="math-container">$P$</span> is the rotation period. Putting some numbers in
<span class="math-container">$$ P \gg 0.0004\left(\frac{M}{2M_\odot}\right)^{-1}\left(\frac{r}{10{\rm km}}\right)^3 \ {\rm seconds}\ .$$</span></p>
<p>Since the fastest spinning pulsar has a period of 0.0014 s and since the vast majority of neutron stars spin down to much slower periods than 1 s very early in their lives, then rotation is going to be unimportant in almost all cases for neutron stars.</p>

CC BY-SA 4.0
2024-03-16T07:34:50.340Z
2024-03-16T17:24:02.507Z
2024-03-16T17:24:02.507Z
ProfRob
<p>This is the article as mentioned above: <a href="https://www.universetoday.com/166172/the-maximum-mass-of-a-neutron-star-is-2-25-solar-masses/" rel="nofollow noreferrer"><em>The Maximum Mass of a Neutron Star is 2.25 Solar Masses</em></a></p>
<p>To repeat, according to this article, the maximum mass of a non-rotating neutron star is around 2.5-3 solar masses.</p>
<ol>
<li><p>What is defined as a non-rotating neutron star? I find it difficult to understand how any such body could not rotate, based on the conservation of energy principle.</p>
</li>
<li><p>Secondly, is the possible maximum mass of a rotating neutron star greater than this value range for a non-rotational body?</p>
</li>
<li><p>According to the article, if I understand correctly, a non-rotating neutron star of greater than this mass will turn into a black hole, albeit one of the smallest mass.</p>
</li>
</ol>

CC BY-SA 4.0
2024-03-16T19:46:02.307Z
"What is defined as a non-rotating neutron star?" is sort of like asking "what is defined as a vacuum?", or "what is defined as absolute zero?". The lack of real, absolute examples of an ideal, should not prevent understanding the meaning of the ideal.
Karl Knechtel
<neutron-star>
According to this article the max mass of a non-spinning neutron star is around 2.5 solar masses. What is defined as a non-rotating neutron star?
Stefan 
# According to this article the max mass of a non-spinning neutron star is around 2.5 solar masses. What is defined as a non-rotating neutron star?

(Asked by: username_0 on 2024-03-16)

This is the article as mentioned above: [*The Maximum Mass of a Neutron Star is 2.25 Solar Masses*](https://www.universetoday.com/166172/the-maximum-mass-of-a-neutron-star-is-2-25-solar-masses/)

To repeat, according to this article, the maximum mass of a non-rotating neutron star is around 2.5-3 solar masses.

1. What is defined as a non-rotating neutron star? I find it difficult to understand how any such body could not rotate, based on the conservation of energy principle.
2. Secondly, is the possible maximum mass of a rotating neutron star greater than this value range for a non-rotational body?
3. According to the article, if I understand correctly, a non-rotating neutron star of greater than this mass will turn into a black hole, albeit one of the smallest mass.

**Tags:** neutron-star
> **Comments:**
> **username_1**: "What is defined as a non-rotating neutron star?" is sort of like asking "what is defined as a vacuum?", or "what is defined as absolute zero?". The lack of real, absolute examples of an ideal, should not prevent understanding the meaning of the ideal.  

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## 2 answers

### Answer by username_2 on 2024-03-16. Score: 18

The paper ([arXiv version](https://arxiv.org/abs/2309.12644)) actually finds a mass limit of $M\_{\mathrm{TOV}}=2.25^{+0.08}\_{-0.07}M\_{\odot}$; it is believed that above this limit, subject to some caveats, a neutron star would further collapse into a black hole. The range $M\sim2.5\mathrm{-}3M\_{\odot}$ really refers to compact objects that have been detected in gravitational wave events; they're mentioned just to indicate that according to the analysis, they are in fact likely the lowest-mass black holes known. LIGO/Virgo gravitational wave data alone could not determine whether that class of objects are neutron stars or black holes.

"Non-rotating" really does mean non-rotating. The equation governing models, the [Tolman-Oppenheimer-Volkoff equation](https://en.wikipedia.org/wiki/Tolman%E2%80%93Oppenheimer%E2%80%93Volkoff_equation), assumes the star is not rotating. Since we would expect all neutron stars to rotate to *some* degree, and since the paper is based on a selection of real neutron stars, it might appear the data isn't applicable! However -- and this is a point the paper and articles maybe don't emphasize enough -- most reasonable spin periods do not significantly change the mass limit of a neutron star. For all but the fastest-spinning neutron stars, the difference would not be more than a percent or so ([Alsing et al. 2018](https://arxiv.org/abs/1709.07889)), and it is only for the very shortest possible spin periods that we would see deviations of up to 20% ([Breu & Rezzolla 2016](https://ui.adsabs.harvard.edu/abs/2016MNRAS.459..646B/abstract)). No known pulsars spin fast enough to change the limit by more than ~2%.

The authors therefore assume that most of the neutron stars in the sample can be modeled as slow rotators. Three (PSR J0952-060, PSR J2215+5135, and PSR J0740+6620) have short enough rotation periods that an adjustment must be made, which they do using relations derived from simulations. They therefore believe the results are valid.

For what it's worth, since the paper's computed uncertainties on the mass limit are ~4%, any variation due to rotation among the other neutron stars in the sample is a factor of a few less, and those uncertainties dominate. It certainly doesn't affect their conclusion regarding the type of objects in the $2.5\mathrm{-}3M\_{\odot}$ range.

> **Comments:**
> **username_3**: Just a note. The fastest rotating pulsar is already only at about 1/3 of  the Keplerian limit. So the formulae in Breu & Rezzolla suggests the TOV limit is ony increased by 2% at most.  
> **username_2**: @ProfRob Thanks, my language in the second paragraph was not ideal; I should have made it clear that the last sentence was referring to models and not specific pulsars.  

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### Answer by username_3 on 2024-03-16. Score: 9

HDE226868's answer is fine and I won't repeat that. However, an answer to your headline question is that a non-rotating neutron star is effectively one where the rotation does not significantly alter the structure.

A criterion to decide if rotation is important is to compare the angular acceleration with the gravitational acceleration at the surface. Rotation is unimportant if
$$ r \omega^2 = \frac{4\pi^2 r}{P^2} \ll \frac{GM}{r^2}\ $$
and thus
$$P^2 \gg \frac{4\pi^2 r^3}{GM}\ , $$
where $r$ and $M$ are the neutron star radius and mass, and $P$ is the rotation period. Putting some numbers in
$$ P \gg 0.0004\left(\frac{M}{2M\_\odot}\right)^{-1}\left(\frac{r}{10{\rm km}}\right)^3 \ {\rm seconds}\ .$$

Since the fastest spinning pulsar has a period of 0.0014 s and since the vast majority of neutron stars spin down to much slower periods than 1 s very early in their lives, then rotation is going to be unimportant in almost all cases for neutron stars.


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