<p>The minimum mass of a neutron star is actually about 0.2 solar masses and has nothing to do with the Chandrasekhar limit (see <a href="https://physics.stackexchange.com/a/143174/43351">this Physics SE answer</a>). It seems unlikely however that there is any astrophysical channel to produce such neutron stars.</p>
<p>Several neutron stars have precisely measured masses that are smaller than 1.44 solar masses. The smallest is currently about 1.17 solar masses (<a href="https://arxiv.org/abs/1509.08805" rel="nofollow noreferrer">Martinez et al. 2015</a>). Note that more massive neutron stars may actually have smaller radii. It depends on the uncertain relationship between pressure and density.</p>
<p>The plot below shows <em>theoretical</em> relationships between masses and radii of neutron stars. The measured mass range for neutron stars is 1.17-2.1 solar masses, so you could estimate the smallest possible radius from your favourite model curve. For the &quot;softest&quot; equations of state (labelled &quot;SQM1,  where quark matter develops at the neutron star core), the smallest radius for a 1.17 solar mass neutron star is about 8.5 km.</p>
<p><a href="https://i.sstatic.net/yXL5W.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/yXL5W.jpg" alt="enter image description here" /></a></p>
<p>Actually <em>measuring</em> the radii of neutron stars is incredibly difficult. The &quot;measurements&quot; that exist are rather indirect inferences and have large uncertainties.</p>
<p>There is however a fundamental limit in General Relativity, that is larger than the Schwarzschild radius, for the minimum radius of an object at a given mass. This <a href="https://en.m.wikipedia.org/wiki/Buchdahl%27s_theorem" rel="nofollow noreferrer">&quot;Buchdahl limit&quot;</a> is 9/8 of the Schwarzschild radius (shown in the picture as a forbidden region, labelled &quot;GR&quot;). It does not matter what type of pressure support is provided, a spherically symmetric object will collapse to a black hole if smaller than that.</p>
<p>For realistic relationships between pressure and density, then the true limit is a bit bigger than the Buchdahl limit - perhaps 1.2 to 1.3 times the Schwarzschild radius (represented by the grey forbidden region labelled &quot;causality&quot;). This is around 5 km, for neutron stars with the smallest measured masses, so presumably those neutron stars are bigger than that.</p>
<p>Rapid rotation could change some of these considerations, but the measured rotation rates of pulsars are too slow to have much effect.</p>
<p>I think the reason for the strange, small radius values you see (especially in Wikipedia, which does not have much quality control - always look at the original sources) is that it is a &quot;fitting parameter&quot; and represents the size of the emitting region and not necessarily the radius of the neutron star.</p>
<p>Finally, the surface temperature of a neutron star does not directly depend on its radius. Neutron stars start their lives very hot and cool down with time. To first order, the surface temperature of a neutron star would depend on its age. It could also depend on the rate at which it was accreting material from the interstellar medium (or a companion) or perhaps even its initial magnetic field strength.</p>

CC BY-SA 4.0
2021-09-26T14:38:46.820Z
As usual, very informative answer! Are you aware of an analogous limit for Kerr black holes as the Buchdal limit?
Daddy Kropotkin
CC BY-SA 4.0
2021-09-26T14:26:50.780Z
2023-01-11T09:39:06.800Z
2023-01-11T09:39:06.800Z
ProfRob
<p>I think your reservations are very well founded</p>
<p>Basing neutron star sizes only on measurements, Capano et al* concluded that a typical neutron star has a radius of around 12-km.  The heaviest known is around 2.35 solar masses, and star density increases with increasing mass, so the smallest conceivabe radius for a neutron star** (based on 0.2-solar masses) will be at least 5.2-km.
Other than measurement uncertainty, these calculations have chosen values that predict a larger-than-reasonable minimum size - and the probability of the measurement  being a factor of four in error is low beyond (at least my) comprehension.</p>
<p>*. <a href="https://www.osti.gov/pages/biblio/1614838-stringent-constraints-neutron-star-radii-frommultimessenger-observations-nuclear-theory" rel="nofollow noreferrer">https://www.osti.gov/pages/biblio/1614838-stringent-constraints-neutron-star-radii-frommultimessenger-observations-nuclear-theory</a></p>
<p>**. So far as we know these don't yet exist; however, assuming Hawking radiation is real, all neutron stars will eventually shring to this mass.</p>

CC BY-SA 4.0
2023-05-22T13:46:16.523Z
Neutron stars do not have an event horizon, so they do not emit Hawking radiation.
PM 2Ring
CC BY-SA 4.0
2023-05-23T14:35:20.993Z
My understanding is that Hawking radiation is the consequence of a non-uniform gravitational field, not the event horizon per se.   In this regard, it is worth noting that pair splitting has to occur at least finitely outside the event horizon, otherwise the energy of the outgoing radiation will be zero.
As an example in the public literature, the Wikipedia article on Hawking radiation only explicitly cover black holes.  Nevertheless, even here the experimental section refers to the small size of the effect under "achievable experimental conditions", which implies it is not entirely absent.
george storm
CC BY-SA 4.0
2023-05-23T14:50:58.320Z
BTW, even in the absence of Hawking-related radiation, the Neutron star will eventually evaporate to its smallest possible size due to low-probability  thermal evaporation. 

https://astronomy.stackexchange.com/questions/529/how-long-would-it-take-for-a-rogue-planet-to-evaporate-in-the-late-stages-of-the/14554#14554.   

At this point it will of course explode

I did not mention the thermal evaporation because (at least while the body retains its density as a Neutron star) the Hawking-type evaporation is likely to be very much less slow
george storm
CC BY-SA 4.0
2023-05-22T12:55:43.290Z
2023-05-22T12:55:43.290Z
george storm
<p>According to the Chandrashekhar limit, the minimum mass of a neutron star is about 1.44 solar masses; however I found some examples of neutron stars less massive than that.</p>
<p>Additionally, I thought that the minimum radius of a neutron star of a given mass would be greater than the Schwarzchild radius of a blackhole with the same mass.</p>
<p>Still, Wikipedia lists some small neutron stars like CXOU J085201.4-461753 as having a radius of 1.2 km which I would not have thought possible?</p>
<p>So my question is: what is the smallest radius a neutron star can have? And, what is the smallest neutron star we have discovered?</p>
<p>Additionally, is the surface temperature of a neutron star in any way dependant on its radius and/or mass?</p>

CC BY-SA 4.0
2021-09-25T16:04:01.200Z
Neutron stars get smaller with larger mass, so you should be looking at the top end of the mass range for neutron stars, not the bottom end.
James K
CC BY-SA 4.0
2021-09-25T19:18:12.963Z
Yeah I was aware the mass was inversely proportional to the radius, but even that makes it feel a bit more complicated since more mass means smaller radius which means even closer to the schwarzchild radius. Since the SR for 1.44 solar masses is 4.2 KM and the neutron stars mentioned on wiki there are smaller
Donald Blake
CC BY-SA 4.0
2023-01-11T09:33:20.047Z
@DonaldBlake Actually, this may not be the case. For some equations of state (particularly if quark matter is present in neutron star cores), the radius get bigger with mass, reaches a maximum and then gets slightly smaller. A diagram is now in my answer.
ProfRob
<black-hole><neutron-star>
What is the smallest possible radius for a neutron star?
Donald Blake
# What is the smallest possible radius for a neutron star?

(Asked by: username_0 on 2021-09-25)

According to the Chandrashekhar limit, the minimum mass of a neutron star is about 1.44 solar masses; however I found some examples of neutron stars less massive than that.

Additionally, I thought that the minimum radius of a neutron star of a given mass would be greater than the Schwarzchild radius of a blackhole with the same mass.

Still, Wikipedia lists some small neutron stars like CXOU J085201.4-461753 as having a radius of 1.2 km which I would not have thought possible?

So my question is: what is the smallest radius a neutron star can have? And, what is the smallest neutron star we have discovered?

Additionally, is the surface temperature of a neutron star in any way dependant on its radius and/or mass?

**Tags:** black-hole, neutron-star
> **Comments:**
> **username_1**: Neutron stars get smaller with larger mass, so you should be looking at the top end of the mass range for neutron stars, not the bottom end.  
> **username_0**: Yeah I was aware the mass was inversely proportional to the radius, but even that makes it feel a bit more complicated since more mass means smaller radius which means even closer to the schwarzchild radius. Since the SR for 1.44 solar masses is 4.2 KM and the neutron stars mentioned on wiki there are smaller  
> **username_2**: @DonaldBlake Actually, this may not be the case. For some equations of state (particularly if quark matter is present in neutron star cores), the radius get bigger with mass, reaches a maximum and then gets slightly smaller. A diagram is now in my answer.  

---

## 2 answers

### Answer by username_2 on 2021-09-26. Score: 12

The minimum mass of a neutron star is actually about 0.2 solar masses and has nothing to do with the Chandrasekhar limit (see [this Physics SE answer](https://physics.stackexchange.com/a/143174/43351)). It seems unlikely however that there is any astrophysical channel to produce such neutron stars.

Several neutron stars have precisely measured masses that are smaller than 1.44 solar masses. The smallest is currently about 1.17 solar masses ([Martinez et al. 2015](https://arxiv.org/abs/1509.08805)). Note that more massive neutron stars may actually have smaller radii. It depends on the uncertain relationship between pressure and density.

The plot below shows *theoretical* relationships between masses and radii of neutron stars. The measured mass range for neutron stars is 1.17-2.1 solar masses, so you could estimate the smallest possible radius from your favourite model curve. For the "softest" equations of state (labelled "SQM1, where quark matter develops at the neutron star core), the smallest radius for a 1.17 solar mass neutron star is about 8.5 km.

[![enter image description here](https://i.sstatic.net/yXL5W.jpg)](https://i.sstatic.net/yXL5W.jpg)

Actually *measuring* the radii of neutron stars is incredibly difficult. The "measurements" that exist are rather indirect inferences and have large uncertainties.

There is however a fundamental limit in General Relativity, that is larger than the Schwarzschild radius, for the minimum radius of an object at a given mass. This ["Buchdahl limit"](https://en.m.wikipedia.org/wiki/Buchdahl%27s_theorem) is 9/8 of the Schwarzschild radius (shown in the picture as a forbidden region, labelled "GR"). It does not matter what type of pressure support is provided, a spherically symmetric object will collapse to a black hole if smaller than that.

For realistic relationships between pressure and density, then the true limit is a bit bigger than the Buchdahl limit - perhaps 1.2 to 1.3 times the Schwarzschild radius (represented by the grey forbidden region labelled "causality"). This is around 5 km, for neutron stars with the smallest measured masses, so presumably those neutron stars are bigger than that.

Rapid rotation could change some of these considerations, but the measured rotation rates of pulsars are too slow to have much effect.

I think the reason for the strange, small radius values you see (especially in Wikipedia, which does not have much quality control - always look at the original sources) is that it is a "fitting parameter" and represents the size of the emitting region and not necessarily the radius of the neutron star.

Finally, the surface temperature of a neutron star does not directly depend on its radius. Neutron stars start their lives very hot and cool down with time. To first order, the surface temperature of a neutron star would depend on its age. It could also depend on the rate at which it was accreting material from the interstellar medium (or a companion) or perhaps even its initial magnetic field strength.

> **Comments:**
> **username_3**: As usual, very informative answer! Are you aware of an analogous limit for Kerr black holes as the Buchdal limit?  

---

### Answer by username_4 on 2023-05-22. Score: -1

I think your reservations are very well founded

Basing neutron star sizes only on measurements, Capano et al\* concluded that a typical neutron star has a radius of around 12-km. The heaviest known is around 2.35 solar masses, and star density increases with increasing mass, so the smallest conceivabe radius for a neutron star\*\* (based on 0.2-solar masses) will be at least 5.2-km.
Other than measurement uncertainty, these calculations have chosen values that predict a larger-than-reasonable minimum size - and the probability of the measurement being a factor of four in error is low beyond (at least my) comprehension.

\*. <https://www.osti.gov/pages/biblio/1614838-stringent-constraints-neutron-star-radii-frommultimessenger-observations-nuclear-theory>

\*\*. So far as we know these don't yet exist; however, assuming Hawking radiation is real, all neutron stars will eventually shring to this mass.

> **Comments:**
> **username_5**: Neutron stars do not have an event horizon, so they do not emit Hawking radiation.  
> **username_4**: My understanding is that Hawking radiation is the consequence of a non-uniform gravitational field, not the event horizon per se.   In this regard, it is worth noting that pair splitting has to occur at least finitely outside the event horizon, otherwise the energy of the outgoing radiation will be zero. As an example in the public literature, the Wikipedia article on Hawking radiation only explicitly cover black holes.  Nevertheless, even here the experimental section refers to the small size of the effect under "achievable experimental conditions", which implies it is not entirely absent.  
> **username_4**: BTW, even in the absence of Hawking-related radiation, the Neutron star will eventually evaporate to its smallest possible size due to low-probability  thermal evaporation.   https://astronomy.stackexchange.com/questions/529/how-long-would-it-take-for-a-rogue-planet-to-evaporate-in-the-late-stages-of-the/14554#14554.     At this point it will of course explode  I did not mention the thermal evaporation because (at least while the body retains its density as a Neutron star) the Hawking-type evaporation is likely to be very much less slow  

---

