# What is the smallest possible radius for a neutron star?

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?

• 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. Sep 25, 2021 at 16:04
• 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 Sep 25, 2021 at 19:18
• @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. Jan 11 at 9:33

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). 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). 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. 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" 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.

• As usual, very informative answer! Are you aware of an analogous limit for Kerr black holes as the Buchdal limit? Sep 26, 2021 at 14:38

In order to get the Radius limit, We would have to take the upper limit of the mass of a neutron star not the lower limit because Neutron stars have less radius when the mass is increased, and the upper limit is the Tolman-Oppenheimer-Volkov limit which is the upper limit for cold, non-rotating Neutron stars.

So even a Neutron star has Hydrostatic equilibrium i.e the internal pressure which in a Neutron star's case is Neutron degeneracy pressure given by the Pauli exclusion principle (which states that any 2 particles can't have the same Quantum state) is the same as the self gravity however a Black hole's singularity i.e the main point of a Black hole does not have Hydrostatic equilibrium. So in order to get the max Neutron star radius limit (before it becomes a Black hole ), We would have to find the max limit of radius using Hydrostatic equilibrium which can be done by solving EOS/Equations Of State which in this case is the Hydrostatic equilibrium equation more specifically there is the Tolman-Oppenheimer-Volkoff equation which requires more variables but at the same time but is a bit more accurate and is made specifically for Neutron stars. So using Hydrostatic equilibrium equation/Tolman-Oppenheimer-Volkoff equation, you might find the answer

Also since we're including Neutron Degerency pressure, the EOS might be a bit complex since a Neutron star is not an Ideal fermi gas

Moreover the Tolman-Oppenheimer-Volkov limit is not as accurate because of the no-hair theorem which states that a Black hole remembers the charge, mass, and the angular momentum i.e spin of the Star and and since most of the Neutron stars are spinning thus the "spinning Neutron" star will result in a Kerr black hole (and not a Swarzchild Black hole) so if the Neutron star is spinning the Tolman-Oppenheimer-Volkov Limit will be a bit smaller because it explicitly states that it is for "non-rotating Neutron stars"

• Neutron stars are not supported by ideal neutron degeneracy pressure and therefore the relationship between mass and radius is more complex than white dwarfs. It is more likely that the radii of most neutron stars is almost the same, regardless of their mass and some equations of state lead to increasing radius with increasing mass. Jan 11 at 8:47
• @ProfRob I think the difference between the mean radius of Neutron star and the least radius might be a few kilometers, So I shall try to keep it the most accurate possible
– user47732
Jan 11 at 9:22

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.

**. 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.

• Neutron stars do not have an event horizon, so they do not emit Hawking radiation. May 22 at 13:46
• 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. May 23 at 14:35
• 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. astronomy.stackexchange.com/questions/529/…. 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 May 23 at 14:50