# Are neutrons in a neutron star relativistic?

## Main Question or Discussion Point

As the title of the thread suggests, I'm interested in estimating the velocity distribution of neutrons in neutron star cores. Putting T ~ 10^12K gives v ~ 15%c or more under Boltzmann statistics. Could someone provide more information or a second opinion on this estimate? Thanks.

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Simon Bridge
Homework Helper
By "relativistic" you mean "is the mean speed of neutrons close enough to lightspeed that relativistic corrections to Newtonian mechanics are needed?". Something like that?

Or do you just want to know if the thermal energy, by itself, is sufficient to require these corrections?

Generally: you need General Relativity for neutron stars - don't forget the gravity.

By "relativistic" you mean "is the mean speed of neutrons close enough to lightspeed that relativistic corrections to Newtonian mechanics are needed?". Something like that?
Yes, I'm asking if the thermal energy is enough to consider relativistic corrections (i.e., v ~ c)

Simon Bridge
Homework Helper
Only you can figure if the relativistic correction is needed - it depends on context.
If you don't make a relativistic correction, then how big would the resulting systematic error be?

i.e. if you were to calculate mean kinetic energy?

Is this significant compared with the measurements you can make?

Better: how fast does something have to be going to get a significant systematic error for the quantity of interest?

Ken G
Gold Member
One way to proceed is to simply compare the radius of a 1.4 solar-mass neutron star with the event horizon radius of a black hole of the same mass. We do not know exactly the radius of a neutron star, so your question cannot be answered exactly, but a mass of 1.4 solar masses implies a Schwarzschild radius between 3 and 4 km, and neutron stars are thought to be perhaps about 3 times that. The escape speed goes like the inverse square root of the radius, so it seems likely that the escape speed from a typical neutron star could be about half the speed of light. The particles on average move slower than the escape speed, but not a lot slower, so a reasonable estimate should be that the neutron speed is about a third the speed of light, though of course this will vary with mass and the location in the star. Note there could even be places in the star where the neutrons break up into free quarks, and those would be even more relativistic. Note we should probably not call that "thermal speed", because a neutron star is degenerate so its thermal properties are significantly different from the more basic properties like speed and kinetic energy per particle.

Simon Bridge
Homework Helper
The particles on average move slower than the escape speed, but not a lot slower...
... we already know that the gravity is enough to make for relativity. But for the speed of the surface particles to be comparable to escape velocity (not close - since that is not what the word "surface" means) would imply that the gravitational collapse is balanced mostly by thermal pressure. Is this the case for neutron stars? Still, that seems to be the usual approach for coursework:
i.e. http://es.ucsc.edu/~glatz/astr_112/lectures/notes18.pdf (bottom p3).

I don't know where the T~1012K (post #1) figure comes from. A quick look gives me ball-park figures from 105K to 1011K (ref above gives 106K). This puts expected speeds much much slower than 0.5c ... so I suspect that the apparent discrepancy (boltzman speed vs escape velocity estimate) here is what the question is about.

Thanks for your replies, Simon & Ken. It looks like your estimate for v < 0.5c seems convincing enough. In any case, allow me to be more specific. I want to consider the problem of a neutron trapped in the core of a NS. I want to consider an extremely oversimplified treatment: my neutron is confined by an extremely deep potential within the core. In this situation, I want to make sure that the dynamics of this neutron will be governed by Klein-Gordon or Dirac equation rather than Schrodinger's, hence my question.

Given the extreme densities and temperatures, would it be correct to consider a quantum relativistic treatment of such a problem? I hope I was clear enough.

Simon Bridge
Homework Helper

I think I've seen Schrodinger treatments in the "simplistic" regime - treating as each neutron confined to an infinite square well with dimensions based on the density. A rigorous treatment will certainly be relativistic though.

General relativity definitely applies for neutron stars. Nobody knows how matter behaves under the kind of density found in the core of neutron stars. http://www.astronomy.ohio-state.edu/~ryden/ast162_5/notes21.html

I'm not aware of neutron stars having neutrons in their cores. The neutrons are generally in the mantle, with other stuff in the atmosphere, crust and core. The physics of the core is so complex that theories describing it are usually considered "exotic" meaning no way to test them. Quark-matter is sometimes discussed, but only speculatively.

Although the word "relativistic" frequently occurs in discussions of neutron stars, anything to do with the neutrons usually has the word "general" preceding "relativistic." The neutrons tend to form superfluid vortices (thousands of them per square centimeter). Even though they carry a great deal of kinetic energy, which they occasionally transfer to the star as a whole by rearranging themselves and producing "glitches," their velocity is typically described as "slow." However, that seems to refer more to the models that describe them (Newtonian) rather than to their actual velocities. I've seen at least one paper that describes them in (special) relativistic terms.

stevebd1
Gold Member
I recall reading somewhere that 'pressure is confined kinetic energy', so while neutrons may or may not be travelling at some velocity within the neutron star (not counting if the star had angular momentum), the pressure itself could make the neutrons relativistic. The algebraic interpretation for Einsteins equation for gravity is $g=\rho c^2+3P$ which means pressure contributes to the overall mass/energy and time dilation of the neutron star pretty much in the same way that kinetic energy would contribute to the mass/energy and time dilation of an object at relativistic speed.

Ken G
Gold Member
And note that for any relativistic fluid, 3P is nothing but the kinetic energy density, so when the gas starts to go relativistic, that's when the second term starts to matter to the gravity. So a general way to frame the OP question is, "how much of the rest mass of a neutron star has been converted into kinetic energy."

Simon Bridge
Homework Helper
... recall reading somewhere that 'pressure is confined kinetic energy', so while neutrons may or may not be travelling at some velocity within the neutron star (not counting if the star had angular momentum), the pressure itself could make the neutrons relativistic.
You should be able to recall more than that from your earlier education in the kinetic theory of gasses.
The velocity distribution gives rise to the pressure through impacts with the surrounding material.

$P=\frac{1}{3}\rho \bar v^2$ ... this is the Newtonian calculation.

Plug in the numbers for pressure and density inside a neutron star.

You should be able to recall more than that from your earlier education in the kinetic theory of gasses.
The velocity distribution gives rise to the pressure through impacts with the surrounding material.

$P=\frac{1}{3}\rho \bar v^2$ ... this is the Newtonian calculation.

Plug in the numbers for pressure and density inside a neutron star.
The degeneracy of the matter is what prevents the star from collapsing. so the gas equation doesn't apply. The density deep in the core increases more slowly than you might expect. The core is a superfluid of neutrons and superconductor of protons once the temperature gets below 500,000,000 degrees. For a while I have wondered how temperature is defined in such an environment, where the individual particles have to some degree lost their identity.

I once looked into how much time would slow down in the center of the most massive known neutron star (1.97 stellar masses). It was somewhere between 5% and 90%, the uncertainty because the precise radius is unknown.

Simon Bridge
Homework Helper
Does the fact that neutron stars are not held up by kinetic pressure account for the difference between the boltzman mean velocity (post #1) and that determined by other means?

Ken G
Gold Member
The degeneracy of the matter is what prevents the star from collapsing. so the gas equation doesn't apply.
This seems to be widely believed, but it is not correct. The gas equation does apply, degeneracy pressure is just gas pressure. (In general the coefficient is a bit less than 1/3 due to relativistic effects, but it's never less than 1/6.)
For a while I have wondered how temperature is defined in such an environment, where the individual particles have to some degree lost their identity.
The definition of the temperature is not hard, because it is always defined by the zeroth law of thermodynamics. Thus you could use more mundane reservoirs whose temperature you already understand, and find the hottest reservoir that would spontaneously draw heat from the one you are wondering about. Not saying the calculation would be easy, only the definition.
I once looked into how much time would slow down in the center of the most massive known neutron star (1.97 stellar masses). It was somewhere between 5% and 90%, the uncertainty because the precise radius is unknown.
Yes, a lot of the numbers bandied about seem to place the degree of importance of relativistic effects to be exactly in the range of uncertainty of the neutron star equation of state. Perhaps that is not a coincidence, it gets hard to know what is going on when you are not sure which limit is the better one to use.

Ken G
Gold Member
Does the fact that neutron stars are not held up by kinetic pressure account for the difference between the boltzman mean velocity (post #1) and that determined by other means?
You were right about the kinetic pressure equation, so that's not the reason the Boltzmann mean doesn't matter. The reason is just that the kinetic energy is not partitioned among the particles in the Maxwell-Boltzmann way, but rather in the Fermi-Dirac way. Indeed, in the limit of complete degeneracy, the kinetic energy is huge but the temperature is zero.