The size of neutron stars compared to cities

[repoman]

I keep on seeing this comparison and it is a bit annoying. Yes, as far as radius is concerned it is accurate. But as far as volume is concerned it is way off.

I wonder if you were to take the volume of every building (also roads, bridges, dams...) and vehicle that is on earth now, how it would compare to a neutron star.

looking on wiki:

The equation of state for a neutron star is still not known. It is assumed that it differs significantly from that of a white dwarf, whose equation of state is that of a degenerate gas that can be described in close agreement with special relativity. However, with a neutron star the increased effects of general relativity can no longer be ignored. Several equations of state have been proposed (FPS, UU, APR, L, SLy, and others) and current research is still attempting to constrain the theories to make predictions of neutron star matter. This means that the relation between density and mass is not fully known, and this causes uncertainties in radius estimates. For example, a 1.5 M☉ neutron star could have a radius of 10.7, 11.1, 12.1 or 15.1 kilometres (for EOS FPS, UU, APR or L respectively).

Take the middle ground of 12.1 km and round to 12km...

The volume is 4/3*pi*r^3 =~ 7.2*10^12 m^3

I looked to see how make people could fit inside it assuming average weight of 50kg (lots of kids) assuming density of water so, each person is 0.05 m^3.

Doing the simple math: that is about 140 trillion people could fit into a neutron star volume without compression. about 20,000 times the human population.

So in human terms neutrons stars are still pretty damn big.

 

[skepticalbip]

You are doing very different comparisons than they are. They are generally trying to describe the size of the neutron star in a way people can understand by giving its diameter sorta like describing the diameter of a basketball rather than its volume. You somehow jumped to volume.

But to use your "how many people can be jammed in", the area of the city of Jacksonville, Florida is 885 sq. miles. If we assume that the average person covers an area of 3 sq. feet then we could put over 8 trillion people (more than 1,000 times the current population of Earth) within the city limits of Jacksonville.

 

[repoman]

I only get 8 billion people in Jacksonville. Jacksonville is close to the size of the cross section of a neutron star.

That is 17,500 times less people than in the volume of the neutron star from the numbers I used.

 

[lpetrich]

[1205.6871] The Neutron Star Mass-Radius Relation and the Equation of State of Dense Matter

The equation of state (EOS) of dense matter has been a long-sought goal of nuclear physics. Equations of state generate unique mass versus radius (M-R) relations for neutron stars, the ultra-dense remnants of stellar evolution. In this work, we determine the neutron star mass-radius relation and, based on recent observations of both transiently accreting and bursting sources, we show that the radius of a 1.4 solar mass neutron star lies between 10.4 and 12.9 km, independent of assumptions about the composition of the core. We show, for the first time, that these constraints remain valid upon removal from our sample of the most extreme transient sources or of the entire set of bursting sources; our constraints also apply even if deconfined quark matter exists in the neutron star core. Our results significantly constrain the dense matter EOS and are, furthermore, consistent with constraints from both heavy-ion collisions and theoretical studies of neutron matter. We predict a relatively weak dependence of the symmetry energy on the density and a value for the neutron skin thickness of lead which is less than 0.20 fm, results that are testable in forthcoming experiments.

That's a diameter of 21 - 26 km or 13 - 16 mi -- a bit longer than Manhattan in New York City.

To compare to other cities, you can use this map: Map Shows How Big New York Is Compared To Other Cities | Mental Floss

 

[barbos]

Yes, cities are mostly flat and empty. So is ordinary matter which consists of point particles which have zero size.

 

[repoman]

I am not sure what you mean, Barbos...

Anyway, I did a rough estimate, and 140 trillion people (which will roughly pack into a neutron star without being compressed) will fit onto the land mass of Africa and Australia combined. That is really large amount of people to conceive of. I guess it shows how small we are compared to astronomical bodies.

hmm, I get that a neutron star will be about 7*10^18 times the mass of the human population. Guess that makes sense given that neutron stars are ~50 trillion times as dense as iron and ~400 trillion times denser than water.

It is interesting playing with these numbers. I get that for Iron to compress to neutron star density it needs to shrink to a ratio of about 1 to 37,000 on each linear dimension. That is like shrinking 1 mile down to ~ 1.7 inches. Of course 3 times each for the x,y and z axes. 1 cubic mile down to ~5 cubic inches or two golf balls.

 

[barbos]

I am saying You confuse size with volume.

 

[repoman]

So, I have been watching PBS Space Time channel on Youtube recently. The new host really condenses it down for a lay audience in a very impressive way.



Anyway, in this video he talks about neutron stars and the Pauli Exclusion Principle given that neutrons are fermions. I already knew that. Then he talks about the Heisenberg Uncertainty Principle where when the position is getting very constrained the momentum is inversely very undefined.

momentum gives velocity and that gives kinetic energy which is directly related temperature.

This is kind of scattered in my thinking, but given a massive amount of time where the neutron star will equilibrate to the background temperature (say 3K for sake of example) it seems hard to think that the neutrons won't be buzzing around with less energy than would be dictated by the momentum from HUP.

Maybe it would become a sort of neutron crystal where the neutron vibrate just as quickly but not relative to each other - but with each other?

What kind of frequency do neutrons vibrate in neutron stars?

- - - Updated - - -

you mean area with volume?

 

[barbos]

Temperature corresponds to movement for simple classical systems.
In quantum system it is bit more complicated and "movement" itself is not well defined.
Neutron stars have millions degrees temperature but as far as quantum mechanics concerned they are very cold degenerate fermi gas and people suggested superconductivity in neutron stars.

And neutron stars are not going to cool down to 3K probably in trillions of years.

 

[repoman]

ah, play along barbos.

I am just trying to understand what would happen to a neutron star as it cools. I don't even care about a realistic full model of it.

Basically, I am trying to get an idea of having just pure neutrons (at neutron star pressure) that "somehow" you can lower the temperature of.

from the HUP they have a very large uncertainty of momentum. This means they are vibrating like crazy. But how fast? I can't find any info on this despite looking for over an hour. I am too rusty on my math to figure it out either.

But what I was thinking is that as temperature drops they will have the same velocity but neighboring neutron will start to move more in alignment with each other. But they are fermions, so am I wrong?

 

[barbos]

ah, play along barbos.

I am just trying to understand what would happen to a neutron star as it cools. I don't even care about a realistic full model of it.

Basically, I am trying to get an idea of having just pure neutrons (at neutron star pressure) that "somehow" you can lower the temperature of.

from the HUP they have a very large uncertainty of momentum. This means they are vibrating like crazy. But how fast? I can't find any info on this despite looking for over an hour. I am too rusty on my math to figure it out either.

Oh, now I get your logic. No, HUP has no effect in this case. You assume that neutrons are confined to their own tiny cells within the whole volume. That's completely wrong. They are free and actually move through whole volume. Nothing really prevents them to move. Neutron stars is a fermion gas and stay that way at any temperature.

But what I was thinking is that as temperature drops they will have the same velocity but neighboring neutron will start to move more in alignment with each other. But they are fermions, so am I wrong?

Nothing special will happen. You are too concerned with these velocities. Electrons in metals never stop "moving" even at very cold temperature.

 

[repoman]

So a neutron star is like the electrons in a metal?

I would think it would be more like a very cold noble gas element under extremely high pressure. How can the neutron have a free path to go anywhere when it is just about as dense as a nucleus? Maybe I am taking the classical ball model too far. But in the video I posted he said that uncertainty of the position was very low and momentum high. That seems like a bunch of packed vibrating neutrons to me.

 

[barbos]

So a neutron star is like the electrons in a metal?

I would think it would be more like a very cold noble gas element under extremely high pressure. How can the neutron have a free path to go anywhere when it is just about as dense as a nucleus? Maybe I am taking the classical ball model too far. But in the video I posted he said that uncertainty of the position was very low and momentum high. That seems like a bunch of packed vibrating neutrons to me.

No, they are not vibrating. Cold neutron star is a degenerate fermion gas, not a crystal.

 

[repoman]

hmm, I was reading again about it and finally went to the wiki on degenerate fermi gases.

If a plasma is cooled and compressed repeatedly, it will eventually not be possible to compress the plasma any further. This is by the Pauli exclusion principle, which states that two fermions cannot share the same quantum state. When in this highly compressed state, since there is no extra space for any particles, a particle's location is extremely defined. This is due to the Heisenberg uncertainty principle, ΔpΔx ≥ ħ/2, where Δp is the uncertainty in the particle's momentum and Δx is the uncertainty in position. This implies that the momentum of a highly compressed particle is extremely uncertain, since the particles are located in a very confined space. Therefore, even though the plasma is cold, such particles must be moving very fast on average. This leads to the conclusion that, in order to compress an object into a very small space, tremendous force is required to control its particles' momentum.

So what it seems like is that the neutrons are just about kissing each other and the also have very large momenta/energy which I assume would be determined by the Fermi-Dirac statistics. So it you had a gazillion neutrons all of the gazillion momentum levels would be full (give or take random jostling) from the lowest to highest. so they would be packed into something that looks like a solid maybe, but there is other stuff happening.

Would it not be right to call neutronium solid, like we call sodium chloride solid?

So the pressure that is applied because of the gravity force is resisted internally by the weighted average of the neutron momenta hitting neighbor neutrons and bouncing back - similar to the particles in a bow model for ideal gases?

But you can't know which neutron has which momentum, right?

ETA: I think I was hung up on the word "gas". It means a different thing to physicists than other people like layman especially. Yes, you are right it is indeed a gas that is described by Fermi-Dirac statistics.

Ok, I see that the "cold" means well below the Fermi Temperature, same for metals and so on.

The fermi temp for neutron stars is about 6*10^11 K and neutron stars are only about 10^8 K. So they are cold.

 

[barbos]

You can't know which neutron is which regardless of what you have. But if the system is not degenerate this fact does not affect anything, it will look the same as the classical case where you can label particles. In solid materials atoms are not degenerate and kept in place (in a classical sense) through chemical interaction. There is no such thing in neutron star which is fermion gas.

 

[repoman]

yeah, but solid in the everyday sense (which is not scientific) means compact and maybe also hard to compress. I read that the young's modulus for a neutron star is like 20 magnitudes of order higher than diamond. It is kind of weird to think that is not "solid"

 

[barbos]

yeah, but solid in the everyday sense (which is not scientific) means compact and maybe also hard to compress. I read that the young's modulus for a neutron star is like 20 magnitudes of order higher than diamond. It is kind of weird to think that is not "solid"

Water is hard to compress too.
Also young's modulus should be normalized on particle density before you start comparing neutron star to ordinary matter. It's extreme form of gas therefore there is no surprise it has extreme properties, but nevertheless it's a gas.

 

[skepticalbip]

yeah, but solid in the everyday sense (which is not scientific) means compact and maybe also hard to compress. I read that the young's modulus for a neutron star is like 20 magnitudes of order higher than diamond. It is kind of weird to think that is not "solid"

Not sure how you could compare the two for hardness or indeed if the normal meaning of hardness even applies to a neutron star. The neutron star's "hardness" is a matter of gravitation holding the star together while the hardness of diamonds (and other normal solids) is a matter of the strength of the bonds between the atoms. There are no bonds that I am aware of between neutrons.