Neutron Star Stability: Does Proton Crust Matter?

[twofish-quant]

One other thing, I thought of a quick test to see if the assumption of "timescale decoupling" works.

Is it round?

If you have a round object it means that the atomic processes are in local equilibrium so that you can do pressure calculations assuming local equilibrium. If it's not round, that means that the atomic processes aren't allowing the object to go into pressure equilibrium.

 

[zonde]

I just did a crash course in quantum field theory. Here is another crash course in computational astrophysical modelling.

One thing about astrophysics is that a lot of astrophysical phenomenon can be modelling through what I call "time scale decoupling." What happens is that you look at the time scales over which two different processes happen and if they are very different, you can "decouple" them by assuming that process one is in local equilibrium and then process two interacts with process two by changing the equilibrium.

You squeeze a tube of toothpaste. You don't have to calculate how your actions affect the protons and electrons because the time scale of the atomic process is very different than that of the squeezing process.

Now in neutron stars, electron processes happen over the course of nanoseconds whereas pressure processes happen over the course of milliseconds. That means that the two processes decouple, and if the time scale you are interested in is the pressure timescale, the atomic processes are in equilibrium.

Put another way any charge imbalance is going to resolve itself in a few nanoseconds, which means that if you look at processes over the course of milliseconds, the material is going to be in charge equilibrium.

So the question is at what timescales electrons will arrive at distribution described by Fermi-Dirac statistics and if that process will happen at speed of sound or speed of light. Now if we look at atomic orbitals I suppose there is no question that electrons can't occupy lower already occupied quantum state despite electrostatic attraction. If it wouldn't be so all electrons would fall into 1s orbital before they could be "bounced" out of it (at speed of sound).

Another point. Pressure arise from elastic collisions between particles. Now we can ask what energy determines outcome of elastic collision - is it all the energy of the particle or is it only the portion of energy that it can give up by falling into another available quantum state. And I say it's the later. So it seems that fully degenerate particles will not exert any pressure on other kind of particles. So it is rather confusing to call "degeneracy pressure" a "pressure".

Quantum states of electrons and muons are independent as far as we can tell. Now it turns out that because neutrinos have mass that quantum states of electron neutrinos and muon neutrinos aren't independent.

Also, it's not a vague problem. Given particles X1, X2, X3, with momentum vectors p1, p2, and p2 into a system, what pops out?

Now that I think about it. Brehstrahlung won't work. It's an electromagnetic process so you aren't going to get muons from that. What might work is the reaction

electron + electron antineutrino -> muon + muon antineutrino

The trouble with this is that you need a source of 100+ MeV antineutrinos. This is difficult because the main source of antineutrinos is pair production, which means that you are going to get 20-30 MeV ones.

Trying to work out a particle/nuclear reaction chain is a lot like doing a crossword puzzle.

Just to be clear about my hesitation.
Let's say we have ion without any electrons. Now we let muon occupy some (muon) orbital of that ion. And you say that electron orbitals will remain exactly the same around that ion. Say we add some electron to that ion and then let muon decay and we won't observe any anomalous radiation as electron hypothetically will fall into "proper" electron orbital. Right? And I am not sure about that.

But if we assume that you are right then we can discuss it further.
We can look at it from different side. Let's say that there appears some muon inside electron degenerate core of the star. And it is at rest meaning that all available electron quantum states have equally high energy in any direction. In order for it to decay it should emit electron into very high energy quantum state and in order to conserve momentum in it's rest frame it should emit very high energy neutrinos in other directions. So theoretically we can reach level where muon rest energy is simply not enough to do that i.e. it will be stable.
I am speaking now only on qualitative level. I do not say that it should work on quantitative level.

There's "I don't know"-doubt and "this doesn't exist"-doubt. If someone just asks "so how do muons work in neutron stars" that's one thing. For someone to assert "physicists have not considered the role of muons in neutron stars therefore the whole concept is suspect" is another.

Okay, point taken.