Neutron Star Stability: Does Proton Crust Matter?

[iced199]

Ok, I know neutron stars are mainly composed of neutrons. But also, they have some protons and normal nuclei at their surfaces. Is this crust of protons needed to keep the neutrons below stable? As in, if it disappeared, would the neutrons below start decaying back to protons to form its 'protective' layer? Or would the neutrons be stable and not decay - the protons are just leftovers from the creation of the neutron star.

 

[ImaLooser]

Ok, I know neutron stars are mainly composed of neutrons. But also, they have some protons and normal nuclei at their surfaces. Is this crust of protons needed to keep the neutrons below stable? As in, if it disappeared, would the neutrons below start decaying back to protons to form its 'protective' layer? Or would the neutrons be stable and not decay - the protons are just leftovers from the creation of the neutron star.

The neutrons and protons are in a stable dynamic equilibrium. If the crust somehow suddenly disappeared pressure would be greatly reduced, the equilibrium would change drastically, and many of the neutrons would decay. My guess is that there would be a cataclysmic explosion.

The crust is made of iron nuclei polymerized by the intense magnetic field.

 

[iced199]

Thanks. I didn't know there was an equilibrium between the core and surface of the neutron star. :)

 

[zonde]

Ok, I know neutron stars are mainly composed of neutrons. But also, they have some protons and normal nuclei at their surfaces. Is this crust of protons needed to keep the neutrons below stable? As in, if it disappeared, would the neutrons below start decaying back to protons to form its 'protective' layer? Or would the neutrons be stable and not decay - the protons are just leftovers from the creation of the neutron star.

Idea of neutron star is based on some rather shaky speculations. You have to realize that idea of neutron star was proposed around the same time as discovery of beta plus decay.

Basically all the observations we have say that proton decays into neutron when it leads to stable configuration of nucleons. And electrons have no role in determining that stability. But idea of neutron star is based on reasoning that you can "push" electrons into protons and that way make them decay into neutrons.

Yet another thing is that heavy version of electron - "muon" was recognised as such long after idea of neutron star was proposed. And the problem here is that when electrons become relativistic and very very energetic then speculation that they will decay into muons would be much more reasonable than speculation about them being "pushed" into protons. And muons are much more compressible than electrons as they have much bigger mass (and experience less degeneracy pressure).

 

[ImaLooser]

Thanks. I didn't know there was an equilibrium between the core and surface of the neutron star. :)

You misunderstand me. You can think of it that way, but I think it is an inferior view. The "interaction" of the crust and core is due to gravity and pressure. The extra pressure due to the mass of the crust affects the equilibrium in the core. If somehow there were an impermeable barrier between the crust and core I think it wouldn't make very much difference.

While there is surely some sort of nuclear equilibrium between the two, I would expect it to be insignificant. There are protons and electrons in the core but they come from the decay of neutrons, not from elsewhere.

 

[Drakkith]

Yet another thing is that heavy version of electron - "muon" was recognised as such long after idea of neutron star was proposed. And the problem here is that when electrons become relativistic and very very energetic then speculation that they will decay into muons would be much more reasonable than speculation about them being "pushed" into protons. And muons are much more compressible than electrons as they have much bigger mass (and experience less degeneracy pressure).

Have you heard of electron capture? Your post seems to suggest that you either haven't heard of it or don't believe it. Electrons are not "pushed" into the protons, at least not in any sense of the way I use the word "push".
See here: http://en.wikipedia.org/wiki/Electron_capture

 

[zonde]

Have you heard of electron capture? Your post seems to suggest that you either haven't heard of it or don't believe it. Electrons are not "pushed" into the protons, at least not in any sense of the way I use the word "push".
See here: http://en.wikipedia.org/wiki/Electron_capture

Yes, I know about electron capture proton decay. And I have read this wikipedia article.
It describes quite specific case of proton decay. Say couple of quotes from wikipedia to illustrate this:
"Note that a free proton cannot normally be changed to a free neutron by this process: The proton and neutron must be part of a larger nucleus."
"Radioactive isotopes that decay by pure electron capture can, in theory, be inhibited from radioactive decay if they are fully ionized ("stripped" is sometimes used to describe such ions)."
So it is believed that this kind of decay can not happen in plasma with fully ionized ions.

This of course might change for high densities but there are just not enough experimental facts to make reliable speculations IMHO.

 

[Drakkith]

This of course might change for high densities but there are just not enough experimental facts to make reliable speculations IMHO.

We work with the knowledge we have at the time. It's simply the best explanation we have at the moment. If in the future we learn something new, then we will apply that knowledge to neutron stars and see if something changes. Call it "shaky" if you want to.

 

[twofish-quant]

But also, they have some protons and normal nuclei at their surfaces. Is this crust of protons needed to keep the neutrons below stable?

Not really. The neutrons at the center are that way because of the pressure and temperature.

As in, if it disappeared, would the neutrons below start decaying back to protons to form its 'protective' layer?

If you take something at the center of the neutron star and then reduced the pressure and temperature, it would turn into iron nuclei.

 

[twofish-quant]

Idea of neutron star is based on some rather shaky speculations

This really isn't the case. You can do nuclear experiments to see what happens with the material. There might be some weird stuff happening in the core, but it's mostly neutrons. There is still a lot of "unknown stuff" when it comes to quark physics, but the physics of leptons is pretty well known.

Yet another thing is that heavy version of electron - "muon" was recognised as such long after idea of neutron star was proposed. And the problem here is that when electrons become relativistic and very very energetic then speculation that they will decay into muons would be much more reasonable than speculation about them being "pushed" into protons. And muons are much more compressible than electrons as they have much bigger mass (and experience less degeneracy pressure).

This won't work. In order to change an electron to a muon, you need a source of muon neutrinos. Electrons just don't spontaneously turn into muons. It violates conservation of electron number and muon number.

Now you *can* get muons via things like reactions with protons and neutrons, but people have already added them into the calculations

http://arxiv.org/abs/nucl-th/9510045

 

[twofish-quant]

This of course might change for high densities but there are just not enough experimental facts to make reliable speculations IMHO.

There are. The densities we are talking about are nuclear densities and you can simulate the reactions by throwing nuclei at each other. The other thing is we are talking about lepton processes, and the theory behind that (i.e. electroweak theory) is very well established. One thing that is very well established is the electrons don't spontaenously turn into muons. They just don't. We don't see this happening in our experiments.

There are also some very well established princples like conservation of mass, and conservation of energy, and similar conservation laws. One conservation law is conservation of electron number and conservation of muon numbers. Also you can ask "what happens if those laws aren't conserved". If electrons were spontaenously changing into muons, this would radically increase the Chandraseakar mass to about 400 solar masses rather than 1.4. This isn't what we see.

One reason I like neutron star physics is that we are talking for the most part about known nuclear physics, and you can't make up random stuff. Also, if you change the physics it has some pretty dramatic changes, so that "we don't know' also doesn't work.

 

[zonde]

We work with the knowledge we have at the time. It's simply the best explanation we have at the moment. If in the future we learn something new, then we will apply that knowledge to neutron stars and see if something changes. Call it "shaky" if you want to.

Is it the "best explanations" or the "only speculation"? But please do not misunderstand me, I am not saying that we shouldn't have baseline for our research. Instead I am saying that if we haven't established that part as scientifically confirmed solid theory then we shouldn't build further speculations on top of it.

 

[zonde]

This really isn't the case. You can do nuclear experiments to see what happens with the material. There might be some weird stuff happening in the core, but it's mostly neutrons. There is still a lot of "unknown stuff" when it comes to quark physics, but the physics of leptons is pretty well known.

Yes, for weak interactions we have tested model that works quite fine. But is there updated model of white dwarf collapse into the neutron star that takes into the account all we know about weak interactions?

This won't work. In order to change an electron to a muon, you need a source of muon neutrinos. Electrons just don't spontaneously turn into muons. It violates conservation of electron number and muon number.

Well, but muons do turn into electrons spontaneously. And in the process appropriate neutrinos are emitted. Actually W bosons decay into any of the three leptons (+ appropriate neutrino) with almost equal branching ratio. At least that is what wikipedia says about it - http://en.wikipedia.org/wiki/W_boson#W_bosons

Now you *can* get muons via things like reactions with protons and neutrons, but people have already added them into the calculations

http://arxiv.org/abs/nucl-th/9510045

As I understand this paper speaks about hypothetical neutron star cooling by emission of neutrino/antineutrino pairs. So it's very specific calculation where they have added muons.

 

[zonde]

There are. The densities we are talking about are nuclear densities and you can simulate the reactions by throwing nuclei at each other. The other thing is we are talking about lepton processes, and the theory behind that (i.e. electroweak theory) is very well established. One thing that is very well established is the electrons don't spontaenously turn into muons. They just don't. We don't see this happening in our experiments.

I believe that the sentence you where commenting was about protons not turning into the neutrons when they are bombarded by fast electrons.
So do you think that there are experimental facts that would indicate the opposite i.e. that protons turn into the neutrons when they are bombarded by fast electrons?

 

[Drakkith]

Is it the "best explanations" or the "only speculation"? But please do not misunderstand me, I am not saying that we shouldn't have baseline for our research. Instead I am saying that if we haven't established that part as scientifically confirmed solid theory then we shouldn't build further speculations on top of it.

Considering that we may never be able to make it a "confirmed solid theory" since we may never be able to visit a neutron star itself, what are our options? We have a very good understanding of atomic and subatomic particles and how they interact with each other and using this knowledge to make predictions about neutron stars doesn't seem like too big a step to me. What's the difference between this and guessing that Jupiter has a metallic hydrogen layer underneath it's outer atmosphere? We haven't been inside Jupiter that far, so we "don't know for sure", but since we know how hydrogen behaves we can make predictions based on our knowledge.

And this is by no means just something done in astronomy and astrophysics. This is done, to some extent, in all areas of science. We use our knowledge to make predictions about the universe. When we find out something new, or that something we thought we understood is incorrect our predictions change accordingly.

 

[twofish-quant]

Yes, for weak interactions we have tested model that works quite fine. But is there updated model of white dwarf collapse into the neutron star that takes into the account all we know about weak interactions?

Most models used Steve Bruenn's electroweak reaction rates and the Lattimer-Swesty equation of state.

The Appendix C of this paper goes into the gory details...

http://articles.adsabs.harvard.edu/full/1985ApJS...58..771B

It turns out that we don't know how the collapse process works, but it's something other than particle physics processes and EOS. You can change those, and within the limits of known physics, it doesn't make much of a difference.

Well, but muons do turn into electrons spontaneously. And in the process appropriate neutrinos are emitted.

Right. But electrons don't turn into muons without a source of muon neutrinos. The only physical process that breaks electron/muon number conservation is the MSW process in which free streaming neutrinos change flavor. That's important for solar neutrinos, but it's not for supernova because the neutrinos get reabsorbed pretty quickly after emission.

Bruenn's paper also doesn't include massive neutrinos. It's not hard to change the equations to include those, but it doesn't make a difference.

As I understand this paper speaks about hypothetical neutron star cooling by emission of neutrino/antineutrino pairs. So it's very specific calculation where they have added muons.

All these calculations are very specific. For leptons we have a firm theory that gives cross sections and we have experiments that confirm the theory. For baryons, there is a lot more that is unknown because quark-quark interactions are hard to measure.

In cosmology, you can get away with inventing new particle processes, but with neutron stars, we are talking about energies that can be simulated in particle accelerators, so inventing a new reaction process is like inventing a new element. You have to argue *really* hard to get people to believe you.

 

[twofish-quant]

I believe that the sentence you where commenting was about protons not turning into the neutrons when they are bombarded by fast electrons.
So do you think that there are experimental facts that would indicate the opposite i.e. that protons turn into the neutrons when they are bombarded by fast electrons?

These reaction rates are very well studied. If you throw an electron with energy X1 at angle Y1 at proton with energy X2 and angle Y2, then event Z will happen with probability Z1. Appendix C of Bruenn's paper has the gory details on how to calculate this, and the reaction rates are in fact what we see when we in fact throw electrons at protons.

It's high energy chemistry.

 

[twofish-quant]

Considering that we may never be able to make it a "confirmed solid theory" since we may never be able to visit a neutron star itself, what are our options?

Our theories make some very strong statements on some things. For example, lepton interactions have a very firm theory, and that theory just says that "nothing weird will happen with electrons, muons, and tau leptons" in neutron stars. You ask electroweak theory what happens to stuff at neutron star densities, you get firm numbers.

The theory might be wrong, but there is no reason to think that the theory is wrong. One thing is that we can get electrons up to 90 GeV on earth, whereas the energies for neutron stars is 100 MeV.

For stuff to do with quarks, we can't get numbers that are quite as firm. When you increase things to neutron star densities, you get all sorts of effects that we don't know how to calculate, so we don't get firm numbers.

What's the difference between this and guessing that Jupiter has a metallic hydrogen layer underneath it's outer atmosphere? We haven't been inside Jupiter that far, so we "don't know for sure", but since we know how hydrogen behaves we can make predictions based on our knowledge.

Some things we have strong knowledge of, some things we don't. Lepton processes are something that we have strong knowledge of. Baryon processes we don't, and then magnetic fields and convection are total mysteries.

And this is by no means just something done in astronomy and astrophysics. This is done, to some extent, in all areas of science. We use our knowledge to make predictions about the universe. When we find out something new, or that something we thought we understood is incorrect our predictions change accordingly.

And sometimes it doesn't matter. If you build a dog house using Newtonian physics, then it turns out that GR effects don't matter. Same with neutron stars. In a lot of situations, you can ask "how wrong do we have to be before it makes a difference?" and it turns out that there is a huge room for error.

 

[zonde]

Most models used Steve Bruenn's electroweak reaction rates and the Lattimer-Swesty equation of state.

The Appendix C of this paper goes into the gory details...

http://articles.adsabs.harvard.edu/full/1985ApJS...58..771B

It turns out that we don't know how the collapse process works, but it's something other than particle physics processes and EOS. You can change those, and within the limits of known physics, it doesn't make much of a difference.

This paper does not have Appendix C - it ends with references.

And we know about particle physics in "open" environment. We have theoretical ideas about particle physics in a volume of dense plasma. These ideas might work just fine if we have reasonably correct model for degeneracy pressure. And I doubt that.

Right. But electrons don't turn into muons without a source of muon neutrinos. The only physical process that breaks electron/muon number conservation is the MSW process in which free streaming neutrinos change flavor. That's important for solar neutrinos, but it's not for supernova because the neutrinos get reabsorbed pretty quickly after emission.

Electrons do not turn into muons directly with or without neutrinos. Just like muons do not turn into electrons directly with or without neutrinos. They can do that only through intermediate state of W^- boson. Are we on the same line so far?

 

[zonde]

These reaction rates are very well studied. If you throw an electron with energy X1 at angle Y1 at proton with energy X2 and angle Y2, then event Z will happen with probability Z1. Appendix C of Bruenn's paper has the gory details on how to calculate this, and the reaction rates are in fact what we see when we in fact throw electrons at protons.

It's high energy chemistry.

Give some reference that works. Or alternatively we can try to go trough the details ourselves.

 

[twofish-quant]

This paper does not have Appendix C - it ends with references.

It's on page 822. Appendix C - Derivation of the Weak Interaction Rates

And we know about particle physics in "open" environment. We have theoretical ideas about particle physics in a volume of dense plasma. These ideas might work just fine if we have reasonably correct model for degeneracy pressure. And I doubt that.

Any particular reason? Degeneracy pressure is just Fermi-Dirac statistics. You count the number of states, count the number of particles, put it into a canonical ensemble, and boom, you have the equation of state. It's pretty well covered in any graduate level textbook on solid state physics or thermodynamics (i.e. Kerson Huang's or Kittel/Kromer).

If there is a particular solid state/HEP phenomenon that you think isn't being modeled that will make a big difference, then it would be useful to state what that is.

Electrons do not turn into muons directly with or without neutrinos. Just like muons do not turn into electrons directly with or without neutrinos. They can do that only through intermediate state of W^- boson. Are we on the same line so far?

I just care about the cross sections, and conservation of electron/muon number makes that cross-section zero for the energies that are relevant here.

I'd be *very* interested if you can dig up a reference for someone that has done the calculation that Bruenn did which shows a significant generation of muons in neutron stars.

 

[twofish-quant]

Give some reference that works. Or alternatively we can try to go trough the details ourselves.

I just checked

http://articles.adsabs.harvard.edu//full/1985ApJS...58..771B

It's on pages 822 to 834.

There is one important process that Bruenn doesn't include which is the effects of Bremsstrahlung and that's calculated in gory detail at

http://arxiv.org/abs/astro-ph/9711132

 

[twofish-quant]

Also here is a paper that describes what happens if you put in "new physics" (i.e. flavor violation lepton processes)

http://arxiv.org/abs/1010.0883

IMHO, it doesn't make much of a difference...

 

[zonde]

It's on page 822. Appendix C - Derivation of the Weak Interaction Rates

Thanks, I will look what I can make out of it. And thanks for other papers too.

Any particular reason? Degeneracy pressure is just Fermi-Dirac statistics. You count the number of states, count the number of particles, put it into a canonical ensemble, and boom, you have the equation of state. It's pretty well covered in any graduate level textbook on solid state physics or thermodynamics (i.e. Kerson Huang's or Kittel/Kromer).

If there is a particular solid state/HEP phenomenon that you think isn't being modeled that will make a big difference, then it would be useful to state what that is.

Yes there is particular reason.
When degeneracy pressure is calculated for astronomical bodies it is assumed that distribution of charged particles is homogeneous. But electrons might (should) be partially squeezed out of the body due to larger degeneracy pressure.
Another thing is that degeneracy pressure should be higher in the middle of the body.

I just care about the cross sections, and conservation of electron/muon number makes that cross-section zero for the energies that are relevant here.

I'd be *very* interested if you can dig up a reference for someone that has done the calculation that Bruenn did which shows a significant generation of muons in neutron stars.

Look, I am trying to get us on the same page. If I read about muon decay in wikipedia the things you say do not make much sense.

Can you read that page (the part about muon decay)? It does not speak about any cross sections. It speaks about "decay width" and "Fermi's golden rule" and these do not seem to be related to any cross sections.

So what I am missing?

 

[twofish-quant]

Also, if you are interested in the nuclear equation of state...

http://www.astro.sunysb.edu/dswesty/lseos.html

And several other EOS that are mentioned in this paper

http://arxiv.org/abs/1108.0848
New Equations of State in Simulations of Core-Collapse Supernovae

And this one

http://arxiv.org/abs/1202.5791
Equation of State for Proto-Neutron Star

And if you want to take a walk on the wild side (hyperions! strange quark matter!)

http://arxiv.org/abs/1205.3621
Hadron-Quark Crossover and Massive Hybrid Stars with Strangeness

I'm a lot less familiar with that physics, since I was able to get numbers out by using Lattimer-Swesty as a "black box". For the radiation hydro part, I had to tinker with the particle cross sections and implement some of these equations, so I know a bit more about what goes into that part of the code.

 

[twofish-quant]

Yes there is particular reason. When degeneracy pressure is calculated for astronomical bodies it is assumed that distribution of charged particles is homogeneous.

Nope. It's assumed that there is no charge at scales that are larger than atomic scales. You can show that any sort of non-zero charge above atomic scales is very, very quickly going to get canceled out.

Astrophysical plasma are conductive, but they have zero net charge.

But electrons might (should) be partially squeezed out of the body due to larger degeneracy pressure.

Nope. The pressure is going to work on everything in the same way. Pressure forces don't cause charge separation. Once you have any sort of charge separation, the particles are going to correct that at near the speed of light, whereas pressure forces act at the speed of sound.

Another thing is that degeneracy pressure should be higher in the middle of the body.

Pressure is going to be higher. Whether degeneracy pressure is going to be higher depends on a lot of things, but it usually is. One reason why we think that the middle of neutron stars are "quark soup" is that if you have more particles you have less degeneracy pressure and that reduces the mass you need before everything goes into a black hole.

But this is something that goes into the calculations.

Look, I am trying to get us on the same page. If I read about muon decay in wikipedia the things you say do not make much sense.

That's because wikipedia is missing information. It would be nice if that article were linked to a intro textbook on quantum field theory.

Can you read that page (the part about muon decay)? It does not speak about any cross sections. It speaks about "decay width" and "Fermi's golden rule" and these do not seem to be related to any cross sections.

OK. Here's some intro quantum field theory...

What you end up with when you do a QFT calculation is to answer the equation, if particles X1, X2, X3 with energies E1, E2, E3, go into a region with angles theta1, theta2, theta3 etc. What is the probability that you will come up with particles X4, X5, X6. You calculate these by writing down the Feyman diagrams, and going through a lot of nasty math.

Now if you want to calculate decay rates, you are asking if I go in with a muon with any energy at any angle, what is the probability that I will come out with an electron, neutrino pair with any energy. To do the calculation, you use a math trick called Fermi's golden rule, and then you also, end up with a probability distribution that contains a number called a decay width.

So the article is talking about cross sections...

Once you have cross sections, you can use symmetry arguments to figure out stuff. For example, every muon that has ever been seen to decay to an electron has emitted a muon neutrino, that means that in order to produce a muon from an electron, you need to reverse the reaction and add some extremely highly energetic muon neutrinos, and you don't have these present in neutron stars. The only mechanisms you can use to create muon neutrinos are bremstrahlung and pair production. Both of these are thermal which means that the muons you make are going to be at the temperature of the surroundings, and you don't get enough high energy muon neutrinos to make muons.

It's not that people are being stupid when they ignore muons in neutron stars. It's that people have thought about the problem and concluded that you aren't going to get them (except at the core where everything goes wild). At the core things are weird enough so that you can create heavy baryons and let those decay into muons. But the cross sections are so high, that the muons are trapped and only get out with difficulty.

 

[zonde]

Nope. The pressure is going to work on everything in the same way. Pressure forces don't cause charge separation. Once you have any sort of charge separation, the particles are going to correct that at near the speed of light, whereas pressure forces act at the speed of sound.

Well, if we would speak about thermal pressure then yes, you would be right. But we are not speaking about thermal pressure, we are speaking about degeneracy pressure instead. And so you are wrong. Fermi-Dirac statistics describes population of identical fermions. So electron degeneracy pressure is acting inside population of electrons and proton degeneracy pressure is acting inside population of protons.

That's because wikipedia is missing information. It would be nice if that article were linked to a intro textbook on quantum field theory.



OK. Here's some intro quantum field theory...

What you end up with when you do a QFT calculation is to answer the equation, if particles X1, X2, X3 with energies E1, E2, E3, go into a region with angles theta1, theta2, theta3 etc. What is the probability that you will come up with particles X4, X5, X6. You calculate these by writing down the Feyman diagrams, and going through a lot of nasty math.

Now if you want to calculate decay rates, you are asking if I go in with a muon with any energy at any angle, what is the probability that I will come out with an electron, neutrino pair with any energy. To do the calculation, you use a math trick called Fermi's golden rule, and then you also, end up with a probability distribution that contains a number called a decay width.

And if because of degeneracy pressure outcomes with low energy electron are forbidden the probabilities should come out different.

Once you have cross sections, you can use symmetry arguments to figure out stuff. For example, every muon that has ever been seen to decay to an electron has emitted a muon neutrino, that means that in order to produce a muon from an electron, you need to reverse the reaction and add some extremely highly energetic muon neutrinos, and you don't have these present in neutron stars. The only mechanisms you can use to create muon neutrinos are bremstrahlung and pair production. Both of these are thermal which means that the muons you make are going to be at the temperature of the surroundings, and you don't get enough high energy muon neutrinos to make muons.

Hmm, no.
I have different symmetry argument. In order for electron to decay into muon we must have very high energy electron and we produce muon, electron neutrino and muon antineutrino.

 

[twofish-quant]

Well, if we would speak about thermal pressure then yes, you would be right. But we are not speaking about thermal pressure, we are speaking about degeneracy pressure instead.

Pressure is pressure. More precisely pressure is - d(internal energy) / d (volume).

Thermal pressure, degeneracy pressure. It's all the same thing. There is a standard way of calculating pressure of bulk matter. There are some assumptions, and it's known to break in some situations, but high density/high pressure situations make the method work even better.

And so you are wrong. Fermi-Dirac statistics describes population of identical fermions. So electron degeneracy pressure is acting inside population of electrons and proton degeneracy pressure is acting inside population of protons.

Yup and neutrino degeneracy pressure acts on neutrinos. You end up dividing up different species of particles. Count the number of states. If the lower states are filled up, you have to put the particle into the lowest available state. That gives you the internal energy. You then take the derivative with respect to volume, and that gives you the pressure.

And if because of degeneracy pressure outcomes with low energy electron are forbidden the probabilities should come out different.

Yup, and that's in there.

If you look on page 822, equation (C2), there is a expression for the occupation probability for a given state. Equations (C4) (C5) and (C6) gives you the emissivity, absorption and the scattering cross sections in terms of the raw reaction rates, and you'll see expressions (F(E)) and (1-F(E)) scattered about those terms.

I have different symmetry argument. In order for electron to decay into muon we must have very high energy electron and we produce muon, electron neutrino and muon antineutrino.

Won't work.

One of the constraints of quantum field theory is that everything has to work the same in any reference frame, so if I do my calculations in the rest frame of the electron, everything should end up the same.

So I transform everything into the rest frame of the electron where it has zero momentum. At that point, all you are left with is the rest mass of the electron, and since that's less than the muon, you don't have the energy to generate a muon, electron neutrino and muon antineutrino. In the absence of another particle, you can't get a particle to decay by accelerating it. Special relativity won't allow it.

(Now if you accelerate an electron, and then put it into an electric field, *then* you can get particle creation through Bremstrahlung, it's not enough to generate muons unless you are very deep in the core at which point all heck breaks loose anyway.)

One of my points here is that physicists aren't idiots. A lot of times you get posts about how stupid scientists are that they haven't thought of this ridiculously simple idea, when it fact lots of people have thought very deeply about that topic.

If you can come up with a mechanism by which you can get a non-trivial muon flux in the outer layers of a neutron star, *that* would be worth a paper, but people have thought about it, and the conclusion is that you can't. Also, you can invoke the tooth fairy. If invoking muons, would resolve some sort of observational difficulty, that *also* would be worth a paper. (i.e. I have no clue how to make muons in neutron stars, but if we wave our magic wand, and produce muons, then something cool happens.)

As it is, the theory says that you aren't going to get muons. The observations provide no reasons why we think muons are important. If you want to work on figuring out ways of getting muons in neutron stars, you are welcome to do so, but there are probably more promising areas of research else where.

 

[zonde]

Pressure is pressure. More precisely pressure is - d(internal energy) / d (volume).

Thermal pressure, degeneracy pressure. It's all the same thing. There is a standard way of calculating pressure of bulk matter. There are some assumptions, and it's known to break in some situations, but high density/high pressure situations make the method work even better.



Yup and neutrino degeneracy pressure acts on neutrinos. You end up dividing up different species of particles. Count the number of states. If the lower states are filled up, you have to put the particle into the lowest available state. That gives you the internal energy. You then take the derivative with respect to volume, and that gives you the pressure.

Then let me rephrase my statement. Charges will get separated in electron degenerate plasma because electrons will move faster.

Won't work.

One of the constraints of quantum field theory is that everything has to work the same in any reference frame, so if I do my calculations in the rest frame of the electron, everything should end up the same.

So I transform everything into the rest frame of the electron where it has zero momentum. At that point, all you are left with is the rest mass of the electron, and since that's less than the muon, you don't have the energy to generate a muon, electron neutrino and muon antineutrino. In the absence of another particle, you can't get a particle to decay by accelerating it. Special relativity won't allow it.

Yes, you are right of course. It would have to be at least two particle collision to speak about electron with very high energy.

But I would like to drop this line about muons. It occurred to me that I am not sure if quantum states of electrons and muons are completely independent. And without that assumption the problem becomes too vague.

One of my points here is that physicists aren't idiots. A lot of times you get posts about how stupid scientists are that they haven't thought of this ridiculously simple idea, when it fact lots of people have thought very deeply about that topic.

Doubt is very important thing in science, wouldn't you say? And it doesn't mean that physicists are idiots.

 

[twofish-quant]

Then let me rephrase my statement. Charges will get separated in electron degenerate plasma because electrons will move faster.

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.

Now, if you can come up with an argument in which a charge imbalance can remain for several milliseconds so you can't assume equilibrium, that gets interesting. This happens a lot in the interstellar medium. As density goes down, so does the sound speed so you end up in situations you can't "separate" pressure processes and atomic processes.

The reason I got sucked into neutrino physics is that I'm interested in radiation hydrodynamics. You can show that neutrino processes will happen over the course of milliseconds which means that the neutrinos are not going to be in equilibrium, which means that I have to go over the details of those processes, whereas because nuclear processes are in equilibrium, I can just load in a file and I don't have to think about them,

This is because neutrinos interact through weak nuclear forces which are much weaker than EM. Electrons are going to be in equilibrium.

But I would like to drop this line about muons. It occurred to me that I am not sure if quantum states of electrons and muons are completely independent. And without that assumption the problem becomes too vague.

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.

Doubt is very important thing in science, wouldn't you say? And it doesn't mean that physicists are idiots.

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.

 

[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.