Neutron star warm matter - temperature?

[Vrbic]

In calculations of quantities in Neutron stars with degenerate matter is usual to set temperature zero. If I'm right it means that pressure of this matter is negligible against pressure due to Pauli principle. But what about situation when the matter is in neutron star locally compressed. How can I resolve which density started to be important and pressure of the matter starting to be equivalent to pressure from degenerate neutron gas?
How can I calculate a temperature in such place?

If I'm wrong with something or this topic isn't for this section please let me also know.
Thank you all.

 

[Ken G]

In calculations of quantities in Neutron stars with degenerate matter is usual to set temperature zero. If I'm right it means that pressure of this matter is negligible against pressure due to Pauli principle.

It is a common misconception that the Pauli principle causes pressure. But it does not-- it simply means the system has reached its ground state. The ground state contains a huge kinetic energy, and the kinetic energy is what causes pressure, in a perfectly normal way. All the Pauli principle does is prevent further rise in the pressure, by preventing further rise in the kinetic energy per unit volume that gives the pressure. Remember, the history of a contracting star is a story of ever-rising pressure and kinetic energy density, so we certainly never need the Pauli principle to be a cause of any new pressure-- the star has plenty already!

So what you really mean is, the Pauli principle drives the temperature way down, compared to what you might expect for that pressure, if it were an ideal gas. But the temperature is not doing anything other than making the star refuse to lose very much more heat, and that's the whole reason that the contraction ceases and the pressure stops rising.

But what about situation when the matter is in neutron star locally compressed. How can I resolve which density started to be important and pressure of the matter starting to be equivalent to pressure from degenerate neutron gas?

I'd say the first thing to do is forget about temperature, it is not playing any role in the pressure. Just look at the kinetic energy density, and what happens to it when you compress the gas locally. All gases respond exactly the same way in that situation, it makes no difference if they are degenerate-- that only affects the temperature. The only tricky thing is if the neutrons go relativistic, as that changes the connection between pressure and kinetic energy density. If you can ignore that, the pressure is always 2/3 of the kinetic energy density, independent of temperature, and independent of the Pauli principle.

How can I calculate a temperature in such place?

Why would you want to? The temperature only affects heat transfer, and most likely you are not including that anyway, as it is quite difficult.

 

[mfb]

Remember, the history of a contracting star is a story of ever-rising pressure and kinetic energy density, so we certainly never need the Pauli principle to be a cause of any new pressure-- the star has plenty already!

The Pauli principle limits the drop of energy and pressure that can occur via cooling processes.

 

[Ken G]

That would have been correct had you said it limits the rise in pressure. This is the common misconception!

 

[mfb]

Let me make it more explicit: for a given size and mass, the Pauli principle limits the drop of energy and pressure that can occur via cooling processes.

 

[Ken G]

Yes that's true, but in a star, you cannot specify the size, it must come from hydrostatic equilibrium. So what actually happens is as the star loses heat, it contracts, and its pressure always rises. When Pauli exclusion kicks in, all that happens is the heat loss is stopped, nothing happens to the pressure, and the PEP does not prevent the pressure from falling, it stops any further rise. But this is widely misunderstood, because the common language is to frame the PEP as either a kind of floor to the pressure, or a kind of new source of pressure. But in the conditions of a star, where hydrostatic equilibrium determines the size, neither of those scenarios are really what is going on. The PEP isn't directly doing anything with the pressure, it is merely inhibiting heat loss, and has no other global effects.