When will neutronization occur in the black hole formation process?

[Ken G]

To clarify this, a reference might help. For example, look at the second page in http://jila.colorado.edu/~pja/astr3730/lecture17.pdf, which derives degeneracy pressure by starting with the expression for kinetic pressure. The nonrelativistic substitution of mv where they have p shows by inspection that the pressure they derive is 2/3 of the kinetic energy density, which is a standard aspect of nonrelativistic kinetic pressure in three dimensions. This means that degeneracy pressure is not unusual in how it relates pressure to kinetic energy density, it is unusual only in how it connects the average kinetic energy per particle with a temperature. I say this means degeneracy pressure is a thermodynamic, not mechanical, effect, so its common label can be quite misleading.

 

[PeterDonis]

As to how relativistic the electrons are, yes, they are very relativistic. All the same, they are not completely relativistic

Well, "completely relativistic" would mean "lightlike", correct?

To clarify this, a reference might help.

Thanks, I'll take a look.

 

[WannabeNewton]

This may just be an issue of semantics. For a non-relativistic ideal gas the energy of each mode is $E_k = \frac{\hbar^2 k^2}{2m}$ which is just the spectrum of the free particle Hamiltonian and as such constitutes the kinetic energy of each mode. The number density of the Fermi-Dirac distribution is then $n = \frac{g}{6\pi^2}k_F^3$ where $g$ is the degeneracy of states and $k_F$ is the Fermi momentum. From this one obtains the degeneracy pressure $P_F = \frac{2}{5}nE_F$ where as usual $E_F \equiv \lim_{T\rightarrow 0} (\frac{\partial F}{\partial n})_{T,V}$ in the canonical ensemble, with $F$ the Helmholtz free energy.

In this sense the degeneracy pressure of an ideal gas is certainly due to kinetic energy. I don't see where the existence of pressure due to kinetic energy in the degeneracy limit implies some kind of motion in this limit as this is a quantum mechanical effect after all and the concept of motion is moot in quantum mechanics, not to mention there always exists kinetic energy in quantum mechanical ground states.

However the statement

I say this means degeneracy pressure is a thermodynamic, not mechanical, effect, so its common label can be quite misleading.

doesn't make sense to me as thermodynamic degrees of freedom such as pressure are obtained from averages over statistical ensembles of mechanical degrees of freedom. Again it may just be an issue of semantics.

 

[Ken G]

Well, "completely relativistic" would mean "lightlike", correct?

Correct-- at which point the adiabatic index is 4/3, the value needed for only marginal stability against contraction.

Thanks, I'll take a look.

Sorry for not providing it sooner.

 

[Ken G]

In this sense the degeneracy pressure of an ideal gas is certainly due to kinetic energy. I don't see where the existence of pressure due to kinetic energy in the degeneracy limit implies some kind of motion in this limit as this is a quantum mechanical effect after all and the concept of motion is moot in quantum mechanics, not to mention there always exists kinetic energy in quantum mechanical ground states.

I agree that the meaning of "motion" is rather ambiguous in the quantum limit, but the meaning of "kinetic" is not, i.e., we still use the term "kinetic energy", and its informal meaning as "energy of motion." My point is simply that degeneracy pressure is a thermodynamic, not mechanical, effect-- it relates to the temperature, not the pressure, if the conserved quantities of mass and energy are being tracked in the evolution of some object.

However the statement [that I've repeated here] doesn't make sense to me as thermodynamic degrees of freedom such as pressure are obtained from averages over statistical ensembles of mechanical degrees of freedom. Again it may just be an issue of semantics.

What I mean is that it is something that affects the temperature we associate with the energy, not the pressure we associate with that same energy. Is that not the standard distinction between a "thermodynamic" vs. "mechanical" effect?

 

[Ken G]

On the question of when neutronization sets in, I've looked up some hard numbers. If you take the maximum energy an electron can get from a beta decay, and equate that to the Fermi energy of a fully degenerate electron gas, you find that beta decay is suppressed at a density of about 20 million g/cc (for example, in Kippenhahn and Wiegert, page 135). The density of a solar mass of material in the volume of the Earth is about 1/10 that amount, suggesting that if you compress such a radius by anything more than a factor of 2, you will begin neutronization. Yet a crucial question is still, what is the timescale? Since neutronization does not appear in the force equation, it appears in the composition and energy equations, if the neutronization timescale is way longer than the sound crossing time, you still have a quasi-steady force equilibrium, so you still have dynamical stability (for the reasons given above-- the highly relativistic electron adiabatic index is still a little above 4/3). Thus my point is, you must wait for something that eats up the kinetic energy, and that will be whichever of these eats up that energy the fastest: neutronization, the Urca process, or photodistintegration.

So collapse must occur as soon as one of these processes plays out on the sound crossing time, since then you will have an instability in the energy equation which runs away on timescales shorter than you can establish a force balance. It's not obvious how small the radius needs to get before this sets in, but it has to happen when the object is still much larger than a neutron star (so for a mass appreciably less than the Chandra mass), or there would not be enough gravitational energy left for the sudden release needed in a supernova. My point is, that release must happen, not by virtue of the force equation alone, but by virtue of something happening in the energy equation which eats up the kinetic energy that goes into the pressure needed in the force equation. That is not what would be called a dynamical instability, like a pencil on its point, even though core collapse is sometimes explained that way.