Re: Relativistic velocities of electrons and the formation of Neutron stars

You've asked a number of questions, so this answer will be somewhat lengthy.

Let me start with a definition. It will be easier to think about energies rather than velocities. How does a particle's energy compare to its rest energy (which is given by mc²)? If a particle's energy is much less than its rest energy, it's nonrelativistic, while if a particle's energy is much more than its rest energy, it is relativistic. For reference, the rest energy of an electron (the mass of which is 9.11 x 10^-31 kg) is 511 kiloelectron volts (keV).

An important initial point is that, even within atoms under normal conditions (say, here on Earth), electron energies can be significant. They are not so large that the electrons are highly relativistic, but they might still be 10% of the electron's rest energy.

The second initial point is that electrons (and protons and neutrons) obey certain quantum mechanical rules so that they are called fermions. Among the important rules that they obey is one called the exclusionary principle, which says (crudely) that no two electrons can be in the same place doing the same thing at the same time. A consequence of the exclusionary principle is that electrons produce a pressure.

(As a simple analogy to this electron pressure, consider a group of people. No two people can be in the same place at the same time. Thus, if you try to get people to move closer together, they can move closer together initially. However, at some point, you will pack the group of people so close together that they will begin to resist any closer packing.)

Consider the electrons in the core of a massive star. Suppose we conduct the following thought experiment. Initially, there is only one electron in the core. By the exclusionary principle, there are no other electrons around so it can do whatever it wants. Now suppose we add another electron. This second electron is limited by the movements of the first. If we continue this thought experiment (more precisely by applying the uncertainty principle), each new electron that we add must have a slightly higher energy so that it is not violating the exclusionary principle by doing something that one of the already present electrons is doing. The result is that some electrons can have extremely high energies. Moreover, these electrons produce a pressure, and they are so tightly packed that the pressure is known as a degeneracy pressure. (The electrons are degenerate not because they are immoral but because they are as tightly packed as they can be without violating the exclusionary principle.)

It's worth noting that this degeneracy pressure is what is responsible for supporting a white dwarf. Effectively the pressure provided by the electrons obeying the exclusionary principle is so large within the white dwarf that their combined allegiance to quantum mechanics supports the mass of the white dwarf (about 1 solar mass or 2 x 10³⁰ kg) from collapsing under its own weight.

The final piece that we need to understand the collapse is the temperature of the core of a massive star. The temperature is roughly several billions degrees. It needs to be this hot so that various fusion reactions, like the fusion of silicon, can occur. The nuclei of atoms contain protons, which are positively charged. Thus, two nuclei will repel each other ("opposites attract, likes repel"). Temperature is a measure of the velocity of particles, so only at very high temperatures (high velocities) can silicon atoms overcome their electronic repulsion to other nuclei and engage in fusion. (This high temperature also means that there are no atoms in the core of a star. Actually, there are no atoms in the core of even a low-mass star like the Sun. Stellar core temperatures are high enough that all one has is a soup of protons, neutrons, atomic nuclei, electrons, and photons.)

This high temperature is a double-edged sword. It is so high that a product of silicon fusion, namely iron, can be destroyed. Atomic nuclei can be split by gamma-ray photons. The high temperatures and fusion reactions occurring in the core of a massive star on the verge of collapse mean that there are abundant, extremely energetic gamma-ray photons. In fact, the core of a massive star on the verge of collapse contains roughly equal parts of iron nuclei and helium nuclei, the helium nuclei resulting from the destruction of iron nuclei.

We now have all of the pieces required to understand the collapse of the core of a massive star into a neutron star. The core is a soup of protons, neutrons, atomic nuclei, electrons, and photons. The atomic nuclei range from helium nuclei all the way to iron nuclei, with some of the lighter element nuclei resulting from the destruction of iron nuclei. The core temperature is high, several billions of degrees. Finally, the electrons are degenerate and partially or fully supporting the core with their degeneracy pressure, and they are relativistic because they are so energetic with energies of perhaps a few thousand keV.

The electrons become so degenerate that some of them have energies sufficient to drive a neutronization process. (There can be other important processes as well, such as pair annihilation, but for simplicitly I'll stick to neutronization.) For a free neutron, this process usually happens

                    n -> p + e + neutrino + energy

meaning that a neutron decays to form a protron, electron, and neutrino and typically energy as well. If electrons are sufficiently energetic, the process can be driven "in reverse"

                    e + p -> n + neutrino

In the core of a massive star, the degeneracy of the electrons means that some electrons are sufficiently energetic to drive this process. (The electrons can be driven either into free protons, or in extreme cases, into atomic nuclei.) Neutronization is a double whammy for the star. First, remember that the electrons are providing pressure to help support the core. Removing electrons means that one is removing pressure from the core making it more difficult for the core to support its own weight. Second, the neutrinos escape from the core, because they interact only weakly with matter. By escaping from the core, the neutrinos carry energy with them, effectively cooling the core. The loss of the electrons might not be so bad, if the core could remain sufficiently hot, because heat produces pressure, which might be able to support the core. (Indeed, it is the heat from the nuclear reactions produced in the Sun's core that is producing the pressure that supports the Sun's weight.) The double whammy is now apparent. The neutrino losses require some heat/pressure source to make up for the energy losses, and the electrons are disappearing, further removing a source for pressure. (Actually, from the fusion reactions, neutrino losses have been problematic for the star for some time, but once neutronization begins, the neutrino losses become catastrophic.) Once neutronization starts, the core can no longer resist collapse and the star is doomed.

I realize that this is a long answer, but the gist of it is the following. Yes, the electrons become relativistic, but they are forced to become relativistic because of their quantum mechanical properties. Once relativistic, they then have enough energy to drive a nuclear reaction that robs the star's core of its support, and it collapses under its own weight.

There are various summaries of the late stages of stellar evolution on the Web, in various professors' lecture notes. One succint summary, including the lifetimes of the various fusion processes is this set. Much of this information was drawn from (the rather technical reference) Black Holes, White Dwarfs, and Neutron Stars, by Shapiro and Teukolsky. You might also want to search for information on supernova SN 1987A, from which neutrinos were detected.