----- The following copyright 1991 by Dirk Terrell
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 Lecture #10  "Dengenerate But Not Reprehensible"

   So, what happens to this iron core? At this point the core has reached a 
density of around 10E10 (10 billion) grams per cubic centimeter (10 million 
kilograms per cc which is 22 million pounds per cubic centimeter or 360 
million pounds per cubic inch). Obviously, this is a very dense regime. At 
these densities, the electrons are "packed" very closely and quantum 
mechanics must be used to describe them. In particular, we must make use of 
what is known as the Pauli Exclusion Principle which says that no two 
electrons (in our case, but any identical particles in general) can have the 
same "quantum state." An electron's quantum state is determined by its 
position and momentum (mass times velocity). As the density increases, the 
number of unoccupied quantum states decreases. Just as the electrons in an 
atom fill the lowest energy levels first, the electrons in the core fill the 
lowest momentum states first. The remaining electrons must fill higher 
momentum (and hence higher velocity) quantum states. The collisions between 
high-velocity electrons and other electrons or nuclei creates a strong 
pressure resisting compression. In this state, the electrons are said to be 
degenerate, and the pressure is referred to as electron degeneracy pressure.

   Electron degeneracy pressure can support the star even if there is no 
fusion going on to assist it. There is, however, a limit to how massive a 
core can be and still hold itself up in this manner. If the core is more 
massive than about 1.4 times the mass of the sun, the gravitational force 
will be too high and a collapse will occur. This limit is known as the 
Chandrasekhar limit after S. Chandrasekhar who first derived it. It is a 
beautiful derivation. For those of you who know some quantum mechanics and 
are a little more mathematically inclined, look it up. I believe it is done 
in his book "Stellar Structure." 

   Well, what happens to this core if it is close to, but less than the 
Chandrasekhar limit, and a silicon shell source continues to dump ash on it? 
If the ash causes the mass to exceed the limit, what happens? The core must 
collapse. What can stop it? The contraction of the core causes 
photodisintegration to occur at an even higher rate because the core 
temperature continues to increase. Iron group nuclei are torn apart and the 
process of electron capture becomes important. In this process an electron 
and a proton combine to form a neutron plus a neutrino. Neutrinos do not 
interact with the other particles very much and essentially escape from the 
star "untouched", leaving the neutrons behind. This process only serves to 
accelerate the collapse, because it removes the electrons, which were the 
main source of the pressure support. Is there anything to save the star now?

   It turns out that there is. Very soon after the collapse begins (less 
than a second), the density in the core will reach approximately 10E14 grams 
per cubic centimeter. At this point the core can support itself by neutron 
degeneracy pressure. Is there a limit for neutron degeneracy pressure, like 
there was for electrons? It turns out that there is, but its exact value is 
not known as accurately as the Chandrasekhar limit because of our limits in 
understanding how matter behaves at these high densities. A couple of years 
ago, I did some simulations based on the best available information on the 
properties of neutron matter and found the value to be 1.76 solar masses 
(although I suspect it won't become known as Terrell's limit.) As we learn 
more, the number will be better determined, but it will probably be in the 
range of 1.8 to 3 solar masses. (By the way, I uploaded the FORTRAN program 
I used to do the simulations into the astronomy library.) 

   Our 15 solar mass star will reach this stage and the collapse will be 
finally halted by nuclear degeneracy pressure. But what about a star that 
starts out with a mass of say 30 or 50 solar masses. The cores of such stars 
will easily exceed 3 solar masses. What happens now to the core? Why, it 
collapses of course. But what other kind of degeneracy is going to kick in 
to stop the collapse? In our present understanding of things, there is 
nothing else to stop the collapse and the core will collapse to a radius of 
zero.

   While all this has been going on in the core, what is happening to the 
envelope of the star? We'll talk about what it looks like from the outside 
next time.

 Dirk