Implications of Nuclear Forces 1-16-02
Energy Scale
The binding energy of nuclei is roughly 8 MeV/nucleon (Fig. 3.1 in
Supernovae and Nucleosynthesis).
The rest mass of a nucleon is about 931 MeV, so nuclear
burning will release only about 0.008 of the rest mass energy.
For comparison, the gravitational binding energy of a neutron
star is about 10% of the rest mass. For a black hole the
corresponding energy release might rise by a factor of a few
above this.
Conversion of 4H to He4 releases about 7 Mev/nucleon, and
conversion of He4 to C12 or O16 releases about 0.7 Mev/nucleon.
There burning stages are cooled by photon escape, so that about
90% of the stars we see are burning hydrogen to helium.
Note that 1 MeV/nucleon is about 1e18 ergs/gram. Carbon burning
and oxygen burning both release about 0.5 MeV/nucleon, so that
a supernova energy of 1e51 ergs corresponds to burning a mass
of 1e51/0.5e18 = 2e33 grams (1 solar mass) of such fuel. This
is much less than the binding energy of a neutron star,
1.5(2e33)grams*(3e10)**2 0.1 = 3e53 ergs.
Types of Reactions
There are two nuclear forces, the strong and the weak.
Both are short range. Similarity in n-p and p-p interactions
(after the coulomb effects are taken out) gave rise to the
notion of "charge invariance of strong interactions." The proton
is one "isospin" state of the nucleon, and the neutron the other
(isospin=1/2 has two states). The strong interaction is isospin
invariant. It does not change n to p or p to n. The strong
interaction provides nuclear binding.
As the name implies, the weak force is associated with weaker stellar
processes. However, while the strong force cannot change neutrons
into protons, and vice versa, the weak force can. To burn the
Big Bang fuel (H) to He4, some protons must be converted to neutrons,
and a weak interaction is required. A determining parameter for
thermonuclear synthesis is the neutron excess,
which is governed by the rate of the weak interactions (beta decay,
positron decay, electron capture).
Neutronization
The rest mass of the electron is only 0.511 Mev. Charge balance
requires that each electron be associated with a proton (either
free or in a nucleus). Light particles like electrons have large
compton wavelengths, h/mc. Fermions obey the Pauli exclusion
principle. Putting light particles in a small box is harder than
for heavier particles. Squeezing the light particles in requires
that we give them a lot of energy to make their wavelength small
enough. At high densities it is energetically advantageous to
replace the electron and its associated proton by a neutron.
Thus, star tend to "neutronize" by electron capture. Why must
stars contract? If they are too massive, above the Chandrasekhar
limit (1.45 solar masses), they can balance their cooling only
so long as they have fuel to burn. Otherwise cooling leads to
contraction to higher density.
Neutrino cooling
In addition to nuclear weak interactions, such as
<(A,Z)+ e- = (A,Z-1)+nu,
there are a set of reactions coupling the leptons (electrons and neutrinos)
only. This coupling is a consequence of the nature of the weak interaction,
and was predicted to be astrophysically important long before it was
measured experimentally (see Chapter 10).
An important example is
e- + e+ = nu + nubar,
that is, electron positron annihilation
which goes to a neutrino antineutrino pair instead of two photons.
While much weaker than conventional pair annihilation (1e-20 times
less probable), the neutrino can escape the star with almost no
further interaction, while phontons would have to diffuse out slowly.
For example, a photon diffusion time for the sun is the order of
1e5 YEARS!
These processes have a high temperature threshold (annihilation requires
at least the rest mass energy of the pair, 2m(e)c**2=1.022Mev). They
do not become effective below about temperatures of 5e8 K, but above
that they rapidly overwhelm radiative diffusion as a cooling process.
As we shall see, an implication is that the yields of C12 and O16 are
significantly enhanced (these fuels avoid being burnt) because of
effects from neutrino cooling on stellar structure.