Stars as Laboratories for Fundamental Physics - MPP Theory Group

600 Appendix D
D.2 Nuclear Matter
D.2.1
The Ideal p n e ν e Gas
For the topics discussed in this book, the properties of hot nuclear
matter in a young supernova core are of great interest. The relevant
range of densities and temperatures is about 3×10 12 −3×10 15 g cm −3
and 3−100 MeV, respectively. The properties of matter at such conditions
is determined by its equation of state which takes the nuclear
interaction fully into account. However, in order to gain a rough understanding
of the behavior of the main constituents of the medium
(protons, neutrons, electrons, and electron neutrinos) it is worthwhile
to study a simple toy model where these particles are treated as ideal
Fermi gases.
To this end, neutrinos and electrons are treated as massless. Their
dispersion relation is dominated by the interaction with the medium.
Because they interact only by electroweak forces their “effective mass”
is always much smaller than their energies.
D.2.2
Kinetic and Chemical Equilibrium
The reaction e p ↔ n ν e which establishes β equilibrium is fast compared
to other relevant time scales. Therefore, the relative abundances
of n, p, e, and ν e are determined by the conditions of kinetic and chemical
equilibrium. The physical condition of the medium is then determined
by the baryon density n B , the temperature T , and the condition
of electric charge neutrality
n B = n n + n p Baryon density,
n p = n e Charge neutrality, (D.18)
µ e + µ p = µ n + µ νe β equilibrium.
Here, the µ j are the relativistic chemical potentials of the fermions
which determine their number densities n j according to the Fermi-Dirac
distribution Eq. (D.5). Recall that n j is the difference between fermions
and antifermions of a given species.
In addition, one of two extreme assumptions is made. In a young
SN core the neutrinos are trapped so that the local lepton number is
conserved. In this case the lepton fraction Y L is the fourth required
input parameter,
Y L n B = n e + n νe Lepton conservation. (D.19)