TD6 Statistical Physics (M1)
White Dwarfs
I.
Preliminary part:
Reminder: in the relativity's theory, the relation between energy, momentum, mass and light’s speed
is: E2=p2c2+m2c4. It means that, for ultra-relativistic particles such as p>>mc, E=pc with p  k .
White dwarfs correspond to the end of life of stars like sun. These stars are little (the typical radius R
is of the same order as the radius of the earth, about 500 km, far from the 700 000 km for the actual
sun). However their mass is of the same order as the mass of the sun (about 1030 kg). The central
regions of white dwarfs are mainly composed of Carbon and Oxygen (isotopes 12 and 16
respectively). The star is not sufficiently heavy to burn them - that is to initiate nuclear reactions with
C and O. The temperature inside central regions is about 107 °K.
1) Give the approximate density of a white dwarf (kg/m3 and then as number of nuclei per m3). What
is the volume available per nucleus?
We will consider in the following a simplified model of white dwarf, as a gas of free electrons
submitted to the gravitational attraction due to the nuclei.
2) Gas of electrons. The free electrons are Ne fermions with spin ½ contained in a box of volume V.
Their density is Ne/V=1038.m-3
2-a) Calculate the momentum pF of the Fermi level of electrons, assuming that the temperature T=0°K
and that the electrons are not relativistics. Prove that the electrons close to the Fermi level are
indeed ultra-relativistics, that is pF>>mec (where me is the electron's mass and c is the light’s speed).
Calculate the energy of the Fermi level of the ultra-relativistic electrons at T=0°K (we consider that
the energy is zero when the particles are at rest).
2-b) The temperature of the system is T0 with kBT0=5.104 eV. Prove that the approximation T=0°K is
good in the calculation of the pressure P and of the internal energy U. Calculate the pressure due to
the electrons. Give its order of magnitude.
3) Protons. The protons have a mass mp=2.103me. Their number per unit volume is Np/V=1038.m-3.
3-a) Prove that for T=T0, the protons satisfy the Maxwell-Boltzmann statistics.
3-b) Give the order of magnitude of the pressure due to the protons.
3-c) Compare the average energy of protons and of electrons at T0. Idem for the pressure due to
protons and due to electrons applied on the walls of the box with volume V. What is the role of
protons?
II.
Equilibrium of the White Dwarf.
The previous part I. has shown that the white dwarf is composed of a mixture of 2 independent
gases: a gas of electrons and a gas of nuclei. The kinetic energy of the electrons is sufficiently high to
neglect the kinetic energy of nuclei, as well as the electrostatic interactions between charged
particles. The nuclei are however very important for the gravitational interactions, as well as for
electric neutrality. The white dwarf is described here as a star with radius R and volume V containing
N electrons due to the total ionization of hydrogen atoms for example. There are Np=N protons with
mass mp in the star. We will see that white dwarfs are stable only if their total mass obeys some
condition that we will determine.
In the following, we will consider T=0°K as a good approximation for the temperature.
We will consider two limit cases: the case of non-relativistic electrons, and the case of ultrarelativistic electrons (pF>>mec).
1)
Electrons.
1-a) In case of non-relativistic electrons compute the internal energy density (U/V) and the electronic
pressure P of the star as a function of the density N/V. Remember the expression of F and pF.
1-b) In case of relativistic electrons, give the general expressions for the internal energy density (U/V)
and for the electronic pressure P as a function of xF=pF/mec and x=p/mec, where p is the momentum
of the electrons with mass me.
1-c) In the ultra-relativistic case where x>>1, use an expansion as a function of 1/x2 to the second
order inside the integral, to compute the energy density U/V and the pressure P.
Give the expression of F and pF.
2)
Equilibrium within a uniform mass density. We consider here that the mass density of the
star is uniform. We will study the equilibrium of the star for a given mass M, and different radii R. The
gravitational pressure allows to counteract the effect of the pressure due to the degeneracy of
electrons. The gravitational interactions give rise to an energy Eg=-3/5.G.M2/R, where G is the
gravitational constant.
2-a) Give the expression of pF as a function of R and M.
2-b) Give the expression of the total energy E of the star. The equilibrium condition is given by
dE/dR=0. Deduce from that condition, the expression of the radius R at equilibrium for nonrelativistic, and for ultra-relativistic electrons.
2-c) What happens when the electrons are ultra-relativistcs and when the mass of the star is larger
than a characteristic mass MR that you will determine?
3)
Equilibrium within non-uniform mass density.
3-a) The equilibrium condition is given by the chemical potential cste. The chemical potential is the
energy given to the system when one electron (and then one proton) is added. The protons are
assumed to give only a gravitational contribution to the total energy. What is the expression of ?
3-b) The gravitational energy uG satisfy the equation uG=mP.G. where  is the mass density of the
star, and mP the mass of a proton,is the Laplacian operator. By computing =mp.N/V in the nonrelativistic and in the ultra-relativistic case as a function of F, write the equation that must satisfy the
chemical potential e of the electrons in both cases.
4)
Neutron’s stars. For high mass density, the electrons can be captured by the nuclei. Each
captured electron combines to a proton to produce a neutron (plus a neutrino). The atomic number
goes from Z to (Z-1). The variation of energy of the nucleus is then =V(Z-1)-V(Z)>0 where V(Z)
denotes the total bonding energy. We consider in this question the mass density as uniform.
4-a) What is the chemical potential e of electrons, when the chemical equilibrium is imposed by the
chemical reaction
nucleus(Z)+electron → nucleus (Z-1)
4-b) What equation satisfy the density of ultra-relativistic electrons in this case?
4-c) What happens to the star when this reaction of production of neutrons starts? Comment.