Stars as Laboratories for Fundamental Physics - MPP Theory Group

8 Chapter 1
forms us that a decrease of ⟨E kin +E grav ⟩ causes the gravitational energy
to become more negative, corresponding to a more tightly bound and
thus more compact system. At the same time the average kinetic energy
goes up which corresponds to an increased temperature if the system
can be considered to be locally in thermal equilibrium. Therefore, as
the system loses energy it contracts and heats up. Self-gravitating systems
have a negative specific heat!
As a protostar contracts it becomes more opaque and soon the energy
loss is limited by the speed of energy transfer from the inner parts
to the surface, i.e. essentially by the photon diffusion speed. Before an
internal energy source for stars was known it was thought that gravitational
energy provided for their luminosity. Stellar lifetimes seemed to
be given by the “Kelvin-Helmholtz time scale” for thermal relaxation
which is fixed by the speed of energy transfer. The total reservoir of
gravitational energy of the Sun is estimated by G N M 2 ⊙/2R ⊙ and its
luminosity is L ⊙ = 3.85×10 26 W = 3.85×10 33 erg/s so that the thermal
relaxation scale is about τ KH ≈ 1G 2 NM 2 ⊙R⊙ −1 L −1
⊙ = 1.6×10 7 yr. Pressed
by thermodynamic theory and the authority of Lord Kelvin, geologists
tried to adjust the age of the Earth to this short time scale against
sound evidence to the contrary. At the beginning of our century the
discovery of radioactivity and thus of nuclear processes revealed that
stars had another source of energy and consequently could live much
longer than indicated by τ KH .
A further contraction is thus intercepted by the onset of hydrogen
burning which commences when the temperature is high enough for
protons to penetrate each other’s electrostatic repulsive potential. The
hydrogen-burning reactions are more fully discussed in Chapter 10; the
bottom line is that four protons and two electrons combine to form
a helium nucleus (α particle), releasing 26.73 MeV of energy. A few
percent are immediately lost in the form of two neutrinos which must
emerge to balance the electron lepton number, but most of the energy
is available as heat. Because the nuclear reaction rates have a
steep temperature dependence, a further contraction and heating of
the star leads to much more nuclear energy generation, which quickly
increases the average E kin of the nuclei and thus leads to an expansion
and cooling by the same virial-theorem logic that led to contraction and
heating when energy was lost. Thus a stable configuration of “thermal
equilibrium” is reached where the energy lost is exactly balanced by
that produced from nuclear reactions. Stars as fusion reactors are perfectly
regulated by the “negative specific heat” of a self-gravitating
system!