Stars as Laboratories for Fundamental Physics - MPP Theory Group

The Energy-Loss Argument 7
As a simple example for the beauty and power of the virial theorem
one may estimate the solar central temperature from its mass and
radius. The material is dominated by protons which have a gravitational
potential energy of order −G N M ⊙ m p /R ⊙ = −2.14 keV where
M ⊙ = 1.99×10 33 g is the solar mass, R ⊙ = 6.96×10 10 cm the solar
radius, and m p the proton mass. The average kinetic energy of a
proton is equal to 3 2 T (remember, k B has been set equal to unity),
yielding an approximate value for the solar internal temperature of
T = 1 3 2.14 keV = 0.8×107 K. This is to be compared with 1.56×10 7 K
found for the central temperature of a typical solar model. This example
illustrates that the basic properties of stars can be understood from
simple physical principles.
1.2.2 Generic Cases of Stellar Structure
a) Normal Stars
There are two main sources of pressure relevant in stars, thermal pressure
and degeneracy pressure. The third possibility, radiation pressure,
never dominates except perhaps in the most massive stars. The pressure
provided by a species of particles is proportional to their density,
to their momentum which is reflected on an imagined piston and thus
exerts a force, and to their velocity which tells us the number of hits on
the piston per unit time. In a nondegenerate nonrelativistic medium
a typical particle velocity and momentum is proportional to T 1/2 so
that p ∝ (ρ/µ) T with ρ the mass density and µ the mean molecular
weight of the medium constituents. For nonrelativistic degenerate
electrons the density is n e = p 3 F/3π 2 (Fermi momentum p F ), a typical
momentum is p F , and the velocity is p F /m e , yielding a pressure which
is proportional to p 5 F or to n 5/3
e and thus to ρ 5/3 .
The two main pressure sources determine two generic forms of behavior
of overall stellar models, namely normal stars such as our Sun
which is dominated by thermal pressure, and degenerate stars such as
white dwarfs which are dominated by degeneracy pressure. These two
cases follow a very different logic.
A normal star is understood most easily if one imagines how it initially
forms from a dispersed but gravitationally bound gas cloud. It
continuously loses energy because photons are produced in collisions
between, say, electrons and protons. The radiation carries away energy
which must go at the expense of the total energy of the system. If it is
roughly in an equilibrium configuration, the virial theorem Eq. (1.2) in-