Stars as Laboratories for Fundamental Physics - MPP Theory Group

The Energy-Loss Argument 9
The most salient feature of a normal stellar configuration is the interplay
of its negative specific heat and nuclear energy generation. Conversely,
if the pressure were dominated by electron degeneracy it would
be nearly independent of the temperature. Then this self-regulation
would not function because heating would not lead to expansion. Thus,
stable nuclear burning and the dominance of thermal pressure go inseparably
hand in hand. Another salient feature of such a configuration is
the inevitability of its final demise because it lives on a finite supply of
nuclear fuel.
b) Degenerate Stars
Everything is different for a configuration dominated by degeneracy
pressure. Above all, it has a positive heat capacity so that a loss
of energy no longer implies contraction and heating. The star actually
cools. This is what happens to a brown dwarf which is a star
so small (M < 0.08 M ⊙ ) that it did not reach the critical conditions
to ignite hydrogen: it becomes a degenerate gas ball which slowly
“browns out.”
The relationship between radius and mass is inverted. A normal
star is geometrically larger if it has a larger mass; very crudely R ∝
M. When mass is added to a degenerate configuration it becomes
geometrically smaller as the reduced size squeezes the electron Fermi
sea into higher momentum states, providing for increased pressure to
balance the increased gravitational force. The l.h.s. of Eq. (1.1) can be
approximated as p/R where p ∝ ρ 5/3 is some average pressure. Because
ρ ≈ M/R 3 one finds p/R ∝ M 5/3 /R 6 while the r.h.s. of Eq. (1.1) is
proportional to Mρ/R 2 and thus to M 2 /R 5 . Therefore, a degenerate
configuration is characterized by R ∝ M −1/3 .
Increasing the mass beyond a certain limit causes the radius to
shrink so much that the electrons become relativistic. Then they move
with a velocity fixed at c (or 1 in natural units), causing the pressure
to vary only as p 4 F or ρ 4/3 . In this case adding mass no longer leads to a
sufficient pressure increase to balance for the extra weight. Beyond this
“Chandrasekhar limit,” which is about 1.4 M ⊙ for a chemical composition
with Y e = 1 (number of electrons per baryon), no stable degenerate
2
configuration exists.
In summary, the salient features of a degenerate configuration are
the inverse mass-radius relationship R ∝ M −1/3 , the Chandrasekhar
limit, the absence of nuclear burning, and the positive specific heat
which allows the configuration to cool when it loses energy.