Stars as Laboratories for Fundamental Physics - MPP Theory Group

42 Chapter 2
2.2 White-Dwarf Cooling
2.2.1 Theoretical and Observed White-Dwarf Properties
After this general survey of how stars evolve it is time to study individual
aspects in more detail, and notably, how stellar evolution is affected
by the emission of weakly interacting particles. I begin with the conceptually
most transparent case of stars for which the loss of energy
simply accelerates their cooling, i.e. white dwarfs and neutron stars.
The former represent the final state of the evolution of stars with
initial masses of up to several M ⊙ , perhaps up to 8 M ⊙ (Sect. 2.1).
For reviews of the theory and observed properties see Hubbard (1978),
Liebert (1980), Shapiro and Teukolsky (1983), Weidemann (1990), and
D’Antona and Mazzitelli (1990). Because white dwarfs (WDs) are supported
by electron degeneracy pressure the hydrostatic and thermal
properties are largely decoupled. In Sect. 1.2.2 we had encountered
their inverted mass-radius relationship; in a polytropic approximation
of the WD structure one finds quantitatively (Shapiro and Teukolsky
1983)
R = 10,500 km (0.6M ⊙ /M) 1/3 (2/µ e ) 5/3 . (2.1)
Here, M is the stellar mass and µ e = ρ m −1
u n −1
e = Ye
−1 the “mean
molecular weight of the electrons” with ρ the mass density, m u the
atomic mass unit, n e the electron density, and Y e the number of electrons
per baryon. WDs do not contain any hydrogen in their interior—it
would immediately ignite—so that µ e = 2. Typically they consist of
carbon and oxygen, the end products of helium burning in the core of
the progenitor star. The central density of a polytropic model is
ρ c = 1.46×10 6 g cm −3 (M/0.6 M ⊙ ) 2 (µ e /2) 5 , (2.2)
assuming nonrelativistic electrons.
If the mass is so large (the radius so small) that the electrons become
relativistic there exists no stable configuration, i.e. the masses of WDs
must lie below the Chandrasekhar limit (Shapiro and Teukolsky 1983)
M Ch = 1.457 M ⊙ (2/µ e ) 2 . (2.3)
Observationally the WD mass distribution is strongly peaked near M =
0.6 M ⊙ (Weidemann and Koester 1984) so that a nonrelativistic treatment
of the electrons is justified. The low mass of observed WDs is
understood by the large rate of mass loss during the red giant and