Stars as Laboratories for Fundamental Physics - MPP Theory Group

46 Chapter 2
In order to translate this equation into the observable luminosity
function I assume a constant WD birthrate B so that the total number
density of degenerates is N = B t gal (age of the galactic disk t gal ).
Taking the above values N ≈ 10 −2 pc −3 and t gal ≈ 9 Gyr one finds
B ≈ 10 −3 pc −3 Gyr −1 . Because the number density of WDs in a given
magnitude interval dM bol is proportional to the time interval dt it takes
to cool through this magnitude range one readily obtains
dN
dM bol
= B
dt
dM bol
= −B
dU/dM bol
L γ + L ν + L x
. (2.5)
The photon luminosity is L γ = 78.7 L ⊙ 10 −2Mbol/5 in terms of the bolometric
magnitude, equivalent to log(L γ /L ⊙ ) = (4.74 − M bol )/2.5. L γ is
related to the internal temperature T by the thermal conductance of
the surface layers so that one may derive a function T (L γ ). The quantities
U, L ν , and L x are given in terms of T so that they can be expressed
in terms of L γ and hence of M bol .
In hot WDs the thermal energy is largely stored in the nuclei which
form a nearly classical Boltzmann gas. At low T the ideal-gas law
breaks down and eventually the nuclei arrange themselves in a lattice.
The internal energy is then a more complicated function of temperature.
The heat capacity per nucleon, which is 3 for an ideal gas, rises to
2
3 near the Debye temperature Θ D and then drops approximately as
(16π 4 /5) (T/Θ D ) 3 to zero (Shapiro and Teukolsky 1983). However, the
observed WDs have a relatively small ρ because of their small mass
around 0.6 M ⊙ so that even the oldest WDs have not yet crystallized.
Therefore, as a reasonable first approximation the internal energy is
U = C T with the ideal-gas heat capacity for the entire star of
C = 3 2
M
m u
∑
j
X j
= 3.95×10 −2 L ⊙ Gyr
A j 10 7 K
M
M ⊙
∑
j
X j
A j
, (2.6)
where X j is the mass fraction of the element j, atomic mass A j , and
m u = 1.661×10 −24 g is the atomic mass unit.
The thermal conductance of the surface layers is calculated by solving
the stellar structure equations. Using a Kramers opacity law,
κ = κ 0 ρ T −7/2 , one finds (van Horn 1971; Shapiro and Teukolsky 1983)
( )
L = 1.7×10 −3 M T 7/2
L ⊙
≡ K T 7/2 , (2.7)
M ⊙ 10 7 K
where T is the internal temperature. The observed WD luminosities of
Tab. 2.1 vary between 0.5×10 −4 and 0.5 L ⊙ , corresponding to a range
0.4−6 × 10 7 K of internal temperatures.