Stars as Laboratories for Fundamental Physics - MPP Theory Group

Anomalous Stellar Energy Losses Bounded by Observations 47
With these results the luminosity function is found to be
dN
= 4 ln(10) B C (L γ/K) 2/7
. (2.8)
dM bol 35 L γ + L ν + L x
With B 3 ≡ B/10 −3 pc −3 Gyr −1 this is numerically
dN
dM bol
= B 3 2.2×10 −4 pc −3 mag −1
×
10 −4M bol/35 L ⊙
78.7L ⊙ 10 −2M bol/5
+ L ν + L x
( M
M ⊙
) 5/7 ∑
j
X j
A j
. (2.9)
If one ignores L ν and L x this is
( ) 5/7
dN
= B 3 2.9×10 −6 pc −3 mag −1 10 2M bol/7 M ∑
dM bol M ⊙ j
X j
A j
.
(2.10)
Taking M = 0.6 M ⊙ and an equal mixture of 12 C and 16 O one finds
log(dN/dM bol ) = 2 7 M bol − 6.84 + log(B 3 ) , (2.11)
a behavior known as Mestel’s cooling law. For B 3 = 1 this function
is shown as a dotted line in Fig. 2.10. Detailed cooling curves and
luminosity functions have been calculated, for example, by Lamb and
van Horn (1975), Shaviv and Kovetz (1976), Iben and Tutukov (1984),
Koester and Schönberner (1986), Winget et al. (1987), Iben and Laughlin
(1989), Segretain et al. (1994), and Hernanz et al. (1994).
From Fig. 2.10 it is evident, however, that Mestel’s cooling law provides
a surprisingly good representation for intermediate luminosities
where it is most appropriate. At the bright end, the luminosity function
is slightly depressed, providing evidence for neutrino cooling. It
rapidly falls off at the faint end, presumably indicating the beginning
of WD formation as discussed above.
2.2.3 Neutrino Cooling
For the hottest WDs volume neutrino emission is more important than
surface photon cooling. The photon luminosity of Eq. (2.7) can be
expressed as an effective energy-loss rate per unit mass of the star, ϵ γ =
L γ /M = 3.3×10 −3 erg g −1 s −1 T7 3.5 with T 7 = T/10 7 K. For the upper
relevant temperature range, neutrinos are emitted mostly by the plasma