Stars as Laboratories for Fundamental Physics - MPP Theory Group

80 Chapter 2
from the number of clump giants in open clusters and from the old
galactic disk population.
In Sect. 1.3.1 it was shown that the main impact of a nonstandard
energy-loss rate on a star is an acceleration of the nuclear fuel consumption
while the overall stellar structure remains nearly unchanged.
The temperature dependence of the helium-burning (triple-α reaction)
energy generation rate ϵ 3α ∝ T 40 is much steeper than the case of hydrogen
burning discussed in Sect. 1.3.1 and so the adjustment of the
stellar structure is even more negligible. With L 3α the standard heliumburning
luminosity of the core of an HB star and L x the nonstandard
energy-loss rate integrated over the core, t He will be reduced by an approximate
factor L 3α /(L x + L 3α ). Demanding a reduction by less than
10% translates into a requirement L x ∼ < 0.1 L 3α . Because this constraint
is relatively tight one may compute both L x and L 3α from an
unperturbed model. If the same novel cooling mechanism has delayed
the helium flash and has thus led to an increased core mass only helps
to accelerate the HB evolution. Therefore, it is conservative to ignore
a possible core-mass increase.
The standard value for L 3α is around 20 L ⊙ ; see Fig. 2.4 for the
properties of a typical HB star. Because the core mass is about 0.5 M ⊙
the core-averaged energy generation rate is ⟨ϵ 3α ⟩ ≈ 80 erg g −1 s −1 . Then
a nonstandard energy-loss rate is constrained by
⟨ϵ x ⟩ ∼ < 10 erg g −1 s −1 . (2.40)
Previously, this limit had been stated as 100 erg g −1 s −1 , overly conservative
because it was not based on the observed HB/RGB number
ratios in globular clusters. However, in practice Eq. (2.40) does not
improve the constraints on a novel energy-loss rate by a factor of 10
because the appropriate average density and temperature are somewhat
below the canonical values of ρ = 10 4 g cm −3 and T = 10 8 K.
For a simple estimate the energy-loss rate may be calculated for
average conditions of the core. Typically, ϵ x will depend on some small
power of the density ρ, and a somewhat larger power of the temperature
T . For the HB star model of Fig. 2.4 the core-averaged values ⟨ρ n ⟩ and
⟨T n ⟩ are shown in Fig. 2.24 as a function of n. The dependence on n
is relatively mild so that the final result is not sensitive to fine points
of the averaging procedure.
In order to test the analytic criterion Eq. (2.40) in a concrete example
consider axion losses by the Primakoff process. The energy-loss rate
will be derived in Sect. 5.2.1. It is found to be proportional to T 7 /ρ
and to a coupling constant g 10 = g aγ /10 −10 GeV −1 . For a typical HB