Stars as Laboratories for Fundamental Physics - MPP Theory Group

Anomalous Stellar Energy Losses Bounded by Observations 83
2.5.2 Helium Ignition
Another powerful constraint arises from the agreement between the
predicted and observationally inferred core mass at the helium flash
(Sect. 2.4.4). An energy-loss mechanism which is efficient in a degenerate
medium (ρ ≈ 10 6 g cm −3 , T ≈ 10 8 K) can delay helium ignition.
To establish the core-mass increase as a function of nonstandard particle
parameters one needs to evolve red giants numerically to the helium
flash.
However, for a simple analytic estimate one observes that the core
mass of a red giant grows by hydrogen shell burning. Because it is
a degenerate configuration its radius shrinks and so the core releases
a large amount of gravitational binding energy which amounts to an
average energy source ⟨ϵ grav ⟩. If a novel energy-loss rate ⟨ϵ x ⟩ is of the
same order then helium ignition will be delayed.
In order to estimate ⟨ϵ grav ⟩ one may treat the red-giant core as
a low-mass white dwarf. Its total energy, i.e. gravitational potential
energy plus kinetic energy of the degenerate electrons, is found to be
(Chandrasekhar 1939)
E = − 3 G N M 2
7 R . (2.41)
The radius of a low-mass white dwarf (nonrelativistic electrons!) may
be expressed as R = R ∗ (M ⊙ /M) 1/3 with R ∗ = 8800 km so that
⟨ϵ grav ⟩ = − Ė
M = G ( ) 1/3
NM ⊙ M M ˙
. (2.42)
R ∗ M ⊙ M ⊙
From the numerical sequences of Sweigart and Gross (1978) one finds
that near the helium flash M ≈ 0.5 M ⊙ and M ˙ ≈ 0.8×10 −15 M ⊙ s −1
so that ⟨ϵ grav ⟩ ≈ 100 erg g −1 s −1 . Therefore, one must require ⟨ϵ x ⟩ ≪
100 erg g −1 s −1 in order to prevent the helium flash from being delayed.
In order to sharpen this criterion one may use results from Sweigart
and Gross (1978) and Raffelt and Weiss (1992) who studied numerically
the delay of the helium flash by varying the standard neutrino losses
with a numerical factor F ν where F ν = 1 represents the standard case.
The results are shown in Fig. 2.25. Note that for F ν < 1 the standard
neutrino losses are decreased so that helium ignites earlier, causing
δM c < 0. It is also interesting that for F ν = 0 helium naturally
ignites at the center of the core while for F ν > 1 the ignition point
moves further and further toward the edge (Fig. 2.26). This behavior is