Stars as Laboratories for Fundamental Physics - MPP Theory Group

The Energy-Loss Argument 11
against the force of gravity. In Eq. (1.5) c p is the heat capacity at
constant pressure. Further, the quantity ∇ ad ≡ (∂ ln T/∂ ln p) s , taken
at constant entropy density s, is the “adiabatic temperature gradient.”
It is not really a gradient. It is a thermodynamic quantity characteristic
of the medium at the local conditions of ρ, p, T , and the chemical
composition.
The calculation of ϵ x for a number of hypotheses concerning the
existence of novel particles or novel properties of neutrinos will be a
major aspect of this book. Even for a given interaction law between
the particles and the medium constituents this is not always a straightforward
exercise because the presence of the ambient medium can have
a significant impact on the microscopic reactions.
This is equally true for the nuclear energy generation rates. Nuclear
reactions are slow in stars because the low temperature allows only few
nuclei to penetrate each other’s Coulomb barriers. Therefore, screening
effects are important, and the nuclear cross sections need to be known at
energies so low that they cannot be measured directly in the laboratory.
Much of the debate concerning the solar neutrino problem revolves
around the proper extrapolation of certain nuclear cross sections to
solar thermal energies.
1.2.4 Energy Transfer
The transfer of energy is driven by the radial temperature gradient. In
the absence of convection heat is carried by photons and electrons moving
between regions of different temperature, i.e. by radiative transfer
and by conduction. In this case the relationship between the energy
flux and the temperature gradient is
L r = − 4πr2
3κρ
d(aT 4 )
, (1.6)
dr
where aT 4 is the energy stored in the radiation field (a = π 2 /15 in
natural units) and κ is the opacity (units cm 2 /g). It is given by a sum
κ −1 = κ −1
γ
+ κ −1
c
+ κ −1
x , (1.7)
where κ γ is the radiative opacity, κ c the contribution from conduction
by electrons, and κ x was included for a possible contribution from novel
particles. The quantity (κ γ ρ) −1 = ⟨λ γ ⟩ R is the “Rosseland average”
of the photon mean free path—its precise definition will be given in
Sect. 1.3.4.