Stars as Laboratories for Fundamental Physics - MPP Theory Group

The Energy-Loss Argument 15
M ′ (r ′ ) = M(r), and the chemical composition at r ′ is the same as
that at r. The density is transformed by ρ ′ (r ′ ) = y −3 ρ(r), and from
Eq. (1.1) one finds that the pressure scales as p ′ (r ′ ) = y −4 p(r). The
equation of state for a nondegenerate, low-mass star is approximately
given by the ideal-gas law where p ∝ ρT/µ, where µ is the average
molecular weight of the electrons and nuclei. Since µ ′ (r ′ ) = µ(r) by
assumption, the temperature is found to scale as T ′ (r ′ ) = y −1 T (r), and
the temperature gradient as dT ′ (r ′ )/dr ′ = y −2 dT (r)/dr.
The assumption that the star reacts to new particle emission by a
homologous contraction imposes restrictions on the constitutive relations
for the effective energy generation rate and the opacity. In particular,
for a chemically homogeneous star one needs to assume that
ϵ ∝ ρ n T ν and κ ∝ ρ s T p . (1.8)
For the opacity, Frieman et al. took the Kramers law with s = 1
and p = −3.5 which is found to be a reasonable interpolation formula
throughout most lower main-sequence interiors. Hence, the local
energy flux scales as
L ′ (r ′ ) = y −1/2 L(r) . (1.9)
The hydrogen-burning rate ϵ nuc also has the required form with n = 1,
and for the pp chain ν = 4−6; it dominates in the Sun and in stars
with lower mass. It is assumed that the new energy-loss rate ϵ x follows
the same proportionality; the standard neutrino losses ϵ ν are ignored
because they are small on the lower main sequence. If the star is not
in a phase of major structural readjustment one may also ignore ϵ grav
in Eq. (1.4) so that in Eq. (1.3)
ϵ = (1 − δ x ) ϵ nuc , (1.10)
where δ x < 1 is a number which depends on the interaction strength of
the new particles. From Eq. (1.3) one concludes that
leading to
L ′ (r ′ ) = y −(3+ν) (1 − δ x ) L(r), (1.11)
y = (1 − δ x ) 2/(2ν+5) . (1.12)
Assuming δ x ≪ 1, Frieman et al. then found for the fractional changes
of the stellar radius, luminosity, and interior temperature,
δR
R = −2δ x
2ν + 5 , δL
L = δ x
2ν + 5 , δT
T = 2δ x
2ν + 5 . (1.13)
Therefore, the star contracts, becomes hotter, and the surface photon
luminosity increases—it overcompensates for the new losses. Moreover,