Stars as Laboratories for Fundamental Physics - MPP Theory Group

552 Chapter 15
The main impact of the assumed G N variation is a change in the
time it takes for a star to burn out hydrogen at its center and thus to
leave the main sequence (MS). The subsequent fast evolution (ascending
the RGB, HB evolution, etc.) is determined by the present-day
value of G N . The stellar evolutionary tracks in the color-magnitude diagram
can look significantly different from the standard ones if the
G N variation was sufficiently severe. However, within the range of
possibilities left open by the above ĠN/G N bounds, the present-day
isochrone cannot be observationally distinguished from the standard
case (Degl’Innocenti et al. 1995). Apparently, then, the only significant
consequence of a time-varying gravitational constant is that the
true age τ of a globular cluster is different from its apparent age τ ∗
which is inferred from its color-magnitude diagram in the framework of
a constant-gravity scenario.
The change of the MS lifetime can be estimated by Teller’s (1948)
homology relation L ∝ G γ N. Based on a specific assumption for the
opacity variation with temperature and density Teller found γ = 7
while a more appropriate value for low-metallicity globular-cluster stars
is γ = 5.6 (Degl’Innocenti et al. 1995). Either way, one can easily
show that the true (τ) and apparent age (τ ∗ ) at the MS turnoff are
approximately related by
τ ∗ =
∫ t0
t 0 −τ
dt [ G N (t)/G N (t 0 ) ] γ
(15.3)
(Prather 1976; Degl’Innocenti et al. 1995). For all practical purposes
this analytic result can be considered to be exact because it agrees with
numerical calculations surprisingly well.
Unless one wishes to probe the very early universe it is fairly generic
to assume a linear G N variation of the form
G N (t) = [ 1 + Γ 0 (t − t 0 ) ] G N (t 0 ), (15.4)
where Γ 0 = ĠN(t 0 )/G N (t 0 ) is the present-day rate of change of Newton’s
constant. Then one finds explicitly
τ
τ ∗
=
γ 1 Γ 0 τ
1 − (1 − Γ 0 τ) γ 1 = 1 − (1 − γ 1Γ 0 τ ∗ ) 1/γ 1
Γ 0 τ ∗
, (15.5)
where γ 1 ≡ γ + 1. Given a present-day rate of change Γ 0 one can
thus determine the modification of the globular-cluster age if a certain
apparent age or a certain true age is assumed.
The observed color-magnitude diagrams of globular clusters yield
apparent ages τ ∗ in the range 14 to 18 Gyr. With these values one