Stars as Laboratories for Fundamental Physics - MPP Theory Group

62 Chapter 2
fast. Therefore, the stars along the red-giant branch (RGB), horizontal
branch (HB), and asymptotic giant branch (AGB) in a globular
cluster have almost identical initial masses whence for these phases a
single-star track is practically identical with an isochrone. On the other
hand, the TO region requires the construction of detailed theoretical
isochrones to compare theory and observations and thus to determine
the ages of globular clusters.
In order to associate a certain stellar mass with the TO in a cluster
one needs to know the absolute brightness of the stars at the TO, i.e.
one needs to know the precise distance. All else being equal, the inferred
age varies with the TO luminosity as ∂ log(age)/∂ log L TO = −0.85 or
∂ log(age)/∂V TO = 0.34 (Iben and Renzini 1984). Therefore, a 0.1 mag
error in V TO leads to an 8% uncertainty in the inferred cluster age.
Put another way, because L ∝ (distance) 2 a 10% uncertainty in cluster
distances leads to an 18% age uncertainty. This is the main problem
with the age determination of globular clusters.
A particularly useful method to measure the distance is to use
RR Lyrae stars as standard candles. As discussed in Sect. 2.1, their
luminosity is determined almost entirely by their core mass (apart from
a dependence on chemical composition), which in turn is fixed by helium
ignition on the RGB which, again, depends only on the chemical
composition and not on the red-giant envelope mass. Therefore, the
brightness of the HB is nearly independent of stellar mass. Consequently,
the brightness difference ∆VHB
TO between the HB and the TO
is a distance-independent measure of the TO mass and thus of the cluster
age. Moreover, because the color of RR Lyrae stars coincides with
that of the TO region it is not necessary to convert from the measured
brightness with a certain filter (e.g. visual brightness V ) to a bolometric
brightness, i.e. there is no need for a bolometric correction (BC).
Also, RR Lyrae stars are bright and easily identified because of their
pulsations. Therefore, ∆VHB
TO is one of the most important observables
in the color-magnitude diagram of globular clusters (Iben and Renzini
1984; Sandage 1986).
As an example the recent ∆VHB
TO
determinations of Buonanno, Corsi,
and Fusi Pecci (1989) in 19 globular cluster are shown in Fig. 2.17 as
a function of metallicity; the logarithmic metallicity measure [Fe/H] is
defined in Eq. (2.15). The best linear fit is
∆V TO
HB = (3.54 ± 0.13) − (0.008 ± 0.078) [Fe/H], (2.13)
so that the HB brightness varies with metallicity almost exactly as the
TO brightness. The measured points are in agreement with a Gaussian