The horizontal-branch stars in globular clusters. I. The period-shift effect, the luminosity of the horizontal branch, and the age-metallicity relation

Synthetic models of the horizontal branches in globular clusters have been constructed from a new grid of standard horizontal-branch (HB) evolutionary tracks. These models have been used to investigate the period shifts at constant T_eff between the RR Lyrae variables in globular clusters of different metallicities and the variation in HB luminosity with [Fe/H]. We find that two effects are responsible for the disagreement between the HB luminosity-[Fe/H] relationship found observationally by Sandage in his investigation of the Oosterhoff effect and that given by calculations of zero-age horizontal-branch (ZAHB) stars. One is that the evolution away from the ZAHB will play a role in the Oosterhoff group II clusters by increasing the mean luminosity and lowering the mean mass of the stars in the instability strip over the values predicted by ZAHB models. This leads to longer periods at a given T_eff in the group II clusters than in the group I clusters, as indicated by observations. The second is the realization that Sandage's assumption that light-curve shape (rise time) and amplitude are unique functions of T_eff is at odds with the observations of the RR Lyrae variables in the globular cluster ω Cen and of a sample of field variables, which show that both quantities depend on [Fe/H] as well as T_eff. When these two effects are taken into account, the observed relationship between period shift and [Fe/H] matches the theoretical model calculations to within the errors. Consequently, there is no reason to hypothesize the existence of a sizable anticorrelation between the abundances of helium and metals or that standard models of HB stars are substantially in error, as had been suggested previously. The synthetic HB calculations suggest that the observed differences in the mean periods of the ab variables and the fraction of c-type variables between the two Oosterhoff groups are due to a difference in mean luminosity of the ab variables of ∼0.18 in M_bol and to the uneven distribution of variables across the instability strip in the group II clusters. Our models for nine clusters predict that for a primordial helium abundance of 0.23, the luminosity of the HB varies as M_V^RR ≈0.17[Fe/H] + 0.82. This relationship and a similar one for Y = 0.20 are compared with other determinations in the literature. When combined with the result of Buonanno, Corsi, and Fusi Pecci (1989) that in the mean there is no variation in the difference in V-magnitude between the HB and the mainsequence turnoff (TO), specifically ΔV(TO - HB) = 3.59, these relationships yield a significant age-metallicity relationship for halo globular clusters: 17.0 Gyr at [Fe/H] = -2.3 and 12.9 Gyr at [Fe/H] = -0.8, for Y = 0.23.