A&A 442, 1041–1048 (2005) Astronomy DOI: 10.1051/0004-6361:20053390 & c ESO 2005 Astrophysics On constructing horizontal branch models A. Serenelli1,2 and A. Weiss1 1 Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, 85748 Garching, Germany e-mail: [email protected] 2 Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA Received 9 May 2005 / Accepted 28 July 2005 ABSTRACT We investigate different methods used to construct (zero-age) horizontal branch models and compare the resulting horizontal branch evolution with that of models resulting from the calculation of the complete stellar evolution from the main sequence and through the core helium flash. We show that the approximate methods may lead to small but discernible effects, but that some methods, which are as simple, reproduce the complete evolution very well. Key words. stars: evolution – stars: horizontal-branch – methods: numerical primarily by the composition (the “first parameter”) and a “sec- 1. Introduction ond parameter”, the nature of which is so far unknown, al- The horizontal branch (HB) in the Hertzsprung-Russell- though age is considered to be a likely candidate (see, for exam- Diagram (HRD) has a number of important applications. For ple Lee et al. 1994; Sweigart & Catelan 1998; Catelan 2000). example, it allows distance-independent age determinations One should add that here secondary age effects are considered, of globular clusters, when the difference between the time- because age itself is a primary parameter, due to the fact that dependent turn-off brightness and the almost time-independent the turn-off mass is already influencing the HB morphology be- one of the HB is used (e.g., Salaris & Weiss 1997). On the cause younger populations, i.e. more massive stars will become other hand, its brightness or that of RR Lyr stars (Feast 1997) more massive, therefore cooler, HB stars. populating it can be used to determine distances. The number The evolution to an HB star goes through the violent core of stars on the HB relative to the number of stars on the Red helium flash. For several decades, only very few calculations Giant Branch (RGB) above the HB is an indicator for the he- have been able to follow stellar models through this event. The lium content of the cluster (Buzzoni et al. 1983; Salaris et al. reason is that typical timesteps decrease to the level of seconds 2004). because of the rapid change in helium luminosity, which may 10 Physically, HB stars are in the core helium burning phase flare up to 10 L for a few days. Because other parts of the following the RGB. The ignition of helium burning takes place star remain completely unchanged during this time, the differ- under degenerate conditions and is a very energetic and par- encing schemes become numerically unstable and no longer > tially dynamic event. Stars of initial mass Mi ∼ 2.3 M ig- converge. Therefore, full flash calculations were successful in nite Helium under non-degenerate conditions and populate the most cases only for higher masses, where degeneracy of the equivalent of the HB, the so-called clump, which is observa- RGB stellar core is lower and the flash milder (see Thomas tionally a region of increased star density close to or on top of 1967, for a pioneering study). With a few exceptions, complete the RGB at the level of the HB. stellar evolution tracks into the HB phase were absent. The distribution of post-flash low-mass stars along a hor- For this reason most HB models were calculated by starting izontal sequence can be explained by a constant core mass, new sequences on the HB, where the initial structure is taken which is closely related to the luminosity of the star, and a from that at the tip of the giant branch, and solutions of the spread in envelope and thus total mass (Faulkner 1966; Dorman structure equations are sought where the effect of the helium 1992). This determines the effective temperature, but only flash is taken into account. This implies a reduced core density, weakly the luminosity. In a globular cluster, due to the fact that hydrostatic helium burning, a lower surface luminosity and an all stars have the same age, the initial mass of the stars now effective temperature depending on the envelope mass. Details populating the HB must have been the same and the spread of such approximate methods will be given later in the paper. along the HB must be the result of different mass loss histories. Overall, such starting models represent the true zero-age HB The detailed distribution of HB stars as function of Teff, (ZAHB) and the following evolution quite well, but there is the so-called HB morphology, appears to be determined no systematic investigation of the effects of the approximation. Article published by EDP Sciences and available at http://www.edpsciences.org/aa or http://dx.doi.org/10.1051/0004-6361:20053390 1042 A. Serenelli and A. Weiss: On constructing horizontal branch models The underlying assumption of such methods is that during the Table 1. List of models for which we did full evolutions through the helium flash no events take place that modify the internal chem- helium flash. The columns list metallicity, initial mass, the parameter η ical structure significantly. in Reimers’ mass loss formula, total mass Mf after the core helium Since a few years an increasing number of stellar evolu- flash, core mass Mc, and convective core mass Mcc at the ZAHB. tion codes have been improved numerically to a level where / η / / / full flash calculations have become possible. Still, the compu- ZMM Mf M Mc M Mcc M tational effort is large, such that grids of HB models derived 0.0001 0.80 0.0 0.7971 0.5011 0.131 from complete evolution sequences are not truely practicable. 0.3 0.6788 0.5014 0.134 However, it is now possible to compare full with approximate 0.5 0.5863 0.5015 0.133 HB models and to design schemes for the construction of HB 0.85 0.0 0.8471 0.4996 0.132 models which are both economical and accurate. Given the fact 0.5 0.6574 0.4999 0.131 that the requirements of accuracy of stellar models is steadily 0.7 0.5632 0.5001 0.132 0.003 0.82 0.0 0.8170 0.4892 0.121 increasing, either from observational data about stars, or – as in 0.2 0.6650 0.4895 0.121 the case of the cluster ages – because of more detailed questions 0.3 0.5734 0.4897 0.122 to be answered, we must consider the accuracy of HB models. 0.90 0.0 0.8970 0.4878 0.121 Recently, Piersanti et al. (2004) published a similar 0.3 0.6906 0.4879 0.120 and independent investigation testing their own method for 0.5 0.5085 0.4880 0.119 ZAHB model construction. They showed results for two cases, to which we will refer. In Sect. 2 we will discuss briefly our stellar evolution code obviously violate the basic assumption of the approximate and present full core helium flash calculations and the resulting models of an unmodified composition profile between the onset ZAHB models and HB evolution. In Sect. 3 we present sev- of the flash and the ZAHB. eral typical schemes to create approximate HB models without We note that convection is always treated in the most following the complete flash. Section 4 is devoted to critically simple way, i.e. using mixing length theory and employing analyzing results for approximate models by using results of the Schwarzschild criterion for convective stability. We thus Sect. 2 as a reference. We will not investigate methods that rely consider neither overshooting nor semiconvection. Both ef- on a completely new construction of models, since today they fects play a role during the HB evolution, but need addi- are of only historical interest (Rood 1970; Sweigart & Gross tional assumptions and introduce free parameters, which usu- 1976). All calculations have been done with the same code to ally are determined from comparison with observations. The avoid differences resulting from details in the numerical pro- inclusion would not change our results significantly, because grams. Section 5 provides a short summary. we are mainly interested in the quality of the constructed ZAHB models. 2. Complete evolution from the ZAMS to the HB 2.1. Code 2.2. Calculations . Throughout this paper we use the Garching stellar evolution We have evolved low-mass stars between 0 8and09 M,and = . code, which has been described in some detail in Weiss & compositions of Z 0 0001 and 0 003. The initial helium con- = . Schlattl (2000). The numerical aspects of this code have been tent was always Y 0 25. A list of cases is given in Table 1. ff improved such that the evolution of Asymptotic Giant Branch With these parameters we cover turn-o ages between 8 and (AGB) stars can be followed through many thermal pulses 12 Gyr years and the lower metallicity range of galactic glob- without convergence problems and intervention (Wagenhuber ular clusters. The evolution was followed from the zero-age & Weiss 1994). Due to the same improvements in the algo- main-sequence (ZAMS), starting from a homogeneous model rithms controlling the temporal and spatial discretization the up to the lower AGB, that is, prior to the onset of thermal helium core flash also became feasible (Wagenhuber 1996), be- pulses. For mass loss we used the standard Reimers formula cause the numerical problems encountered in both phases are (Reimers 1975), of a similar nature.