Horizontal-branch stars play an important rôle in defining the absolute distance scale in the universe. For that one has to know M_V at relevant B-V with well defined [M/H] for each type of horizontal-branch star. The nature of the horizontal-branch stars is reviewed with emphasis on all the open questions regarding evolution and structure, questions which show that these stars are not the easiest objects in the distance scale discussions.

Using the HB stars with the best Hipparcos parallaxes, the HB at (B-V)₀= 0.20 and [Fe/H] = -1.5 has M_V = +0.71. Assuming the [Fe/H] to M_V relation for RR Lyrae stars to be valid for HB stars too, one finds M_V =~1.00 for [Fe/H] = 0.

The stars

Stars of the horizontal branch (HB) are post red giant (RG) stars which burn helium in their core. The mass of the He in the core is thought to be =~0.5 M_{sun}. Along the HB in the colour-magnitude diagram (CMD) one recognizes (from hot to cool) the types sdOB, sdB, HBB, HBA, RR Lyr, and RHB (for definitions see nomenclature review by de Boer et~al. 1998). Depending on the initial parameters of the progenitor main sequence star (initial mass, metal content, etc.) as well as on aspects of the RG mass loss phase, the star will retain a hydrogen shell of a certain mass. If sufficient hydrogen remained there is also a hydrogen burning shell.

The HB stars have a low to normal atmospheric metal content. Naturally, the colour of the star is determined by the atmospheric structure (T_eff, logg, [M/H]). HB stars show no or only very slow rotation (Peterson 1983).

A large variety of models for HB stars have been calculated (e.g., Sweigart & Gross 1976; Sweigart 1987; Dorman 1992) all indicating that total mass is positively correlated with luminosity and redness. At the very blue end of the HB the mass of the stars is =~0.5 M_{sun} (vanishingly thin H shell, no shell burning), at the red end it may amount to =~1.0 M_{sun} or more (thick shell and well established H shell burning).

In metal poor globular clusters the HB is populated at the blue end, while in more metal rich globular clusters the HB stars group together in a small colour range at the red end, there forming a `red clump' seemingly associated with the red giant branch. However, this picture is rather simple and observations show that further parameters are required to also explain those globular clusters deviating from that scheme.

Evolutionary history and the HB stars as a group

What is the evolutionary history of stars now being in the HB phase? With which mass did these stars start on the main sequence? Models indicate that all stars initially having 0.8 <=M <=3 M_{sun} do become HB-like stars. This range immediately implies that the HB stars intrinsically span a wide range in age and thus probably also in metal content. A star starting with 3 M_{sun} will evolve into an HB star within =~10⁹ y. It must thus be regarded as `young' and have formed in the disk from material with most likely `solar' composition. Stars starting with 0.8 M_{sun} will need >=10¹⁰ y to become HB like, can thus be regarded as `old' objects and therefore must have formed from material substantial poorer in metals than the Sun.

Since it is virtually impossible to determine the age of an individual field star with any certainty, one cannot discriminate a young from an old field HB star, except perhaps when assuming some age-metallicity relation. We must acknowledge that the field HB stars we have access to are a mix of young and old stars. With a more or less constant star formation rate in the Milky Way the HB star group forms a continuum in age but the older ones are likely to dominate in number.

HB star metal content and its consequences

The determination of the metal content in the atmospheres of stars is with present day technology in principle not very difficult. Spectroscopic investigations show that the sdB stars have near normal Si with somewhat low He (Heber et~al. 1984). The HBA stars, among which are the long known classical field HB stars, range in metal content from solar to [M/H] =~-2.0 using photometric methods (e.g. Gray et~al. 1996), while spectroscopic determinations (naturally for just a few stars) indicate generally low metal abundances, such as small [Fe/H], in part as low as in metal poor globular cluster giants (Kodaira & Philip 1984; Adelman & Philip 1994). RR Lyr stars are known to spread over a large range in metal content (see e.g. Lambert et~al. 1996, Layden 1994) with a predominance of metal poor ones. The RHB stars have not been studied intensively yet.

Many metal abundance values are based on the measurement of some line index, like the Ca index, or are based on the {Delta}S method, giving a metallicity index to be denoted with [M/H]. The index is then calibrated with the help of spectroscopic [Fe/H] values (Lambert et~al. 1996, Layden 1994). Such studies lead, for RR Lyrae stars, to a relation of M_V versus [Fe/H] in which M_V is taken from Baade-Wesselink methods (see Fernley 1994). However, one has to acknowledge that such metallicity indices are often not very accurate. When metallicities are given   Only when Fe lines are seen and analysed indeed, the use of [Fe/H] is justified one always should trace their origin (are they photometric or spectroscopic   Also spectroscopic abundances may have substantial uncertainties; when the T_eff choosen for the analysis is different from reality, the line excitation model is off too, leading to errors in metal abundances easily of the order of 0.1 dex values?) and one should be wary of too many decimals which often come only from a numerical transformation using rather uncertain input values.

Also for globular clusters the photometric indices are problematic. This is of relevance here, since spectroscopic abundance determinations are available only for a limited number of clusters. Also the Q₃₉-index gives ambiguous results (de Boer 1988). This can be seen in Fig. 12 of Zinn & West (1984), in which the discrepancy between the metallicity from the Q₃₉-index and that from spectroscopy is seen to range from {Delta}[Fe/H] of -0.3 to +0.3, depending on the metallicity and type of cluster. The discrepancies for clusters with blue HBs show a clear trend. The reason is, of course, that the photometry uses the entire cluster in which the various star types (red giants, main-sequence stars, red and blue HB stars) contribute according to their brightness and their number in the cluster (in particular a blue or a red HB), thus mixing stars of widely differing temperature and gravity into one photometric index measurement. The widely used Zinn & West (1984) metallicity scale clearly is of limited accuracy (de Boer 1988).

Can one use the abundance values to estimate the age of each star? Unfortunately, these abundances are not necessarily giving the intrinsic metal content of the stars. HB-like stars have a rather stable atmosphere. This means that in particular in conditions of high surface gravity, such as in sdOB and sdB stars, the heavy elements can diffuse downward and sink out of the atmosphere so that the star looks more metal poor than it is intrinsically. The effect was fog g and mass problem of the HB stars. Using the 8 best Hipparcos observed field HBA and sdB stars de Boer et~al. (1997a) showed that they apparently have the low mass of =~0.4 M_{sun} as well. The similarity of the mass of the cluster HB stars and the field HB stars can be seen in Fig. 2 of de Boer et~al. (1997b).

It has been claimed that there is a difference in M_V between cluster RR Lyr stars and field RR Lyr stars (see Gratton 1998). Catelan (1998) doubts that such a difference exists and speculates that uncertainties in the basic stellar parameters and in the methods to derive them lie at the base of such claims.

The absolute magnitude of the blue HB stars depends in a very sensitive way on the luminosity and the surface temperature. When going to hotter stars, M_V will be fainter dramatically, leading to `drooping' HBs. Thus the blueest part of the HB is not useful to calibrate M_V of the HB.

Kinematics indicates two populations of HB stars

Using spectroscopic distances, radial velocities, and proper motions, de Boer et~al. (1997c) showed that the sdB stars have, by and large, disk like orbits. As a sample the stars rotate along with the disk rather well with an asymmetric drift of only -36 km~s^-1. The scale height of the sdB stars is of the order of 300 pc (Aguilar Sanchez 1998).

Altmann et~al. (1998) investigated the kinematics of HBA stars. The orbits of several of these extend to large z. The asymmetric drift of their sample is nearly -200 km~s^-1, indicating these stars are rather on halo like orbits.

RR Lyrae stars as a group show the wide range of disk like to halo like orbits. The velocity in the direction of galactic rotation can be correlated with metallicity, giving the trend that metal rich RR Lyrae partake in the rotation of the disk, the metal poor stars (the majority) do not (Layden et~al. 1996).

The existence of differences in kinematic behaviour of RR Lyr and HBA stars on one hand and that of sdB stars on the other hand may indicate that the field HB star population differs significantly from the globular clusters. So is it allowed to assume that the field HB stars and the cluster HB stars are identical? Would it be possible to take the kinematic behaviour as an indication for [M/H] in the stars, and thus as indication for what M_V ought to be?

Brightness bias in data samples due to evolution and metallicity

HB stars become brighter when evolving from the zero-age HB (ZAHB). Slowly the luminosity gets larger while the surface temperature stays almost the same. This means that HB stars cover a range in absolute magnitude. Inspection of the evolutionary tracks of Dorman (1992) shows the rise in M_V amounts to about 0.07 mag. Thus, when studying a sample of field HB stars one is not dealing with a sample of the same M_V for a given B-V. A nice example of the significance of this aspect can be found in the ultraviolet CMD and the derived L, T_eff diagram for M 13 (Parise et~al. 1998).

The metal content of the stars has an effect on M_V. Lambert et~al. (1996), basing themselves on numerous previous studies, arrive at a metal dependence of M_V = 0.93 + 0.17 ×[Fe/H] for RR Lyr stars (in which [Fe/H] really is only [M/H]). Thus M_V may vary (for -2< [Fe/H] <0) over =~0.34 mag, also for field HB stars.

The sum of the effects of evolution and of metallicity (assuming that the metallicity relation applies also to the star types adjacent to the RR Lyrae) is that the metal poor evolved HB stars are the brightest, brighter by 0.4 mag than the metal rich ZAHB stars. The latter may thus be underrepresented in samples.

What do models tells us?

The end of the RGB phase is marked by the ignition of He in the core of the star, leading to the transformation of the star into a ZAHB star. The He mass is then thought to be =~0.5 M_{sun}. The reddest HB stars have retained a rather heavy hydrogen shell and thus a non-negligible amount of hydrogen shell burning.

Depending on the overall metallicity the H-burning contributes more or less to the overall luminosity. Here the details of CNO to Fe variations lead to considerable variations in luminosity (Dorman 1992). At T_eff = 10^4 K, e.g., one finds (see his Fig. 7) a 0.55 M_{sun} star of [M/H]=-0.47 dex to have logL = 1.48 while for a 0.68 M_{sun} star existing at the same temperature with [M/H]=-2.26 dex the luminosity is logL = 1.58.

The core luminosity itself depends on the total mass of the star, as well as on the He-core mass. Rotation of the RGB star may lead to higher core masses (Mengel & Gross 1976) with later a larger HB star core luminosity. However, this does not directly translate into a larger overall luminosity since the consequent larger mass loss at the RGB tip will lead to a lower H-shell luminosity. Modelling the effect of a larger He core mass (Caloi et~al. 1997; Sweigart & Catelan 1998) one finds for metal poor clusters brighter and bluer ends of the HB.

Diffused He will alter the structure of the HB stars and the shape of the horizontal branch. Metal rich RHB stars with enhanced He will evolve away from the ZAHB into very extended blue loops, thereby producing a HB sloping up to brighter stars toward the blue (Sweigart & Catelan 1998). Here the HB may be as bright as M_V = 0 mag.

At the cooler end (logT_eff = 3.85) Cassisi et~al. (1997) investigated the brightness of the RHB and the RGB bump. The absolute magnitude of the ZAHB varies from M_V = 0.51 mag for [M/H]=-2.04 to M_V = 0.86 mag for [M/H]=-0.57. The M_V of the RGB bump is then -0.20 and 1.12 mag, respectively. It shows that the brightness difference between ZAHB and RGB bump changes over more than 1 mag in this metallicity range (Cassisi et~al. 1997). Such models make clear that, due to the possible confusion of RHB stars with clump stars, candidate field RHB stars are utterly useless to calibrate the HB.

Field Horizontal Branch stars and RR Lyr

Name    V    B-V  AV  [Fe/H]  {pi}  
   {sigma}{pi}  d  MV ± {Delta}MV 
6-7 
            (mas)  (pc)

HD   86986        7.99    +0.12 
         0.09  -1.9  3.78  0.95  265  +0.79 ± 0.55
HD 109995       7.62    +0.04 
         0.00  -1.8  4.92  0.89  205  +1.08 ± 0.40
HD 130095       8.13    +0.08 
         0.31  -2.0  5.91  1.08  170  +1.68 ± 0.40
HD 139961       8.85    +0.10 
         0.31       4.50  1.19  220  +1.81 ± 0.60
HD 161817       6.96    +0.16 
         0.06  -1.6  5.81  0.65  170  +0.72 ± 0.25
RR Lyr          7.66    +0.25  
         0.09  -1.4  4.38  0.59  228  +0.78 ± 0.29

HB star data from de Boer et~al. (1997a), RR Lyr from Fernley et~al. (1998)
{Delta}M_V

given is due to the uncertainty in the parallax only

What does Hipparcos tell us about the HB?

With Hipparcos about a dozen blue HB stars were measured of which only few have parallaxes accurate enough to be used to determine M_V for the individual stars. These are essentially only the long known prototype HBA stars whose M_V were presented by de Boer et~al. (1997a). Their data ({pi}/{sigma}_{pi} \lower4pt~>4) are given in Table 1.

Several stars with colour and brightness like RHB stars are present in the Hipparcos data base. Given the risks of confusion with stars in different states of evolution (see Sect. 5) we will ignore these objects.

Stars of the blue HB, the red HB and RR Lyrae type have been used in an analysis by Gratton (1998). Many of these stars have less accurate parallaxes and thus deteriorate the quality of the ultimate averaged M_V. He assumes the HB should have a shape like that of M 5 (does M 5 have the same history as the field stars?) and looks at the brightness differences between the field and the cluster HB shape. Then, his error weighting procedure is flawed (Popowski & Gould 1998), and the analysis should have resulted in an absolute magnitude =~0.1 mag fainter.

A large number of RR Lyr type stars have been observed with Hipparcos. Only RR Lyr has a parallax large enough ({pi}/{sigma}_{pi} \lower4pt~>4) to be included in Table 1. The absolute magnitude for the RR Lyraes as a group is not based on parallaxes but on the detected proper motions together with a model for their kinematics (Fernley et~al. 1998). Since RR Lyrae stars have a range of metallicity and other intrinsic parameters, it is not proper to treat them as a single kinematic group. Thus, only RR Lyr is of use.

Clearly, working directly with the best parallaxes (Table 1) provides the least ambiguous result. Plotting these best M_V values one can see the HB for the field stars (Fig. 1).

An extrapolation to the blue edge of the RR Lyr strip, taking the plotted data at face value, gives M_V =~0.75 mag (Seggewiss 1998).

Assuming that the [M/H] values for these protoype HBA's are accurate, the M_V's can be translated (using the relation found for RR Lyr stars, see Sect. 8) into the solar metallicity HB. The observed stars so indicate that for [M/H]=0 and (B-V)₀= 0.20 the M_V =~1.00 mag, and for [M/H]=-2 (at (B-V)₀= 0.20) that M_V =~0.65 mag (see Fig. 1).

Consequences for studies of field HB stars

From all of the above one must conclude that the assumption to be able to find one ultimate M_V for HB-like stars is most likely wrong.