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 first explained for White Dwarfs by Michaud et~al. (1984). In the RHB stars on the other hand, with low surface gravity, the radiation field may levitate the heavy elements (see Cassisi et~al. 1997), so that their atmospheres may look richer in metals than the star really is. Furthermore, due to convection during the RGB phase He may be mixed into the surface layers of the star, possibly producing He-rich HB stars. For such He enriched HB star atmospheres, Sweigart (1997) speculates that the mixing in of He results in a bluer HB morphology (see also Sect. 9), possibly an enlarged RR Lyrae period shift, and lower surface gravities in blue HB stars.

According to model calculations, the absolute magnitude M_V of HB-like stars depends on [M/H] in the sense that metal poor stars are brighter than metal rich stars. This is caused by an intricate interplay between internal opacity and the luminosity of the H-burning shell (Dorman 1992) and has no detailed relation with the metal content in the photosphere (see further Sect. 9).

Summarizing, the determination of the abundance of the elements leads to knowledge about the composition of the stellar surface. For inhomogeneous envelopes these atmospheric abundances are not related with the overall [M/H] of the star. The stellar structure models must take these inhomogeneities into account, lest they predict brightnesses and colours which do not conform with reality. Vice-versa, interpretation of the observed colour and brightness with current models may lead to faulty atmospheric and structural parameters.

Are `red HB stars' really red HB stars?

The location in the CMD where red HB stars appear contains numerous other kinds of stars. Therefore, knowing if a red field star is a RHB star is not easy, because the appearance of a star with colour and magnitude like that of a red HB star is not sufficient proof.

The red part of the HB crosses the red giant branch (RGB). This is well known from the study of the redder (less metal poor) globular clusters, where in some cases the RHB stars group together into the `red clump'. For the field stars the above must be true, too.

It has become clear from models that the evolution on the RG branch has temporal irregularities (depending on which parameter one considers). This is related to the passage of the H-burning shell through the chemical discontinuity left by the convective envelope during the first dredge up phase (see e.g. Cassisi et~al. 1997). It results in a so called RGB bump, a location where the RGB is (relatively) overpopulated (at M_V =~+0.5 mag).

Due to the above coincidence, the RHB and the RGB bump appear to lie almost at the same L and T (depending on the age and the metallicity of the star group) and thus at virtually the same position in the CMD. This implies that it is for field stars almost impossible to discriminate between a RG star and a RHB star.

A further problem is that blue loop stars (more massive stars which land in a more luminous core He burning phase) appear in colour magnitude diagrams very close to where the HB is. At the reddest and faintest point, these stars lie at M_V = -0.5 mag.

The mentioned evolutionary states lead in complicated ways to an enhancement of stars in the CMD, as nicely illustrated by Gallart (1998). A summary of evolutionary details is given by Girardi et~al. (1998). Clearly, stars in the RHB domain of a colour magnitude diagram for field stars cannot simply be taken to be RHB stars indeed.

Gaps and other structure on the HB

Newell (1970) presented the first evidence for an irregular distribution of stars along the horizontal branch. In fact, he showed that there is evidence for two `gaps', regions on the HB with reduced numbers (or even devoid) of stars. This has been corroborated since then many times in newer and more extensive data sets, such as those collected for the sdB studies in Bonn (Aguilar Sánchez 1998). The gaps are also present in the T_eff versus logg diagram. Globular cluster HB's show gaps as well. A recent detailed study is that by Ferraro et~al. (1998). The cause for the gaps is largely unclear and several possibilities are being investigated.

One possibility is that, in globular clusters, HB stars are present with two different mass ranges. Catelan et~al. (1998) have simulated globular cluster HB distributions using both unimodal and bimodal mass distributions. The gaps do seem to be present but not with great significance. Even in the unimodal mass distribution sparse regions can occur.

Another possibility for the occurrence of gaps is that the HB stars behave such that certain ranges of T_eff (or of B-V) are not present. I speculate that a gap may result from a small discontinuity in the burning in the hydrogen shell. If the energy production in a marginally burning shell is temporarily decreased (e.g. by stochastic fluctuations) the ensuing drop in temperature may be sufficient for the burning not to come up to the original level any more, and the hydrogen burning may extinguish altogether. The atmosphere then would become more compact and the colour of the star more blue than before. Thus at that HB location a gap in the smooth distribution along the HB may be created.

A further possibility for the gaps to emerge is when the HB is populated by stars having different evolutionary origins. If the genesis of HB stars follows different routes, such as one kind directly from the RGB, the other through some binary star evolution (see e.g. Iben & Tutukov 1985), the two routes may lead to different final masses and possibly to preferred locations on the HB.

HB stars do seem to deviate in colour from expected values in certain cases, too. Grundahl et~al. (1998) noted from Strömgren photometry that the shape of the distribution of the stars on the HB in the u-y colour in the globular clusters M 13, NGC 288 and 6752, deviates from that of theoretical models. Either the stars are brighter than expected over a part of the blue HB, or they are bluer than expected. They speculate that there are two populations even for HB stars within a globular cluster.

New HST data on a few metal rich globular clusters have shown that these clusters do have a population of blue HB stars, in contrast to established beliefs for metal rich clusters (Rich et~al. 1997). Not only that, the HB's appear to become brighter toward the blue, in stark contrast to what has thusfar been found.

Summarising, the shape of the observed HBs has large variations which the standard models cannot explain. What consequences have the gaps, the in-cluster differences, and the different possibe origins of the HB stars for our understanding of field HB stars?

HB star mass and absolute magnitudes

Models indicate that the mass of HB stars runs from 0.50 M_{sun} at the blue end to =~1.0 M_{sun} at the red end of the HB. The luminosity runs from logL = 1.2 to 1.7 L_{sun}, respectively. Both mass and luminosity can only be found for stars with known distance.

In spectroscopic studies, which result in T_eff, logg, and the apparent luminosity l, masses of the HB stars in globular clusters (distance known) have been determined. It was found that the HBB stars in NGC 6397 (de Boer et~al. 1995) and the sdB stars in M 15 (Moehler et~al. 1995) have a mass of =~0.4 M_{sun}, much lower than the canonical value of 0.5 to 0.6 M_{sun} for such stars. Both NGC 6397 and M 15 have the very low metallicity ([M/H] =~-2 dex).

Masses for RR Lyrae stars can be calculated from the relations of period, luminosity and mass. Here the luminosity is based on the distance from Baade-Wesselink methods. Carney et~al. (1992) found masses as low as 0.45 M_{sun} for low metallicity ([M/H] =~-1.8 dex) field RR Lyr stars, but M =~0.55 M_{sun} for larger [M/H], still all below the theoretical expectation.

In a first study using Hipparcos data, de Boer et~al. (1997a) tried to resolve the T_eff, \ls 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.