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The HR Diagram and the Galactic distance scale after Hipparcos
I. Neill Reid
Palomar Observatory, 105í24, California Institute of Technology, Pasadena, CA 91125, eímail:
inr@astro.caltech.edu
ABSTRACT
The completion and publication of the Hipparcos astrometric catalogue has
revitalised studies in many fundamental areas of Galactic structure and stellar
evolution. This article reviews the impact of the new parallax results on our
understanding of the location of the mainísequence as a function of abundance, of
the luminosity calibration of primary distance indicators and of the Galactic distance
scale. Many of these issues remain to be resolved.
Subject headings: Stars: subdwarfs, parallaxes, RR Lyraes; Globular clusters:
distances; Galactic structure

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1. Introduction
Calibration of the extragalactic distance scale rests on our ability to determine accurate
locations in the observational HertzsprungíRussell diagram for specific types of stars at particular
stages in their evolution. Whether one uses Cepheids, RR Lyraes, planetary nebulae or type
I supernovae as links in the distance chain, the underlying assumptions are that in matching
similar types of stars in different stellar systems, one is comparing like with like; and that if the
calibrators are not identical, systemic variations can be traced to systematic trends with changes
in physical characteristics. Given that those criteria are satisfied, in principle all distanceíscale
investigations can determine relative distances between different external galaxies. However,
determining absolute distances requires absolute luminosities for individual calibrators. Thus, the
accuracy with which one can derive parameters such as the Hubble constant, H 0 , depends on the
accuracy with which one can measure distances within the Galaxy so as to calibrate the various
distance indicators.
Trigonometric parallax measurements offer the only method of directly measuring distances
to almost all single stars or muliple star systems. The application of CCD detectors to astrometry
(Monet, 1988) has shown that subímilliarcsecond (mas) precision can be achieved from the
ground. However, the need for adequate reference stars within the small field of view allowed by
current CCDs limits observations to faint apparent magnitudes (VÖ 14), and hence to nearby
stars of low intrinsic luminosities. Moreover, the accuracy of the final parallax is dependent on the
transformation from relative to absolute reference frames.
The ESA Hipparcos satellite was designed to address this issue. Full details of the scope
of this mission and the subsequent analysis are given by van Leeuwen (1997) and Kovalevsky
(1998). Two 13ícm diameter optical telescopes were used to image 1 squareídegree regions of the
sky, separated by 58 o , onto the same focal plane. A photodiode timed stellar transits across a
reference grid as the satellite rotated on its axis. Groundíbased parallax observations are limited
to small angular fields of view, and reference stars share the same parallactic motion as the target
í hence the necessity for correction from relative to absolute systems. Hipparcos measured angular
separations of stars with significantly different parallax factors, allowing direct determination of
absolute parallaxes in the final astrometric solution.
Hipparcos was a targeted mission, rather than a sky survey, with programme objects limited
generally to surface densities of eight per square degree. Photonícounting statistics rendered
impracticable observations of sources fainter than 13th magnitude. The final catalogue includes
positions, proper motions and parallax measurements, as well as BV photometry, for 118,000
stars, including nearly every star brighter than 7th visual magnitude, but for only 45,000 stars
with 9 ! V ! 11 and 4,000 stars between 11th and 13th magnitude.
These data have formed the basis for a large number of investigations over the last two years
(160 refereed publications as of May, 1998). The present article does not aim at a comprehensive
review of all issues raised in those papers. Rather, our intention is to consider the impact of the

-- 3 --
Hipparcos results on investigations of the distance scale. This leads us to focus on two issues:
the location of the mainísequence as a function of age and chemical abundance, based partly
on observations of nearby stars and partly on distance estimates to clusters; and direct distance
measurements of primary distance indicators. We consider how these reícalibrations affect distance
estimates to the Magellanic Clouds and M31, but carry extragalactic distance scale arguments no
further.
2. Astrometric accuracy and systematic biases
2.1. Measurement precision
The formal precision of the astrometry for an individual star depends primarily on the number
of observations and the signalítoínoise of each observation. Scans were made with the satellite
rotational axis precessing at 6.4 revolutions yr \Gamma1 at an angle of 43 o to the direction to the Sun.
As a result, stars at ecliptic latitudes fi ? 47 o (particularly those at fi ¦ 47 o ) were observed more
frequently than those at lower latitudes, with a corresponding ¦ 50% increase in accuracy (ESA,
1997). Figure 1 illustrates the typical distribution of formal uncertainties (oe ï(ff) ; oe ï(ffi) ; oe ‹ ) as a
function of apparent magnitude.
Data reduction was undertaken independently by two consortia, NDAC and FAST, with the
final catalogue consisting of the merged astrometric parameters. Both undertook extensive tests
and comparisons to verify the accuracy of the satellite astrometry, particularly the reliability of
the absolute zeropoints in ï and ‹. Those tests are discussed by Arenou et al (1995), Lindegren et
al (1995) and Lindegren (1995) for the initial 30ímonth analysis, and in the first and third volumes
of the Hipparcos catalogue (ESA, 1997). In brief, the positional data show clear evidence for
distortions in the FK5 groundíbased frame at the 60í100 milliarcsecond level, while a comparison
of the proper motions against groundíbased data, which have longer time baselines and comparable
accuracy, confirms random uncertainties of ffl H (ï) ¦ 1 \Gamma 2 mas. Similarly, comparisons of the
Hipparcos parallax data against the most accurate available groundíbased measurements (by
the US Naval Observatory) generally confirm the 1í2 mas quoted uncertainties of individual
observations, as does Lindegren's (1995) analysis of the distribution of negative parallaxes.
Determining the reliability of the absolute zeropoints, notably in parallax, is a more complex
issue. Coíordinates are on the J2000 system (epoch 1991.25), referenced to the International
Celestial Reference System (ICRS) via secondary standards (no extragalactic reference sources
were observable by Hipparcos). The zeropoint for the properímotion system is tied to the
extragalactic frame through reference sources with absolute motions, either directly from VLBI
or indirectly, from photographic surveys (see Johnson, 1999, this volume). The zeropoint of the
Hipparcos parallaxes, ‹H , was tested by matching against independent parallax estimators (e.g.
photometric parallax), primarily for distant stars where oe ‹ (other) !! oe ‹ (H). In particular,
Hipparcos data for 46 Magellanic Cloud stars (‹ ‹ 0:02 mas) give ï
‹ = \Gamma0:16 \Sigma 0:26 mas and

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oe ‹ = 1:72 \Sigma 0:18 mas. Arenou et al conclude that the overall results are consistent with a global
zeropoint error of ! 0:1 mas in ‹H . However, this does not exclude larger deviations on smaller
angular scales, as discussed further below.
These tests of the parallax zeropoint rest primarily on relatively faint stars, since few
stars brighter than 7th magnitude have highíaccuracy groundíbased parallaxes. Harris et al
(1997) and Gatewood et al (1998) present comparisons based on 23 and 63 stars respectively. A
weighted mean of their results (Hanson, priv. comm.) gives ‹H \Gamma ‹ other = 0:30 \Sigma 0:24mas, where
the uncertainty is the standard error of the mean. While this is consistent with the absolute
zeropoint defined at fainter magnitudes, they fail to provide a stringent test of the presence of any
magnitudeídependent systematics.
2.2. Systematic biases in parallax determination
LutzíKelker bias
The measured parallax provides the best (trigonometric) estimate of the distance to a
particular star system. If, however, one considers that same object in a statistical sense í as a
representative of the disk main sequence, for example í then one must allow for possible systematics
introduced by the sampleíselection criteria adopted. This potential for bias in statistical analysis
of parallaxíselected datasets was noted early this century by Russell and is also discussed by
Trumpler & Weaver (1953). Lutz & Kelker (1973) provide the first quantitative description of this
effect.
Under the standard assumption that the measured parallax, ‹ o , is an unbiased estimate of
the true parallax, ‹, the conditional distribution C o (‹ o j‹) has a mean value ‹. However, we are
aiming to determine ‹ given ‹ o , and are therefore concerned with the conditional distribution
C(‹j‹ o ). In that case,
C(‹j‹ o ) ¦ C o (‹ o j‹) \Lambda ND (‹) (1)
where ND (‹) is the true parallax distribution of stars in dataset D. Note that ND (‹) is often not
identical with the parallax distribution of the parent Galactic stellar population, since D is usually
preíselected based on a combination of magnitude, color and proper motion. The dependence on
ND (‹) leads to systematic bias in the mean parallax if one selected a subísample, S, from dataset
D on the basis of ‹ o .
LutzíKelker corrections
\DeltaM LK = hM true \Gamma M o i = h5log
‹
‹ o
i (2)
provide a means of correcting this bias in the absolute magnitude plane. The bias is not present
if there is no selection based on ‹ o , as in the methods of mean parallaxes (e.g. averaging open
cluster parallaxes) or reduced parallaxes (cf. Feast & Catchpole's (1997) Cepheid analysis and

-- 5 --
Feast (2000)). The LK bias, however, is always present when one averages absolute magnitudes,
since the latter cannot be calculated if ‹ o is negative.
Lutz & Kelker computed \DeltaM LK for the specific case of a uniform distribution, ND (‹) / ‹ \Gamma4 ,
and are limited to oe ‹
‹ ! 0:175. Hanson (1979) showed that common observational selection effects
(magnitude limits, proper motion limits) lead to less steep powerílaw density distributions, and
used series expansion to derive general solutions for ND (‹) / ‹ \Gamman . This results in both smaller
LK corrections, and reliable corrections for higher values of oe ‹
‹ (see also Koen, 1992).
With an apparent magnitude limit of V ¦ 12, the Hipparcos dataset is magnitudeílimited for
stars with M true ? 5:0. In that case, the powerílaw exponent n of the density distribution and the
resultant LutzíKelker bias are reduced significantly (Lutz, 1983), since imposing a magnitude limit
modifies ND (‹) without direct reference to ‹ o . The situation is analagous to observations toward
an opaque molecular cloud at ‹MC : no stars have ‹ ! ‹MC , reducing the number of stars with
smaller ‹, and the probability that, for any given star, ‹ ! ‹ o . This emphasises that there is no
unique \DeltaM LK for a given star: the correction depends on context. If one modifies ND (‹) based on
parallaxíindependent criteria before selecting the ‹ídefined subsample, one also modifies \DeltaM LK .
Effects on small angular scales
The surface density of Hipparcos targets was set to match the limits of the observing mode.
As a result, the same objects were generally observed on successive scans of a particular region.
This leads to correlations in the derived astrometric data over scales of up to ‹2 degrees (twice
the field of view). One notable effect is that the precision of the mean parallax of stars with
separations within this r'egime is oe ‹
n 0:35 rather than oe ‹
n 0:5 (Lindegren, 1988).
These correlations may lead to systematic bias in ‹H at those angular scales. A comparison
between the results from NDAC and FAST (Hipparcos catalogue, ESA, 1997; vol 3, ch 16 & 17)
shows that differences of up to ¦ 2 mas can occur within 2 \Theta 2 degree regions near the ecliptic,
corresponding to potential offsets of ‹ 1 mas in the final, merged catalogue. This result does not
contradict the finding that the global zeropoint in ‹H is reliable at the 0.1 mas level, but it serves a
cautionary note for studies which require subímilliarcsecond accuracy over 1í2 degree scalelengths.
Temporal effects in parallax measurement
In simple terms, the aim of the Hipparcos data reduction is to solve the following equations
of condition for each target:
X = X 0 + ïX (T \Gamma T 0 ) + FX ‹ (3)
Y = Y 0 + ï Y (T \Gamma T 0 ) + F Y ‹ (4)
where (X, Y) are the observations, (X 0 ; Y 0 ) the coíordinates, (ï X ; ï Y ) the proper motions and
(F X ; F Y ) the parallax factors. If observations are not wellídistributed in time, the unknowns are
correlated. Hipparcos observations span the period January, 1990 to March, 1993, but no data
were obtained between September and November, 1992, due to the loss of a gyro and subsequent
reíconfiguration of the control system. As a result, January to March scans are available at four

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epochs, but September to November at only two. This imbalance leads to correlation coefficients,
ae, of up to \Sigma0:6 (Hipparcos catalogue, ESA, 1997; volume 1, ch 3.2), particularly in ae(‹; ff).
Groundíbased parallax analyses almost always achieve ae ! 0:1 since one of the usual criteria is
equal numbers of morning and evening observations. The existence of such high correlations does
not guarantee bias, but the data are clearly vulnerable to such effects.
3. Distances to open clusters
The typical Galactic open cluster is populated by from a few tens to a few hundred stars,
the product of a single staríforming region, spanning a narrow range in age and in chemical
composition. Several fiducial clusters fall within the range of Hipparcos (Mermilliod et al, 1997;
van Leeuwen et al, 1997). Since open clusters have a small lineíofísight depth and since cluster
membership can be decided based on photometric, proper motion or radial velocity criteria,
individual parallax measurements can be combined without incurring significant LutzíKelker bias,
although other systematic sources of error may be present. In particular, the small angular size of
the more distant clusters renders the astrometry susceptible to the smallíscale systematic problems
outlined in section 2. Indeed, Lindegren (1988) identified observations of open cluster stars as a
powerful diagnostics of the extent of biases on small angular scales.
3.1. The Hyades
The metalírich Hyades is near enough to permit groundíbased trigonometric and, in resolved
binaries, orbital parallax measurements, as well as statistical convergentípoint analysis of the
proper motions and radial velocities. The most recent latter study, by Schwann (1991), derived
a mean modulus of 3.40\Sigma0:04 mag. (47.9\Sigma0:9 pc.) based on 145 stars with proper motion
measurements from the FK5/FK4Sup, N30 and PPM catalogues. Perryman et al (1998) have
analysed the Hipparcos astrometry, complemented by additional radial velocity data. Conventional
convergentípoint analyses derive the cluster distance from the equation
ï
d = \Sigma N
i ( V S sinÖ i
ßï i
) = N (5)
where ß = 4:74, V S is the cluster space velocity, ï i the individual proper motions toward the
convergent point and Ö i the angular distance between each star and the convergent point. As
Perryman et al point out, the exact location of the last parameter depends on the extent to
which one allows for random cluster motions (oe cl ¦ 0:2 to 0.4 kms \Gamma1 ) in defining the membership
list. The Hipparcos data provide full positional and velocity information, and allow iterative
membership analysis. Following that approach, Perryman et al derive a distance of 46.34\Sigma0.27
pc. ((míM)=3.33) to the center of mass defined by 134 stars within a radius of 10 parsecs of the
cluster center.

-- 7 --
The main source of the 3.4% discrepancy between the Hipparcos result and Schwan's analysis
is the value adopted for the cluster space motion: Schwan's datum is V S = 46:6kms \Gamma1 , based on
Detweiler et al (1984), while Perryman et al derive V S = 45:72kms \Gamma1 from CORAVEL data, a 2%
difference. Most of the remaining discrepancy rests with the proper motions, and those differences
are also responsible for the larger mean distance derived by Torres et al (1997) based on precise
orbital parallax determination. Thus, the new Hyades distance determination is fully consistent
with previous observations.
3.2. The Pleiades and other clusters
The same consistency with previous work is not apparent in initial analyses of the distance
to the Pleiades cluster, which offered potentially the most significant result of the Hipparcos
mission. The concensus of preíHipparcos studies located the Pleiades at a distance of ¦ 130pc.,
(míM) 0 =5.6 mag. However, both van Leeuwen & HanseníRuiz (1997) and Mermilliod et al (1997)
derive a mean parallax from Hipparcos observations of ï ‹ = 8:6 \Sigma 0:24 mas, or r=116\Sigma3 pc. and
(míM) 0 =5.33\Sigma0:05 mag., placing the (solaríabundance) Pleiades mainísequence ¦0.2 magnitudes
fainter than the relation defined by nearby field stars (h[F e=H]i ¦ \Gamma0:15 dex). Such a result can
be accommodated within standard stellar evolution models, but only at the expense of invoking
substantial anomalies, such as a helium abundance YMS ¦ 0:35, and casting severe doubt on the
global utility of distance determination by mainísequence fitting.
Pinsonneault et al (1998) reíexamine this issue and show that plotting parallax against
the ae(‹; ff) correlation (see section 2) for individual Pleiads reveals a clear trend, with stars
having higher ae(‹; ff) also having larger ‹ (their figure 18). As noted in section 2, observations
with high ae(‹; ff) correlation indices may also have biased parallax determinations. Since the
relevant Pleiads are also predominantly bright stars near the cluster center, carrying most of the
weight in deriving ï ‹, the small angularíscale correlation inherent in Hipparcos astrometry leads
one to expect correlated bias. Given that Pleiads with ae(‹; ff) ! 0:2 give ï
‹ = 7:49 \Sigma 0:50mas,
(míM) 0 =5.63\Sigma0:14 mag, and that there is an absence of any [Fe/H]¦ 0 field stars which reproduce
the van Leeuwen/Mermilliod mainísequence (Soderblom et al, 1998), it is reasonable to reject the
initial Pleiades distance estimates as likely to be biased. The current best estimate of the distance
of the Pleiades remains at (míM) 0 =5.6 magnitudes, or 132 parsecs (although see van Leeuwen,
1999, for a dissenting viewpoint).
Mermilliod et al (1997) and Robichon et al (1997) estimate distances to other nearby
clusters, including Coma, Praesepe, IC 2602 and ff Persei. Most are significantly more distant
than the Pleiades and, in general, the Hipparcos distance moduli are consistent with previous
determinations. An exception is the sparse, nearby cluster Coma Berenices, where Pinsonneault
et al find a discrepancy of ¦ 0:2 magnitudes with respect to isochroneífitting, and trends in
(‹; ae(‹; ff)) similar to those in the Pleiades.

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In summary, Hipparcos astrometry does not provide the major challenge to stellar evolution
theory suggested by the preliminary analyses. Systematic biases, correlated over scales of ! 1
degree, limit the reliability of results derived for most open clusters. To echo Pinsonneault et al,
it seems more prudent to regard these comparisons as tests of smallíscale zeropoint errors, rather
than as measurements of cluster distances.
4. The HR diagram defined by field stars
The technique of distance determination through mainísequence fitting rests on the
hypothesis that the absolute magnitude (luminosity) of normal, single stars of a given color
(effective temperature) is a monotonic function of metallicity. Hipparcos provides the first
opportunity for a thorough empirical test of that hypothesis.
4.1. The abundance scale
In principle, fine analysis of highíresolution optical spectra, coupled with accurate temperature
determination, provides the best basis for measuring stellar metallicity. However, it is only recently
that such studies have been extended to significant numbers of even moderately faint stars. As a
result, most investigations of abundance distributions are based on a variety of loweríresolution
spectroscopic or photometric techniques, each anchored to the highíresolution scale through a
limited (and noníunique) set of standards. The latter step can lead to systematic zeropoint offsets
between different studies, particularly given the relativelyírecent revision in the accepted value of
the solar iron abundance (Bi'emont et al, 1991).
Advances in atmosphere modelling are opening the door for the first quantitative abundance
analyses of lateítype (K, M) dwarfs, notably using metal hydride bandstrengths (Allard et al,
1997; Gizis, 1997). However, most investigations continue to concentrate on F, G and earlyíK
stars on the upper mainísequence. Sandage & Eggen (1959) originally devised ffi(UíB), the
ultraviolet excess index, as a means of using Johnson broadband photometry to measure relative
lineíblanketing shortward of 4000 Ú A. Intermediateíband systems, such as Strèomgren or Geneva
photometry, have refined that measurement to some extent, while spectroscopic calibrations center
on Fe, CH, CN and Mg features in the 4000í5300 Ú A range (Rose, 1991; Carney et al, 1994; Jones
et al, 1996).
Two of the most extensive abundance catalogues are Schuster et al's (1993) Strèomgreníbased
dataset and Carney et al's (1994: CLLA) spectroscopic calibration. Schuster & Nissen (1989: SN)
derive relations, tailored for mainísequence stars, between the (bíy), c 1 and m 1 indices and [Fe/H],
while the CLLA calibration is based on high dispersion, but low signalítoínoise, spectra covering
¦ 50 Ú A centerd on the Mgb triplet. Abundances are derived by crossícorrelating the latter spectra
against observations of standard stars. Jones et al (1996) adopt a similar approach, basing their

-- 9 --
calibration on the CLLA system, while Flynn & Morell (1997) have devised an abundance index
for G and K dwarfs which combines Geneva photometry with a Cousins (RíI)íbased temperature
scale.
All of these calibration (save the last) are tied to standards with highíresolution abundance
analyses which predate the revision in [Fe/H] fi . This leads to an offset between those abundance
scales and metallicities defined in the more recent highíresolution studies by Axer et al (1994:
AFG) and Gratton et al (1997a: GCC), while there are further systematic differences between the
individual datasets. Reid (1998) finds
[F e=H]GCC ¦ [F e=H]AFG ¦ [F e=H] SN + 0:15 ¦ [F e=H]CLLA + 0:3
with dispersions of oe ¦ 0:2 dex. The offsets in the latter two scales become less pronounced at
nearísolar abundances. There is also evidence for a color term (temperature scale difference)
between the Schuster & Nissen and Carney et al calibrations, while Axer et al clearly underestimate
logg for a number of Hipparcos stars (see also Nissen et al, 1997). Clementini et al (1999) generally
confirm these offsets,
[F e=H]GCC = [F e=H] SN + 0:102 \Sigma 0:012
[F e=H]GCC = (0:935 \Sigma 0:032)[F e=H]CLLA + 0:181 \Sigma 0:173
while comparison with data from Ryan & Norris (1991) gives
[F e=H]GCC = [F e=H]RN + 0:40 \Sigma 0:04
Since the Jones et al scale is based on CLLA standards, one expects similar systematic errors in
their abundances.
4.2. The HR diagram of the local disk
Stars within the immediate Solar Neighbourhood can be used to probe the general properties
of the disk, since the overall velocity dispersion permits stars born at Galactic radii from 3 to 13
kpc to migrate through the Solar Radius. However, one of the early results from the Hipparcos
survey was an indication that a significant fraction of the stars included in the most recent
incarnation of the Nearby Star catalogue (CNS3: Jahreiss & Gliese, 1991) are not, in fact, within
the nominal distance limits. analysis of the final catalogue confirms that ¦ 40% of the CNS3
stars observed by Hipparcos lie beyond 25 parsecs. This has obvious implications for statistical
investigations which require unbiased, volumeílimited samples. For example, figure 2 plots
Hipparcos data for 106 nominallyísingle stars from the ''Gídwarf'' sample which are used by Wyse
& Gilmore's (1995) to analyse the disk abundance distribution. Approximately 10 percent of the
stars lie well above the mainísequence, and are likely to be evolved subgiants rather than dwarfs,
while a further 5í10 stars are probably previouslyíunrecognised binaries. To date, no comparable
study has been undertaken based on a volumeílimited sample drawn from the Hipparcos catalogue.

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While the Wyse/Gilmore stars do not constitute a volumeílimited sample, these observations
can be used to examine the variation in mainísequence location with changing abundance. Wyse
& Gilmore use Strèomgren photometry (from Schuster & Nissen, 1989) to estimate metallicities.
Dividing the stars into four subsets, figure 2 compares the (M V , (BíV)) distribution against
1íGyr and 5íGyr isochrones for solar abundance and [Fe/H]=í0.4 dex as computed by Bertelli
et al (1994). Ignoring evolved stars, the relative ranking is consistent with expectations, with
M V generally becoming fainter with decreasing [Fe/H] SN at a given color. However, there is a
systematic offset with respect to the isochrones, in the sense that the loweríabundance stars are
brighter than expected given the Strèomgren metallicity estimates. This is consistent with the
abundanceíscale comparisons given in the previous section.
The data plotted in figure 2 are consistent with the hypothesis of a monotonic change in
the location of the mainísequence with varying metallicity. Moreover, the comparison with the
isochrones suggests that most disk stars have abundances within \Sigma0:2 dex of the solar value (see
also figure 4 of Pinsonneault et al, 1998). Jimenez et al (1998) find a comparable systematic offset
between the loweríabundance (í0.6 ! [F e=H] ! \Gamma0:4) stars in the Flynn & Morell sample and
their [Fe/H]=í0.5 isochrones. The implication is that previous studies, such as Wyse & Gilmore's
analysis, overestimate the extent of the metalípoor tail to the disk abundance distribution, an
inference consistent with Reid's (1998) analysis of Hipparcos data for Lowell properímotion
stars. Definitive results await more extensive observations, particularly detailed abundance
determinations, of an Hipparcosíbased volumeílimited sample.
4.3. The HR diagram for halo subdwarfs
The low space density of the local halo coupled with the apparent magnitude limits of the
Hipparcos survey lead to the final catalogue including only a few tens of metalípoor subdwarfs
or subgiants with parallaxes of even moderate accuracy. As a result, those data offer only weak
constraints on theoretical calculations. Cayrel et al (1997) compare the observed (M bol , log T eff )
distributions for stars with \Gamma1:8 ! [F e=H] ! \Gamma1:2, with temperatures taken from Alonso et al
(1996), against (asíyet unpublished) isochrones computed by Lebreton and Vandenberg 1 They
find a systematic offset of 0.01 in logT eff , in the sense that the models are too hot. Adjusting the
isochrones, Cayrel et al deduce an age of ¦ 14 Gyrs for the field halo, although it should be noted
that that result rests on the location of two subgiants.
Figure 3 matches D'Antona et al's (1997) theoretical isochrones for 12 Gyríold populations
against data for subdwarfs from the AFG and GCC analyses, restricting the latter to stars with
no known binary companion having \Deltam ! 4 mag., with oe ‹
‹ ! 0:15 and with [Fe/H]! \Gamma0:7.
1 As discussed further in section 5, the D'Antona et al (1997) models predict a similar (M bol , log T eff ) relation
for mainísequence stars with MV ? MV (TO)1 .

-- 11 --
As with the disk stars, the overall trend matches theoretical expectations. At metallicities
exceeding [Fe/H] = í1.7 there is reasonable agreement between the observations and the predicted
isochrones, a circumstance also noted by Chaboyer et al (1998). There is a suggestion that the
lowestíabundance subdwarfs are more luminous (redder?) than expected, but this may reflect
smallínumber statistics or an inherent selection effect rather than intrinsic stellar properties.
These data can also be used to estimate \DeltaY
\DeltaZ , the proportional increase in helium and heavy
element abundance due to stellar nucleosynthesis í an important quantity in determining the
primordial helium abundance, Y P . Helium abundance cannot be measured directly for stars on the
lower mainísequence, but can be inferred by matching the observed distribution in the (M bol , log
T eff ) plane against theoretical isochrones. Based on preíHipparcos data for nearby disk Gídwarfs,
Fernandes et al (1996) deduce that \DeltaY
\DeltaZ ? 2. Pagel and Portinari (1998) extend this analysis by
matching Hipparcos data for stars in the abundance range 2 \Gamma3 ! [F e=H] ! 0 against isochrones
computed for \DeltaY
\DeltaZ = 0; 3; 6. Limiting analysis to stars with M V ? 5:75, Pagel & Portinari deduce a
bestífit value of \DeltaY
\DeltaZ = 3 \Sigma 2, with the large uncertainty reflecting primarily the small sample size.
5. Distances to globular cluster
While globular clusters lie well beyond the scope of present parallax measurement, highí
accuracy astrometry which Hipparcos provides for metalípoor stars allow reíexamination of the
cluster distance scale through indirect techniques, notably mainísequence fitting. In particular,
new data for field subdwarfs offer the chance to retrace the observer's route to globular cluster
distances (Sandage, 1970; 1986). A preírequisite for the successful application of those techniques
is a consistent metallicity scale for both clusters and field stars.
5.1. Cluster abundances
The mostífrequently referenced source of cluster metallicities remains the extensive study
by Zinn & West (1984). Their calibration is based on largeíaperture broadí and narrowíband
photometry, with individual magnitudes combined to give reddeningífree color indices; and on
integrated lowíresolution spectrophotometry, isolating the Ca II Kíline, Gíband and Mgb features.
Those measurements are scaled and averaged to give a composite estimate of the Q 39 index,
measuring the lineíblanketing in the 3820í4010 Ú A region. That index was calibrated itself in terms
of [Fe/H] using Cohen's (1983) analysis of highíresolution, imageítube spectra of giants in sixteen
clusters.
Cohen's observations were, of necessity, limited to stars near the tip of the redígiant
2 Note that the stellar abundances are drawn either from Carney et al (1994) or the compilation by Cayrel de
Strobel et al (1997), and can differ by 0.2í0.3 dex with more recent analyses.

-- 12 --
branch. Advances in detector technology over the last decade have allowed high signalítoínoise,
highíresolution spectroscopy of larger samples of cluster giants, notably by the Lick/Texas group
(Sneden et al, 1997 and refs within: hereinafter LT) and by Gratton and coíworkers (Carretta
& Gratton, 1997, and refs within: CG97). Both of the latter groups tie their analyses to the
revised value of the solar iron abundance, adopting log ffl(Fe)=7.52, ¦ 0:15 dex lower than the
previouslyíaccepted value. The LT analysis, aimed primarily at determining [O/Fe] ratios, spans
stars in seven clusters (Table 1), with homogeneous observations, temperatures derived from either
(BíV) [M15, M92] or (VíK) colors, using MARCS model atmospheres (Gustafsson & Bell, 1979)
as a reference.
Carretta & Gratton analyse both their own CASPEC echelle spectra and equivalentíwidth
data from the literature, including the LT dataset. Temperatures are based primarily on (VíK)
colors, while CG97 adopt the Kurucz (1992) model atmosphere grid as their reference. The
resulting abundances are ¦ 0:1 dex more metalírich than previous analyses, including the LT
determinations. CG97 ascribe this systematic discrepancy to the different model atmospheres
used in the different studies (although one should note that JR King et al (1998) derive a lower
abundance in their reíanalysis of LT's M92 data).
Both LT and CG97 compute true iron abundance, [Fe/H], based on equivalent widths of Fe I
and Fe II lines. As with field subdwarfs, population II globulars are expected to have nonísolar
elemental abundance ratios, with enhanced oxygen and ffíelements. The LT observations,
however, reveal more complex behaviour among the cluster giants than in the field, with [O/Fe]
values ranging from +0.3 dex to ! \Gamma0:4dex in some clusters (notably M13). [ff/Fe] ratios are
typically +0.2 to +0.3, typical of the halo. The oxygen abundance anomalies are coupled with an
observed antiícorrelation of NaíO and correlation of NaíN, and are probably explicable in terms
of deepímixing nucleosynthesis (Kraft et al, 1993), but possible inconstancy in the total CNO
abundance may reflect primordial variations. The latter would represent a potential complication
for MSífitting. If [O/Fe]=+0.3 is adopted as the standard value for [Fe/H]! \Gamma1, then following
Salaris et al (1993),
logZ = [M=H] \Gamma 1:66 = [F e=H] \Gamma 1:45 (6)
All of these analyses center on stars on the giant branch, whereas MSífitting demands (by
definition) matching cluster subdwarfs against field subdwarfs of comparable abundance. While
there is no a priori reason to distrust the existing cluster [Fe/H] data, the variations in [O/Fe]
emphasise the complications in dealing with evolved stars. Spectroscopy of mainísequence cluster
members would allow abundance calibration using the same criteria as used in fieldísubdwarf
calibration. Such observations are within the scope of the new generation of 8ímetre class
telescopes and are underway (J. Cohen, priv. comm.), but no studies have been completed. JR
King et al (1998) have obtained the first highíresolution spectra of six subgiant stars near the
M92 turnoff. Their analysis of those data, based on the Kurucz (1992) atmospheres and with
temperatures estimated from (BíV) colors, imply [Fe/H]¦ \Gamma2:5, significantly lower than either the

-- 13 --
LT or CG97 analyses, but matching Peterson et al's (1990) earlier analysis of two M92 red giants.
One should note, however, that they derive a higher reddening (EB \GammaV =0.07) than any other study,
suggesting possible systematic errors in the colors and the temperature scale. Observations of a
larger sample of stars, preferably with accurate infrared colors, in this and other clusters should
be given high priority.
5.2. Cluster distances
Empirical mainísequence fitting, the observer's route to cluster distances, is simple in
principle. Observations of nearby stars with known trigonometric parallax are used to define the
location of the mainísequence in the absolute magnitude/color plane; photometry of cluster stars
allows one to determine a fiducial cluster mainísequence in the apparent magnitude/color plane;
and the offset between the two sequences is the distance modulus of the cluster. In practise, this
idealised concept is vulnerable to a number of complications.
1. Photometry: the fieldístar and cluster observations must both be of high accuracy and
tied to a consistent photometric system. Crowding is clearly a problem in analysing cluster data,
although the high spatial resolution offered by the Hubble Space Telescope (HST) addresses that
issue. Most field subdwarfs have photometry in only the Johnston B and V passbands, and
individual (BíV) measurements drawn from the literature can differ by up to 0.04 magnitudes.
Since the slope of the mainísequence is @M V
@(B \GammaV ) ‹ 5, systematic uncertainties at that level lead
to errors of up to 0.2 magnitudes in (míM). Accurate, homogeneous UBVRIJHK photometry for
both field and cluster subdwarfs is eminently desirable.
2. Abundances: the metallicity of the calibrating stars should be either close to that of the
individual cluster under study, or the stellar parameters at least adjusted reliably to match the
appropriate abundance. Variations in the [O/Fe] and [ff/Fe] ratios from the nominal values of
¦ +0:2= + 0:3 amongst field subdwarfs (e.g. HD 134439 & 134440: King, 1997) further complicate
this issue. The most recent stellar models (e.g. D'Antona et al, 1997) indicate @(B \GammaV )
@[Fe=H] ¦ 0:14 at
[Fe/H]= \Gamma1 and @(B \GammaV )
@[Fe=H] ¦ 0:05 at [Fe/H]= \Gamma2for M V ? 5.
3. Binary and multiple stars: unrecognised duplicity can lead not only to erroneous absolute
magnitude estimates for calibrating subdwarfs, but can distort the position of the fiducial
(magnitude, color) relation for the lower mainísequence in binaryírich clusters, such as NGC 288
(Bolte, 1992).
4. Reddening: most clusters are subject to foreground Galactic obscuration, as are a few
calibrating subdwarfs. Qualitatively, if the cluster reddening is overestimated (cluster sequence
too blue), the distance modulus is also overestimated. On the other hand, if the reddening of
the calibrating subdwarfs is overestimated (reference sequence too blue), the cluster distance is

-- 14 --
underestimated.
5. Distribution in (M V , color): even with the addition of Hipparcos data, few subdwarfs are
available over the full range of M V /(BíV) and [Fe/H]. Typically, the distance to a given cluster is
estimated using only five to ten subdwarf calibrators, so decisions on whether to include individual
field stars can influence the derived (míM) by ¦ 0:1 magnitude.
6. Evolution off the mainísequence: theoretical isochrones show that at ages of from 10 to 16
Gyrs, metalípoor ([Fe/H]!í1) stars within 1.5 magnitudes of the turnoff have luminosities which
lie above the zeroíage mainísequence, although the offset is less than 0.1 magnitude for \DeltaM ? 1
magnitude. This reduces further the potential number of calibrators. One should note, however,
that evolution will bias MSífitting to larger distances only if the calibrating field turnoff stars are
significantly older (i.e. more luminous) than the cluster.
The preíHipparcos cluster distance scale
The sparse numbers of local subdwarfs with accurate groundíbased parallax measurements
discouraged purelyíempirical MSífitting analyses, and most preíHipparcos analyses either match
model isochrones directly against cluster data (the theoretician's route to distances), or adopted
a semiíempirical approach. In some studies (e.g. Richer & Fahlman, 1986) isochrones are used
to adjust subdwarf colors to those appropriate to the cluster abundance, and the MSífitting
distance derived using the resultant monoímetallicity sequence; alternatively, the subdwarfs
calibrate zeropoint offsets in the isochrones, and the latter are adjusted and matched to the cluster
colorímagnitude data. In one of the more influential studies, Bolte & Hogan (1995) adopted the
latter approach in deriving the distance and age of M92, using data for HD 103095 (Groombridge
1830), to derive an offset of 0.015 magnitude in the (BíV) colors predicted by Bergbusch &
Vandenberg's (1992 í BV92) models. Clearly, this singleípoint adjustment assumes that the
models have no systematic errors as a function of luminosity and/or effective temperature (i.e. the
isochrones have the correct shape in the colorímagnitude plane).
Bolte & Hogan's results were not unchallenged. D'Antona, Caloi & Mazzitelli (1997 í DCM)
derive a distance modulus of 14.80\Sigma0:1 to M92, based on weighted MSífitting against eight nearby
subdwarfs with accurate groundíbased parallaxes, rather than relying on only Groombridge 1830.
Distance estimates to the extreme metalípoor clusters M30 and M68 exceed previous analyses by
a similar factor.
RR Lyrae stars provide an alternative to MSífitting in cluster distance determination, and
we discuss their luminosity calibration in section 6.3. The other main options are astrometric
distances and, recently, white dwarf sequence fitting. In the former case, measurement of the
properímotion dispersion of cluster members is combined with observations of the radial velocity
dispersion and a dynamical model to derive a distance estimate. This method has the advantage
of being reddeningíindependent, but requires relative astrometry of extremely high precision. Rees
(1997) has summarised results for eight clusters, and typical uncertainties are from \Sigma0:2 to \Sigma0:4
mag in (míM) 0 . As yet, those results offer only weak constraints on the distance scale.

-- 15 --
Accurate photometry for white dwarfs in globular clusters has become possible only with the
existence of HST. Deep imaging has led to the detection of degenerate sequences in at least three
clusters: NGC 6752 (Renzini et al, 1996), NGC 6397 (Cool et al, 1996) and M4 (Richer et al,
1997). Renzini et al use their observations to estimate a distance modulus of 13.05 to NGC 6752,
choosing as calibrators nearby disk white dwarfs with spectroscopic masses of ¦ 0:5M fi , close
to the expected value for halo white dwarfs. The latter mass estimates, however, are based on
white dwarf models with thiníhydrogen atmospheres: thickíhydrogen atmosphere models provide
a better match to masses derived using gravitational redshifts (Reid, 1996a), and imply higher
masses for these disk calibrators. Lowerímass white dwarfs have larger radii, and therefore higher
luminosities at a given effective temperature, so Renzini et al are likely to have underestimated
(míM) 0 in their analysis. Concordant with this hypothesis, Richer et al note an offset of 0.1
magnitude between their MSífitting distance estimate to M4 (1.73 kpc í in good agreement with
the astrometric estimate) and the WDífitting result. Finally, Cool et al do not use their data to
estimate directly a distance to NGC 6397, but inspection of their figure 6 shows a systematic offset
of ¦ +0:3 magnitudes between the 0.5M fi theoretical white dwarf sequence and their data for the
adopted (míM) 0 = 11:7 magnitudes.
Table 2 presents distance estimates, derived through these various techniques, for a
representative sample of clusters. The 'standard' preíHipparcos distance scale is characterised
using the values listed in Harris' (1996) compilation. In each case we list both the true distance
moduli and the reddening, EB \GammaV , (from Harris). The latter values are derived using a variety of
techniques, and as a selfíconsistent reference we list reddenings derived from Schlegel et al's (1998)
100 ïm COBE/IRAS allísky analysis. In general, increasing the adopted cluster reddening by
\DeltaE B \GammaV leads to an increase of 2 \Theta \DeltaE B \GammaV in (míM) 0 . In addition, we list the apparent magnitude
(uncorrected for reddening) at the mainísequence turnoff. Those values are taken from the fiducial
cluster sequences referenced in Reid (1997, 1998).
PostíHipparcos cluster distance determinations
Even with the completion of the Hipparcos catalogue, there are few lower mainísequence
subdwarfs which are unambiguously single, have reliable abundance determinations and accurate
parallaxes. Extreme metalípoor clusters, such as M92 and M15, are illíserved with calibrators
for distance measurement. Table 2 lists distance modulus determinations from the several
postíHipparcos MSífitting analyses Apart from the choice of abundance calibration and foreground
reddening, each analysis employs a different set of subdwarf calibrators and uses different
colorícorrections and residualíminimisation techniques in MSífitting. Uncertainties in (míM) 0 are
at least \Sigma0:1 magnitude.
The largest discrepancies between the preí and postíHipparcos distance estimates listed in
Table 2 lie among the extreme metalípoor clusters, notably M92, where Harris' distance modulus
is 0.1 magnitude shorter than even Bolte & Hogan's result. Reid's (1997) initial Hipparcos analysis
produces the largest distances, but, apart from being based on the Carney et al (1994) abundance
scale, Pont et al (1998) point out that several of metalípoor calibrators used in that survey are

-- 16 --
binaries. 3 In their M92 analysis, Pont et al apply statistical corrections to adjust M V for known
binary subdwarfs, a procedure criticised by both Chaboyer et al (1998) and Reid (1998); excluding
those stars increases their derived modulus to (míM) 0 =14.74, in close agreement with other recent
studies.
NGC 6397 represents a special case. The preíHipparcos distance estimate rests on one
subdwarf, HD 64090, whose parallax is reduced by 20% by Hipparcos astrometry. Lying at low
latitude, this cluster is subject to significant fieldístar contamination, but deep HST observations
span a sufficient baseline in time to allow properímotion identification of members, and the
resultant colorímagnitude diagram extends to luminosities approaching the hydrogeníburning limit
(IR King et al, 1998). Exploiting these results, Reid & Gizis (1998) have used nearby Mísubdwarfs
with accurate groundíbased parallax data to undertake MSífitting on the lower mainísequence,
and derive a distance modulus of 12:12 \Sigma 0:15.
Given a distance to NGC 6397, that cluster can be used as a template to estimate distances
to more metalípoor systems by extending the differential upperíMS matching technique devised
by Vandenberg et al (1990). Theoretical isochrones can be used to make explicit allowance for
intrinsic abundanceídependent variations in the color and luminosity at the turnoff, corrections
analagous to those required in constructing a monoímetallicity sequence of subdwarf calibrators.
This requires that the models are accurate in a differential sense over a limited range in luminosity
and effective temperature. The resultant distance estimates (Table 2 í from Reid & Gizis, 1998)
are in good agreement with more direct analyses. Overall, MSífitting based on Hipparcos subdwarf
data points to a modest increase of 0.1 to 0.2 magnitudes in the distance moduli, or 5 to 7% in
distance, of the majority of globular clusters.
5.3. Globular cluster ages
Much of the interest in globular cluster distances centers on their use as Galactic age
indicators. The luminosity at the mainísequence turnoff remains the most effective absoluteíage
determinant, although there are observational difficulties in defining the parameter. The
mainísequence is vertical near the turnoff, so M V (TO) can be defined with a typical accuracy
of only \Sigma0:1 magnitude, equivalent to an uncertainty of ¦ 1:5 Gyrs in age. Matching M92
observations against the available stellar models, Bolte & Hogan set a firm lower bound of 16 \Sigma 2
Gyrs on the age of the Galaxy, identifying a strong conflict between stellar evolutioníbased and
(short distanceíscale) cosmological estimates of the age of the Universe (see also Chaboyer et al,
1996). As in all classical drama, hubris stepped forth on cue, not only in the guise of a longer
distance scale, but also in the form of refined physics of stellar interiors.
3 Note that Reid (1997) derives (míM)0=14.80 for M92 based on preíHipparcos data, in good agreement with
D'Antona et al, and (míM)0=14.82 by direct calibration against the metalípoor Fítype subdwarf, HD 19445.

-- 17 --
Mainísequence fitting based on Hipparcosícalibrated subdwarfs implies M V (TO) values that
are brighter by 0.1 to 0.15 magnitudes, corresponding to an age reduction of 1.5 to 2 Gyrs.
However, revisions in the (M V (TO), [Fe/H], age) relation lead to significant changes in the age
calibration. The last decade has seen substantial improvement in defining many of the physical
parameters underlying stellar interior theory, prompted primarily by theoretical inconsistencies
revealed through helioseismological observations. In particular, while most model isochrones
predating 1994 are based on the equation of state derived by either Eggleton et al (1973) or
Straniero (1988), and opacities from the Los Alamos calculations (Huebner et al, 1977), recent
calculations employ the new generation of OPAL opacity data (Rogers & Iglesias, 1992; Seaton et
al, 1994; Iglesias & Rogers, 1996), and the corresponding equations of state (Rogers et al, 1996).
The first extensive set of models to incorporate these new data were computed by Mazzitelli,
D'Antona & Caloi (1995 í MDC; see also DCM), and those calculations predict significantly lower
turnoff luminosities for a given age and abundance: there is a ¦ 0:2 magnitude (¦ 3 Gyr) offset
in M V (TO) between the DCM calibration and the preíOPAL Straniero & Chieffi (1991) models
(figure 4a). 4 Matched against the DCM calibration (figure 4b), the Hipparcosíbased distance scale
implies cluster ages of 11 to 13 Gyrs.
Figure 4a shows that the BV92 models predict turnoff absolute magnitudes within 0.1
magnitude of the OPALíbased DCM calculations. This is due partially to the highlyíenhanced
[O/Fe] ratios adopted in the former models í both DCM and Straniero & Chieffi employ
scaledísolar abundances. Given this concordance in M V (TO), one might ask why Bolte &
Hogan deduced an age of 16 Gyrs for M92. The answer lies with the second parameter of
the colorímagnitude diagram. Figure 5 compares 12íGyr isochrones from BV92 and DCM for
[Fe/H]=í2.26 and í2.03. respectively. The isochrones in (M bol , log T(eff) are almost identical on
the lower mainísequence, but diverge near the turnoff, with Bergbusch & Vandenberg predicting
higher temperatures. However, there is a significant, luminosityídependent color offset between
the isochrones in the (M V , (BíV)) observational plane. These discrepancies reflect several factors:
different effectiveítemperature/color transformations; different bolometric corrections; different
treatment of convection; and inclusion or exclusion of helium sedimentation. In the last context,
Straniero et al (1997) demonstrate that allowing for helium diffusion in their OPALíbased models
moves the turnoff redward by up to 0.06 magnitudes in (BíV). Matching their models directly
against Walker's (1994) M68 data, they estimate (míM) 0 =15.10 (for EB \GammaV = 0:04 cf. Table 2)
and an age of ¦ 13 Gyrs.
Clearly, even if one aims to use M V (TO) as the age indicator, there is a strong tendency
to select an isochrone which matches the observed color of the turnoff í particularly given the
observational ambiguity in locating M V (TO). Figure 5 shows that that choice drives one to
choosing an older age if the BV92 isochrones are taken as the reference set than if one matches
4 Calculations by Cassisi et al (1998),incorporating further modifications in the input physics, predict turnoff
luminosities lower by \DeltalogL ¦ 0:05 than even the MDC calibration.

-- 18 --
against the DCM calculations: the former predict an intrinsically bluer turnoff for a given age
and composition. In general, the agreement in age estimates derived from M V (TO) and fullíCMD
comparisons against OPALíbased stellar models favors cluster ages in the range 11 to 13 Gyrs.
5.4. Summary
The changes in the cluster distance scale resulting from the addition of accurate Hipparcos
parallax data for halo subdwarfs are relatively minor: no more than 0.15 magnitudes, or ¦ 7% in
distance. Nonetheless, when combined with the recent revisions in the physics of stellar interiors,
the net result is a reduction of 30 to 40% in the likely ages of even the most metalípoor globular
clusters. A final resolution of the age question must always rest with the stellar models, but
the availability of accurate parallaxes for a larger number of unambiguouslyísingle subdwarfs,
particularly at [Fe/H]!í1.5, would go a long way toward setting distance estimates of a firm
foundation. This requires subímilliarcsecond astrometry to at least 15th magnitude, and probably
the next astrometric space mission. In the interim, more accurate photometric and spectroscopic
data for both subdwarfs and a larger number of globular clusters (including CCD photometry of
stars above the turnoff) would not be amiss.
6. Distance indicators
6.1. Cepheid variables
The Cepheid periodíluminosity (PíL) relation constitutes the backbone of the extragalactic
distance scale. Descendants of intermediateímass stars, Cepheids have typical ages of ¦ 10 8
years and therefore are found predominantly near the parent staríforming region. As a result,
only a handful of variables lie within 1 kpc. of the Sun, and most Cepheids are subject to
considerable foreground reddening. However, the continued spatial association with coeval
lowerímass stars offers a potential means of distance determination, albeit for a limited number of
stars. Until recently, the primary calibrators of the Galactic PíL relation were thirteen Cepheids
in clusters and associations (Sandage & Tamman, 1968), whose distances were estimated using
mainísequence fitting. More recent calibrations of the same kind are by Feast & Walker (1987)
and Laney & Stobie (1994) (see later). Theoretical models (e.g. Chiosi et al, 1993) predict a small
metallicityídependence in M bol , which is amplified by differential blanketing at optical wavelengths.
Metalírich Cepheids are cooler and brighter in V and I. The amplitude of variation is currently
under debate (Sasselov et al, 1997; Kochanek, 1997; Sandage, Bell & Tripicco, 1999), but since
M31 Cepheids are nearísolar abundance and LMC Cepheids have [O/H]¦ \Gamma0:3 (ffiM V ¦ \Gamma0:05),
we consider this effect no further.
Hipparcos astrometric analyses

-- 19 --
Observations of over 200 Galactic Cepheids are included in the final Hipparcos catalogue,
with most lying at distances exceeding 1 kiloparsec. As a result, few variables have parallaxes
measured to an accuracy of better than 2oe. Lynga & Lindergren (1998) estimate parallaxes for
two Cepheids associated with open clusters, but with accuracies of no better than 40%. However,
since the Hipparcos Cepheids were not selected for observation based on parallax, the sample as
a whole is amenable to statistical analysis. Feast and Catchpole (1997) have used the method of
reduced parallaxes to complete the most rigorous such study to date.
Given parallax data for an unbiased sample of stars at effectively the same distance (e.g.
open cluster members), h‹i = \Sigma ‹ i
N . Consider an unbiased (as regards ‹) sample of stars spanning
a range of distances with identical, but unknown, M V . The apparent magnitude can be used to
scale each parallax measurement to a single distance, e.g. ‹ 0
i = ‹10 \Gamma0:2\ThetaV 0 , and h‹ 0 i = \Sigma ‹ 0
i
N gives
the appropriate zeropoint. Cepheids do not have identical magnitudes, but follow the PíL relation
hM V i = ffilogP + ae
which can be used as a scaling relation. Feast and Catchpole adopt ffi = \Gamma2:81 from Magellanic
Cloud Cepheid data, and solve for ae using
10 0:2ae = 0:01‹10 0:2(hV 0 i\GammaffilogP )
The Cepheid periodícolor relation is used to estimate foreground reddening, and hence hV 0 i,
for each star. Since Cepheids follow a periodíluminosityícolor (PLC) relation, the PíL relation has
intrinsic scatter. However, the slope of the color dependence in (BíV) (¦ 2:5) is close to the ratio
between total and selective reddening (¦ 3:15). Thus, using the periodícolor relation to estimate
EB \GammaV compensates for a substantial proportion of the intrinsic scatter in the PíL relation. Feast
and Catchpole derive ae = \Gamma1:43 \Sigma 0:1.
Feast & Whitelock (1998) follow similar precepts to derive the zeropoint, ae 2 , of the PLC
relation as ae 2 = \Gamma2:38 \Sigma 0:1. This calibration allows the derivation of Galactic rotation
constants from proper motion data, and Feast and Whitelock good agreement with radial velocity
analyses. Inverting the process, they note that Pont et al's (1997) Galactic rotation study implies
ae 2 = \Gamma2:42 \Sigma 0:13.
Madore & Freedman (1998) and Oudmaijer et al (1997) have criticised the Feast/Catchpole
analysis and derive zeropoints fainter by ¦ 0:1 magnitude. However, both calibrations are based
on variables with higheríaccuracy Hipparcos parallaxes; that is, the samples are parallaxíselected,
and therefore liable to bias. Oudmaijer et al apply estimated LutzíKelker corrections to individual
stars, but the substantial values of oe ‹
‹ mean that those corrections are sensitive to the assumed
spatial distribution. In contrast, the individual weights in the Feast/Catchpole reduced parallax
analysis are determined primarily by the apparent magnitude, rather than ‹, giving an estimate
of ae which is effectively unbiased by parallax uncertainties (Feast, Pont & Whitelock, 1998).

-- 20 --
The highestíweight Cepheids in the Feast/Catchpole analysis have near nakedíeye apparent
magnitudes, significantly brighter than the average amongst stars used to test the Hipparcos
absolute zeropoints, and the possibility of lowíamplitude magnitude terms has not yet been
eliminated fully. However, Sandage & Tamman (1998) point out that Feast & Catchpole's
PíL calibration is no more than 0.1 magnitudes brighter than previous Galactic calibrations,
notably their 1968 clusteríCepheid calibration (with the cluster distances given by Kraft's (1961)
mainísequence fitting). Those discrepancies lie at the 1oe level, even without allowing for possible
unrecognised systematics in the Hipparcos analysis.
Pulsational parallax analyses
The concept of using Cepheid radial pulsations to estimate distances originated with Baade
(1926) and was elaborated further by Wesselink (1946). In the simplest terms, the difference in
apparent magnitude at phases with identical colors (temperatures) is interpreted as a difference
in angular size; integrating over the velocity curve gives the corresponding change in linear
diameter; and the distance follows from the ratio of those quantities. analysing the velocity
curve requires correcting the observed velocities, integrations over the full stellar disk, to give
pulsational velocities. Recent studies are based primarily on variations on the surfaceíbrightness
method introduced by Barnes & Evans (1976), who derived a linear relation between visual surface
brightness, S V , and Johnson (VíR) color. The technique has been extended to nearíinfrared
colors (cf. Di Benedetto, 1997), with the zeropoints for these relations set through observations
of nonívariable stars, generally lateítype giants, whose diameters can be measured directly using
interferometric techniques (Fouqu'e & Gieren, 1997).
The most recent analyses are by Di Benedetto (1997), based on an (S V , (VíK) 0 ) calibration,
and Gieren et al (1997, 1998), who use both (S V , (VíK) 0 ) and (S K , (JíK) 0 ) relations. Gieren et
al (1996 and refs within) have also applied an (S V , (VíR) 0 ) to analysis of individual Cepheids
associated with Galactic open clusters and associations. Comparing the derived distances, Di
Benedetto finds excellent agreement between his BaadeíWesselink estimates and mainísequence
fitting distances (from Feast & Walker, 1987, and Laney & Stobie, 1994) for cluster Cepheids: 25
variables give
\Deltaï 0 = (m \Gamma M) DB
0 \Gamma (m \Gamma M) ZAMS
0 = \Gamma0:05 \Sigma 0:23
All of these studies are based on a Hyades distance modulus of 3.27, and hence the zeropoint
should be made brighter by 0.06 magnitudes to match the Hipparcos analysis (Perryman et al,
1998). Comparison between the Di Benedetto and Gieren et al analyses (17 stars in common)
gives
\Deltaï 0 = (m \Gamma M) DB
0 \Gamma (m \Gamma M) GFG
0 = 0:01 \Sigma 0:21
Many of these Cepheids have substantial foreground reddening: ten of the 34 stars in Gieren
et al's sample have EB \GammaV ? 0:5 mag., while only three have EB \GammaV ! 0:25 magnitudes. Thus,

-- 21 --
the deduced M V is subject to uncertainties in both EB \GammaV and in R, the ratio of total to selective
absorption. 5 In general, the slope (ffi) adopted for the periodíluminosity relation is taken from
observations of LMC variables. Gieren et al derive ffi = \Gamma2:76, ae = \Gamma1:29, a relation less steep than
the Laney & Stobie LMC calibration, and 0.16 fainter than Feast & Catchpole's relation at logP
= 0.5, and 0.2 magnitudes fainter at logP = 1.5. Di Benedetto finds an intermediate zeropoint
(again adjusted to (míM)Hyades =3.33) of ae = \Gamma1:41 from his analysis of Galactic Cepheids.
6.2. Red clump stars
Population II horizontal branch stars are at the coreíhelium/Híshell burning stage of
evolution. Their higheríabundance counterparts form the 'red clump' evident on the giant branch
in intermediateíage disk open clusters, such as NGC 2420 and 2243, and in the field population of
the Galactic Bulge and the Magellanic Clouds. Hipparcos provides the first accurate definition of
the Solar Neighbourhood population, with over 1000 evolved stars having both oe ‹
‹ ! 10% and BVI
photometry. analysing that sample, Paczy'nski and Stanek (1998) derive hM I i = \Gamma0:185 \Sigma 0:016,
with the majority of the clump stars lying in the range 0:85 ! (V \Gamma I) C ! 1:1, \Gamma0:4 ! M I ! 0:1.
Paczy'nski & Stanek propose using these stars as distance indicators. Neither the Solar
Neighbourhood stars nor the Bulge population (1:0 ! (V \Gamma I) C ! 1:4) show evidence for significant
trends in hM I i with color. Bolstered by evolutionary models (Jimenez et al, 1998), and despite
the nearísolar abundance derived McWilliam & Rich (1994), the difference in mean colors is
interpreted as a metallicity effect. The invariance in hM I i then implies at most a weak dependence
on abundance. Building on that inference, Udalski et al (1998) extend analysis to the Magellanic
Clouds.
The lynchpin of this technique is the assumption that, over the age range ¦ 2 \Gamma 10 Gyrs, hM I i
is invariant with metallicity. However, Cole (1998), Girardi et al (1998) and Beaulieu & Sackett
(1998) show that evolutionary models predict that red clump stars spanning a range of ffiI ¦ 0:5
magnitudes, depending on age and abundance. Cole, in particular, points out that the small
dispersion in M I within individual systems can be understood as reflecting internal homogeneity,
rather than global uniformity. Moreover, contrary to expectations, Paczy'nski (1998) finds that
individual Bulge stars show no correlation between (VíI) and [Fe/H] (assessed from Washington
photometry). Finally, one should note that none of the stellar models used in these analyses
are based on the new OPAL opacities, rendering the relative zeropoints uncertain. Given these
complications, and the lack of consistency with other distance indicators, it is prudent to defer full
acceptance of red clump stars as distance indicators until a reliable independent calibration has
been obtained.
5 These considerations are of lesser importance an nearíinfrared wavelength. Extensive JHK observations of both
Galactic and Magellanic Cloud Cepheids are being undertaken by SE Persson and collaborators.

-- 22 --
6.3. RR Lyraes and blue horizontal branch stars
Although less luminous than Cepheids, RR Lyraes are identifiable in most Local Group
galaxies, providing an independent test of the distance scale. The most common, and most
easily identified, are the Bailey RRabítype fundamentalímode pulsators, with asymmetric,
largeíamplitude (\DeltaV ¦1 mag) light curves. Firstíovertone RRcítype variables have nearísinusoidal
light curves, \DeltaV ¦ 0:4mag, while a growing number of doubleímode RRdítype stars and a few
secondíovertone RReítype variables are being identified, primarily through highíaccuracy CCD
photometric monitoring of globular clusters. Discrepancies persist, however, amongst the various
techniques used to calibrate the mean luminosity of both variable and nonívariable horizontal
branch stars, centering particularly on the metallicity dependence (see Smith, 1995). While only a
subset of those analyses rely directly on astrometry, we review recent results from each method to
set the Hipparcosíbased results in context.
Direct trigonometric parallax measurements
Like Cepheids, RR Lyraes have short evolutionary lifetimes (œ ¦ 10 8 years) and the local
space density is correspondingly low (¦ 2 \Theta 10 \Gamma8 stars pc \Gamma3 ). Indeed, the eponymous star of this
class (RR Lyrae) is the only star with an Hipparcos parallax determined to a precision better
than 15% (‹ = 4:38 \Sigma 0:59 mas), implying an uncorrected absolute magnitude of M V =0:82 +0:28
\Gamma0:31 at
[Fe/H]=í1.37. Both de Boer et al (1997) and Gratton (1998) have analysed Hipparcos astrometry
of larger samples of field horizontal branch stars. de Boer et al note that the eight blue horizontal
branch (BHB) stars in their sample have individual absolute magnitudes in agreement with
Dorman's (1992) model predictions, although the inferred masses are only ¦ 0:4M fi , significantly
lower than expected. Gratton estimates a mean absolute magnitude for his larger sample of 22
stars using a variation of the reduced parallax method, with the M5 horizontal branch taken as
a template of the absolute magnitude distribution. He derives a mean absolute magnitude of
hM V i = 0:69 \Sigma 0:1 for h[F e:H]i = \Gamma1:4. Finally, Tsujimoto et al (1998) have analysed data for 125
RR Lyraes (\Gamma2:5 ! [F e=H] ! 0:06) using the maximum likelihood technique proposed by Smith
(1988), and derive
M V (RR) = (0:59 \Sigma 0:37) + (0:20 \Sigma 0:63)([F e=H] + 1:60) (7)
Statistical parallaxes
The mean absolute magnitude of a kinematicallyídistinct group of stars can be estimated
using statistical methods if one has radial velocity and proper motion data (and photometry) for
a representative sample, and field RR Lyraes are well suited to this type of analysis. Most studies
are based substantially on stars drawn from the Shanghai observatory catalogue (Wan et al, 1980),
and almost all use the maximum likelihood method outlined by Murray (1983). As Table 3 shows,
the results show a remarkable consistency, with the most recent analyses agreeing to within 0.15
magnitudes with the mean absolute magnitudes derived at the outset by Strugnell et al (1986),
Barnes & Hawley (1986) and Hawley et al (1986).

-- 23 --
Layden et al (1996) supplement the Shanghai catalogue with data for field RR Lyraes included
in the Lick Northern Proper Motion sample, almost doubling the number of metalípoor stars in
the sample. Their analysis also has the benefit of improved radial velocity determinations for
many stars. Using Monte Carlo simulations to test the robustness of their solution, they find no
suggestion of systematic bias due to inherent sample properties.
Fernley et al (1998) and Tsujimoto et al (1998) have undertaken similar analyses based on
Hipparcos data. While the proper motions are typically a factor of two more accurate than in the
groundíbased catalogue, there is no evidence for any systematic differences. Thus, the derived
mean absolute magnitudes are little changed. Gould and Popowski (1998) have also examined
critically the photometry, radial velocities and reddening determinations, and find no indication
of potential systematic biases in hM V i. Similar results follow if they constrain the kinematics
using data for spectroscopicallyíselected field halo stars from Beers & SommeríLarsen (1995).
In summary, statistical parallax analysis of field RR Lyraes consistently leads to faint absolute
magnitudes, and provides no empirical evidence for the trend in hM V i with [Fe/H] which is
predicted theoretically and derived empirically by other analysis methods, as discussed in the
following sections.
Theoretical analyses
Luminosities for both RR Lyraes and nonívariable HB stars can be estimated directly from
theoretical evolutionary models. Building on the fundamental calculations by Sweigart & Gross
(1976), extensive grids of `standard' models (solar CNO/Fe, no core rotation) were computed by
Lee & Demarque (1990 í LD90) and Castellani et al (1991 í CCP), while Dorman (1992) extended
calculations to include enhanced CNO abundances. In broad terms, the location of a zeroíage
horizontal branch (ZAHB) star depends on the mass of the helium core, MC (‹ 0:50M fi for
globulars); the mass of the residual envelope, which is predominantly hydrogen, but with a helium
fraction enriched through dredgeíup to 0.01í0.02 above YMS ; and the metallicity. Increasing MC
leads to higher luminosities, while the effective temperature is highest (and log L lowest) at the
minimum envelope mass: that is, BHB stars are lower mass than RHB stars at given abundance,
Z.
Increasing Z for a given total mass leads to lower temperatures, with the effect becoming more
pronounced with increasing envelope mass. Hence, while the LD90 models predict only a modest
change in the effective temperature range spanned by [Fe/H]=í2.25 and í1.25 ZAHB models (4.4
? logT eff ? 3:75 versus 4.32 ? logT eff ? 3:72), the mass distribution is markedly less uniform,
with the higher mass stars concentrated at low temperatures. This is the firstíorder explanation
for the change in HB morphology with abundance, and there are three important consequences
for the expected properties of RR Lyrae stars. First, Bono & Stellingwerf (1994) have shown that
the location of the instability strip is relatively insensitive to abundance, with (to within ¦ 150K)
overtone pulsation stable for 7200 ? T eff ? 6500K and fundamental pulsation viable over the

-- 24 --
range 6900 ? T eff ? 5900. 6 Combined with evolutionary models, the prediction is that RR
Lyraes in intermediateíabundance clusters (such as M5) are lowerímass than their counterparts
in metalípoor systems. In particular, the LD90 models predict 0.65í0.70 M fi and 0.75í0.90 M fi
respectively.
These predictions can be tested using data for RRd variables in globulars. analysing
doubleímode Cepheids, Petersen (1973) demonstrated that, since the pulsation constants depend
only weakly on luminosity, T eff and composition, the overtone/fundamental period ratio, P 1 /P 0 ,
is determined primarily by the mass and radius (or haei). Hence, stars of a given mass occupy a
narrow locus in the (P 1 /P 0 , P 0 ) plane í the Petersen diagram. The same technique can be applied
to RRd stars, and initial results indicated masses ¦ 0:1M fi lower than the evolutionary predictions:
¦ 0:55M fi at [Fe/H]¦ \Gamma1:3 (M3) and ¦ 0:65M fi at [Fe/H]¦ \Gamma2:1 (M15). However, both Kovacs
et al (1991) and Cox (1991) showed that adopting Iglesias and Rogers' (1992) opacities, rather
than the Los Alamos calculations used previously, led to increases of ¦ 0:1M fi in the mass derived
from linear pulsational analysis. Full nonlinear calculations by Bono et al (1996) give even higher
mass estimates, with ¦ 0:70M fi for M3 variables and ¦ 0:82M fi for M15 and M68 stars. These
results are therefore in good agreement with direct evolutionary calculations.
The second consequence is that as RR Lyrae stars evolve, the predominant direction of
evolution is expected to be a function of abundance. In M5ílike systems, the majority of variables
should evolve from low to high temperature, while the reverse holds for M15ílike systems. Recent
observations of variable star periodíchanges (Silbermann & Smith, 1995; Reid, 1996b) tend to
support this hypothesis. In particular, Clement et al (1997) have shown that the M3 star V79 has
evolved from a fundamental pulsator to a mixedímode RRd variable, while the strength of the
overtone pulsations in V68, an RRD star, have increased. Both results imply blueward evolution.
In contrast, there is evidence for more dominant fundamentalímode oscillations in the M15 RRd
variable V30, implying redward evolution.
The third consequence concerns the average luminosity of RR Lyraes, which is expected to
exceed the level of the ZAHB. In most clusters the offset is ¦ 0:1 magnitudes, but Lee et al (1990)
emphasise that in clusters with an extremely blue horizontal branch, those stars which reach the
instability strip have evolved significantly, can be brighter than M V (ZAHB) by ¦ 0:3 magnitudes.
This is only a serious consideration for clusters where Lee's HBítype parameter exceeds 0.9 (M13,
NGC 6397, etc).
Quantitative predictions of the luminosity of the ZAHB have been influenced significantly
by the inclusion of the revised OPAL opacities (Iglesias & Rogers, 1996) and equation of state
(Rogers et al, 1996). Mazzitelli et al (1995) demonstrated that the revised physical parameters
predict an increase over the LD90 and CCP models of 0.01 to 0.015 M fi in the core mass at the
tip of the red giant branch. Cassisi et al (1998) further extend the analysis by including revised
6 Note the overlap in temperature, considered further in discussion of the Oosterhoff dichotomy.

-- 25 --
Heíburning (ff) rates and updated neutrino energyíloss calculations. They derive slightly higher
core masses than even MDC, and HB lifetimes ¦ 23% shorter than the CCP models (‹ 80 Myrs
vs. 100 Myrs). The latter result has important repercussions, beyond the scope of the current
article, for estimating YMS . Including the effects of helium diffusion, luminosities on the ZAHB at
log T eff =3.85 (i.e. the center of the instability strip) are predicted as log(L)=1.68 at [M/H]=í1.3
and log(L)=1.74 at [M/H]=í2.0. Given bolometric corrections of 0.03 mag. (Flower, 1996) and
M bol (fi)=4.70 (see above), these imply M V (ZAHB)=0.53 and 0.38 mag., with hM V (RR)i brighter
by up to 0.1 magnitude.
The higher HB luminosities predicted by these most recent models are also in accord with
earlier spectroscopic analysis of BHB stars. Stellar atmosphere models can be used to match
spectral line profiles to derive (T eff , log(g)) and, given a distance estimate, the mass. The average
mass derived in previous studies of metalípoor clusters (M15: Moehler et al, 1995; NGC 6397: de
Boer et al, 1995) is no more than 0.5 M fi , significantly below theoretical expectations. However,
this discrepancy is mitigated substantially if one adopts the longer distance scale implied by either
the Hipparcos MSífitting, the OPALíbased ZAHB calibration (Heber et al, 1997), the brighter
absolute magnitudes advocated by Walker (1992), Sandage (1993b) and McNamara (1997), and
the models calculated by Caloi, D'Antona & Mazzitelli (1997) and Cassisi et al (1998).
BaadeíWesselink and pulsational analyses
Early studies, based primarily on BV photometry, derived inconsistent results, largely because
shocks in the RR Lyrae atmospheres (higher density than Cepheids) invalidate the assumption of
equal surface brightness at equal color. Later analyses center on longer wavelength colors, notably
(VíI) and (VíK), which are less susceptible to these effects, and use more sophisticated model
atmospheres. Even so, most are limited to wellíbehaved phases of the lightícurve. Fernley et al
(1989) have also combined this technique with the infrared flux method, directly determining m bol
at all phases.
Results from field star studies are summarised by Cacciari et al (1992), Skillen et al (1993)
and McNamara (1997). Direct analyses of cluster variables (e.g. Cohen, 1992; Storm et al, 1994)
give results consistent with the field star data, but with typical uncertainties of \Sigma0:15 to 0.2
magnitudes. Skillen et al's final calibration, based on combined infrared flux/BaadeíWesselink
analysis of 29 stars, gives a mean relation of
hM V i = (0:21 \Sigma 0:05)[F e=H] + (1:04 \Sigma 0:10) (8)
However, McNamara has reíanalysed these same stars using more recent Kurucz model
atmospheres, and finds a systematic offset of 200í300K between temperatures derived from optical
photometry (including (VíI) and (VíR)) and from (VíK) data. Adopting the optical scale,
McNamara derives a steeper, more luminous calibration,
hM V i = (0:29 \Sigma 0:05)[F e=H] + (0:98 \Sigma 0:04) (9)
Further systematic uncertainties remain in analysing the spectroscopic data. The observed radial
velocity is an integration over the stellar disk, and therefore underestimates the true pulsational

-- 26 --
velocity by a factor, p. Fernley (1994) argues that that factor is underestimated in standard
analyses; hence the change in radius, linearly dependent on p, is also underestimated; and the
zeropoint should be brighter by 0.07 magnitudes. Adjusting McNamara's relation, this implies
hM V i = 0:58 \Sigma 0:1 for M5 variables and 0:29 \Sigma 0:1 in M15. On the other hand, Bono et al (1994)
point out that velocity gradients and changes in the opacity within the atmosphere during the
pulsational cycle can lead to crossícorrelation techniques overestimating the true velocity.
Another absoluteímagnitude calibration method tied to the pulsational characteristics of RR
Lyraes is Fourier decomposition of the light curves. Simon & Clement (1993) have applied this
technique to RRc variables, concentrating on the Fourier term, OE 31 . Matched against hydrodynamic
models, their results indicate a steep gradient and bright zeropoint, M V ‹ 0:36[F e=H] + 0:96,
close to McNamara's recent BaadeíWesselink results. In contrast, Kovacs & Jurcsik (1996) use
similar techniques to analyse RRab cluster variables, deriving M V ‹ 0:19[F e=H] + 1:04.
Period shifts and the Oosterhoff dichotomy
Following Bailey's discovery of the first cluster variables, globular clusters became the targets
of many photographic studies during the early 20th century. Collecting the results from those
studies, Oosterhoff(1939, 1944) computed the mean period for the RRabítypes in each cluster,
and identified an apparent bimodality in that distribution, with peaks at hP ab i ¦ 0 d :55 (type
I) and 0 d .65 (type II). Arp (1955) first noted that the division between type I and II was also
a division by abundance, with type II systems limited to the most metalípoor clusters, such as
M15 and M30. Finally, in a detailed study of the RR Lyraes in several clusters, notably M3
and M15, the archetypical type I and II clusters, Sandage (1981) presented evidence for both
systematic clusterítoícluster offsets in the mean periodíamplitude and periodítemperature (but
not amplitudeítemperature) relation, and starítoístar offsets at fixed temperature. Interpreting
and understanding these observational results is crucial to determining the RR Lyrae (M V /[Fe/H])
relation.
Most interpretations of these effects are based on the pulsation equation for RR Lyraes. The
fundamental equation, P
p
haei = Q, where P is the period, ae the density and Q the pulsation
constant, holds for homologous systems. Substituting for ae allows one to write this equation in
terms of mass, luminosity and effective temperature. In the case of RR Lyraes, as with all stars,
differences in chemical composition mean that the stars are not homologous, but van Albada &
Baker (1971), from analysis of stellar models, showed that a very similar relation holds
logP ab = \Gamma0:34M bol \Gamma 0:68log M
M fi
\Gamma 3:48logT eff + const: (10)
Kovacs et al (1991: see also Fernley, 1993) repeat this analysis, using models based on the revised
Livermore opacities (Rogers & Iglesias, 1992), and find that these coefficients vary by up to ‹ 10%
over the abundance range spanned by the Galactic halo. Irrespective of those variations, this
equation indicates that a difference in period at the same effective temperature implies either a
difference in mass, or a difference in luminosity.

-- 27 --
In qualitative terms, the Oosterhoff dichotomy is generally agreed to result from the change
in morphology of the horizontal branch with decreasing abundance in old (? 10 Gyr) stellar
systems (Catelan, 1992; Sandage 1993a and refs within). In metalírich clusters, such as 47 Tuc,
the horizontal branch lies redward of the instability strip, and there are no 'traditional' RR
Lyraes. Decreasing abundance leads to a migration to higher temperatures, until, at abundances of
between ¦ \Gamma1:7 and í1.9 dex, the horizontal branch lies blueward of the instability strip. At lower
abundances, the trend reverses, and the instability strip is well populated in clusters such as M15
and M68, although less so in M92. The physical origin of the Oosterhoff phenomenon, however,
remains a subject of debate, with two main explanations proposed: evolutionary hysteresis and an
increased luminosity with decreasing abundance.
van Albada & Baker (1973) originated the hysteresis hypothesis. As discussed above,
evolutionary models predict (and observations tend to confirm) that metalípoor RR Lyraes
originate from the blue HB and evolve from high to low temperatures, while the opposite holds
for intermediate abundance systems. Hence Oo I HB stars evolve initially from longíperiod to
shortíperiod as RRab variables before becoming higherítemperature RRcítypes. In contrast,
type II variables evolve from RRcítype to fundamentalímode pulsators. van Albada and Baker
proposed the existence of an 'eitheríor' temperature zone, where both fundamental and first
overtone pulsations were stable. Under this scenario, Oo II variables switch to fundamental
pulsation at a lower temperature than their type I counterparts, leading to a longer average period
ï
P ab and a higher fraction of cítype stars in metalípoor systems.
The alternative model (Sandage, 1958, 1982) is that HB luminosity (hence ï
P ab ) increases
as [M/H] decreases. analysing both cluster data and Lub's (1987) observations of field stars,
Sandage (1993a) argues that this is a continuous relation, and that the apparent dichotomy
amongst cluster variables arises solely from the change in HB morphology with [M/H]. A similar
trend is evident amongst RRc cluster variables, although Simon & Clement (1993) identify
temperature differences as the likely source of ffihP c i between M5 and M68. Sandage, however, also
finds systematic differences in period when comparing individual stars at the same temperature
in clusters of different abundances. The last is the Sandage periodíshift effect (SPSE). Given
a gradient of @log(P ab )
@[Fe=H] = \Gamma0:12 for stars at the blue fundamental edge (BFE), combined with
abundanceídependent relations for mass, T eff (BFE) ( @T
@[Fe=H] = 0.012) and bolometric correction,
Sandage (1993b) deduces a mean relation for the horizontal branch of hM v i / 0:30[F e=H].
Both hysteresis and luminosity variation likely play a role in the Oosterhoff dichotomy. As
noted above, Bono et al (1995) predict that the fundamental and overtone pulsation zones overlap
in temperature, while RRc variables are more than twice as common in type II than type I clusters
(45% vs 20%). Hysteresis, however, cannot account for the SPSE. Lee et al's (1990) proposed
explanation of higher luminosities due to evolution above the ZAHB is not viable in Oo II clusters
with both BHB and RHB stars (M15, M68). Caputo & de Santis (1992) have challenged both the
cluster reddenings adopted by Sandage and his temperature scale for RRab variables. Catelan
(1992), however, has shown that uncertainties in the former are insufficient to account fully for the

-- 28 --
SPSE, while Fernley (1993) finds excellent agreement between the latter scale and his (VíK)íbased
temperature calibration.
Catelan and Fernley both derive shallower gradients than Sandage's periodíabundance
relation, @P
@[Fe=H] ¦ \Gamma0:06 to í0.09. However, a shallower periodíshift relation does not necessarily
imply a shallower gradient @hM V i
@[Fe=H] Fernley uses his nearíinfrared SPSE analysis and the theoretical
pulsation relations to derive
hM V i = 0:19[F e=H] + 0:84 (11)
That analysis, however, is based on a value of M bol (fi)=4.75 and assumes little variation in
mass between Oo I and Oo II RR Lyraes. If one adopts the most recent RRd mass calibration
(0.65 and 0.80 M fi respectively) and M bol (fi)=4.70, then Fernley's nearíinfrared data imply
hM V (RRab)i ¦ 0:45 for M15 and ¦ 0:8 for M5 i.e. hM V i / 0:35[Fe/H].
Mainísequence fitting calibration of cluster variables
The RR Lyrae (hM V i, [Fe/H]) relation can be estimated directly from globular cluster
colorímagnitude diagrams given a distance scale calibration, such as the Hipparcosíbased results
tabulated in the previous section. The main limitation rests with the availability of highíaccuracy
CCD photometry, since most studies concentrate on lowíluminosity mainísequence stars, rather
than evolved stars. Timeíresolved data are particularly sparse. To date, the clusters with sufficient
observations to defined hV 0 (RRab)i are the Oo I cluster M5 (Reid, 1996b) and the three Oo II
clusters M15 (Silbermann & Smith, 1995), M68 (Walker, 1994) and M92 (Carney et al, 1992). The
corresponding values of hM V (RRab)i, given the distance moduli and reddening listed in the final
column of Table 2, are 0.5, 0.32, 0.32 and 0.26 í implying M V / 0:2[F e=H] and a bright zeropoint.
Summary
The various techniques outlined above can be divided into two categories based on the deduced
RR Lyrae absolute magnitude calibration: direct trigonometric parallax and statistical parallax
analyses favor hM V i ¦ 0:75 \Sigma 0:15, with little or no variation with abundance; other methods favor
a steeper (M V , [Fe/H]) relation and, usually, a brighter zeropoint (figure 6). Deciding between
these alternatives is not straightforward. The local RR Lyrae sample is dominated by relatively
metalírich stars ([Fe/H]? \Gamma1:4), but a brighter zeropoint in M V also implies higher velocity
dispersions, at odds with other tracers of the halo population. Gould & Popowski (1998) suggest
that the problem lies in a mismatch between the cluster and fieldísubdwarf metallicity scales, with
the clusters significantly lower abundance than the CG97 giantíbased scale. However, reducing
the M15 distance modulus by 0.4 magnitudes to accommodate hM V i(RR) ¦ 0:8 requires adjusting
the (BíV) colors of the field subdwarfs by í0.07 mag., placing those stars well blueward of even the
most extreme metalípoor isochrones.
Demanding that the field and cluster calibrations agree assumes that the two sets of variables
have identical absolute magnitude distributions. Cluster red giants exhibit abundance anomalies
which have yet to be detected in halo giants in the field (Shetrone, 1997). Sweigart (1997) points

-- 29 --
out that if those anomalies stem from deep mixing, the atmospheric helium content can also
be increased, leading to higher luminosities and hotter temperatures on the horizontal branch.
Catelan (1998), however, suggests that the good agreement between the periodítemperature
distributions of field and cluster RR Lyraes argues against significant luminosity differences.
Currently, the majority of analysis techniques favor the steep (M V , [Fe/H]) relation derived
originally by Sandage (1982).
6.4. Mira variables
Observations of longíperiod asymptotic giantíbranch variables (Miras) in the Magellanic
Clouds have established that, like Cepheids, those stars follow a periodíluminosity relation,
bestídefined in the 2.2ïm Kíband (Feast et al, 1989). Hipparcos obtained astrometry for 180
Galactic Miras, with the highestíprecision measurement having an accuracy of 10.5% (R Car). van
Leeuwen et al (1997) follow the precepts outlined by Feast & Catchpole in the latter's Cepheid
study in analysing a subísample of 16 Miras, preíselected using parallaxíindependent criteria.
Fixing the slope of the periodíluminosity relation using LMC variables, the preferred relation
(( 1
oe ‹
) 2 solution) is
MK = \Gamma3:47 logP + 0:88 \Sigma 0:18
7. The local distance scale
Revised calibrations for distance indicators inevitably lead to revised distance estimates. We
summarise those results in the context of other recent analyses, concentrating on Cepheids and
horizontal branch stars. More extensive reviews of measurements of R 0 and distance estimates
to the Magellanic Clouds are provided by Reid (1993) and Walker (1998) respectively. Given the
continuing uncertainties, as far as possible we avoid the temptation to rationalise discrepancies
and arrive at preferred values.
7.1. The distance to the Galactic center
Rotation constants: the Solar Radius can be derived from the Oort constants, A = \Gamma 1
2 R 0 (
d\Omega dR ) 0
and B =
\Gamma\Omega 0 \Gamma 1
2 R 0 (
d\Omega dR ) 0 ,
where\Omega 0 is the local angular rotation velocity, \Theta 0
R 0
. Feast and Whitelock
(1998) combine Hipparcos Cepheid properímotion data with Pont et al's (1997) radial velocities
to
derive\Omega 0 = 27:19 \Sigma 0:87kms \Gamma1 kpc \Gamma1 and R 0 = 8:5 \Sigma 0:5 kpc. They derive similar results if they
scale Metzger et al's (1998) Cepheid distance scale to match the Hipparcos PíL relation zeropoint;
Metzger et al's own determination is 7.66\Sigma0.54 kpc.
Feast and Whitelock's analysis implies \Theta 0 ¦ 230kms \Gamma1 . Olling & Merrifield (1998), however,

-- 30 --
have modelled the local rotation curve, and find that variations in the surface density lead to
nonílinear behaviour in the Oort functions, A(R) and B(R). They argue that those variations
account for the range of values for A & B derived in studies sampling sources at different
heliocentric distances. Their bestífit solution has a significantly lower local velocity of rotation,
\Theta 0 = 184 \Sigma 8kms \Gamma1 and R 0 =7.1\Sigma0.4 kpc.
Geometric determinations: Properímotion and radial velocity data for H 2 O masers allow the
derivation of an expansion parallax. analysis of the Sgr B2 (North) complex, ‹ 300 pc. from
the Galactic center, leads to a value of R 0 = 7:1 \Sigma 1:5kpc, while averaging several other studies
gives R 0 = 7:2 \Sigma 0:7 kpc (Reid, 1993). Alternatively, the measured proper motion of Sgr A*
(ï l = \Gamma6:55 \Sigma 0:17 mas yr \Gamma1 , ï b = \Gamma0:48 \Sigma 0:06 mas yr \Gamma1 ; Backer, 1996, interpreted as reflex Solar
motion, can be used to constrain \Theta 0
R0 . Adopting V fi = 12 kms \Gamma1 relative to the Local Standard of
Rest, then if 170 ! \Theta 0 ! 240 kms \Gamma1 , R 0 is constrained to lie between 5.9 and 8.1 kiloparsecs.
Red clump stars: The OGLE and MACHO microlensing surveys have fueled extensive studies
of the colorímagnitude distribution of both the Magellanic Cloud and Bulge stellar populations.
Paczyinski and Stanek (1998) derive a mean magnitude for hI 0 i = 14:32 for red clump stars in
the loweríreddening (A V ! 1:5) regions of Baade's window. Locally, the red clump star observed
by Hipparcos have hM I i = \Gamma0:18, which Paczynski and Stanek correct to hM I i = \Gamma0:28. As
noted in the previous section, the local clump stars are bluer than their Bulge counterparts
(h(V \Gamma I) 0 i ¦ 1:0 versus h(V \Gamma I) 0 i ¦ 1:25), but if one assumes that the stellar populations are
compatible, then this implies a distance modulus of (míM) 0 = 14.60 magnitudes to the centroid
of the Baade's window population. Adjusting that estimate to the Galactic center, Paczynski &
Stanek derive (míM) 0 =14.62, or R 0 = 8:4 \Sigma 0:4 kpc.
RR Lyraes: the Galactic bulge variables have an average abundance of [Fe/H]¦ \Gamma1 (Walker
& Terndrup, 1991), comparable with the globular cluster M5. Both cluster and Bulge RR Lyraes
describe wellídefined relations in the (logP, K) plane, with an offset in zeropoint in the sense
that the Bulge stars are fainter (Reid, 1998). If both sets of RR Lyraes have identical absolute
magnitude distributions, then a distance modulus of 14.5 to M5 implies (míM) 0 =14.8 to the
centroid of the Bulge variables, or R 0 = 9:1 \Sigma 0:4 kiloparsecs.
7.2. The distance to the Large Magellanic Cloud
SN 1987A: The fortuitous existence of a gaseous ring surrounding the progenitor of SN 1987A
permits a geometric estimate of its distances. As originally discussed by Panagia et al (1991), the
time delay between the onset of fluorescence in forbidden lines of C and N, and the maximum
luminosity in those lines (observed by the IUE satellite) provides a a measure of the linear
dimensions of the ring, while imaging by HST allows measurement of the angular dimensions.
Gould & Uza (1998) employ sophisticated lightícurve analysis to determine an upper limit to the
distance modulus of (míM) 0 = 18:44 (for an elliptical ring, b
a ¦ 0:95). However, Panagia (1998)

-- 31 --
points out that their analysis omits a crucial factor: the SN 1987A ring is expanding; hence the
diameter measured by HST in 1990 overestimates the angular size at maximum fluorescence in
1988. Taking the expansion into account, Panagia derives a distance modulus of 18.57\Sigma0:1, or a
distance of 51.7\Sigma2:3 kiloparsecs.
Cepheids and Miras: Laney & Stobie's (1995) preíHipparcos Galactic calibration (scaled to
a Hyades modulus of 3.33 magnitudes) gives a true distance modulus of 18.58 to the LMC. In
comparison, Feast & Catchpole use their Galactic calibration to estimate (míM) 0 = 18:70. As both
Sandage & Tamman (1998) and Feast & Whitelock (1998) emphasise, these estimates depend on
both the assumed foreground reddening toward the LMC and metallicity corrections. Madore &
Freedman (1991) adopt E V \GammaI =0.10 for LMC Cepheids lacking direct measurement, while Feast &
Whitelock argue for E V \GammaI = 0:075. This translates to ffi(m \Gamma M) 0 ¦ 0:1 mag. for identical Galactic
calibrations. Gieren et al (1998) adopt the same reddening corrections as Feast & Catchpole,
and no metallicity corrections (Feast & Catchpole adopt +0.042 mag.), and use their Galactic
calibration to derive a shorter distance modulus of (míM) 0 =18.46 magnitudes, while scaling
Di Benedetto's calibration gives (míM) 0 =18.64 (Walker, 1998). Finally, van Leeuwen et al's
preferred calibration of the Mira periodíluminosity relation leads to an estimate of (míM) 0 =18.54
magnitudes. Averaging these three estimates gives (míM) 0 =18.57, matching the preíHipparcos
Cepheid value.
Horizontal branch and red clump stars: Population differences between the Galaxy and the
LMC complicate the use of these stars, particularly the red clump. The LMC clusters with the
highest RR Lyrae frequency (such as NGC 1466) have abundances of [Fe/H]¦ \Gamma1:8, at which
abundance the horizontal branch lies exclusively blueward of the instability strip in the (older)
Galactic clusters. Theoretical models predict relatively small luminosity differences, but one must
assume that a similar (M V , [Fe/H]) relation holds for both systems. Given hV 0 i = 18:98 for the
LMC cluster variables (Walker, 1994), the Galactic RR Lyrae statistical parallax calibration
(M V = 0:77 \Sigma 0:15) implies (míM) 0 ¦ 18:2, compatible with the value derived from red clump
stars (Udalski et al, 1998). On the other hand, the steeper (M V , [Fe/H]) relation favored by the
preponderance of Galactic cluster calibrations implies hM V i ¦ 0:35 to 0.4 at [Fe/H]=í1.8, and an
LMC modulus of (míM) 0 ¦ 18:60. The latter is in better accord with estimates based on other
distance indicators, notably SN1987A.
7.3. Distances to the SMC and M31
The SMC
The relative distance modulus between the LMC and the SMC can be determined through
observations of both Cepheids and RR Lyraes, making due allowance for the significant depth
of the latter system over much of its area. RR Lyraes in NGC 121 ([Fe/H]¦ \Gamma1:9) have
hV 0 i = 19:46 \Sigma 0:07 (Walker & Mack, 1988), implying ffi(míM) 0 = 0:48 \Sigma 0:07 relative to the
average of the LMC clusters. Udalski's (1998) analysis of field RR Lyraes leads to a slightly higher

-- 32 --
relative distance, ffi(míM) 0 = 0:52 \Sigma 0:05, although reddening and possible abundance differences
are a complication in this comparison. analyses of LMC and SMC Cepheids (Laney & Stobie,
1995; di Benedetto, 1997), on the other hand, favor ffi(míM) 0 = 0:42 \Sigma 0:04. Thus, the available
estimates span a only limited range.
M31
Distance estimates for M31 are based primarily on observations of fundamental distance
indicators, with the high spatial resolution provided by HST rendering globular cluster HB
stars accessible. As with the SMC, direct determinations based on Galactic calibrations can be
complemented by measuring the relative distance modulus with respect to the LMC.
Cepheids: Freedman & Madore (1990) derive a relative distance modulus of
hffi(m \Gamma M) M31\GammaLMC
0 i = 5:89mag. using observations of variables in Baade's fields I, III
and IV, while Kochanek (1997) derives 5:8 ! ffi(m \Gamma M) 0 ! 5:97 for a range of metallicity relations.
One should note, however, that Freedman & Madore derive ffi(m \Gamma M) 0 = 6:10mag. from data
for ¦ 50 Cepheids in Baade's field IV, the outermost field, which has both negligible foreground
reddening and a metallicity ( Z
Z fi
¦ 0:3) close to that of the LMC. Comparative results for RR
Lyraes are also consistent with a longer modulus.
RR Lyraes: Pritchet & Van den Bergh (1987) obtained the first photometry of RR Lyraes in
the halo of M31, deriving hBi = 25:68 \Sigma 0:06. FusiíPecci et al (1996) and Holland et al (1997) have
taken advantage of HST to determine colorímagnitude diagrams for ten M31 globulars ranging
from [Fe/H]=í0.2 to í2.3 dex. Assuming a linear relation, V 0 (HB) = ff[F e=H] + fi, a leastísquares
fit to the mean HB magnitudes gives ff = 0:13 \Sigma 0:07, fi = 25:38 \Sigma 0:09, slightly shallower than
deduced for Galactic cluster variables. Considering individual clusters, the three lowestíabundance
systems are G219 ([Fe/H]=í2.28), G351 (í1.95) and G302 (í1.85), with V 0 (HB) = 25.10, 25.16 and
respectively. Adopting V 0 (HB)=25.13 at [Fe/H]=í2.1 gives ffi(m \Gamma M) M31\GammaLMC
0 = 6:15 to 6.25
for 0 ! ff ! 0:35, or (míM) M31
0 = 24:75 to 24.85 for (míM) LMC
0 = 18:6. Direct comparison with
data for the Galactic clusters M68 and M15 gives ffi(m \Gamma M) M31\GammaM68
0 = 9:70 ((míM) M31
0 = 24:81
if the M68 modulus is taken from the final column of Table 2) and ffi(m \Gamma M) M31\GammaM15
0 = 9:59
((míM) M31
0 = 24:82) respectively. Similarly, one can average Fusi Pecci et al's result for G1
([Fe/H]=í1.33, V 0 (HB)=25.15) and G105 (í1.25, 25.30) to estimate V 0 (HB)=25.22 at [Fe/H]=í1.3.
Matching against M5 gives a relative distance modulus of 10.22 to 10.29 (0 ! ff ! 0:35) and
(míM) M31
0 = 24:72 to 24.79, consistent to within 0.1 magnitude with the estimates based on the
metalípoor RR Lyraes.
Red clump stars: Stanek & Garnavich (1998) have used HST observations to measure
hI 0 i = 24:27 for red clump stars in three fields in M31. Coupled with their Hipparcosíbased
Galactic calibration, they derive (míM) 0 = 24:47 \Sigma 0:05, a value identical with Freedman
& Madore's (1990) result. However, the latter value is based on (míM) LMC
0 = 18:50 and
ffi(m \Gamma M) M31\GammaLMC
0 = 5:89mag., while Udalski et al (1998) derive hI 0 i = 17:85 for LMC red
clump stars, giving ffi(m \Gamma M) M31\GammaLMC
0 = 6:42mag and (míM) LMC
0 = 18:04. These inconsistencies

-- 33 --
underline current uncertainties in using red clump stars as distance indicators.
8. Conclusions and prognosis
From the preceding discussion it should be clear that the Hipparcos results do not, in and
of themselves, provide definitive resolution of any of the major questions associated with the
determination of the Galactic distance scale. This is not surprising, since the main achievement
lies in extending greatly the sample of stars with accurate astrometry, rather than pushing that
accuracy beyond the best groundíbased limits. Nonetheless, these data have catalysed a general
reappraisal of several issues of fundamental importance and have underlined the potential of
spaceíbased astrometry. Several satellite missions are under consideration for launch over the next
decade to capitalise on this potential.
Among the acronymírich proposed future projects are DIVA, GAIA and SIM. DIVA (Rèoser et
al, 1997) is proposed for launch as early as 2002, and marks an extension of the Hipparcos project,
with twiníbeamed Fizeau interferometers used to determine positions to 0.5 mas for all stars with
V! 10:5 and a limited sample extending to 15th magnitude. GAIA (Lindegren & Perryman,
1996) is an interferometric satellite observatory, proposed for ESA's Horizon 2000+ plan. The
aims include observations of astronomical targets to 16th visual magnitude, including all sources
brighter than 15th magnitude (a total of 50 million objects), with an accuracy of 10ïarcsec at
V=15.
Finally, and most ambitiously, the Space Interferometry Mission (SIM), a 5í to 10íyear mission
scheduled for launch in 2006, aims to achieve a precision of 1 ïarcsec in relative measurements
over small (! 5 o ) angles, and an overall accuracy of 4 ïarcsec in absolute astrometry. The latter
capability will permit trigonometric parallax measurements of not only all primary distanceíscale
calibrators, but also direct distance determinations for stars in the nearer globular clusters. If
achieved, those observations should finally settle the remaining discrepancies in at least the local
distance scale.
Bob Hanson provided helpful comments and insight on systematic problems associated with
parallax measurements.
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Wyse RFG, Gilmore G 1995. AJ110:2771í2787
Zinn R, West MJ 1984. ApJS55:45í66
This preprint was prepared with the AAS L A T E X macros v4.0.

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Table 1: Globular cluster abundances
Cluster [Fe/H] C [Fe/H] ZW [Fe/H] LT [O/Fe] LT [Fe/H] CG
M71 í0.7 í0.58 í0.79 0.2 í0.70
M5 í1.4 í1.40 í1.17 0.35/í0.25 í1.11
M3 í1.8 í1.66 í1.47 0.3/í0.2 í1.34
M13 í1.6 í1.65 í1.49 0.3/í0.15/í0.6 í1.39
M10 í1.60 í1.52 0.3 í1.41
M92 í2.35 í2.24 í2.25 0.3/í0.05 í2.16
M15 í2.20 í2.17 í2.40 0.2 í2.12

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Table 2: Mainísequence fitting distance determinations
Cluster EB\GammaV (IR) [Fe/H] CG Harris Rees/WD DCM R97 G97 CDKK 1 R98
V TO EB\GammaV EB\GammaV EB\GammaV EB\GammaV EB\GammaV EB\GammaV
47 Tuc 0.03 í0.70 13.17 13.47 13.56
17.65 0.05 0.055 0.04
M71 0.34 í0.70 12.92 13.19
18.00 0.25 0.28
NGC 288 0.015 í1.07 14.55 14.85 15.00
19.02 0.03 0.03 0.01
M5 0.045 í1.11 14.32 14.4\Sigma:4 14.45 14.51 14.42 14.52
18.48 0.03 0.03 0.035 0.03 0.02
NGC 362 0.035 í1.15 14.60 14.88 14.79 2
18.80 0.05 0.055 0.04
M4 3 0.495 í1.19 11.41 11.2\Sigma:2
16.90 0.36
M3 0.015 í1.34 15.01 14.9\Sigma:6 15.10 4
19.14 0.01 0.01
M13 0.02 í1.39 14.22 14.06\Sigma:23 14.48 14.41 14.45
18.50 0.02 0.02 0.02 0.02 0.02
NGC 6752 0.06 í1.41 12.96 13.05 WD 13.17 13.24 13.20 13.16
17.30 0.04 0.04 0.02 0.035 0.04 0.04
NGC 6397 0.18 í1.82 11.75 12.24
16.60 0.18 0.18
M30 0.06 í1.91 14.46 14.80 14.95 14.84 14.85 5
18.60 0.03 0.04 0.05 0.04 0.04
M68 6 0.07 í1.99 15.01 15.10 15.24 15.15 15.15 5
19.05 0.04 0.06 0.05 0.04 0.06
M15 0.10 í2.12 15.03 15.38 15.23 5
19.40 0.09 0.09 0.09
M92 7 0.02 í2.16 14.54 14.76\Sigma:3 14.80 14.93 14.74 14.76 14.78 5
18.60 0.02 0.02 0.02 0.02 0.02 0.02
Globular cluster distance moduli and reddening:
Column 2 lists EB \GammaV deduced from 100 ïm maps (Schlegel et al (1998)) and the apparent magnitude
(uncorrected for reddening) at the turnoff;
column 3 gives the abundance on the Carretta & Gratton (1997 íCG97) scale.
Succeeding columns list derived distance moduli and adopted reddenings from individual
investigations: column 4, from the preíHipparcos compilation by Harris (1996); column 5,
astrometric distances from Rees (1996) and WDífitting distance for NGC 6752 (Renzini et al,
1996); column 6, from D'Antona, Caloi & Mazzitelli (1997 í DCM); column 7, Reid (1997 í R97);
column 7, Gratton et al (1997b í G97); column 9, Chaboyer et al (1998 í CDKK); and column 10,
data from Reid (1998) and Reid & Gizis (1998 í RG98).
The G97, CDKK, R98 and RG98 analyses are tied to the CG97 abundance scale and use subdwarf
metallicities from GCC and AFG. The remaining studies adopt the Zinn & West cluster abundances
and, in most cases, CLLA's subdwarf metallicities.
Notes:
1 Mainísequence fitting distance estimates only
2 Relative distance modulus (Vandenberg et al, 1990) with respect to M5
3 R=3.8 is assumed for M4
4 Based on matching the turnoff region against M13 (Reid & Gizis, 1998)
5 Relative distance moduli with respect to NGC 6397 (Reid & Gizis, 1998)
6 MSífitting results adjusted to match photometry from Walker (1994)
7 Pont et al (1998) derive (míM) 0 =14.67 for EB \GammaV =0.02 for M92.

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Table 3: RR Lyrae statistical parallax analyses
Source hM V i n \Lambda h[F e=H]i U V W oe U oe V oe W
Hawley et al a 0.79\Sigma0:21 65 í0.7 5 í120 í14 128 120 78
0.73\Sigma0:18 65 1.5 í21 í184 í4 166 114 91
Strugnell et al b 0.90\Sigma0.21 64 í0.68 4 í110 í17 124 109 70
0.85\Sigma0.19 75 í1.43 í2 í185 í3 161 108 94
Layden et al 0.94\Sigma0:30 c 51 í0.76 6 í45 í16 52 48 29
0.71\Sigma0:12 d 162 í1.61 9 í210 í12 168 102 97
Fernley et al 0.77\Sigma0:15 84 í1.66
0.69\Sigma0:21 60 í0.85
Tsujimoto et al 0.69\Sigma0:10 99 í1.58 í12 í200 1 160 104 86
Gould & Popowski 0.77\Sigma0:13 e 147 í1.60 6 15 1 171 99 90
a Reddenings based on Sturch's (1966) method
b Reddenings based on Burstein & Heiles (1982) HI maps
c Diskí3 sample
d Haloí3 sample
e Subset of Haloí3 sample

-- 46 --
Fig. 1.--- Formal uncertainties in Hipparcos astrometry as a function of apparent magnitude í the
sample consists of stars drawn from the Lowell Observatory proper motion survey.
Fig. 2.--- Hipparcosíbased (M V , (BíV)) data for nominallyísingle nearby stars from Wyse &
Gilmore's analysis of the local abundance distribution. The dashed lines are 1í and 5íGyr isochrones
from Bertelli et al (1994) for [Fe/H]=0.0 and í0.4 dex; the solid line is the mean relation described
by local stars (Reid & Murray, 1992).
Fig. 3.--- (M V , (BíV)) data for single subdwarfs with abundance determinations by Gratton et al
(1997) or Axer et al (1994) compared against 12íGyr isochrones predicted by D'Antona et al (1997)
for [Fe/H]=í0.7, í1.0, í1.5 and í2.0.
Fig. 4.--- a: the predicted relation between M V (TO) and [Fe/H] for 12, 14 and 16 Gyrs given by
Bergbusch & Vandenberg (solid lines), Straniero & Chieffi (dashed lines) and D'Antona, Caloi &
Mazzitelli (dotted); b: M V (TO) deduced for clusters in Table 2 (except the highlyíreddened M4)
compared to the DCM predictions for ages of 10, 12, 14 and 16 Gyrs. The models have [ff/Fe]=0
and are therefore offset by í0.2 dex.
Fig. 5.--- (M V , T eff ) and (M V , (BíV)) isochrones predicted for 12íGyr old metalípoor systems
by Bergbusch & Vandenberg (BV92) and D'Antona et al (DCM). The former models adopt
[O/Fe]=+0.4 while the latter have scaledísolar abundance ratios. To allow for this we compare
the BV92 [Fe/H]=í2.26 model against the [Fe/H]=í2.03 DCM isochrone.
Fig. 6.--- A summary of the various RR Lyrae (M V , [Fe/H]) relations. The five solid points mark
hM V i for M5, M13, M15, M68 and M92 variables, using the distance moduli given in the final
column of Table 2. The dotted horizontal lines mark the 1oe limits deduced in the Layden et al
statisticalíparallax analysis, while the upper histogram plots the abundance distribution of field
RR Lyraes.

-- 47 --
4 5 6 7 8 9 10 11 12
0
1
2
3
4
5
V magnitude
4 5 6 7 8 9 10 11 12
0
1
2
3
4
5
V magnitude
4 5 6 7 8 9 10 11 12
0
1
2
3
4
5
V magnitude
Fig. 1.---

-- 48 --
.3 .4 .5 .6 .7 .8 .9
7
6
5
4
3
2
[Fe/H] > í0.2
í0.2 to í0.4
í0.4 to í0.7
[Fe/H] < í0.7
Fig. 2.---

-- 49 --
.2 .3 .4 .5 .6 .7 .8 .9 1 1.1
7
6
5
4
3
2
(BíV)
í0.7 to í1
í1 to í1.3
í1.3 to í1.7
< í1.7
Fig. 3.---

-- 50 --
í2.2 í2 í1.8 í1.6 í1.4 í1.2 í1 í.8
4.4
4.2
4
3.8
3.6
3.4
[Fe/H]
í2.4 í2.2 í2 í1.8 í1.6 í1.4 í1.2 í1 í.8
4.4
4.2
4
3.8
3.6
3.4
[Fe/H]
10 Gyrs
16 Gyrs
Fig. 4.---

-- 51 --
.2 .4 .6 .8 1
8
7
6
5
4
3
2
1
BV92: í2.26
DCM: í2.03
3.85 3.8 3.75 3.7 3.65 3.6
8
7
6
5
4
3
2
1
Fig. 5.---

-- 52 --
í.4 í.6 í.8 í1 í1.2 í1.4 í1.6 í1.8 í2 í2.2
1
.5
0
í.5
[Fe/H]
Skillen et al
McNamara
Sandage
Caloi et al Fernley
Fig. 6.---