The RR Lyrae Period-Luminosity Relation. I. Theoretical Calibration

Abstract

We present a theoretical calibration of the RR Lyrae period-luminosity (PL) relation in the UBVRIJHK Johnson-Cousins-Glass system. Our theoretical work is based on calculations of synthetic horizontal branches (HBs) for several different metallicities, fully taking into account evolutionary effects besides the effect of chemical composition. Extensive tabulations of our results are provided, including convenient analytical formulae for the calculation of the coefficients of the period-luminosity relation in the different passbands as a function of HB type. We also provide "average" PL relations in IJHK, for applications in cases where the HB type is not known a priori; as well as a new calibration of the MV-[M/H] relation. These can be summarized as follows:
and

Full text

arXiv:astro-ph/0406067v1 2 Jun 2004
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THE RR LYRAE PERIOD-LUMINOSITY RELATION.
I. THEORETICAL CALIBRATION
M. Catelan
Pontificia Universidad Cat´olica de Chile, Departamento de Astronom´ıa y Astrof´ısica,
Av. Vicu˜na Mackenna 4860, 782-0436 Macul, Santiago, C hile
Barton J. Pritzl
Macalester College, 1600 Grand Avenue, Saint Paul, MN 55105
and
Horace A. Smith
Dept. of Physics and Astronomy, Michigan State University, East Lansing, MI 48824
ApJ Supplement Series, in press
ABSTRACT
We present a theoretical calibration of the RR Lyrae p e riod-luminosity (PL) relation in the
UBV RIJHK Johnsons-Cousins-Glass system. Our theoretical work is based on calculations of syn-
thetic horizontal branches (HBs) for several different metallicities, fully taking into account evolu-
tionary effects besides the effect of chemical composition. Extensive tabula tions of our results ar e
provided, including convenient analytical formulae for the calculation of the coefficients of the period-
luminosity relation in the different passbands as a function of HB type. We also provide “average”
PL relations in IJHK, for applications in cases where the HB type is not known a priori; as well a s
a new calibration of the M
V
− [M/H] relation. These can be summarized as follows:
M
I
= 0.471 − 1.132 log P + 0.205 log Z,
M
J
= −0.141 − 1.773 log P + 0.190 log Z,
M
H
= −0.551 − 2.313 log P + 0.178 log Z,
M
K
= −0.597 − 2.353 log P + 0.175 log Z,
and
M
V
= 2.288 + 0.882 log Z + 0.108 (log Z)
2
.
Subject headings: stars: horizontal-branch – stars: variables: other
1. INTRODUCTION
RR Lyrae (RRL) sta rs are the cornerstone of the Pop-
ulation I I distance scale. Yet, unlike Cepheids, which
have for almost a century been known to present a tight
period-luminosity (PL) relation (Leavitt 1912), RRL
have not been known for presenting a particularly note-
worthy PL relation. Instead, most researchers have uti-
lized an average relation between absolute visual mag-
nitude and metallicity [Fe/H] when deriving RRL-based
distances. This relation pos sesses several potential pit-
falls, including a str ong dependence on evolutionary e f-
fects (e.g., Demarque e t al. 2000), a possible non-
linearity as a function of [Fe/H] (e.g., Castellani, Chieffi,
& Pulone 199 1), and “pathological outliers” (e.g., Pritzl
et al. 2002).
To be sure, RRL have also b e e n noted to follow a PL
relation, but only in the K band (Longmore, Fernley, &
Jameson 1986). This is in sharp contrast with the case
of the Cepheids, which fo llow tight PL relations both
Electronic address: mcatelan@astro.puc.cl
Electronic address: pritzl@macalester.edu
Electronic address: smith@pa.msu.edu
in the visua l and in the near-infrared (see, e.g., Tanvir
1999). The reason why C e pheids present a tight PL re-
lation irre spective of bandpass is that these stars cover
a large range in luminosities but only a modest ra nge
in tempera tur es. Conversely, RRL stars are restricted
to the horizontal branch (HB) phase of low-mass stars,
and thus necessarily cover a much more modes t range in
luminosities—so much so that, in their c ase, the range
in temper ature of the instability strip is a s important as,
if not more important than, the range in luminosities of
RRL stars, in deter mining their range in pe riods. There-
fore, RRL stars may indeed present PL relations, but
only if the bolometric corrections are such as to lea d to a
large range in abs olute magnitudes when going from the
blue to the red sides of the instability strip—as is indeed
the case in K.
The purpose of the present paper, then, is to perform
the first systematic analysis of whether a useful RRL
PL relation may also be pre sent in other bandpasses
besides K. In particular, we expect tha t, using band-
passes in which the HB is not quite “horizontal” at the
RRL level, a PL relation should indeed be present. Since

2 M. Catelan, B. J. Pritzl, H. A. Smith
Fig. 1.— Upper panels: Morphology of the HB in different bandpasses (left: B; middle: V ; right: I). RRL variables are shown in
gray, and non-variable stars in black. Lower panels: Corresponding RRL distributions in the absolute magnitude—log-period plane. The
correlation coefficient r is shown in the lower panels. All plots refer to an HB simulation with Z = 0.002 and an intermediate HB type, as
indicated in the upper panels.
the HB around the RRL region becomes distinctly non-
horizontal both towards the near-ultraviolet (e.g., Fig. 4
in Ferraro et al. 1998) and towards the near-infrared
(e.g., Davidge & Courteau 1999), we present a full anal-
ysis of the slope and ze ro point of the RRL PL relation
in the Johnsons-Cousins-Glass system, from U to K, in-
cluding also BV RIJH.
2. MODELS
The HB simulations employed in the present paper a re
similar to those described in Catelan (2004a), to which
the reader is referred for further details and references
about the HB synthesis method. The evolutionary tracks
employed here are those computed by Catelan et al.
(1998) for Z = 0.001 and Z = 0.0005, and by Sweigart
& Catelan (1998) for Z = 0.002 and Z = 0.006, and as-
sume a main-sequence helium abundance of 23% by mass
and scaled-solar compositio ns. The mass distribution is
represented by a normal deviate with a mass dispersion
σ
M
= 0.020 M
⊙
. For the purposes of the pr e sent pa-
per, we have added to this code bolometric corrections
from Girardi et al. (2002 ) for URJHK over the rele-
vant rang e s of temperature and gravity. The width of
the instability strip is taken as ∆ log T
eff
= 0.075, which
provides the temperature of the red edge of the instabil-
ity strip for each star once its blue edge has been com-
puted on the basis of RRL pulsation theory results. More
sp e c ifically, the instability s trip blue edge ado pted in this
paper is based on equation (1) of Caputo et al. (19 87),
which provides a fit to Stellingwerf’s (1984) results—
except that a shift by −200 K to the temperature va l-
ues thus derived was applied in order to improve agree-
ment with more recent theore tical prescriptions (see §6
in Catelan 2004a for a detailed discussion). We include
both fundamental-mode (RRab) and “fundamentalized”
first-overtone (RRc) variables in our final PL relations.
The computed periods a re based on equation (4) in Ca-
puto, Marconi, & Santolamazza (1998), which represents
an updated version of the van Albada & Baker (1971)
period-mean density relation.
In order to study the dependence of the zero po int and
slope of the RRL PL relation with both HB type and
metallicity, we have computed, for each metallicity, se-
quences o f HB simulations which produce from very blue
to very red HB types. These simulations are standard,
and do not include such effects as HB bimodality or the
impact of second parameters other than mass loss on the
red giant branch (RGB) or age. For each such simula-
tion, linear relations of the type M
X
= a + b log P , in
which X repres e nts any of the UBV RIJHK ba ndpa sses,
were obtained using the Isobe et al. (1990) “OLS bisec-
tor” technique. It is crucial that, if these rela tions are to
be compared a gainst empirical data to derive distances,
precisely the same recipe be employed in the analysis of
these data as well, particularly in cases in which the cor-
relation coefficient is not very close to 1. The final result
for each HB morphology actually represents the average
a, b values over 100 HB simulations with 500 stars each.
3. GENESIS OF THE RRL PL RELATION
In Figure 1, we show an HB simulation c omputed
for a metallicity Z = 0.00 2 and an intermediate HB
morphology, indicated by a value of the Lee-Zinn type
L ≡ (B − R)/(B + V + R) = −0.05 (where B, V ,

The RRL PL relation in U BV RIJHK 3
Fig. 2.— PL relations in several different passbands. Upper panels: U (left), B, V , R (r ight). Lower panels: I (left), J , H, K (right).
The correlation coefficient i s shown in all panels. All plots refer to an HB simulation with Z = 0.001 and an intermediate HB type.
and R are the numbers of blue, variable, and red HB
stars, respectively). Even using only the mor e usual BV I
bandpasses of the Johnson- C ousins system, the change
in the detailed morpholo gy of the HB with the passband
adopted is obvious. In the middle upper panel, one can
see the traditional display of a “horizontal” HB, as ob-
tained in the M
V
, B−V plane. As a consequence, one
can see, in the middle lower panel, that no PL relation re-
sults using this bandpass. On the other hand, the upper
left panel shows the same simulation in the M
B
, B −V
plane. One clearly sees now that the HB is not anymore
“horizontal.” This has a clear impact upon the resulting
PL relation (lower left panel): now one does see an indi-
cation of a correlation between period and M
B
, though
with a large scatter. The rea son fo r this scatter is that
the effects of luminosity and tempe rature variations upon
the ex pected periods are almost orthogonal in this plane.
Now one can also see, in the upper right panel, that the
HB is also not quite horizo ntal in the M
I
, B−V plane—
only that now, in comparison with the M
B
, B−V plane,
the s tars that look brighter are also the ones that are
cooler. Since a decrease in temperature, as well as an
increase in brightness, both lead to longer periods, one
exp ects the effects of brightness a nd temperature upon
the periods to be more nearly parallel when using I. This
is indeed what happens, as can be seen in the bottom
right panel. We now find a quite reasonable PL relation,
with much less scatter than was the case in B.
The same concepts explain the behavior of the RRL PL
relation in the other passbands of the Johnson-Cousins-
Glass system, which becomes tighter both towards the
near-ultraviolet and towards the near-infrared, as com-
pared to the visual. In Figure 2, we show the PL relations
in all of the UBV R (upper panels) and IJHK (lower
panels) bandpasses, for a synthetic HB with a morphol-
ogy similar to that shown in Figure 1, but computed for
a metallicity Z = 0.001 (the results are qualitatively sim-
ilar for all metallicities). As one can see, as one moves
redward from V , where the HB is effectively horizontal
at the RRL level, an increasingly tighter PL relation de-
velops. Conversely, as o ne moves from V towards the
ultraviolet, the ex pectatio n is also for the PL relation to
become increasingly tighter—which is confirmed by the
plot for B. In the case of broadband U, as can be seen,
the expe c ted tendency is not fully confirmed, an effect
which we attribute to the complicating impact of the
Balmer jump upon the predicted bolometric corrections
in the region of interest.
1
An investigation of the RRL PL
relation in Str¨omgren u (e.g., Clem et al. 2004), which
is much less affected by the Balmer discontinuity (and
might accordingly produce a tighter PL relation than
in broadband U), as well as of the UV domain, should
thus prove of interest, but has not been attempted in the
present work.
4. THE RRL PL RELATION CALIBRATED
In Figure 3, we show the slope (left panels) and zero
point (right panels) of the theoretically-calibra ted RRL
PL relation, in U BV R (from top to bottom) and for four
different metallicities (as indicated by different symbols
and shades of gray; see the lower right panel). Each dat-
apoint corresponds to the average over 10 0 simulations
with 500 stars in each. The “err or bars” correspond to
the standard deviation of the mea n over these 100 sim-
ulations. Figure 4 is analogous to Figure 3, but shows
instead our results for the IJHK pas sbands (from top
1
The Balmer jump occurs at around λ ≈ 3700
˚
A, marking the
asymptotic end of the Balm er li ne series—and thus a discontinuity
in the radiative opacity. The broadband U filter extends well red-
ward of 4000
˚
A, and is thus strongly affected by the detailed physics
controlling the size of the Balmer jump. In the case of Str¨omgren
u, on the other hand, the transmission efficiency is practically zero
already at λ = 3800
˚
A, thus showing that it is not severely affected
by the size of the Balmer jump.

4 M. Catelan, B. J. Pritzl, H. A. Smith
Fig. 3.— Theoretically calibrated PL relations in the U BV R passbands (from top to bottom), for the four indicated metalli ci ties. The
zero points (left panels) and slopes (right panels) are given as a function of the Lee-Zinn HB morphology indicator.
to bottom). I t should be no ted that, for all bandpasses,
the coefficients of the PL relations are much more sub-
ject to statistical fluctuations at the extremes in HB type
(both very red and very blue), due to the smaller num-
bers of RRL variables for these HB types. In terms of
Figures 3 and 4, this is indicated by an increa se in the
size of the “error bars” at both the blue and red ends of
the relations.
The slopes and zero points for the UBV RIJHK ca l-
ibrations are given in Tables 1 through 8, respectively.
Appropriate values for any given HB morphology may
be obtained from these tables by direct interpolation, or
by using suitable interpolation for mulae (Catelan 2004b),
which we now proceed to describe in more detail.
4.1. Analytical Fits
As the plots in Figures 3 and 4 show, all bands
show some dependence on bo th metallicity and HB type,
though some of the effects clearly bec ome less pro-
nounced as one goes towards the near-infrared. Analy sis
of the data for each metallicity shows that, except for
the U and B cases, the coefficients of all PL relations (at

The RRL PL relation in U BV RIJHK 5
Fig. 4.— As in Figure 3, but for I J HK (from top to bottom).
a fixed metallicity) can be well described by third-order
polynomials, as follows:
M
X
= a + b log P, (1)
with
a =
3
X
i=0
a
i
(L )
i
, b =
3
X
i=0
b
i
(L )
i
. (2)
For all of the V RIJHK passbands, the a
i
, b
i
coefficients
are provided in Table 9.
5. REMARKS ON THE RRL PL RELATIONS
Figures 3 and 4 reveal a complex pattern for the varia-
tion of the coefficients of the PL r e lation as a function of
HB morphology. While, as anticipated, the dependence
on HB type (particularly the slope) is quite small for
the redder passbands (note the much smaller axis scale
range for the corresponding H and K plots than for the
remaining ones), the same cannot be said with respect
to the bluer passbands, particularly U and B, for which
one does see marked variations as one moves from very
red to very blue HB types. This is obviously due to the
much more impor tant effects of evolution away from the

6 M. Catelan, B. J. Pritzl, H. A. Smith
Fig. 5.— Variation in the M
U
− log P relation as a function of HB type, for a metallicity Z = 0.001. T he HB morphology, indicated by the L value, becomes bluer from upper left to
lower right. For each HB type, only the first in the series of 100 simulations used to compute the average coefficients shown in Figures 3 and 4 and Table 1 was chosen to produce this
figure.

The RRL PL relation in U BV RIJHK 7
Fig. 6.— As in Figure 5, but for the M
B
− log P relation.

8 M. Catelan, B. J. Pritzl, H. A. Smith
Fig. 7.— As in Figure 5, but for the M
V
− log P relation.

The RRL PL relation in U BV RIJHK 9
Fig. 8.— As in Figure 5, but for the M
R
− log P relation.

10 M. Catelan, B. J. Pritzl, H. A. Smith
Fig. 9.— As in Figure 5, but for the M
I
− log P relation.

The RRL PL relation in U BV RIJHK 11
Fig. 10.— As in Figure 5, but for the M
J
− log P relation.

12 M. Catelan, B. J. Pritzl, H. A. Smith
Fig. 11.— As in Figure 5, but for the M
H
− log P relation.

The RRL PL relation in U BV RIJHK 13
Fig. 12.— As in Figure 5, but for the M
K
− log P relation.

14 M. Catelan, B. J. Pritzl, H. A. Smith
TABLE 1
RRL PL Relation in U: Coefficients of the Fits
L σ(L ) a σ(a) b σ(b)
Z = 0.0005
0.934 0.013 0.497 0.042 −0.863 0.129
0.877 0.018 0.519 0.034 −0.849 0.110
0.776 0.022 0.547 0.024 −0.827 0.079
0.627 0.027 0.552 0.055 −0.860 0.192
0.414 0.028 0.799 0.267 −0.037 0.949
0.167 0.029 1.045 0.014 0.825 0.058
−0.102 0.031 1.013 0.010 0.706 0.048
−0.358 0.031 0.993 0.010 0.630 0.046
−0.590 0.025 0.983 0.009 0.595 0.041
−0.765 0.021 0.980 0.010 0.586 0.050
−0.883 0.014 0.977 0.013 0.587 0.068
−0.950 0.010 0.979 0.038 0.605 0.200
Z = 0.0010
0.940 0.011 0.537 0.070 −1.027 0.191
0.873 0.018 0.580 0.070 −0.952 0.217
0.744 0.025 0.609 0.083 −0.923 0.275
0.556 0.034 0.844 0.277 −0.151 0.962
0.282 0.033 1.134 0.109 0.833 0.376
−0.037 0.035 1.136 0.012 0.818 0.055
−0.342 0.036 1.117 0.011 0.741 0.047
−0.603 0.025 1.107 0.012 0.698 0.055
−0.789 0.022 1.098 0.017 0.659 0.081
−0.906 0.015 1.094 0.017 0.646 0.078
−0.963 0.008 1.104 0.034 0.701 0.169
Z = 0.0020
0.965 0.009 0.675 0.224 −0.890 0.696
0.910 0.015 0.708 0.207 −0.826 0.643
0.794 0.025 0.853 0.286 −0.405 0.920
0.594 0.029 1.155 0.250 0.541 0.816
0.307 0.036 1.255 0.117 0.847 0.383
−0.023 0.034 1.268 0.015 0.865 0.066
−0.356 0.035 1.258 0.017 0.820 0.069
−0.630 0.032 1.248 0.017 0.771 0.069
−0.822 0.021 1.247 0.021 0.768 0.089
−0.928 0.012 1.242 0.030 0.739 0.127
−0.974 0.008 1.236 0.077 0.709 0.326
Z = 0.0060
0.922 0.015 1.032 0.369 −0.439 0.985
0.810 0.021 1.243 0.383 0.128 1.034
0.601 0.034 1.445 0.285 0.648 0.792
0.298 0.039 1.535 0.186 0.871 0.520
−0.070 0.039 1.574 0.077 0.959 0.210
−0.420 0.036 1.582 0.024 0.958 0.075
−0.693 0.025 1.586 0.032 0.954 0.095
−0.868 0.018 1.571 0.079 0.887 0.229
−0.951 0.012 1.564 0.136 0.866 0.398
zero-age HB in the bluer passbands. In order to fully
highlight the changes in the PL relations in each of the
considered bandpasses, we show, in Figures 5 through 1 2,
the changes in the absolute magnitude–log-pe riod distri-
butions for each bandpass, from U (Fig. 5) to K (Fig. 12),
for a representative metallicity, Z = 0.001. Each figure
is comprised of a mosaic of 10 plots, each for a different
HB type, from very red (upper left panels) to very blue
(lower right panels ). In the bluer passbands, one can see
the stars that are evolved away from a position on the
blue zero-age HB (and thus brighter for a given period)
gradually becoming more dominant as the HB type gets
bluer. As already discussed, the effects of luminosity and
TABLE 2
RRL PL Relation in B: Coefficients of the Fits
L σ(L ) a σ(a) b σ(b)
Z = 0.0005
0.934 0.013 0.553 0.124 −0.763 0.524
0.877 0.018 0.555 0.110 −0.838 0.431
0.776 0.022 0.739 0.257 −0.263 0.940
0.627 0.027 1.093 0.058 0.933 0.199
0.414 0.028 1.104 0.013 0.899 0.044
0.167 0.029 1.107 0.012 0.873 0.040
−0.102 0.031 1.106 0.010 0.851 0.037
−0.358 0.031 1.105 0.010 0.836 0.035
−0.590 0.025 1.105 0.010 0.831 0.036
−0.765 0.021 1.107 0.011 0.839 0.047
−0.883 0.014 1.105 0.013 0.838 0.060
−0.950 0.010 1.108 0.020 0.853 0.100
Z = 0.0010
0.940 0.011 0.568 0.123 −0.966 0.437
0.873 0.018 0.651 0.196 −0.768 0.665
0.744 0.025 0.994 0.271 0.362 0.944
0.556 0.034 1.195 0.016 1.001 0.048
0.282 0.033 1.202 0.013 0.973 0.046
−0.037 0.035 1.206 0.015 0.955 0.051
−0.342 0.036 1.205 0.013 0.935 0.044
−0.603 0.025 1.205 0.013 0.923 0.048
−0.789 0.022 1.203 0.015 0.906 0.067
−0.906 0.015 1.200 0.019 0.895 0.076
−0.963 0.008 1.207 0.036 0.930 0.164
Z = 0.0020
0.965 0.009 0.741 0.300 −0.659 0.974
0.910 0.015 0.822 0.305 −0.442 0.972
0.794 0.025 1.116 0.294 0.466 0.939
0.594 0.029 1.302 0.068 1.022 0.214
0.307 0.036 1.315 0.017 1.024 0.059
−0.023 0.034 1.320 0.017 1.011 0.063
−0.356 0.035 1.319 0.018 0.990 0.066
−0.630 0.032 1.320 0.019 0.969 0.066
−0.822 0.021 1.320 0.021 0.969 0.080
−0.928 0.012 1.321 0.033 0.955 0.125
−0.974 0.008 1.315 0.084 0.923 0.346
Z = 0.0060
0.922 0.015 1.072 0.413 −0.205 1.110
0.810 0.021 1.288 0.377 0.370 1.020
0.601 0.034 1.495 0.212 0.907 0.589
0.298 0.039 1.563 0.079 1.067 0.222
−0.070 0.039 1.574 0.027 1.068 0.069
−0.420 0.036 1.580 0.028 1.060 0.078
−0.693 0.025 1.588 0.035 1.065 0.099
−0.868 0.018 1.573 0.108 0.992 0.317
−0.951 0.012 1.571 0.145 0.987 0.420
temper ature upon the p eriod-absolute magnitude distri-
bution are almost orthogonal in these bluer passbands.
As a consequence, when the number of stars evolved away
from the blue ze ro-age HB becomes comparable to the
number of stars on the main phase of the HB, which oc-
curs at L ∼ 0.4 − 0.8, a sharp break in slope results,
for the U and B passba nds, at around these HB types.
The effect is more pronounced at the lower metallicities,
where the evolutionary effect is expected to be more im-
portant (e.g., Catelan 1993). For the redder passbands,
including the visual, the changes are smoother as a func-
tion of HB morphology.
The dependence of the RRL PL relation in IJHK on

The RRL PL relation in U BV RIJHK 15
TABLE 3
RRL PL Relation in V : Coefficients of the Fits
L σ(L ) a σ(a) b σ(b)
Z = 0.0005
0.934 0.013 0.203 0.038 −0.973 0.126
0.877 0.018 0.231 0.029 −0.947 0.096
0.776 0.022 0.274 0.021 −0.871 0.065
0.627 0.027 0.305 0.017 −0.815 0.049
0.414 0.028 0.338 0.013 −0.749 0.041
0.167 0.029 0.360 0.013 −0.723 0.045
−0.102 0.031 0.374 0.014 −0.714 0.053
−0.358 0.031 0.385 0.017 −0.723 0.069
−0.590 0.025 0.415 0.096 −0.639 0.409
−0.765 0.021 0.448 0.134 −0.530 0.600
−0.883 0.014 0.510 0.158 −0.271 0.745
−0.950 0.010 0.520 0.137 −0.249 0.689
Z = 0.0010
0.940 0.011 0.215 0.065 −1.162 0.188
0.873 0.018 0.265 0.036 −1.060 0.103
0.744 0.025 0.312 0.028 −0.967 0.088
0.556 0.034 0.351 0.019 −0.879 0.057
0.282 0.033 0.383 0.016 −0.815 0.044
−0.037 0.035 0.399 0.016 −0.801 0.057
−0.342 0.036 0.428 0.096 −0.741 0.359
−0.603 0.025 0.468 0.147 −0.627 0.567
−0.789 0.022 0.596 0.205 −0.146 0.824
−0.906 0.015 0.625 0.183 −0.037 0.762
−0.963 0.008 0.664 0.152 0.126 0.678
Z = 0.0020
0.965 0.009 0.276 0.142 −1.195 0.415
0.910 0.015 0.310 0.063 −1.118 0.172
0.794 0.025 0.361 0.041 −1.007 0.107
0.594 0.029 0.401 0.024 −0.919 0.068
0.307 0.036 0.429 0.023 −0.865 0.061
−0.023 0.034 0.464 0.097 −0.784 0.314
−0.356 0.035 0.477 0.092 −0.771 0.312
−0.630 0.032 0.533 0.171 −0.610 0.590
−0.822 0.021 0.630 0.224 −0.293 0.796
−0.928 0.012 0.733 0.196 0.072 0.715
−0.974 0.008 0.737 0.170 0.073 0.654
Z = 0.0060
0.922 0.015 0.428 0.177 −1.103 0.445
0.810 0.021 0.454 0.065 −1.061 0.152
0.601 0.034 0.487 0.046 −1.001 0.113
0.298 0.039 0.513 0.033 −0.957 0.084
−0.070 0.039 0.551 0.074 −0.880 0.201
−0.420 0.036 0.584 0.117 −0.815 0.325
−0.693 0.025 0.642 0.201 −0.677 0.561
−0.868 0.018 0.789 0.273 −0.280 0.781
−0.951 0.012 0.840 0.257 −0.142 0.748
the adopted width of the mass distribution, as well as
on the helium abundance, has been analyzed by comput-
ing additional sets of synthetic HBs for σ
M
= 0 .030 M
⊙
(Z = 0.001) and for a main-sequence helium abundance
of 28% (Z = 0.002). The results are shown in Figure 13.
As can be seen from the I plots, the precise shape of the
mass distribution plays but a minor role in defining the
PL relation. On the other hand, the effects of a signif-
icantly enhanced helium abundance can be much more
impo rtant, particularly in regard to the zero point o f the
PL relations in all four passbands. There fo re, caution
is recommended when employing locally calibrated RRL
PL relations to extragalactic environments, in view of the
TABLE 4
RRL PL Relation in R: Coefficients of the Fits
L σ(L ) a σ(a) b σ(b)
Z = 0.0005
0.934 0.013 −0.132 0.031 −1.195 0.108
0.877 0.018 −0.106 0.021 −1.163 0.068
0.776 0.022 −0.065 0.017 −1.081 0.049
0.627 0.027 −0.028 0.014 −0.996 0.040
0.414 0.028 0.013 0.011 −0.891 0.033
0.167 0.029 0.047 0.011 −0.801 0.033
−0.102 0.031 0.070 0.009 −0.738 0.031
−0.358 0.031 0.087 0.008 −0.689 0.029
−0.590 0.025 0.098 0.008 −0.656 0.032
−0.765 0.021 0.107 0.010 −0.630 0.039
−0.883 0.014 0.110 0.012 −0.619 0.055
−0.950 0.010 0.110 0.020 −0.634 0.102
Z = 0.0010
0.940 0.011 −0.119 0.051 −1.354 0.148
0.873 0.018 −0.070 0.032 −1.248 0.089
0.744 0.025 −0.021 0.026 −1.134 0.078
0.556 0.034 0.023 0.018 −1.017 0.054
0.282 0.033 0.065 0.017 −0.906 0.051
−0.037 0.035 0.098 0.015 −0.815 0.047
−0.342 0.036 0.125 0.014 −0.740 0.045
−0.603 0.025 0.143 0.014 −0.687 0.049
−0.789 0.022 0.159 0.016 −0.639 0.058
−0.906 0.015 0.167 0.019 −0.610 0.070
−0.963 0.008 0.167 0.027 −0.623 0.116
Z = 0.0020
0.965 0.009 −0.071 0.092 −1.401 0.268
0.910 0.015 −0.027 0.059 −1.290 0.163
0.794 0.025 0.027 0.043 −1.158 0.118
0.594 0.029 0.074 0.028 −1.039 0.082
0.307 0.036 0.110 0.029 −0.948 0.082
−0.023 0.034 0.145 0.024 −0.854 0.070
−0.356 0.035 0.165 0.020 −0.803 0.061
−0.630 0.032 0.191 0.022 −0.732 0.067
−0.822 0.021 0.201 0.029 −0.708 0.095
−0.928 0.012 0.224 0.036 −0.640 0.116
−0.974 0.008 0.233 0.049 −0.619 0.183
Z = 0.0060
0.922 0.015 0.060 0.103 −1.332 0.252
0.810 0.021 0.111 0.070 −1.217 0.168
0.601 0.034 0.147 0.051 −1.142 0.127
0.298 0.039 0.179 0.041 −1.076 0.106
−0.070 0.039 0.217 0.041 −0.991 0.103
−0.420 0.036 0.246 0.039 −0.928 0.101
−0.693 0.025 0.270 0.048 −0.876 0.125
−0.868 0.018 0.315 0.055 −0.767 0.146
−0.951 0.012 0.320 0.080 −0.763 0.221
possibility of different chemical enrichment laws. From a
theoretical point of view, a conclusive assessment of the
effects of helium diffusion on the main sequence, dredge-
up on the first ascent of the RGB, and non-canonical
helium mixing on the upper RGB, will all be required
befo re calibrations such as the present ones can be con-
sidered final.
6. “AVERAGE” RELATIONS
In a pplications of the RRL PL relatio ns presented in
this paper thus far to derive distances to objects whose
HB types are not known a priori, as may easily happen
in the case of distant galaxies for instance, some “av-

16 M. Catelan, B. J. Pritzl, H. A. Smith
TABLE 5
RRL PL Relation in I : Coefficients of the Fits
L σ(L ) a σ(a) b σ(b)
Z = 0.0005
0.934 0.013 −0.343 0.023 −1.453 0.082
0.877 0.018 −0.322 0.015 −1.426 0.046
0.776 0.022 −0.291 0.012 −1.364 0.036
0.627 0.027 −0.261 0.010 −1.294 0.030
0.414 0.028 −0.231 0.009 −1.215 0.027
0.167 0.029 −0.207 0.008 −1.150 0.026
−0.102 0.031 −0.192 0.007 −1.109 0.022
−0.358 0.031 −0.182 0.006 −1.080 0.020
−0.590 0.025 −0.176 0.006 −1.060 0.023
−0.765 0.021 −0.171 0.007 −1.044 0.029
−0.883 0.014 −0.170 0.008 −1.038 0.037
−0.950 0.010 −0.171 0.014 −1.050 0.070
Z = 0.0010
0.940 0.011 −0.318 0.037 −1.568 0.107
0.873 0.018 −0.279 0.025 −1.482 0.068
0.744 0.025 −0.239 0.019 −1.385 0.057
0.556 0.034 −0.204 0.014 −1.292 0.042
0.282 0.033 −0.172 0.013 −1.208 0.039
−0.037 0.035 −0.148 0.012 −1.140 0.037
−0.342 0.036 −0.130 0.011 −1.089 0.034
−0.603 0.025 −0.119 0.010 −1.055 0.033
−0.789 0.022 −0.110 0.011 −1.028 0.037
−0.906 0.015 −0.106 0.014 −1.012 0.051
−0.963 0.008 −0.105 0.020 −1.013 0.086
Z = 0.0020
0.965 0.009 −0.265 0.074 −1.592 0.216
0.910 0.015 −0.230 0.047 −1.503 0.128
0.794 0.025 −0.185 0.035 −1.390 0.097
0.594 0.029 −0.148 0.023 −1.295 0.067
0.307 0.036 −0.120 0.022 −1.225 0.062
−0.023 0.034 −0.094 0.018 −1.154 0.052
−0.356 0.035 −0.080 0.014 −1.119 0.044
−0.630 0.032 −0.062 0.016 −1.070 0.048
−0.822 0.021 −0.056 0.020 −1.054 0.066
−0.928 0.012 −0.041 0.025 −1.011 0.080
−0.974 0.008 −0.036 0.037 −0.999 0.137
Z = 0.0060
0.922 0.015 −0.142 0.088 −1.523 0.214
0.810 0.021 −0.098 0.057 −1.423 0.139
0.601 0.034 −0.068 0.042 −1.358 0.104
0.298 0.039 −0.042 0.034 −1.302 0.086
−0.070 0.039 −0.013 0.033 −1.237 0.083
−0.420 0.036 0.010 0.031 −1.188 0.080
−0.693 0.025 0.030 0.037 −1.145 0.098
−0.868 0.018 0.062 0.042 −1.068 0.112
−0.951 0.012 0.066 0.061 −1.061 0.168
erage” form of the PL relation might be useful which
does not explicitly show a dependence on HB morphol-
ogy. In the redder passbands, in particular, a meaningful
relation of that type may be obtained when one consid-
ers that the dependence of the zero points and slopes
of the corresponding relations, as presented in the pre-
vious sections, is fairly mild. Therefore, in the present
section, we present “average” relations for I, J, H, K,
obtaining by simply gathering together the 389,484 stars
in all of the simulations for all HB types and metallicities
(0.0005 ≤ Z ≤ 0.006). Utilizing a simple least-square s
procedure with the log-periods and log-metallicities as
TABLE 6
RRL PL Relation in J : Coefficients of the Fits
L σ(L ) a σ(a) b σ(b)
Z = 0.0005
0.934 0.013 −0.826 0.012 −1.902 0.045
0.877 0.018 −0.815 0.007 −1.890 0.024
0.776 0.022 −0.800 0.007 −1.861 0.020
0.627 0.027 −0.786 0.006 −1.825 0.017
0.414 0.028 −0.772 0.005 −1.788 0.015
0.167 0.029 −0.760 0.005 −1.757 0.015
−0.102 0.031 −0.754 0.004 −1.739 0.012
−0.358 0.031 −0.750 0.003 −1.727 0.011
−0.590 0.025 −0.748 0.004 −1.717 0.014
−0.765 0.021 −0.747 0.004 −1.710 0.018
−0.883 0.014 −0.746 0.005 −1.706 0.021
−0.950 0.010 −0.747 0.008 −1.709 0.040
Z = 0.0010
0.940 0.011 −0.795 0.020 −1.981 0.057
0.873 0.018 −0.774 0.014 −1.934 0.039
0.744 0.025 −0.751 0.010 −1.879 0.030
0.556 0.034 −0.733 0.008 −1.830 0.024
0.282 0.033 −0.717 0.007 −1.786 0.021
−0.037 0.035 −0.705 0.007 −1.752 0.020
−0.342 0.036 −0.696 0.006 −1.726 0.019
−0.603 0.025 −0.691 0.005 −1.710 0.017
−0.789 0.022 −0.687 0.006 −1.698 0.020
−0.906 0.015 −0.685 0.008 −1.689 0.030
−0.963 0.008 −0.684 0.011 −1.685 0.049
Z = 0.0020
0.965 0.009 −0.748 0.042 −2.005 0.123
0.910 0.015 −0.728 0.027 −1.955 0.072
0.794 0.025 −0.702 0.020 −1.890 0.056
0.594 0.029 −0.681 0.013 −1.837 0.038
0.307 0.036 −0.667 0.012 −1.800 0.034
−0.023 0.034 −0.653 0.010 −1.763 0.028
−0.356 0.035 −0.647 0.008 −1.745 0.024
−0.630 0.032 −0.638 0.009 −1.721 0.027
−0.822 0.021 −0.635 0.011 −1.713 0.034
−0.928 0.012 −0.628 0.014 −1.691 0.044
−0.974 0.008 −0.625 0.020 −1.682 0.076
Z = 0.0060
0.922 0.015 −0.649 0.052 −1.989 0.125
0.810 0.021 −0.623 0.033 −1.929 0.080
0.601 0.034 −0.606 0.024 −1.891 0.059
0.298 0.039 −0.591 0.019 −1.860 0.049
−0.070 0.039 −0.575 0.019 −1.826 0.047
−0.420 0.036 −0.563 0.017 −1.800 0.044
−0.693 0.025 −0.553 0.021 −1.777 0.053
−0.868 0.018 −0.536 0.023 −1.739 0.062
−0.951 0.012 −0.534 0.033 −1.733 0.092
independent variables, we obtain the following fits:
2
M
I
= 0.4711 − 1.1318 log P + 0.2053 log Z, (3)
with a correlation coefficient r = 0.967;
2
For all equations presented in this section, the statistical er-
rors in the deri ved coefficients are always very small, of order
10
−5
− 10
−3
, due to the very large number of stars involved in the
corresponding fits. Consequently, we omit them from the equations
that we provide. Undoubtedly, the main sources of error affecting
these relations are systematic rather than statistical—e.g., helium
abundances (see §5), bolometric corrections, temperature coeffi-
cient of the peri od-mean density relation, etc..

The RRL PL relation in U BV RIJHK 17
TABLE 7
RRL PL Relation in H : Coefficients of the Fits
L σ(L ) a σ(a) b σ(b)
Z = 0.0005
0.934 0.013 −1.136 0.003 −2.311 0.013
0.877 0.018 −1.137 0.002 −2.315 0.009
0.776 0.022 −1.137 0.002 −2.317 0.008
0.627 0.027 −1.137 0.002 −2.316 0.007
0.414 0.028 −1.137 0.002 −2.316 0.006
0.167 0.029 −1.138 0.002 −2.315 0.007
−0.102 0.031 −1.139 0.002 −2.316 0.006
−0.358 0.031 −1.141 0.002 −2.320 0.006
−0.590 0.025 −1.142 0.002 −2.320 0.008
−0.765 0.021 −1.143 0.003 −2.323 0.011
−0.883 0.014 −1.144 0.003 −2.322 0.013
−0.950 0.010 −1.146 0.005 −2.327 0.027
Z = 0.0010
0.940 0.011 −1.101 0.006 −2.358 0.019
0.873 0.018 −1.096 0.005 −2.346 0.014
0.744 0.025 −1.090 0.003 −2.332 0.009
0.556 0.034 −1.086 0.003 −2.322 0.009
0.282 0.033 −1.083 0.002 −2.310 0.007
−0.037 0.035 −1.081 0.002 −2.304 0.008
−0.342 0.036 −1.079 0.003 −2.297 0.008
−0.603 0.025 −1.079 0.002 −2.295 0.008
−0.789 0.022 −1.079 0.003 −2.295 0.011
−0.906 0.015 −1.080 0.004 −2.292 0.016
−0.963 0.008 −1.080 0.006 −2.292 0.025
Z = 0.0020
0.965 0.009 −1.058 0.015 −2.380 0.045
0.910 0.015 −1.051 0.009 −2.361 0.025
0.794 0.025 −1.042 0.007 −2.339 0.021
0.594 0.029 −1.035 0.005 −2.321 0.014
0.307 0.036 −1.031 0.005 −2.308 0.013
−0.023 0.034 −1.027 0.004 −2.297 0.012
−0.356 0.035 −1.025 0.003 −2.291 0.010
−0.630 0.032 −1.023 0.004 −2.284 0.012
−0.822 0.021 −1.023 0.004 −2.282 0.014
−0.928 0.012 −1.021 0.006 −2.275 0.019
−0.974 0.008 −1.020 0.009 −2.271 0.034
Z = 0.0060
0.922 0.015 −0.975 0.022 −2.381 0.053
0.810 0.021 −0.964 0.014 −2.355 0.035
0.601 0.034 −0.957 0.010 −2.338 0.026
0.298 0.039 −0.950 0.008 −2.325 0.021
−0.070 0.039 −0.944 0.008 −2.312 0.020
−0.420 0.036 −0.939 0.007 −2.301 0.019
−0.693 0.025 −0.935 0.009 −2.292 0.023
−0.868 0.018 −0.929 0.010 −2.277 0.027
−0.951 0.012 −0.928 0.014 −2.274 0.040
M
J
= −0.1409 − 1.7734 log P + 0.1899 log Z, (4)
with a correlation coefficient r = 0.993 6;
M
H
= −0.5508 − 2.3134 log P + 0.1780 log Z, (5)
with a correlation coefficient r = 0.999 1;
M
K
= −0.5968 − 2.3529 log P + 0.1746 log Z, (6)
with a correlation coefficient r = 0 .9992. No te that the
latter relation is of the same form as the one presented by
TABLE 8
RRL PL Relation in K: Coefficients of the Fits
L σ(L ) a σ(a) b σ(b)
Z = 0.0005
0.934 0.013 −1.168 0.002 −2.343 0.012
0.877 0.018 −1.169 0.002 −2.348 0.009
0.776 0.022 −1.170 0.002 −2.352 0.007
0.627 0.027 −1.171 0.002 −2.352 0.007
0.414 0.028 −1.172 0.002 −2.355 0.006
0.167 0.029 −1.173 0.002 −2.355 0.007
−0.102 0.031 −1.175 0.002 −2.358 0.006
−0.358 0.031 −1.177 0.002 −2.362 0.006
−0.590 0.025 −1.178 0.002 −2.364 0.008
−0.765 0.021 −1.180 0.003 −2.368 0.011
−0.883 0.014 −1.181 0.003 −2.367 0.013
−0.950 0.010 −1.183 0.005 −2.373 0.026
Z = 0.0010
0.940 0.011 −1.133 0.005 −2.388 0.017
0.873 0.018 −1.128 0.004 −2.379 0.013
0.744 0.025 −1.124 0.003 −2.367 0.009
0.556 0.034 −1.121 0.003 −2.359 0.009
0.282 0.033 −1.118 0.002 −2.350 0.007
−0.037 0.035 −1. 11 70.002 −2.345 0.008
−0.342 0.036 −1. 11 50.002 −2.339 0.008
−0.603 0.025 −1. 11 60.002 −2.338 0.008
−0.789 0.022 −1. 11 70.003 −2.339 0.011
−0.906 0.015 −1. 11 70.004 −2.337 0.016
−0.963 0.008 −1. 11 70.006 −2.338 0.024
Z = 0.0020
0.965 0.009 −1.091 0.014 −2.410 0.041
0.910 0.015 −1.084 0.008 −2.393 0.022
0.794 0.025 −1.077 0.007 −2.374 0.019
0.594 0.029 −1.071 0.004 −2.358 0.013
0.307 0.036 −1.067 0.004 −2.347 0.012
−0.023 0.034 −1.064 0.004 −2.337 0.011
−0.356 0.035 −1.063 0.003 −2.333 0.010
−0.630 0.032 −1.061 0.004 −2.327 0.012
−0.822 0.021 −1.061 0.004 −2.325 0.013
−0.928 0.012 −1.059 0.006 −2.320 0.018
−0.974 0.008 −1.059 0.009 −2.317 0.032
Z = 0.0060
0.922 0.015 −1.011 0.021 −2.413 0.049
0.810 0.021 −1.001 0.013 −2.389 0.032
0.601 0.034 −0.994 0.010 −2.374 0.024
0.298 0.039 −0.988 0.008 −2.362 0.019
−0.070 0.039 −0.983 0.008 −2.349 0.019
−0.420 0.036 −0.978 0.007 −2.340 0.018
−0.693 0.025 −0.974 0.008 −2.331 0.021
−0.868 0.018 −0.968 0.009 −2.317 0.025
−0.951 0.012 −0.968 0.013 −2.315 0.037
Bono et al. (2001). The metallicity dep e ndence we derive
is basically identical to that in Bono et al., whereas the
log P slope is slightly steeper (by 0.28), in absolute value,
in our c ase. In terms of zero points, the two relations, at
representative values P = 0.50 d and Z = 0.001, provide
K-band magnitudes which differ by only 0.05 mag, our s
being slightly brighter.
In addition, the same type of exercise can provide us
with an average relation between HB magnitude in the
visual a nd metallicity. Performing an ordinary least-
squares fit of the form M
V
= f (log Z) (i.e., with log Z
as the independent variable), we obtain:

18 M. Catelan, B. J. Pritzl, H. A. Smith
TABLE 9
Coefficients of the RRL PL Relation in BV RIJ HK: Analytical Fits
Metallicity a
0
a
1
a
2
a
3
b
0
b
1
b
2
b
3
V
Z = 0.0060 0.5276 −0.0420 0.1056 −0.2011 −0.9440 −0. 0436 0.3207 −0.5303
Z = 0.0020 0.4470 −0.0232 0.0666 −0.2309 −0.8512 0.0122 0.3107 −0.7248
Z = 0.0010 0.3976 −0.0416 0.0482 −0.2117 −0.8243 −0. 0231 0.3212 −0.7169
Z = 0.0005 0.3668 −0.0314 −0.0053 −0.1550 −0.7455 0.0407 0.1342 −0.4966
R
Z = 0.0060 0.2107 −0.0701 −0.0204 −0.0764 −1.0050 −0.1507 −0. 0390 −0.1723
Z = 0.0020 0.1469 −0.0566 −0.0633 −0.0995 −0.8523 −0.1500 −0. 1519 −0.2521
Z = 0.0010 0.1020 −0.0808 −0.0784 −0.0744 −0.8086 −0.2397 −0. 1809 −0.1608
Z = 0.0005 0.0659 −0.0811 −0.0832 −0.0537 −0.7553 −0.2482 −0. 1769 −0.0677
I
Z = 0.0060 −0.0162 −0.0540 −0.0216 −0.0640 −1.2459 −0.1155 −0.0442 −0.1466
Z = 0.0020 −0.0913 −0.0393 −0.0568 −0.0788 −1.1499 −0.1058 −0.1397 −0.2002
Z = 0.0010 −0.1445 −0.0545 −0.0680 −0.0614 −1.1340 −0.1647 −0.1614 −0.1378
Z = 0.0005 −0.1939 −0.0515 −0.0679 −0.0450 −1.1192 −0.1640 −0.1474 −0.0655
J
Z = 0.0060 −0.5766 −0.0278 −0.0150 −0.0378 −1.8291 −0.0581 −0.0318 −0.0874
Z = 0.0020 −0.6517 −0.0181 −0.0335 −0.0452 −1.7599 −0.0498 −0.0809 −0.1167
Z = 0.0010 −0.7023 −0.0250 −0.0383 −0.0355 −1.7478 −0.0786 −0.0891 −0.0824
Z = 0.0005 −0.7546 −0.0209 −0.0343 −0.0238 −1.7438 −0.0728 −0.0706 −0.0379
H
Z = 0.0060 −0.9447 −0.0110 −0.0070 −0.0160 −2.3128 −0.0231 −0.0149 −0.0375
Z = 0.0020 −1.0262 −0.0041 −0.0124 −0.0152 −2.2953 −0.0138 −0.0292 −0.0414
Z = 0.0010 −1.0798 −0.0037 −0.0108 −0.0079 −2.3019 −0.0170 −0.0242 −0.0186
Z = 0.0005 −1.1385 0.0037 −0.0027 0.0012 −2.3169 0.0016 −0.0029 0.0056
K
Z = 0.0060 −0.9829 −0.0101 −0.0066 −0.0148 −2.3499 −0.0210 −0.0140 −0.0346
Z = 0.0020 −1.0630 −0.0032 −0.0113 −0.0132 −2.3357 −0.0115 −0.0268 −0.0361
Z = 0.0010 −1.1159 −0.0024 −0.0093 −0.0061 −2.3427 −0.0131 −0.0212 −0.0142
Z = 0.0005 −1.1742 0.0051 −0.0012 0.0028 −2.3580 0.0061 −0.0004 0.0090
M
V
= 1.455 + 0.277 log Z, (7)
with a correlation coefficient r = 0.83.
The above equation has a stronger slope than is often
adopted in the literature (e.g., Chaboyer 1999). This is
likely due to the fact that the M
V
− [M/H] relation is ac-
tually non-linea r, with the slope increasing for Z & 0.001
(Castellani et al. 19 91), where most of our simulations
will be found. A quadratic version of the same equation
reads as follows:
M
V
= 2.288 + 0.8824 log Z + 0.1079 (log Z)
2
. (8)
As one can see, the slope provided by this rela tion, at
a metallicity Z = 0.001, is 0.235, thus fully compatible
with the range discussed by Chaboyer (1999).
The last two equations can a lso be placed in their more
usual form, with [M/H] (or [Fe/H]) as the independent
variable, if we recall, from Sweigart & Catelan (1998),
that the solar metallicity corresponding to the (scaled-
solar) evolutionary models utilized in the present paper
is Z = 0.01716. Therefore, the conversion between Z and
[M/H] that is appropriate for our models is as follows:
log Z = [M/H] − 1.765. (9)
The effects of an enhancement in α-capture elements
with respect to a solar-scaled mixture, as indeed observed
among most metal-poor stars in the Ga lactic halo, can
be taken into account by the following scaling relation
(Salaris, Chieffi, & Straniero 1 993):
[M/H] = [Fe /H] + log(0.638 f + 0.362), (10)
where f = 10
[α/Fe]
. Note that such a relation should be
used with due care for metallicities Z > 0.003 (Vanden-
Berg et al. 2000).
With these equations in mind, the linear version of the
M
V
− [M/H] relation becomes
M
V
= 0.967 + 0.277 [M/H], (11)
whereas the quadratic one re ads instead
M
V
= 1.067 + 0.502 [M/H] + 0.108 [M/H]
2
. (12)
The latter equation provides M
V
= 0.60 mag at [Fe/H] =
−1.5 (assuming [α/Fe] ≃ 0.3; e.g., Carney 1996), in

The RRL PL relation in U BV RIJHK 19
Fig. 13.— Effects of σ
M
and of the helium abundance upon the RRL PL relation in the I J H K bands. The lines i ndicate the analytical
fits obtained, f rom equations (1) and (2), for our assumed case of Y
MS
= 0.23 and σ
M
= 0.02 M
⊙
for a metallicity Z = 0.001 (thick gray
lines) or Z = 0.002 (dashed lines). In the two upper panels, the results of an additional set of models, computed by i ncreasing the mass
dispersion from 0.02 M
⊙
to 0.03 M
⊙
, are shown (gray circles). As one can see, σ
M
does not affect the relation in a significant way, so that
similar results for an increased σ
M
are omitted in the lower panels. The helium abundance, in turn, i s seen to play a much more important
role, particularly in defining the zero point of the PL relation.
very good agr eement with the favored values in Chaboyer
(1999) and Cacciari (2003)—thus supporting a distance
modulus for the LMC of (m − M )
0
= 18.47 mag.
To close, we note that the present models, which cover
only a modest range in metallicities, do not provide useful
input regarding the question of whether the M
V
−[M/H]
relation is better described by a parabola or by two
straight lines (Bono et al. 2003). To illustrate this, we
show, in Figure 14, the average RRL magnitudes for each
metallicity value considere d, along w ith equations (11)
and (12). Note that the “error bars” actually represent
the standard deviation of the mean over the full set of
RRL stars in the simulations for each [M/H] value.
7. CONCLUSIONS
We have presented RRL PL relations in the band-
passes of the Johnsons-Cousins -Glass UBV RIJHK sys-
tem. While in the case of the Cepheids the existence
of a PL relation is a necessary c onsequence of the large
range in luminosities encompassed by these variables, in
the case of RRL stars useful PL relations are instead pri-
marily due to the occasional presence of a large range
in bolometric corrections when going from the blue to

20 M. Catelan, B. J. Pritzl, H. A. Smith
Fig. 14.— Predicted correlation between average RRL V -band absolute magnitude and metallicity. Each of the four datapoints represents
the average magnitude over the ful l sample of simulations for that metallicity. The “error bars ” actually indicate the standard deviation of
the mean. The full and dashed lines show equations (11) and (12), respectively.
the red edge of the RRL instability strip. This leads to
particularly useful P L relations in IJHK, where the ef-
fects of luminosity and temperature conspire to produce
tight relations. In bluer passbands, on the other hand,
the effects of luminosity do not reinforce those of tem-
perature, leading to the presence of large scatter in the
relations and to a stronger dependence on evolutionary
effects. We provide a detailed tabulation of our derived
slopes and zero points for four different metallicities and
covering virtually the whole range in HB morphology,
from very red to very blue, fully taking into account,
for the first time, the deta iled effects of evolution away
from the zero-age HB upon the derived PL relations in
all of the UBV RIJHK passbands. We also provide “av-
erage” PL relations in IJHK, for applications in cases
where the HB type is not known a priori; as well as a new
calibration of the M
V
− [Fe/H] relation. In Paper II, we
will provide comparisons between these results and the
observations, particularly in I, where we expect our new
calibration to be especially use ful due to the wide avail-
ability and ease of observations in this filter, in compar-
ison with JHK.
M.C. acknowledges support by Proyecto FONDECYT
Regular No. 1030954. B.J.P. would like to thank the
National Science Foundation (NSF) for support through
a CAREER award, AST 99-84073. H.A.S. acknowledges
the NSF for support under grant AST 02-05813.
REFERENCES
Bono, G., Caputo, F., Castellani, V., Marconi, M., & Storm, J.,
2001, MNRAS, 326, 1183
Bono, G., Caputo, F., Castellani, V., Marconi, M., Storm, J., &
Degl’Innocenti, S. 2003, MNRAS, 344, 1097
Cacciari, C. 2003, in New Horizons in Globular Cluster Astronomy,
ASP Conf. Ser., Vol. 296, ed. G. Piotto, G. Meylan, S. G.
Djorgovski, & M. Riello (San Francisco: ASP), 329
Caputo, F., De Stefanis, P., Paez, E., & Quarta, M. L. 1987, A&AS,
68, 119
Caputo, F., Marconi, M ., & Santolamazza, P. 1998, MNRAS, 293,
364
Carney, B. W. 1996, PASP, 108, 900
Castellani, V., Chieffi, A., & Pulone, L. 1991, ApJS, 76, 911
Catelan, M. 1993, A&AS, 98, 547
Catelan, M. 2004a, ApJ, 600, 409
Catelan, M. 2004b, in Variable Stars in the Local Group, ASP Conf.
Ser., Vol. 310, ed. D. W. Kurtz & K. Pollard (San Francisco:
ASP), in press (astro-ph/0310159)
Catelan, M., Borissova, J., Sweigart, A. V., & Spassova, N. 1998,
ApJ, 494, 265
Chaboyer, B. 1999, in Post-Hipparcos Cosmic Candles, ed. A. Heck
& F. Caputo (Dordrecht: Kluwer), 111
Clem, J. L., VandenBerg, D. A., Grundahl, F., & B ell, R. A. 2004,
AJ, 127, 1227
Davidge, T. J., & Courteau, S. 1999, AJ, 117, 1297
Demarque, P., Zi nn, R., Lee, Y.- W., & Yi, S. 2000, AJ, 119, 1398
Ferraro, F. R., Paltrinieri, B., Fusi Pecci, F., Rood, R. T., &
Dorman, B. 1998, ApJ, 500, 311
Girardi, L., Bertelli, G., Bressan, A., Chiosi, C., Groenewegen, M.
A. T., Marigo, P., Salasnich, B., & Weiss, A. 2002, A&A, 391,
195
Isobe, T., Feigelson, E. D. , Akritas, M. G., & Babu, G. J. 1990,
ApJ, 364, 104
Leavitt, H. S. 1912, Harvard Circular, 173 (reported by E. C.
Pickering)
Longmore, A. J., Fernley, J. A., & Jameson, R. F. 1986, MNRAS,
220, 279
Pritzl, B. J., Smith, H. A., Catelan, M., & Sweigart, A. V. 2002,
AJ, 124, 949
Salaris, M., Chieffi, A., & Straniero, O. 1993, ApJ, 414, 580
Stellingwerf, R. F. 1984, ApJ, 277, 322
Sweigart, A. V., & C atelan, M. 1998, ApJ, 501, L63
Tanvir, N. R. 1999, in Post-Hipparcos Cosmic Candles, ed. A. Heck
& F. Caputo (Kluwer: Dordrecht), 17
van Albada, T. S., & Baker, N. 1971, ApJ, 169, 311
van Albada, T. S., & Baker, N. 1973, ApJ, 185, 477
VandenBerg, D. A., Swenson, F. J., Rogers, F. J., Iglesias, C. A.,
& Alexander, D. R. 2000, ApJ, 532, 430