A&A 404, 423-448 (2003)
DOI: 10.1051/0004-6361:20030354
G. A. Tammann1 - A. Sandage2 - B. Reindl1
1 - Astronomisches Institut der Universität Basel,
Venusstrasse 7, 4102 Binningen, Switzerland
2 -
The Observatories of the Carnegie Institution of Washington,
813 Santa Barbara Street, Pasadena, CA 91101, USA
Received 24 December 2002 / Accepted 5 March 2003
Abstract
321 Galactic fundamental-mode Cepheids with good B, V, and (in
most cases) I photometry by Berdnikov et al. (2000) and with
homogenized color excesses E(B-V) based on
Fernie et al. (1995) are used to determine their period-color
(P-C) relation in the range
.
The agreement with colors from different model calculations is good
to poor. - Distances of 25 Cepheids in open clusters and
associations (Feast 1999) and of 28 Cepheids with
Baade-Becker-Wesselink (BBW) distances (Gieren et al. 1998) are
used in a first step to determine the absorption coefficients
,
,
and
appropriate for Cepheids of intermediate color. The two sets of
Galactic Cepheids with known distances define two
independent P-L relations which agree very well in slope; their
zero points agree to within
.
They are therefore combined into a single mean Galactic
P-LB,V,I relation. The analysis of HIPPARCOS parallaxes by
Groenewegen & Oudmaijer (2000) gives absolute magnitudes which
are brighter by
in V and
in
I at
.
Agreement with P-L relations from different
model calculations for the case [Fe/H] = 0 is impressive to poor.
Galactic Cepheids are redder in (B-V)0 than those in LMC and
SMC as shown by the over 1000 Cloud Cepheids with good standard
B, V, I photometry by Udalski et al. (1999b,c);
the effect is less pronounced in (V-I)0. Also
the (B-V)0, (V-I)0 two-color diagrams differ
between Cepheids in the Galaxy and the Clouds, attributed both
to the effects of metallicity differences on the spectral energy
distributions of the Cepheids and to a shift in the effective
temperature of the middle of the instability strip for LMC and SMC
relative to the Galaxy by about
at
,
hotter for both LMC and SMC.
Differences in the period-color relations between the Galaxy and
the Clouds show that there cannot be a universal P-L relation from
galaxy-to-galaxy in any given color.
The inferred non-uniqueness of the slope of the P-L relations in
the Galaxy, LMC, and SMC is born out by the observations. The Cloud
Cepheids follow a shallower overall slope of the P-L relations in
B, V, and I than the Galactic ones. LMC and SMC Cepheids are
brighter in V than in the Galaxy by up to
at short
periods (
)
and fainter at long periods (
). The latter effect is enhanced by a suggestive break of the
P-L relation of LMC and SMC at
towards still
shallower values as shown in a forthcoming paper.
Key words: Cepheids - Magellanic Clouds - distance scale - supernovae: general - cosmological parameters
This is the first paper of a projected three paper series. The purpose of Paper I here is to begin a new analysis of the properties of classical fundamental-mode Cepheid variables, their two-color, period-color (P-C), and period-luminosity (P-L) relations in the Galaxy. We also give a preliminary comparison of these relations with their counterparts in the LMC and SMC.
The goal of the series is to provide a prelude to a projected final summary paper of our calibration of the absolute magnitudes of type Ia supernovae from the Cepheid distances to their parent galaxies (Sandage et al. 1992; Saha et al. 2001 for the first and last of eleven papers in that series; Parodi et al. 2000 for an interim summary), as these data relate to the value of the Hubble constant.
To maintain consistency throughout the supernovae calibration project that began in 1992, we provisionally adopted the same period-luminosity relation for Cepheids that had been proposed by Madore & Freedman (1991) and used by them in their HST Cepheid program for their new calibration of the Tully-Fisher relation. They assumed that the slope of the P-L relation was universal and was given by the Cepheids in the LMC. They also adopted a zero point based on their assumed LMC distance modulus of (m - M)0 = 18.50.
However, to make a new "finer analysis'' in our projected summary paper that is to come, we necessarily must start with a new discussion of the Cepheid period-luminosity relation itself, its slope, its zero point, and whether the relation is the same from galaxy-to-galaxy. The purpose of this paper and the two that will follow is to begin such a rediscussion of the Cepheid relations.
The question of galaxy-to-galaxy variations of the colors of Cepheids, based on photoelectric observations, was pioneered by Gascoigne & Kron (1965) who found SMC Cepheids to be bluer than Galactic ones. Gascoigne (1974) was also the first to discuss the metallicity effect on Cepheid luminosities. Here we address anew the Cepheid period-luminosity relation centered on the question if that relation is the same from galaxy-to-galaxy. Such an inquiry of uniqueness was beyond adequate solution until recently; the observational data on colors and absolute magnitudes by independent means was too meager for both the Galaxy and other external galaxies.
This has now changed because of four new comprehensive photometric studies of Cepheids in the Galaxy, in the two Magellanic Clouds, and in IC 1613. For the Galaxy, Berdnikov et al. (2000) have published homogeneous B, V, I photometry on the Cape (Cousins) photometric system (as realized by Landolt 1983, 1992) for hundreds of Galactic Cepheids for which Fernie (1994) and Fernie et al. (1995) have determined the E(B-V) color excesses.
The calibration of absolute magnitudes for the Galactic Cepheids has also an excellent new prospect using the 53 Cepheids for which Feast & Walker (1987), updated by Feast (1999), together with Gieren et al. (1998) have determined absolute magnitudes by two independent methods (main sequence fittings in clusters and associations, and the Baade-Becker-Wesselink [BBW] method), with good agreement between them.
In other galaxies, Udalski et al. (1999b,c, 2001) have published photoelectric B, V, I magnitudes, also on the Cape (Cousins) system, for well over 1000 Cepheids in LMC, SMC, and IC 1613, as well as individual values of E(B-V) determined from adjacent red-clump stars.
In the present paper we compare the color and absolute magnitude data for the Galaxy, the LMC and the SMC. The object is to test for either agreement or discrepancies in the slopes of the P-L relations, and in the color-color and period-color relations.
The plan in this paper for the Galactic Cepheids is manifold. In Sect. 2.1 the apparent mean B, V, I magnitudes as published by Berdnikov et al. (2000) for 324 Galactic Cepheids are listed in Table 1. These magnitudes were reduced by these authors from their own observations and from additional sources as listed in the original paper. Known overtone pulsators were excluded. Also listed are the E(B-V) color excess values based on nine studies from the literature as reduced to Fernie's reddening system in Sect. 2.2. This "Fernie system'' is then subjected to additional tests, and a small systematic correction is developed by a new procedure in Sect. 2.3. The same procedure is used in Sect. 2.4 to obtain "corrected'' E(V-I)color excesses.
The new reddening results are applied to the data in Table 1, resulting in a system of adopted reddening-free intrinsic (B-V)0 and (V-I)0 colors, also listed in Table 1. The period-color and color-color relations using these Galactic data are compared in Sect. 3.1 with those in the LMC and SMC. These comparisons all show systematic differences. To this point no absolute magnitude data have been needed.
Preliminary absolute magnitudes are introduced in
Sect. 4 (Tables 3 and 4)
from the independent calibrations of Feast (1999) and
Gieren et al. (1998). These are required before
discussion of a definitive P-L relation can be made because we
must calculate new values of the absorption-to-reddening ratios
required for the Cepheids. These, of course, differ from the
standard ratios that apply for the bluer O,B, and A supergiants
because of the effect of color differences on the values due to
the wide band widths of broad-band photometric systems. The
exact absorption-to-reddening ratios are important for the
Galactic Cepheids because their
median reddening is large at
,
requiring that
the large absorption corrections calculated with the ratios must be
very accurate.
Section 5 sets out how we have redetermined
these absorption ratios in B, V, and I using
distances from Feast and Gieren et al. from Sect. 4,
and then iterated to calculate definitive
ratios
valid for individual Cepheids depending on their individual colors.
With this preparation, we begin in Sect. 6.1 the discussion of the P-L relation using the calibrating data from Feast (1999) and Gieren et al. (1998). The independent P-L relations from cluster fittings and from the BBW method agree well, and are combined to produce our finally adopted Galactic P-L relations in B, V, and I of Eqs. (15), (16), and (17) and Fig. 11.
These data are then compared with the independently
determined slopes of the P-L relations of 650
absorption-corrected Cepheids in LMC and 405 equally treated Cepheids
in SMC from Udalski et al. (1999b,c), with
its zero-point determined by adopting the distance modulus of the LMC
using many methods that are independent of the Cepheids.
The slopes (at
)
differ
between the Galaxy, the LMC, and the SMC. This, and the
differences in the two-color (B-V vs. V-I)
relations between the same galaxies are the main conclusions of the
paper.
In Sect. 6.4 we try to make these differences disappear. We test if errors in the adopted reddenings or in the adopted ratios of absorption-to-reddening could cause the observed slope differences in the P-L relation, and in the zero-points of the period-color relations. We fail, and conclude therefrom that the differences are real if the results of Feast (1999) and Gieren et al. (1998) themselves are without systematic error.
In Sect. 7 we discuss the position of the instability strip in the HR diagram for the Galaxy and the LMC, showing again major differences. Changes in the ridge line of the strip in temperature at the blue (faint) end are proposed to explain the difference in the slopes of the P-L relations between the Galaxy and LMC. The general consequences of using Cepheids as distance indicators, in view of these apparent galaxy-to-galaxy differences, are discussed in the final section.
In view of the seriousness of these results, we set out in the following sections the tedious detail for the methods we have used.
The photometric data in B, V, and I as published by
Berdnikov et al. (2000) are listed in Table 1
for Galactic fundamental-mode Cepheids with periods larger than
.
Only confirmed classical, fundamental-mode Cepheids
(designated by DCEP by the authors) are accepted. The listed
photoelectric magnitudes are intensity means taken over the light
curves. The data in B and V are available for 321 Cepheids.
Of these, 250 also have intensity-reduced I magnitudes
on the Cape-Cousins-Landolt system. (It is to be noted that this system
differs substantially from the Johnson I photometric system,
Sandage 1997.) The Cepheids listed in
Table 1 have been selected such that they also
have E(B-V) color excesses from Fernie (1990), as updated by
Fernie et al. (1995).
The data in Cols. 2-5 in Table 1 are
directly from Berdnikov et al. (2000). The color excesses in
Col. 6 are on the Fernie system as averaged from nine independent
literature sources as calculated in the next section. The notation
denotes the color excess on "Fernie's system''
determined in this way.
We show in Sect. 2.3 that these values are
too large by a small systematic scale error
(Eq. (2) later).
Corrected values of
and the
corresponding
are shown in Cols. 7
and 8. The resulting intrinsic (B-V)0 and (V-I)0 colors
are in Cols. 9 and 10.
The selected sample of 321 Cepheids has a median color excess of E(B-V)=0.58, whereas Cepheids in external galaxies are preferentially detected if they suffer little extinction in their parent galaxies. Therefore, if one wants to compare Galactic and extragalactic Cepheids, and particularly if one wants to derive extragalactic distances, optimal extinction corrections are of paramount importance. Every effort should be made to apply reliable extinction corrections particularly to the Galactic Cepheids.
The largest body of E(B-V) values of Galactic Cepheids has
been derived by Fernie (1990, 1994) which are compiled
together with the data of 15 other authors in Fernie et al. (1995).
Nine of these external sources with more than 10 entries
have mean random deviations from Fernie's extinctions of
,
but they show (small) systematic deviations which can be well
expressed by a linear relation of the form
Table 2: The coefficients ai and bi in Eq. (1).
It is remarkable that the mean zero point of the nine
sources in Table 2 agrees with the one of
Fernie's to within
.
It seems therefore justified to
transform the external sources by means of
Eq. (1) into the system of Fernie. The
homogenized external E(B-V) for 271 Cepheids - averaged where
possible - are compared with the values of
Fernie (1990, 1994) in Fig. 1.
The scatter in Fig. 1 amounts to
.
Considering that the E(B-V)s by
Parsons & Bell (1975) alone carry a scatter of
into
the data, and that four of the homogenized sources deviate from
by
on average, one concludes
that the mean error of the color excesses, averaged over all
sources, is
.
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Figure 1:
The difference of the mean
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The values E(B-V) for 321 Cepheids in
Table 1, 271 of which are based on more than one
determination, will be referred to as
(FS
meaning Fernie's system) in the following.
The present procedure of deriving a consistent set of color
excesses leaves the possibility that the derived values of
still carry a scale error. A small scale
error is indeed detected and is corrected by the procedure in
Sect. 2.3 next.
We have made a test for a possible systematic scale error in
the Fernie system of the E(B-V) reddening values. We ask if the
Fernie reddening system derived in the last section shows
systematic trends such as a correlation of color residuals from
an adopted period-color relation with
values. Are the Cepheids with large
values
systematically redder or bluer in their derived intrinsic colors than
Cepheids of the same period with smaller E(B-V) values?
We have made the test by first deriving an interim
period-intrinsic color relation from the data in
Table 1 by combining the
reddening values (Col. 6) with the Berdnikov et al.
observed apparent B and V magnitudes listed in Cols. 3 and 4 of
Table 1.
This interim period-color-corrected relation (not shown) of
course has an intrinsic scatter (Sandage 1958, 1972)
because the lines of constant period in the HR diagram are sloped
through the instability strip that has a non-zero color width. Hence,
Cepheids will show a range of intrinsic (B-V) colors along that
constant period line as it threads through the instability strip
with a color width of
(Fig. 15
later).
Suppose that the assumed color excess
values on the Fernie system were to have a systematic scale
error such that large excess values are either too large or too
small. The consequence would be that the scatter in the derived
intrinsic (but slightly incorrect) color about a mean
color-period line would show a correlation with the assumed
excess value.
This, of course, cannot be physical. It would signal a scale
error in the assumed
values.
We have made the test, correlating the color residuals about
the initial (interim) color-period relation using the Fernie
system
values. A correlation of these residuals
with
is shown in Fig. 2,
signalling a small scale error in the Fernie system B-Vexcesses. There is a significant correlation of these color residuals
with
.
The Cepheids with large
excess values are too blue,
hence the FS excess values are too large.
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Figure 2:
The color residuals
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Figure 3:
The adopted Galactic P-C relation in (B-V)0 as
obtained from the corrected color excesses
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Laney & Stobie (1994) have derived from 47 Galactic Cepheids a
slightly steeper slope (0.416), but their color (B-V)0 at an
intermediate period of
agrees with
Eq. (3) to within
;
the latter is
redder by
at
and bluer by
at
.
The good agreement with independent data proves in favor of
the adopted zero point of the (B-V)0 colors.
The need for a correction to the Fernie system reddenings
can more directly be seen simply by dividing the
sample into two parts, one using the 100 Cepheids with the
smallest reddenings that range between
of
0.035 and 0.417 and the 100 Cepheids with the largest Fernie
system reddenings that range from 0.750 and 1.764 and comparing
the mean color residuals from the preliminary period-color relation in
these two groups. This is equivalent to reading the ordinate in
Fig. 2 at two mean abscissa points, one for each of
the divided groups. Of course this is the same test as made in
Fig. 2 by the continuous least squares line
calculated using all periods and reddening values.
There is a clear difference in the ordinate of
between these two mean
extremes in the divided
sample, in the sense that the group with the largest
is (artificially) bluer.
This can only be due to too large a reddening correction on the Fernie
system, consistent with Eq. (2).
Observed colors (V-I) follow from Table 1
for 250 Cepheids with known I magnitudes. In a first step it is
assumed that
(cf. Schlegel et al. 1998), although this value is not optimized
for the colors of Cepheids. Indeed, colors derived on this assumption
define a preliminary P-C relation in (V-I), whose residuals
still correlate with the adopted E(V-I) (Fig. 4).
Repeating the process with
leads to a considerably weaker dependence of
the color residuals on E(V-I), and after two more iterations any
such dependence is eliminated, which then requires
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Figure 4:
The color residuals
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Most of the scatter in Figs. 3 and 4,
i.e.
,
must be due to the intrinsic width of the
instability strip (cf. Sect. 7). This
proves in favor of the internal consistency of the color excesses
E(B-V) by Fernie et al. (1995), which together with the
photometry of Berdnikov et al. (2000) form the basis of Sect. 2.
The P-C relation in (V-I)0, derived by
Caldwell & Coulson (1986) is slightly steeper (slope 0.292) than
Eq. (5). Their (V-I)0 colors are redder from
(at
)
to
(at
).
It should be repeated that Table 1 contains that subset of Cepheids by Berdnikov et al. (2000) which are not suspected to be overtone pulsators or otherwise peculiar. Equations (3) and (5) should therefore reflect the true colors of an (almost) clean sample of fundamental pulsators.
Another point may be remarked here, which will become relevant in
Sect. 5 when discussing the absorption coefficients .
Since
Armed now with the intrinsic period-color relations in Figs. 3 and 5 and their mean (ridge) lines in Eqs. (3) and (5), we compare the P-C relations for the Galactic Cepheids with similar data for the LMC and SMC. That the zero point of these relations differ for the three galaxies is well known (C.D. Laney quoted by Feast 1991; Laney & Stobie 1994). The latter authors ascribed these differences to differences in chemical composition, presumably causing differences in the mean temperature and line blanketing. New comparisons are made in Fig. 6a for (B-V)0 and Fig. 6b for (V-I)0.
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Figure 5:
The final Galactic P-C relation in (V-I)0 as obtained
from the corrected color excesses
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Figure 6a:
The Period-Color relation in (B-V)0 for Cepheids in
the Galaxy, LMC, and SMC. The Galaxy data are copied from
Fig. 3. The solid line in each panel is the mean
relation for the Galaxy from Eq. (3). Three
theoretical models by Sandage et al. (1999; SBT),
Baraffe & Alibert (2001; BA), and
Caputo et al. (2000) are compared with the mean Galaxy line in
panel (d). - The paucity of red LMC Cepheids is purely
observational due to the few available B magnitudes and saturation
effects for
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Figure 6b: Same as Fig. 6a but for (V-I)0 colors. The Galaxy data are copied from Fig. 5. The Galaxy line in each panel is Eq. (5) of the text. |
The LMC relation in a forthcoming paper of this series is based on 650 Cepheids with uniform photometry in the standard B, V, I system and with color excesses which are determined from nearby red-clump stars and which are hence independent of the Cepheid metallicity (Udalski et al. 1999b). The P-C relation of SMC is based on 465 Cepheids whose data are equivalent to those of LMC (Udalski et al. 1999c).
The LMC Cepheids are bluer at all periods by
(depending on period) in (B-V)0 than those in the Galaxy.
The SMC Cepheids are still somewhat bluer in (B-V)0.
The color discrepancies between the Cepheids of individual
galaxies are less pronounced in (V-I)0. LMC Cepheids are bluer
than their Galactic counterparts by
(depending on
period), but the still metal-poorer SMC Cepheids are redder than those
in LMC.
The color differences cannot be explained by photometric errors of the required size. They can neither be explained by errors of the color excess corrections because of the contradictory behavior of the P-C relations, i.e. LMC Cepheids are redder in (B-V)0, but bluer in (V-I)0 than those in SMC. They can also not be due to overtone pulsators which have been carefully eliminated in the present samples.
The conclusion is that the P-C relations of the Galaxy, LMC,
and SMC are intrinsically different. If the Cepheids with
periods
had been included here this conclusion would be
further supported by SMC whose short-period Cepheids show a
break of the P-C relation (cf. Bauer et al. 1999).
The differences between the individual P-C relations leads to the unavoidable conclusion that the Cepheids in the Galaxy, LMC, and SMC cannot follow a unique P-L relation. This point will be taken up again in Sect. 6.
Model calculations of the P-C relations for [Fe/H] = 0.0 by Sandage et al. (1999; based on Geneva models; - in the following SBT), Caputo et al. (2000), and Baraffe & Alibert (2001) are shown in panel (d) of Fig. 6a,b. The Galactic P-C relation in (B-V)0 is best matched by the models of SBT and of Baraffe & Alibert (2001), but they are somewhat flat. These authors give also quite similar slopes of the P-C relations in (V-I)0 which, however, are flatter than the observed Galactic one. The (V-I)0 colors of Baraffe & Alibert (2001) are rather red. The model colors of Caputo et al. (2000) are much too red in (B-V)0 and (V-I)0.
A powerful diagnostic for differences in the energy
distributions in stellar atmospheres are two-color diagrams. If
such differences exist for Cepheids in different galaxies, they
can be most easily detected by comparing two-color diagrams,
galaxy-to-galaxy.
The comparison of the (B-V)0 vs. (V-I)0 two-color
diagram for the Galaxy, LMC, and SMC are shown in
Fig. 7a. The data for the Galaxy are from
Table 1. The data for LMC and SMC are from
Udalski et al. (1999b,c) corrected for reddening as
will be discussed in the Papers II and III for these galaxies.
The linear fit to the Galaxy data are repeated in all the panels of Fig. 7a. The theoretical lines in panel (d) are curved. They have been superposed on the data in panels (a) to (c) for [Fe/H] values of 0.0, -0.5 and -1.0 respectively from panel (d). The agreement in panel (a) is excellent between the predicted relation for [Fe/H] = 0 and the Galaxy data. Also the LMC and SMC Cepheids in panels (b) and (c) are remarkably well fit by the theoretical lines with the appropriate metallicity. The trends for different metallicities are clear.
Most important are the clear differences between the
diagrams for the Galaxy and the two Magellanic Clouds. These
differences are real; they cannot be due to errors in the
reddening because the reddening vector, shown in panel (a), has
very nearly the same slope as the correlation line itself for
the well known reason that the law of interstellar reddening of
roughly
is close to the black-body gradient for
temperature reddening. Hence, the differences shown in
Fig. 7a between the Cepheids in these three galaxies must
be real. Again, similar to the consequences of the differences in
Figs. 6a,b, these color differences in
Fig. 7a show that even if the P-L relations for the
Galaxy and the Clouds were to be identical say in I, they cannot be
the same in B or V.
A different presentation of the two-color diagrams is shown in
Fig. 7b. Instead of plotting individual Cepheids the
mean P-C relations in (B-V)0 and (V-I)0 are used
here. The Galactic P-C relations are given in
Eqs. (3) and (5). The preliminary P-C relations of LMC and SMC are taken from Eqs. (23) and (24), respectively
(26) and (27), below. The additional
information gained here over Fig. 7a are the loci of
different periods. These loci are separated by significant amounts in
the order of
in (B-V)0 and/or (V-I)0. The shift
between LMC and SMC is roughly orthogonal to the reddening line,
excluding thus errors of the adopted color excesses as a cause. The
shift of Cepheids in the two-color diagram proves again that the
Cepheids in the Galaxy, LMC, and SMC cannot follow a unique P-L
relation.
The strong shifts of the loci of constant period in
Fig. 7b cannot be explained by blanketing
alone. This conclusion is confirmed by Laney & Stobie (1986) who
concluded that SMC Cepheids are hotter than LMC Cepheids by
K at constant period, - a conclusion based on their
different P-C relations in (J-K) which are little affected by
metallicity. We shall return to the question of temperature
differences in Sect. 7.3.2,
where we require that the differences be a function of absolute
magnitude (Eqs. (37)-(39)).
It should be stressed that the adopted P-C relations of LMC and
SMC are preliminary. It has been suggested that the P-C relations of
LMC in (B-V)0 and (V-I)0 have a break at
(Tammann et al. 2002) turning to a steeper slope for longer
periods. This break also implies a discontinuity of the P-L
relation. The break occurs where also other Cepheid parameters change
abruptly, which is attributed to a fundamental-second overtone
resonance (Simon & Schmidt 1976; Simon & Lee 1981). The gap of the period
distribution at 8-10 days (cf. Figs. 3 and 5) has been related to the same phenomenon
(Buchler et al. 1997). Also the amplitude-color relations change
their behavior near
(Kanbur & Ngeow 2002). We will
persue the break of the P-C and P-L relations of LMC and probably SMC
in Papers II and III.
The simplified assumption of linear P-L relations of LMC and
SMC does not alter the qualitative conclusion here from
Fig. 7b that Cepheids at
fixed period in different galaxies are separated in the two-color
diagram.
The possible break of the P-C and P-L relations at
in LMC and SMC questions of course their linearity in the Galaxy over
the entire period interval. However, we have failed to detect any
significant non-linearity of the Galactic P-C and P-L relations.
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Figure 7b:
A comparison of the mean position in the two-color diagram
of Cepheids in the Galacty, LMC, and SMC.
The loci of differentvalues of ![]() |
To this point we have not needed data on absolute magnitudes
either for the Cepheids in the Galaxy or in the Clouds. However,
in all that follows, we now do require them. Final definitive
values are derived in the Sect. 6, based on new consistent
absorption-to-reddening ratios,
,
in B, V, and I calculated by an iterative method in Sect. 5.
To begin the iteration we have used the absolute magnitudes of
Cepheids in open clusters and associations and those from the moving
photosphere method. We then work from there to our finally adopted
calibrations in Sect. 6 for the P-L relations.
Feast (1999), following Feast & Walker (1987), has compiled a
list of 30 Cepheids with
which are members of open
clusters or associations. He has based their distances on an adopted
Pleiades modulus of
(m-M)0=5.57.
The much smaller HIPPARCOS distance is almost certainly due to a
systematic error of the trigonometric parallaxes. Therefore the best
photometric modulus of
(m-M)=5.61 (Stello & Nissen 2001)
is adopted here.
This value agrees well with the photometric distance of
Pinsonneault et al. (1998,
)
and even with the latest
result from HIPPARCOS (Makarov 2002,
).
The Cepheid luminosities are correspondingly
increased by
.
Not affected by this increase is RS Pup, which
has a geometrical distance.
UMi (Walker 1987a) and CS Vel
(Walker 1987b) were added.
V1726 Cyg was omitted as a probable overtone pulsator
(Turner et al. 1994). - For brevity we will refer to the 31
Cepheids with known distances as "open-cluster Cepheids''.
Table 3: Absolute B, V, I magnitudes of Cepheids in open clusters.
The data for the open-cluster Cepheids are compiled in
Table 3.
Columns 1, 4, and 5 are taken from Feast (1999) with the above
revisions. Columns 2 and 3 give the Galactic longitude and
latitude. The color excess
in Col. 6 is from
Fernie et al. (1995), but corrected by
Eq. (2). The
B, V, I magnitudes of the intensity mean in Cols. 7-9 are from
Berdnikov et al. (2000) and in
three cases from Sandage et al. (1999).
The dereddened colors (B-V)0 are shown in Col. 10. The
color-dependent absorption coefficients
as derived
in Sect. 5, are listed in Col. 11 (Note:
,
always).
The resulting absorption corrected absolute magnitudes
MB,V,I0 are given in the last three columns.
Expansion parallaxes of 34 Galactic Cepheids have been derived by Gieren et al. (1998) from the Baade-Becker-Wesselink (BBW) method as revised by Barnes & Evans (1976).
The data for the 34 Cepheids are laid out in
Table 4. The Table is organized like
Table 3. Columns 1, 4, and 5 are from
Gieren et al. (1998). The Galactic coordinates are in Cols. 2 and 3. The
of Fernie et al. (1995),
corrected by Eq. (2), are in Col. 6. The apparent
magnitudes B, V, I (Cols. 7-9) are from Gieren et al. (1998)
with slight revisions where available from
Berdnikov et al. (2000). The dereddened colors (B-V)0 are
shown in Col. 10. The color-dependent absorption values
(Col. 11) are analogous to those in
Table 3. Again, the finally adopted absolute magnitudes
are in Cols. 12-14.
Table 4:
Absolute B, V, I magnitudes of Cepheids with distances
from Gieren et al. (1998) (cf. notes in Table 3).
In Sect. 2 the color excesses
and
could be derived for
Cepheids. The next step is to derive the absorption coefficients
.
The Cepheid distances
provided by Feast (Table 3) and
Gieren et al. (Table 4) are ideally
suited for this purpose.
However, we have omitted five Cepheids which are known or suspected to be overtone pulsators as well as EV Sct which has a bright companion. We have also omitted the three Cepheids with BBW distances with long, yet variable periods; they do not render well to the BBW method. This leaves 25 Cepheids in Feast's list (Table 3) and 28 Cepheids in Gieren's et al. list (Table 4).
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Figure 8: The provisional P-L relations in B, V, and I for Galactic Cepheids in open clusters ( left panels) and with BBW distances ( right panels). The Cepheids shown as triangles are not used for the solution; at short periods they are probable overtone pulsators (shown here with their inferred fundamental period), the three long period Cepheids with BBW distances have variable periods. |
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As a first step, provisional absolute magnitudes
are formed with the data
in Tables 3 and 4.
The values
are determined from
in Col. 6 and on the assumption that
,
,
and
(cf. Schlegel et al. 1998).
The corresponding values
are plotted against
.
The resulting provisional P-L relations are shown in
Fig. 8. However, the residuals
are still a function of
(Fig. 9).
The brightening of the Cepheids with increasing
cannot be due to the color excesses themselves, which have
been freed of systematic effects. They must be explained by too large
absorption corrections, i.e. by an overestimate of the provisional
values
.
A reduction of the latter improves the
situation. After two more iterations any dependence of the Cepheid
magnitude on
is removed. The corresponding
values are
,
,
and
.
![]() |
Figure 9:
The magnitude residuals
![]() ![]() |
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The absorption coefficients
must also obey the
condition in Eq. (6); it holds in addition that
.
This leads to the best
compromise values of:
The present assumption of
in Eq. (8)
being color-independent is not exact, but it is required here because
of the linear approximation between E(V-I) and
E(B-V) in Eq. (4). This linear
approximation and the corresponding constancy of
are,
however, justified for the restricted color range of the Cepheids
under consideration.
As a test for systematic errors, we inquire in this
section if our
values depend on position in
the Galaxy. It is known
the
values may show variations in
special regions such as the Orion Nebula, as first discovered and
discussed by Baade & Minkowski (1937), and apparently verified
by Sharpless (1952). A comprehensive summary by
Sharpless (1963) reviewed the evidence from many studies up to
1962 where only a few large variations of
with environment
had been found.
However, Johnson (1966), in a controversial summary article,
suggested instead that large systematic variations of
with Galactic longitude exist (his Fig. 41 and Table 30), reaching
values of
AV/E(B-V) = 6 near Galactic longitude
,
with
the variation increasing to this level over the canonical value
of 3.8 over a longitude range from
to
.
If true, this would be most serious for our problem
here, introducing unacceptable errors in our finally adopted absolute
magnitudes in Tables 3 and 4.
We have tested for systematic variations of
with longitude for the Cepheids in Tables 3 and 4 by forming the ratios,
,
of the individual
absolute magnitude residuals from our final ridge-line P-L relations
(Eqs. (15), (16), and (17)
later) to our corrected
excess
values. Figure 10 shows the lack of
correlation of these ratios with Galactic longitude. There is no
obvious correlation.
With the distances of the calibrating Galactic Cepheids of Feast
(Table 3) and Gieren
et al. (Table 4), and the corrected color excesses
from Sect. 2.3 as
well as the new absorption coefficients
in
Sect. 5, one can derive absolute magnitudes
M0B,V,I. They lead to two independent sets of P-L
relations:
Open clusters (n= 25)
Also the scatter of
in
Eqs. (9-11) is as small as can be
expected considering the error of the distances of clusters and
associations, the possibility of the inclusion of non-members of
clusters and particularly of associations, and the large absorption
corrections. The scatter of the BBW P-L relations is even smaller.
Since there is no objective way to weight the two independent sets of P-L relations, the data are combined to
determine the mean P-L relations (n=53):
It is reassuring that the subtractions of Eq. (16)
from (15) and (17) from (16) leads to
the following P-C relations
Other systematic errors which might have been introduced by the
present reduction procedures are discussed in
Sect. 6.3.
![]() |
Figure 10:
The lack of correlation with Galactic longitude for the
ratio of absolute magnitude residuals from the adopted P-L
relations in Eqs. (15-17) to the
corrected
![]() |
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![]() |
Figure 11:
The combined P-L relations in B, V, and I from 25
Cepheids in open clusters (filled symbols) and 28 Cepheids with BBW
distances (open symbols). Triangles are not used for the solution
(cf. caption to Fig. 8). - The square in the
M0V and M0I panels are from the HIPPARCOS
calibration (Groenewegen & Oudmaijer 2000). - Upper and lower
boundaries of the P-L relations are shown on the assumption
that the constant-period lines have slopes of 3.52, 2.52, and
1.94 in B, V, and I, respectively, and that the instability
strip has a width of
![]() ![]() ![]() ![]() ![]() |
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There is one independent method of checking the zero point of the
P-L relation. Groenewegen & Oudmaijer (2000) have used 236 carefully
selected Cepheids with HIPPARCOS parallaxes to calibrate a P-L
relation in V and I with preselected slope. But their
values of
and
at
- the median period of their sample - should be quite independent of
any adopted slope.
This solution is very close to the earlier solution by
Feast & Catchpole (1997) based on fewer, but high-weight Cepheids.
The quoted magnitudes are made here somewhat
fainter for the following reason.The authors have adopted instead of
individual color excesses E(B-V) the mean P-C relations in (B-V)0 by Laney & Stobie (1995) and (V-I)0 by
Caldwell & Coulson (1986) which, averaged over all periods, agree
sufficiently well with Eqs. (3) and (5). The adopted values of E(B-V) have necessarily
considerable scatter, but the systematic errors should be small.
However Groenewegen & Oudmaijer (2000) have adopted variable
absorption coefficients which amount to
and
on average. These values appear now to be too high. With
and
from Eq. (7) and a median extinction of
E(B-V)=0.43 for their sample the magnitudes are overcorrected
for absorption by
on average in V and I. These
corrections have been included in the absolute magnitudes quoted
above.
The HIPPARCOS calibration is hence brighter than the
adopted P-L relations by
in B and
in
I. If one would average this determination with the calibration
through open clusters and BBW distances, giving equal weights, the
zero point would shift to brighter magnitudes by
-
.
This suggests that the adopted P-L relations are
reliable to within
.
Fry & Carney (1997) have measured metallicities of 14 calibrating
Cepheids in Table 3, two of which are
overtone pulsators; they have been excluded in the following. Eight of
the fundamental pulsators have also BBW distances from
Gieren et al. (1998). If one plots the magnitude residuals
off the P-L relations in Fig. 11 one
finds a barely significant correlation with [Fe/H]. The sense is that
metal-poor Cepheids are fainter. This is opposite to all model
calculations and to the evidence of the metal-poor LMC Cepheids
(cf. Sect. 6.3).
The suspicion then is that metal-poor Cepheids are undercorrected
for absorption. This can be if the
are still
inflicted by a metallicity effect, i.e. the color excesses of
metal-poor Galactic Cepheids are too small. This is indeed a constant
worry if E(B-V) is determined from the photometry of the
Cepheids themselves. If this interpretation is correct, the metal-poor
Cepheids must appear to be also redder, which is physically most
unlikely. In order to test this, the color residuals
and
off the P-C relations in
Eqs. (3) and (5) are plotted against
[Fe/H] for the (only) 12 Cepheids with known [Fe/H] in Fig. 12.
![]() |
Figure 12:
Color residuals
![]() ![]() |
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The metal-poor Cepheids appear indeed to be redder. Again, the
only explanation for this is to assume that the adopted color excesses
are too small for metal-poor Cepheids. If the
are underestimated by roughly
It must be added that Eq. (20) does not reflect the full dependence of the color excesses on metallicity. The correction is derived on the assumption that the intrinsic colors of Cepheids were independent of [Fe/H], but actually metal-poor Cepheids are expected to be bluer. The coefficient in Eq. (20) should therefore be larger by a yet undetermined amount.
As a consequence the P-C and P-L relations derived here apply to Galactic Cepheids with the mean metallicity of the calibrators, i.e. a value presumably not far from [Fe/H] =0.
In Table 5 previous Galactic P-L relations are compared with the present results. It is obvious that the earlier adopted P-L relations in B, V, and I are too flat. This is no surprise because it had been assumed that the slope in the Galaxy and the average slope in LMC are the same. The steepness of the Galactic slope is accentuated by the exclusion of EV Sct, which has a bright companion, and of the relatively bright overtone pulsators at short periods.
Table 5:
Galactic P-L relations by different authors. The
coefficients a and b are given for a P-L relation of the form
.
Hand in hand with the steep slope here, the new calibration is
considerably fainter at
by up to
than previous
values in Table 5. The difference is only
partially due to the absorption corrections which are smaller here
than adopted by most authors, because the color excesses E(B-V)by Fernie et al. (1995) had to be reduced here by 5% and the
values appropriate for Cepheids were found to be
somewhat smaller than adopted by some authors.
![]() |
Figure 13: Comparison of the present Galactic P-L relations in B, V, and I with model calculations (see text). BA is Baraffe & Alibert (2001). |
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The discrepancy in absolute magnitude decreases towards longer
periods. At
the MV magnitudes of all authors agree
to within
(except the very bright value of
Gieren et al. 1993). The new MI
magnitude at
is
brighter by
than the one from the widely used P-L
relation of Madore & Freedman (1991).
The P-L relations in Eqs. (15-17)
are compared in Fig. 13 with the results of model
calculations for the case of
.
The observed slopes in
B and V lie between the calculated slopes by
SBT (based on Geneva models) and
Baraffe & Alibert (2001). The agreement of the slopes in I is
nearly perfect. The curved P-L relations of Caputo et al. (2000) are
not realistic. The reason is probably their reliance on the calculated
position of the red edge of the instability strip
(Bono et al. 1999), which is sensitive to the treatment of
convection.
The zero point at
of Baraffe & Alibert (2001) is
somewhat bright, viz. by
in B to
in
I. SBT have calculated the luminosities of
pulsating models by Chiosi et al. (1992),
Schaerer et al. (1993, and references therein), and
Saio & Gautschy (1998). The results are similar; the ones from the
Schaerer et al. models are taken here as an
example. SBT refer to the blue edge of the
instability strip. Assuming half-widths of the instability strip of
roughly
,
,
and
in B, V, and I,
respectively, the calculated luminosities for the middle of the strip
have been reduced by the
corresponding amounts. They turn out to be brighter
than observed; at
the difference is
in B and
decreases to
in I.
The non-linear P-L relations of Caputo et al. (2000)
start out quite bright at
to become fainter than observed
by
-
at
depending on the waveband. In view
of the very faint luminosities at long periods their upward revision
of H0 (
km s-1Mpc-1) is unwarranted.
The conclusion is that it is not yet possible to derive reliable
luminosities of Galactic Cepheids from theory over the entire
period range. The best agreement is provided by
Baraffe & Alibert (2001) whose B, V, I magnitudes at
agree fortuitously well within
with the observed
ones.
The apparent P-L relations of LMC are very well defined because of the
excellent B, V, I photometry in the Johnson-Landolt-Cape-Cousins
systems (cf. Sandage 1997) of 650 Cepheids in the fundamental
mode by Udalski et al. (1999b). These authors have also determined
good color excesses E(B-V) from nearby red-clump giants; they
are independent of the metallicity of the Cepheids. Since the excesses
are relatively small,
,
any of their
possible errors cannot alter the following conclusions.
To show the difference between the P-L relations of the Galaxy and LMC the 650 LMC Cepheids are plotted in Fig. 14a with the mean Galactic P-L relations from Eqs. (15-17) overplotted. The LMC P-L relations are flatter in all three wave bands than the Galactic ones. This and the color differences are the main conclusion of the paper.
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Figure 14a: The position of 650 absorption-corrected fundamental-mode LMC Cepheids from Udalski et al. (1999b) in the P-L plane. The adopted LMC modulus is (m-M)0=18.54. Open symbols are Cepheids excluded by Udalski et al. (1999a). Overplotted are the mean Galactic P-L relations from Eqs. (15-17). |
For the comparison of the LMC and Galactic Cepheids in
Fig. 14a a Cepheid-independent distance modulus
of LMC was adopted. The relevant data are compiled in
Table 6. The straight and weighted means of the
entries are equal to
.
-
Table 6 contains also the latest results from
eclipsing binaries and red-clump stars. These two methods were
formerly suggested to give a significantly lower LMC modulus. The low
luminosities in the literature based on statistical parallaxes of
Galactic RR Lyr stars are very sensitive to sample selection and
unreconcilable with other calibrations; they are given zero weight
(cf. Walker 1999; Zaritsky 1999).
Table 6: The Distance Modulus of LMC independent of the P-L relation of Cepheids.
At
the LMC Cepheids are brighter by
,
,
and
in B, V, and I, respectively, than their
Galactic counterparts; their overluminosity decreases towards
.
Then above at
Galactic Cepheids are
brighter.
Of course, the crossing point can be shifted towards shorter
periods by increasing the Galactic calibration within the errors
(
)
or/and by arbitrarily decreasing the LMC
distance. But it is impossible to reach agreement between the
Galactic and LMC P-L relations, because their slopes are
different.
Any attempt to derive an accurate Cepheid distance of LMC by
means of a calibrated Galactic P-L relation is therefore highly
compromised over the entire period range (see
Sects. 8 and 9). The
resulting distance will be a function of period in any case, the
amplitude in
being
in all three colors,
i.e. larger distances at longer periods.
The difference of the P-L relations in different galaxies does not come as a total surprise. Already Laney & Stobie (1994) had convincingly shown that the SMC Cepheids are bluer in (B-V) than those in LMC and in the Galaxy similar to Fig. 6a. This implies that at least the P-L relation in B or/and in V cannot be the same in the three galaxies.
The comparison of the Galactic P-L relations with those of SMC
is shown in Fig. 14b on the assumption that
(SBT). It is obvious, independent of any distance,
that the P-L relations in B, V, and I of SMC are flatter than
those of the Galaxy.
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Figure 14b: The position of 489 absorption-corrected fundamental-mode SMC Cepheids from Udalski et al. (1999c) in the P-L plane. The adopted SMC modulus is (m-M)0=19.00. Symbols as in Fig. 14a. |
The fist question as to systematic errors concerns the zero points
of the adopted color excesses E(B-V). Fernie's own Galactic zero
point is confirmed to within a mean difference of
by nine external sources (cf. Table 2). As
discussed in Sects. 2.3 and 2.4, the adopted intrinsic colors also agree
with those of Laney & Stobie (1994) in (B-V)0 to
within
.
Hence, the somewhat larger deviation of the present
(V-I)0 colors from the early work of
Caldwell & Coulson (1986) cannot be ascribed to errors in
E(B-V).
In view of this argument, the Galactic colors are believed to be good
to within
.
- Also, in a forthcoming paper it will be
shown that the P-L relations in B and V of LMC used here agree
with those from 99 Cepheids with independent B,V photometry.
The latter are taken from Laney & Stobie (1994) and a compilation by
Sandage (1988); for 29 of them E(B-V) is available from
Caldwell & Coulson (1986), for the remaining ones
was assumed. The two sets of mB0 and mV0 magnitudes agree
to within
and
,
respectively.
This suggests that the E(B-V)s of Udalski et al. are reliable at
the
level (assuming
as in the
Galaxy). Udalski's et al. color excesses for SMC Cepheids are derived
in the same way as in LMC and should therefore be subject to equally
small errors. The conclusion is that the errors in intrinsic color
between the Galaxy, LMC, and SMC are hardly larger than
.
The next question is whether the well determined value of the
Galactic extinction-absorption ratio
is also applicable
to the Clouds. In view of the moderate color excesses of the Cloud
Cepheids any reasonable variation of
remains, however,
without consequences for the present conclusions concerning the
differences of the P-L relations.
The crucial question, however, is whether the different slopes of the P-L relations of the Galaxy, LMC, and SMC could be an artefact. The consequences of the non-uniqueness of the P-L relation in Figs. 14a and b are so severe that we have made several tests to try to make the differences go away.
The slopes in the P-L relation for LMC and SMC are so well determined, based on so many stars, and the reddening corrections for both Clouds are so small that there is scarcely any doubt that the data for the Clouds in Figs. 14a,b are secure. Therefore, we must study the data and procedures for the Galaxy (Tables 3 and 4), not in LMC or SMC in searching for a possible systematic error in Figs. 11 and 14a,b.
What seems so compelling is the near perfect agreement in
the independent results on distances, and therefore absolute
magnitudes, between the listings by Feast (1999) and by
Gieren et al. (1998). The P-L Eqs. (9-11) from the data by Feast (to be sure, as rediscussed by us for reddenings and values) are
nearly identical with Eqs. (12-14) using the rediscussed
data based on Gieren et al. The agreement is emphasized by the
lack of difference in Fig. 11 between the open
and closed circles. Hence, if there is a systematic error in the
Galactic data, its effect must be the same between the independent
methods used by Feast (cluster main sequence fittings) and Gieren et al.
(BBW) even through the methods are so fundamentally different.
We first ask if our new procedures of "correcting" Fernie-
system reddenings by a factor of 0.951 (Eq. (2))
and/or our use of the large value of
in Eq. (8) as 0.44 (Buser 1978) rather than what others
have used as 0.28 (Schmidt-Kaler 1965), or 0.25 (Olson 1975)
could have introduced errors in the Galactic P-L slope.
What at first seemed as a possibility for a systematic
error in the slope was that if E(B-V) was a function of (larger excesses for longer period Cepheids), and if our
adopted E(B-V) values were in error we could produce an effect.
The same type of a priori reasoning could also be made on the
effect of incorrect
in Eq. (8).
However, there is no correlation between the E(B-V) values
in Tables 3 and 4 and the period. Large
and small reddenings occur at all periods between
of 0.6 and
1.8. Furthermore, the reddenings on the Fernie system, reduced by
Eq. (2) and listed in Table 3, are
nearly identical (mean difference
)
with those used by
Feast (1999) that were determined using the early-type
stars in the clusters, entirely independently of the Cepheid
procedures used for the Fernie-system reddenings.
Therefore, the basis for any such a priori argumentation on E(B-V) and
fails.
Nevertheless, we tested directly for the effect of
variations in the reddening and
assumptions by making a
series of new reductions of the absolute magnitudes in
Tables 3 and 4 using a variety of
assumptions. Table 7 shows the results
of first using our corrected reddenings listed in
Tables 3 and 4 but using
values of
0.44, 0.25, and zero. We repeated the
calculations using Fernie-system reddenings (not corrected
by Eq. (2)) and again with the three values of
.
The slopes are extremely stable, not changing by nearly
enough to eliminate the differences in Figs. 14a,b.
The reason is that using
instead of
changes the
ratios to smaller
values, compensating in the absolute magnitudes for the larger
.
Table 7:
The sensitivity of the P-L slope from the
Tables 3 and 4 data to assumptions on
reddening and broad-band effects ("Corr'' refers to
from Eq. (2); "FS''
refers to the
in the Fernie system).
It should also be noted that Gieren et al. (1998) themselves
derived a steeper slope with their Galactic data (
;
), very similar to ours, of
and
(Eqs. (16) and (17)), respectively.
But they abandoned their own slopes and adopted the less steep
LMC slopes of -2.769 in V and -3.041 in I on the
assumption that the LMC slopes are universal.
We conclude that if Figs. 14a,b are wrong,
then both the Tables 3 and 4
absolute magnitudes are wrong by some systematic error that does not
depend on reddening errors or the broad-band color effects corrected
by Eq. (8). There would have to be errors in the
absolute magnitudes derived independently by cluster fitting
and by the BBW method.
However, as Gieren et al. (1998) use secure temperatures derived
from JHK near infrared colors where the effects of any reddening
errors are small (Gieren's et al. E(B-V) reddenings differ from
our corrected values in Table 4 by only
on
average) their route via the Barnes-Evans effect also seems secure.
We finally must comment on why Figs. 14a,b were so
unexpected in view of the earlier result by one of us
(SBT) that the Galactic Cepheids from
Feast & Walker (1987) show nearly the same slope as the P-L
relation for LMC, and that the equations shown there (coded in their
Figs. 10 and 11) differ from Eqs. (15-17)
here. There are two reasons. (1) In Tables 3 and 4 we rigorously exclude possible overtone pulsators,
whereas they were not excluded in the Feast/Walker comparison.
(2) Comparing the Feast/Walker data star-by-star with our
Table 3 values derived by the iterated procedure for the
corrected reddenings and the absorption-to-reddening
values
shows that there are six Cepheids with
that are
brighter in Feast/Walker than in Table 3 by an average of
.
These, together with the five overtone pulsators that are
also brighter, skewed the least-squares correlation lines in SBT to
their less steep values, nearly imitating the LMC slope.
If the P-C relations (Eqs. (3) and (5)) are
used to substitute the
term in the P-LV,I relations
(Eqs. (16) and (17)) one obtains:
![]() |
Figure 15: The Galactic instability strip. Symbols are as in Fig. 11. The full lines correspond to Eqs. (21) and (22), respectively. Dashed lines indicate the adopted blue and red edges of the strip. Three outlying Cepheids are identified. |
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Also shown in Fig. 15 are the Cepheids with
open-cluster and BBW distances from Tables 3 and 4. They define a narrow instability strip,
i.e.
in (B-V)0 and
in
(V-I)0. These values are upper limits considering that
the observational errors are in the order of
in absolute
magnitude and at least
in color.
The 314 Cepheids in LMC from Udalski et al. (1999b) and the 486 Cepheids in SMC from Udalski et al. (1999c) are shown as black dots in the MV-(B-V)0 respectively (V-I)0 plane in Fig. 16. Some additional Cepheids, excluded by Udalski et al. on the basis of their large scatter, are shown as open symbols.
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Figure 16: Comparison of the instability strip position in the Galaxy, LMC, and SMC. The dashed line is the ridge line of the instability strip of LMC and SMC from Eqs. (29-32) using the Cepheids adopted by Udalski et al. (1999a). The dotted lines are from all their fundamental-mode Cepheids. The full drawn lines are the Galactic ridge lines from Fig. 15. |
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To obtain mean relations defined by these Cepheids, i.e. the ridge line of the respective instability strips, we have combined the mean P-C and P-L relations of LMC and SMC.
A linear fit (cf. Sect. 3.2) over the whole
period range of the good LMC Cepheids gives
In spite of the relatively large slope errors there is considerable evidence that the slopes of the instability strips [in (B-V)0 and (V-I)0] decrease from the Galaxy through LMC to SMC, i.e. with decreasing metallicity. (The one exception is the LMC slope in the (V-I)0 diagram which is insignificantly steeper than that of the Galaxy.) The blueward shift of the LMC and SMC Cepheids as compared to the Galaxy is manifest in all panels of Fig. 16.
If we had included also the Cepheids which Udalski et al. (1999a) have excluded (dotted lines in Fig. 16), the above conclusions would remain unchanged.
We repeat the central point made earlier, if the slope of the P-L relation for Galactic Cepheids were to be the same in some color, say in I (nearly as in Figs. 14a,b), for the Galaxy compared with that in LMC and SMC, it clearly would not be the same in B and V because of the color offsets in Fig. 16.
Figures 14a,b emphasizes the main conclusion of this paper; the P-L relations are different between the three galaxies. Figures 15 and 16 provide the clue to a possible explanation for the evident slope difference in the next section.
The differences in the positions of the observed instability strips
between the Galaxy, LMC, and SMC in the color offsets at given
absolute magnitudes in Fig. 16 contain the
reason for the slope differences in the P-L relations in
Figs. 14a,b. In the subsections that follow we first
show that these differences in the P-L relations in
Figs. 14a,b must be due to differences in the effective mean temperature and slopes of the
,
relations in the middle of the strip. We then show in
Sect. 7.3.2 that the data
in Fig. 16 show just the required slope and
zero point differences in the temperatures at given absolute
magnitudes. First consider the theoretical expectations.
The problem has been discussed in tedious detail by
SBT where they investigated the effect of
metallicity on the P-L relation as [Fe/H] was varied from +0.3 to -1.3. The method was brute-force straight-forward by adopting
equations for the blue and red edges (for the fundamental pulsation
mode) in the HR diagram as the edge equations
.
These edges were then placed in the theoretical HR diagrams
where evolution tracks for stellar
masses from 1 to 25 solar masses had been calculated for different
metallicities (Z) and helium (Y) abundances by three independent
groups (Geneva, Padua, and Basel).
Where these tracks enter and exit the Cepheid instability strip
give mass/luminosity and temperature/luminosity relations for
Cepheids. These two relations are all that are needed to enter an
adopted pulsation equation,
,
to
obtain a series of P-L relations for different metallicities and
helium abundances. These bolometric P-L relations were then changed to
P-L relations in B, V, and I passbands by a grid of atmospheric
models giving B - V and V - I colors for various
,
[Fe/H],
,
and microturbulence velocities (SBT, Table 6). All
of this is relevant to the present paper and can be carried over to
the present problem of explaining Figs. 14a,b using the
data in Fig. 16.
Consider first the pulsation equation. There is a vast literature and many versions, the later equations showing explicit dependences on the metal and helium abundances. Remarkably, most versions yield nearly identical final P-L relations. SBT tested five such equations and compared all predicted P-L relations. For the purposes of illustration in this section it is sufficient to write only one of these, that due early to van Albada & Baker (1973).
Although this equation is devised by
van Albada & Baker for RR Lyrae stars, remarkably it
predicts the same P-L relation to within
as the
Cepheid equations of Iben & Tuggle (1975, their Eq. (3)),
Chiosi et al. (1992, their Eq. (5)), and Saio & Gautschy (1998)
for the chemical parameters used by SBT.
The reasons for the near identical results of the equations that
have a chemical dependence and those that do not are discussed in
detail by SBT (their Sect. 2.4) and is of no concern here. (It comes about because of compensating trends if one assumes that Yincreases with increased Z as a result of nucleosynthesis). Our
point is that the first order equation of
van Albada & Baker (1973) has been shown to be entirely adequate
to illustrate what we wish to show in this section.
The pulsation equation of van Albada & Baker is:
We now ask for the slope of the predicted P-L relation, as
,
in terms of magnitude rather than luminosity.
Let the mass/luminosity relation, derived as described above from
the intersection of the evolutionary tracks with the mean of the
blue and red edges in the theoretical HR diagram, be:
Because the bolometric correction to convert
to
MV is zero to within
over the temperature range we are
dealing with (Table 6 of SBT), this value for the slope is nearly
identical of the
relation as well. Remarkably, this is
the slope within the errors of Eq. (16) for the Galactic
Cepheids. We are clearly on the right track.
We next inquire how variations of the constants a and c in
Eqs. (34) and (35)
affect the slopes in Eq. (36). Because the
coefficient c on the temperature is so much stronger than the
coefficient a on the mass, it is clear that temperature variations
of the effective mid-point of the instability strip will be the
principal cause of any slope variations between the Galaxy, LMC, and
SMC in Figs. 14a,b. For example, a variation in the
coefficient c from -0.04 to -0.07 changes the P-L slope from
to -2.84, in the direction that
Eq. (16) differs from Eqs. (25) and (28) for the SMC. An exact correspondence
could be obtained by also changing the value of a that can be found
by reading Tables 1-5 of SBT for the mass/luminosity relation for
different metallicities.
Hence, the next step is to determine if the color difference between the Galaxy, LMC and SMC in Fig. 16 will yield temperature differences in the slope of the respective effective luminosity/temperature relations of the respective instability strips.
We now inquire if the color shifts in Fig. 16
relative to the Galaxy of
for LMC at
MV=-4 and
for SMC also at MV=-4(comparing Eq. (21) with Eqs. (29)
and (31)), and
for LMC and
for SMC also at
MV,I=-4(comparing Eq. (22) with Eqs. (30)
and (32)) can be explained by the metallicity effect
alone due to decreased blanketing, or do we require an additional
shift in temperature as well (toward hotter temperature for LMC and
SMC relative to the Galaxy)?
Assuming [Fe/H] =-0.5 for LMC and -1.0 for SMC we can read
the differences in the - (B-V) and
- (V-I) relations at fixed
for different
metallicities from Table 6 of SBT from their model atmosphere
grids. Although there is also a
and a microturbulence
dependence, they are relatively weak.
Reading the tables at
and a turbulent velocity of
,
the result is that we can account for color shifts of
not more than
in (B-V) for [Fe/H] =-0.5 and
for [Fe/H] =-1.0. The color shift in (V-I) is only
for each of these metallicities. These expected color shifts due to
blanketing are less than the observed shifts in
Fig. 16 by about a factor of 2. A real
temperature shift is required.
With Eqs. (21, 22,
29-32), together with Table 6 of SBT
we have the machinery to calculate the effective strip
equations for the Galaxy, LMC, and SMC, and thereby to calculate the
c slope in Eq. (35), and therefore to
determine the expected change of the P-L slope in
Eq. (36) due to slope differences in Eq. (35).
Using Table 6 of SBT with
and turbulence of
,
and using both the (B-V) and (V-I) data from the Eqs. (21, 22, 29-32),
and averaging the results from the (B-V) and (V-I) data
separately the results are as follows.
For the Galaxy:
What is clear from the excess blueness in LMC and SMC in excess to
what is expected from the blanketing effect alone from the low
metallicities of LMC and SMC is that there must be a temperature
offset toward hotter values in LMC and SMC relative to the Galaxy. The
amount at
from
Eqs. (37-39) is
for LMC and
for
SMC. These, of course translate into a zero point difference in
absolute magnitude (Eq. (34)) of
at
constant P, the LMC and SMC P-L relations being brighter, again in
agreement with Figs. 14a,b.
Hence, the data on colors in Fig. 16 are in qualitative agreement with the requirements of steeper slopes and brighter absolute magnitudes in the mid period range for LMC and SMC compared with the Galaxy.
These temperature shifts refer to the mid points in the HR diagram of the distribution of the Cepheids in the three
galaxies in the entire instability strip which has an intrinsic width
of
.
Hence, the midpoint average temperature
variation of
amounts to only 25% of the
intrinsic width. This need not be an astrophysical effect in the
atmospheres of the Cepheids, but rather could be due to say the
position of the red edge changing as a function of metallicity (Eq. (10)
of Chiosi et al. 1992, but it changes in the opposite direction
with metallicity than is required here), or variations in the filling
factor of the strip with metallicity as suggested by
Simon & Young (1997).
We claim nothing definitive in the discussion in this Sect. 7.3 but only that there are many factors that go into the determination of the slopes and zero points of period-luminosity relations for Cepheids. With this deeper investigation of the possibilities to explain the differences in Figs. 14a,b in slope and zero point, we are left with the impression that the last thing we should expect is now that the P-L relations from galaxy-to-galaxy should ever be the same. Our main purpose in this paper is to suggest from Figs. 14a,b that indeed they are not the same in the Galaxy, LMC, and SMC.
The non-uniqueness of the P-L relation makes the application of Cepheids as distance indicators more complex. The different slopes of the P-L relations in the Galaxy and in LMC make it impossible to determine, for instance, the LMC distance from a Galactic calibration of the P-L relation. The situation is even worse for a period-luminosity-color (P-L-C) relation, because the difference of the P-C relations of different galaxies introduces a still wider diversity. (An additional problem of the P-L-C relation is that the slope of the constant-period lines varies with the luminosity (Saio & Gautschy 1998; see also Paper II)).
The apparently hopeless situation is alleviated in part by the fact that it could be shown that at least the dominant reason for the different P-L relations is due to metallicity differences which affects the colors of Cepheids not only by blanketing but may also be the cause of the different temperatures at constant luminosity for LMC and SMC compared with the Galaxy. This leaves, however, the necessity to find independent ways to calibrate the P-L relation in function of metallicity, and/or to determine the temperature of the ridge line of the strip.
The effect of the new P-L relations on previous Cepheid distances is complex. While distances of galaxies with roughly the metallicity of LMC remain so far unchanged (a detailed discussion is deferred to Paper II), the distances of metal-rich galaxies comparable to the Galaxy should now rather be derived from the Galactic P-L relation.
Most available Cepheid distances are based on V and Iobservations with HST. They lead to the apparent distance moduli
and
.
The true modulus is then given by
A list of 31 available Cepheid distances, relevant for the
calibration of the extragalactic distance scale, is given by
Freedman et al. (2001). They have based their Cepheid distances on a
revised P-L relation in I which is even flatter than that of
Madore & Freedman (1991), and have applied a bulk correction for
metallicity. The 25 metal-rich galaxies in their list with a mean
metallicity of
,
i.e. close to the Galaxy,
should now be rather determined from the steep Galactic P-L relations
in V and I. The resulting corrections depend on the median period
of the Cepheids in each galaxy and amounts for the 25 galaxies to a
distance increase of
on average. This reduces their suggested
value of H0=72 to H0=68, which, however, is still affected
by bias effects of their secondary distance indicators.
More relevant for the determination of the large-scale value of
H0 are the Cepheid distances of the nine galaxies which are used
to calibrate the luminosity of supernovae of type Ia
(cf. Parodi et al. 2000; Saha et al. 2001; Tammann et al. 2002). The
distances have been determined with the traditional P-L relations of
Madore & Freedman (1991), yet adopting the zero point at
.
Six or seven of the nine galaxies are at least as
metal-rich on average as the Galaxy. If their distances are now
based on the new P-L relations of Eqs. (16) and (17), the resulting distances are only slightly larger than
from the Madore & Freedman P-L relation and an adopted zero point of
(Tammann et al. 2002).
Thus the value of
remains virtually unchanged.
The interesting fact is that this value is no longer based on only an
adopted distance of LMC, but mainly on the distances of Galactic
clusters (with the Pleiades zero point at
(m-M)=5.61) and on
purely physical BBW distances by Gieren et al. (1998). If the
HIPPARCOS calibration of the P-L relations by
Groenewegen & Oudmaijer (2000) had been included, H0 would be
decreased by only
3 percent.
A detailed analysis will be given in the summary paper of the HST
program for the luminosity calibration of SNe Ia.
The small effect of the P-L relations in Eqs. (16)
and (17) on H0 is due to the coincidence that
the steep P-L relations of the Galaxy and the flat P-L relations of
LMC cross just about at
,
i.e. close to the median period
of the Cepheids of the calibrating galaxies.
Classical Cepheids pulsating in the fundamental mode in the Galaxy, LMC, and SMC define different P-C relations in (B-V)0 and (V-I)0, LMC Cepheids being bluer than Galactic ones in both colors. SMC Cepheids are still bluer in (B-V)0, but redder in (V-I)0 than those in LMC.
The different P-C relations preclude the possibility of a
universal P-L relation in B, V, and/or I. In fact the Galactic
P-L relation is quite steep in all three colors as shown by 28 Cepheids with BBW distance of Gieren et al. (1998) and independently
by Feast's (1999) 25 Cepheids which are members of
open clusters or associations. The corresponding P-L relations in LMC,
based on 650 Cepheids with standard B, V, I photometry by
Udalski et al. (1999b), have a significantly flatter overall slope,
- an effect which is even more pronounced in SMC
(Udalski et al. 1999c). The consequence is that the Cepheids in the
Clouds are brighter than their Galactic counterparts at short periods
(by up to
), but somewhat fainter at long periods.
The requirement that the residuals of the Galactic P-C and P-L
relations be independent of the size of the (large) color excesses
E(B-V) in Fernie's (1994) system leads to a
slight reduction of the published values of E(B-V) and to a
calibration of the E(B-V)-absorption ratios of
,
,
and
.
The values apply to Cepheids of intermediate color.
varies from 4.06-4.35 for the bluest and reddest Cepheids,
respectively.
As a consequence of the different P-C and P-L relations in the Galaxy and in the Clouds, the ridge line of the instability strip in the MV - (B-V)0 and MI - (V-I)0planes of LMC and SMC is shifted bluewards with respect to the Galaxy. It is shown that these shifts are caused by a metallicity-dependent blanketing effect as well as by intrinsic temperature differences that depend on luminosity. These effects explain at the same time the different slopes of the P-C relations of the three galaxies.
At this point it seems a justified working hypothesis to assume
that the observed differences of Cepheids in the three galaxies for
are principally due to metallicity differences.
If so, the Galactic P-L relation should be applied to metal-rich
galaxies, and the LMC P-L relation to galaxies with [Fe/H]
.
A finer grid of metallicities, extending also to lower
metallicities, is a desideratum of the future.
- However, that metallicity may not be the only parameter causing
variations of the slope of the P-L relation follows from the
possibly different slopes of LMC Cepheids with
and
(see the forthcoming paper II of this series).
The present large-scale value of the Hubble constant, H0=57from SNe Ia depends on the luminosity calibration of the latter, which is provided by nine SNe Ia having occurred in galaxies whose Cepheid distances have been determined (Saha et al. 2001; Tammann et al. 2002) on the basis of the traditional flat P-L relation of Madore & Freedman (1991). Six or seven of these galaxies are as metal-rich on average as the Galaxy. Their distances should now rather be based on the steep Galactic P-L relation. Coincidentally this decreases H0 by only a few percent because the Cepheids involved have median periods close to where the Galactic and LMC P-L relations intersect.
However, the principal difference is that the zero point of H0 depends now only weakly on any adopted distance of LMC (through the SNe Ia in the low-metallicity galaxies NGC 5253 and IC 1613), but mainly on the Galactic P-L relation whose zero point rests on the distances of the Cepheid-bearing Galactic clusters by Feast & Walker (1987) and Feast (1999) and to equal parts on the BBW distances of Gieren et al. (1998), the latter being independent of any adopted distance.
The definitive conclusion is that the notion of a universal slope of the P-L relation cannot be maintained in the presence of metallicity and strip ridge-line temperature variations.
Acknowledgements
G. A. T. and B. R. thank the Swiss National Science Foundation for valuable support. The authors thank Dres. R. Buser, S. Kanbur, C. Ngeow, M. Samland, and A. Udalski for helpful discussions and informations as well as the referee, Dr. M. Feast, for constructive improvements.