A&A 474, 557-563 (2007)
DOI: 10.1051/0004-6361:20065840
M. Marconi1 - S. Degl'Innocenti2,3
1 - INAF, Osservatorio Astronomico di Capodimonte, via Moiariello 16, 80131
Napoli, Italy
2 - Dipartimento di Fisica, Università di Pisa, Largo B. Pontecorvo 3, 56126 Pisa, Italy
3 - INFN, Sezione di Pisa, Largo B. Pontecorvo 3, 56126 Pisa, Italy
Received 15 June 2006 / Accepted 19 July 2007
Abstract
Context. A promising technique to derive the physical parameters and the distance of pulsating stars is the fit of the observed light curves by nonlinear pulsation models.
Aims. We apply this technique to a subsample of the RR Lyrae belonging to the Galactic globular cluster M3. The application of the method to cluster pulsators has the advantage of dealing with a homogeneous sample at the same distance and with the same chemical composition allowing to be checked the internal consistency of pulsational calculations.
Methods. We selected seven pulsators (three RR
and four RR
)
which cover a significant period range and show detailed light curves in the B, V and in some cases I bands. For four of them, with different periods, pulsation modes and light curve properties, we analyze the dependence of the theoretical light curve variations on the model input parameters.
Results. For all selected objects, except the reddest one, we are able to theoretically reproduce the observed light curve morphology for self-consistent ranges of intrinsic stellar parameters, in agreement with the evolutionary predictions for the corresponding metal abundance. It is worth noting that, even if the theoretical reproduction of individual light curves gives for each variable only a range of stellar parameters and distances, the analysis of several variables belonging to the same cluster provides a mean distance modulus, namely
mag, and also checks the self-consistency of the adopted theoretical scenario. Taking also into account the evolutionary constraints, the range of the accepted distance modulus is significantly reduced giving a weighted mean value of
.
Our estimates are in agreement with available results in the literature obtained from independent methods.
Key words: stars: evolution - stars: variables: RR Lyr - stars: distances
During the last few decades many theoretical efforts have been devoted to the construction of nonlinear and convective pulsation codes for the modeling of classical helium burning pulsating stars, namely Cepheids and RR Lyrae (e.g. Gehmeyr 1992, 1993; Bono & Stellingwerf 1994 (BS94); Wood et al. 1997; Feuchtinger 1999 and references therein). One of the most important advantages of this kind of models is the unprecedented possibility to predict the time variations of the relevant stellar properties, such as luminosity, radius, effective temperature and surface gravity, along a pulsation cycle. On this basis one can attempt the reproduction of the observed light and radial velocity curves with those predicted by nonlinear pulsation models. This approach has the advantage of relying on the direct comparison between observed and predicted curves, rather than involving only related parameters such as pulsation amplitudes and Fourier parameters, performing the theoretical reproduction of both general features (period, shape, amplitude) and morphological details (bumps and dips), of the observed curves. Since the morphology of the curves depends on the model input parameters, as well as on the physical and numerical assumptions, the modeling of actual light and radial velocity variations along a pulsation cycle offers a unique opportunity to obtain sound estimates of the intrinsic stellar properties of the variables, providing at the same time a key test of current theoretical predictions.
The method has been successfully applied to both Classical Cepheids and RR Lyrae. Theoretical light curves for Cepheids were first presented by Wood et al. (1997, hereinafter WAS) for the LMC bump Cepheid HV905 and, more recently, by Bono et al. 2002 (BCM02) for two LMC fundamental mode Cepheids from the OGLE catalogue. An independent application to the MACHO V and R light curves of 20 bump Cepheids in the LMC (including HV905) was performed by Keller & Wood (2002), and more recently extended to a further 28 Magellanic objects (19 in the LMC and 9 in the SMC), on the basis of the same nonlinear pulsation code previously adopted by WAS (Keller & Wood 2006). The results obtained by BCM02 and Keller & Wood (2002, 2006) appear in good agreement, suggesting that the model fitting technique is not critically dependent on the adopted pulsation code, as well as on the physical and numerical assumptions in the model computations.
For RR Lyrae, the first attempt to compare the morphology of observed and
predicted light curves was provided by Bono et al. (2000) (hereinafter BCM00).
The authors modeled the multi-filter light curve (and the less accurate radial
velocity curve) of the field first overtone pulsator U Com, obtaining a
"pulsational'' evaluation of the intrinsic stellar parameters and, in turn, of
the distance of the variable. These results were found to be in good agreement
with independent evaluations available in the literature, as well as with the
predictions of stellar evolution theory. Similarly good results were later
obtained by di Fabrizio et al. (2002) for another field RR
pulsator, and
by Castellani et al.(2002) for a fundamental mode RR pulsator in the galactic
globular
Cen. However, recently, Marconi & Clementini (2005,
hereinafter MC05) by analyzing the B,V light curves of 14 RR Lyrae stars (7 fundamental and 7 first overtone pulsators) obtained an estimate of the
distance modulus for the LMC in good agreement with distances derived from
Cepheids, but with the evidence for a serious mismatch between observed and
predicted light curves in the case of the longest period fundamental
pulsators.
In this paper we rely on a suitable sample of RR Lyrae in a Galactic globular cluster. With this choice one has the advantage of dealing with an homogeneous sample of variables which are known to be all at the same distance and to have the same chemical composition, with intrinsic stellar parameters constrained by stellar evolution theoretical predictions. In particular, we choose the globular cluster (GC) M3, which is the Galactic cluster with the largest number of observed RR Lyrae variables (e.g. Clement et al. 2001; Corwin & Carney 2001, hereinafter CC01; Clementini et al. 2004), for which accurate multi-filter light curves are available (CC01, Hartman et al. 2005; Benko et al. 2006).
RR Lyrae in M3 have been analyzed by several authors from the pulsational and evolutionary point of view. Here we quote the multicolor and Fourier study of a wide sample of RR Lyrae in M3 recently presented by Cacciari et al. (2005) and the analysis by Marconi et al. (2003; see also Jurcsik et al. 2003), who discussed the good agreement between theoretical and observed pulsational properties, in particular for the instability strip topology and the period-amplitude relations.
Table 1 reports the ranges of current independent determinations of the
distance modulus and the reddening (e.g. Buonanno et al. 1994; Ferraro et al. 1999; Rood et al. 1999) for this cluster. Metal
abundance values provided by various authors are also reported. Here
we adopt the [Fe/H] value of Carretta & Gratton (1997) and assume an
enhancement of
elements [
(e.g. Salaris
& Cassisi 1996) leading to a total metallicity
.
The distance modulus (
)
reported in the table is an apparent (reddened) distance
modulus; as is well known, it differs from the de-reddened one by
3.1 E(B-V). As for the reddening, E(B-V), inspection of Table 1 suggests that the reddening of M3 RR Lyrae is estimated to be very small.
Table 1:
The range of estimated apparent visual distance modulus (),
obtained with different methods, and the reddening values, E(B-V),
provided in the recent literature for M3. The [Fe/H] values given by Zinn & West (1984: ZW), Harris (2003: H) and Carretta & Gratton (1997: CG) are also reported.
We have considered the accurate database of 170 RR Lyrae
light curves in the V and B bands provided by CC01. After exclusion of
variables with Blazkho effect, with blended images or with an insufficiently
precise determination of the period, one remains with 69 RR
and 19 RR
covering the ranges of
magnitudes
and
,
respectively. To deal with a regular sequence of
pulsators covering the whole color extension of the instability strip, we
first selected from this sample the stars having
in the interval
15.60-15.70. We then chose variables with significantly different
periodicities and light curve morphology. We ended with a sample of six stars
(3 RR
plus 3 RR
)
which cover the range of
-
spanned by the whole variable set. Then, in order to test the method on a significantly brighter object we added to the sample, the RR
variable V152.
Table 2 gives selected quantities for the final sample, whereas Fig. 1 shows the seven light curves. We notice that these variables cover a wide period range and are characterized by a variety of light curve morphologies. The possibility to model all these characteristics for consistent values of the intrinsic stellar parameters and distance represents a relevant test for the soundness of the theoretical pulsational scenario. Moreover the requirement that the distance values obtained for different pulsators are in agreement with each other within the uncertainties, further constrains the cluster distance.
Table 2: Observed characteristics of the selected sample of variables. From left to right one finds the star identification, the period in days, the intensity averaged V magnitude, the intensity averaged B-V color and the RR type.
Model computations were performed by adopting Z=0.001 Y=0.24 and the same numerical and physical recipes that in BCM00 were found to provide an accurate modeling of field RR Lyrae and Classical Cepheid light curves (BCM02; Di Fabrizio et al. 2002; MC05). To transform theoretical light curves in the observational plane we adopted the atmospheric models by Castelli et al. (1997a,b). In Sect. 3 we show that plausible variations of the stellar metallicity and helium abundance does not affect in a relevant way the conclusions of the paper. However, when the evolutionary constraints are taken into account, our approach seems to exclude metal abundances as low as Z=0.0004.
For RR
pulsators, unless otherwise advised, the computations have been
performed adopting for the mixing length parameter, governing the efficiency
of the super-adiabatic convection and closing the system of nonlinear
hydrodynamical equations in the pulsation code, the value
.
This
value has already been found to correctly reproduce the relevant pulsational
properties of first overtone RR
pulsators (e.g. M03; Di Criscienzo et al. 2004).
However previous investigations suggest that the
parameter should be
increased when moving toward the red part of the instability strip reaching
values as large as about 2.0 (BCM02; M03; Di Criscienzo et al. 2004;
MC05). As we discuss later, this paper confirms that a mixing length value of
is more suitable for the reproduction of RR
light
curves, even when the effective temperature is as high as 7000 K. This does
not occur for first overtone pulsators due to the more efficient damping of
convection in the smaller mass involved in the pulsation mechanism (outside the first overtone radial node).
To reproduce an observed light curve with a given period, we assume the mass
of the pulsator as almost a free parameter. As we discuss in
Sect. 2.5, evolutionary calculations predict a quite restricted range for
horizontal branch masses within the instability strip. However, to be very
conservative, as a first step, we allow a much wider mass range, namely
.
For each given value of the mass, we explore the
predicted light curves for selected luminosity levels. Once the mass and
the luminosity are fixed, the effective temperature is determined by the constraint:
Adopting solar units for the stellar mass M and the luminosity L,
the fundamental mode periods can be derived with good precision
from the relation (MCDC03):
Figure 2 shows some results of the comparison among theoretical and
observed light curves as applied to the variable V128, the first overtone
pulsator with the shortest period in our sample. The labeled value of the
distance modulus, ,
gives the vertical shift operated on the theoretical
light curve to match the data, as reported in the various panels of the
figure. From the left panels of Fig. 2 one sees that some
theoretical light curves seem to reproduce quite satisfactorily the observed
one, showing also the same characteristic flattening at the maximum
luminosity. Thus it is worth defining a procedure to quantitatively evaluate
the goodness of the match.
To do this we first evaluate the residuals as the difference between the
observational and the theoretical light curves; these residuals are then
fitted with a 10th order polynomial. The
result is shown in the right panels of Fig. 2. We notice
various cases in which the difference between theory and observations
remains within 0.05 mag at each phase point. On the basis of this occurrence,
for this variable, we decided to define as acceptable models those which
match the observed data within 0.05 mag, also reproducing the observed
morphology of the light curve. Moreover, we require that the predicted
intensity weighted colors (
)
agree with those observed
within the observational uncertainties (
0.04 mag), as estimated by taking into account
the photometric uncertainty and the error on the reddening.
As a whole, in the case of V128 one finds that pulsational theories are in perfect agreement with observations. In order to explore the response of the light curve to the structural parameters, Fig. 2 shows the results when the mass, the luminosity or the effective temperature are kept fixed. In particular, data in the bottom panels show that the shape of the curve is essentially a matter of effective temperature, with the other two parameters (M and L) only slowly affecting the pulsational amplitude. Such an occurrence can be easily interpreted in terms of the relevant role of the surface gravity. On this basis, one can easily understand that the relevant variations reported in both the top and the middle panels are mainly the consequence of variations in the effective temperature. Note that the short period of the first overtone pulsator V128 indicates that it is located close to the first overtone instability strip blue boundary. In order to keep a constant period, when increasing the luminosity we also have to increase the effective temperature (Eq. (4)), with the consequence that the amplitude decreases and the morphology becomes smoother, as expected when moving toward the instability strip blue boundary (Bono et al. 1997a).
The left top panel in Fig. 2 reveals that, if the mass is
fixed at
,
the fitting of the light curve appears able to
formally discriminate the luminosity level within less than
,
i.e., within a few hundredths of magnitude. The right top panel
shows the corresponding behavior of the difference between the observed and
the theoretical curve, revealing that only for
K the
discrepancy remains lower than
mag at all pulsation phases.
The bottom panels show that keeping
constant, one finds
an agreement within
0.05 mag for masses in the range
-
.
To check if masses lower than 0.65
or higher than
0.71
provide worse fits for any possible combination of
the stellar parameters, for each mass value outside these limits we explored
the possible luminosity and effective temperature ranges, with the unique
constraint to reproduce the observed period. The procedure is stopped at the
smallest/largest mass for which no acceptable solution is found. The results
are shown in Fig. 3, where, for fixed mass, the temperature
range for stars inside the instability strip is uniformly spanned and
the luminosity is changed according to Eq. (2).
We found that for masses larger than or equal to
,
either the difference between theory and observation is higher than 0.05 mag. or, even when the observed
pulsation amplitude is matched and the discrepancy remains within
0.05 mag at any pulsation phase, the model is not able to accurately
reproduce the morphology of the curve. On the other hand, for
,
the assumed lowest limit in mass, we still obtain acceptable models for
and
,
respectively. Luminosities lower than
are not
considered because they would imply a stellar mass lower than
.
We can also exclude luminosities higher than
because we have checked that at higher luminosity
level models either do not fit the observed light-curve or do not present
a stable pulsation limit cycle.
Figure 4 shows in the B band the comparison between observed
data and the models reported in the bottom panel of Fig. 2.
The quality of the fit in the B band is worse than in the V band with a
maximum discrepancy of about 0.08 mag for the best fit cases. The
apparent B distance moduli obtained are quite the same of the corresponding V ones and
this result is consistent with the occurrence of a very low reddening.
Figure 5 shows that, as in
the V band comparisons,
is still an acceptable mass, while
masses higher than
0.71
do not reproduce the observations.
However one should not forget that the pulsation amplitude also depends
on the assumed value for the mixing length parameter
.
All these
results have been obtained by adopting the standard mixing length parameter
,
which, as noted above, is expected to correctly reproduce the
relevant pulsational properties of RR
.
However, it is worth investigating
the effect on the comparison between theoretical and observed light curves of
small variations of the
parameter. As shown in Fig. 6,
when the
value is increased at fixed mass, luminosity and effective
temperature, the amplitude decreases (while a decrease of the
value has an opposite effect on the pulsational amplitude). This trend is
due to the quenching effect on pulsation, of the increased efficiency of
convection (e.g. Stellingwerf 1982; BS94). However, the same amplitude
might in principle be obtained with different
values if the stellar
parameters are properly varied. On this basis we searched for additional
solutions for
.
Remembering that the main parameter affecting the
light curve shape is the effective temperature of the star, different models
with
have been calculated (by requiring that the theoretical
period exactly reproduces the observed ones) with a suitable fixed effective
temperature, while the mass, and correspondingly, the luminosity, are varied
according to Eq. (1) (see Fig. 7). We note that the ranges of
acceptable stellar parameters and distance modulus, in the case of
,
are almost the same as for the models with
.
Figure 8 shows,
analogously to Fig. 3, that at least one additional
solution can be found for
,
whereas no acceptable solutions
are found for
.
Moreover, as in the
case, the
theoretical reproduction of the observed data is slightly worse in the B band.
For
outside the range
we did not find any acceptable
solutions. In particular, for
it is possible to reproduce the
light curve amplitude but not its morphology.
The first line of Table 3 shows the accepted ranges of mass,
luminosity, effective temperature, intensity weighted B-V color and apparent
distance modulus. Other solutions with parameters within the ranges of
Table 3 may exist but we verified that no solutions can be found
outside these ranges. Data in Table 3 clearly show that while, as
already discussed, the pulsational method is very sensitive to variations of the
effective temperature, it is much less sensitive to mass changes; as we
discuss in Sect. 2.5, indications to restrict the obtained mass range can come
from evolutionary calculations. The distance moduli obtained are well within the range of the most recent estimates (see Table 1) and the theoretical colors are in agreement, within the
uncertainties, with the empirical colors. The obtained distance modulus has an uncertainty of about 0.1 mag. By taking into account the photometric error on the apparent magnitudes (
0.02 mag), our final estimate for the apparent distance modulus of V128 is
mag.
Table 3: Accepted ranges of the physical parameters (mass, luminosity and effective temperature) for variables V128, V126, V72 and V152, together with the corresponding range of apparent visual distance modulus and intensity weighted (B-V) color. For the V152 variable the range in temperature (color) has not been calculated (see text).
Variable V126 is characterized by a longer period and a redder color than
V128, but the apparent visual magnitude is identical within the photometric
error. Following the same procedure as for V128, in the three panels of
Fig. 9 we show the light curve shape of models calculated by fixing
the mass, the luminosity and the effective temperature separately and varying
the other two parameters, according to Eq. (3), to reproduce the observed
period. We adopted
,
as in the case for V128. All the trends
found for V128 also hold for V126. In particular it is confirmed that the
effective temperature is the key parameter to characterize the light curve
morphology. Even in this case there are some models which reproduce the
observed light curve within 0.05 mag. Therefore, as for V128, we decided to
accept models for which the difference with the observation is within this
limit and which reproduce the observed morphology. Following these criteria,
the bottom panels of Fig. 9 seem to indicate
as the highest acceptable mass. In fact, to check that there are
not possible solutions for higher masses, we need to span, at fixed mass and
with the requirement to reproduce the observed period, the whole range of
possible effective temperatures (and thus luminosities). To check this point
we calculated models for higher masses and several temperatures spanning the
whole instability strip range, while the luminosity is fixed by Eq. (3). As
shown in Fig. 10, it is possible to
find acceptable solutions up to the highest mass we tested in our conservative
approach (
).
This is mainly due to the significantly larger intrinsic scatter of the light curve data with respect to the other selected pulsators, and also to the almost sinusoidal shape, which in principle can be easily reproduced by models close to the first overtone blue boundary.
Even if the
color of the highest mass models are significatively
lower than the observed values (see Table 3), taking into account
the uncertainties in the color estimates, we conservatively decided to include
these models.
The accepted ranges of mass, luminosity, effective temperature, apparent visual
distance modulus and color are reported in the second line of
Table 3. The large range of accepted mass values leads to
a relatively wide range of luminosities and thus of distance modulus:
.
The models in the bottom panels of Fig. 9 are shown in
Fig. 11 in the B band.
As for V128, the quality of the comparison between theory and observations is slightly worse than in the V band, but the obtained results are consistent. The colors of the accepted models for V126 (but the already discussed highest mass models)
are in agreement with the observed values and the difference between the inferred apparent B and V distance moduli is consistent with a very low reddening value.
In the previous section we have shown for V128 that small variations
of the parameter (higher variations are not allowed as a result of the
requirement to reproduce the observed light curve morphology and period) do not
significantly change the accepted range of stellar parameters.
We checked that this is true also for V126.
Regarding the stellar parameters, we find indications for an effective
temperature possibly lower than that of V128, as expected on the basis of
the observed colors and periods; whereas the
predicted luminosity could be brighter than that estimated for
V128. These results could suggest a small evolutionary effect for V126 as
is also possibly indicated by the evidence that this RR
pulsator is redder
than 7 RR
variables in the same sample. However, the observed apparent
visual magnitude is the same as that of V128 within the photometric error (see
Table 2) so that, even taking into account the effect of
bolometric corrections, the two pulsators should have similar intrinsic
luminosities.
Variable V72 is a fundamental pulsator with an intensity weighted B-V color
intermediate between those of the RR
V126 and V128, likely close to the
blue boundary of the fundamental instability strip. This assumption is also
supported by the large amplitude of its visual light curve. In order to find
possible theoretical models for this fundamental variable we adopted the same
method as for the RR
pulsators. Indeed, as expected (see the discussion
in Sect. 2), the best agreement is not obtained for
(see
Fig. 12). The best match is found when
is
increased to 2.0, and by assuming a non-vanishing overshooting efficiency
in the regions in which the superadiabatic gradient is negative (see Eq. (7)
and Sect. 3 in BS94). Given the higher and more asymmetric shape of
fundamental light curves, the agreement between the observed and theoretical
light curve is not within the 0.05 mag level at all phase points, as in the
V128, V126 cases. At most we find an agreement within
0.08 mag. Therefore we consider as
acceptable models for V72, those which show a difference with observations
0.08 mag at all the phase points.
The variations of the light curve characteristics at fixed mass, luminosity and effective temperature are shown in the three panels of Fig. 13. The dependence on the physical parameters is similar to that already discussed for V128 and V126.
We note that for V72 we were not able to find equivalently good solutions
for
lower or higher than 2.0. The accepted ranges of the physical
parameters, apparent distance modulus and color are reported in the third
line of Table 3. In particular we find
.
Also in this case we checked that there are no acceptable
models outside these ranges. In order to check that
really represents the upper limit of the acceptable mass values, we computed
models (see Fig. 14) for this mass, varying the luminosity and the
effective temperature with the constraint of reproducing the observed
period. The plot shows that for
there are no possible
solutions. We also verified that no solution is found for a luminosity
higher than
(see Fig. 15). These
tests allow us to confirm the obtained range for the distance modulus. The
fact that searching either for mass limits or for luminosity limits one
obtains the same range of physical parameters is a consequence of Eqs. (1) and (3).
Figure 16 shows the models of the
bottom panel of Fig. 13 in the B band. As for the RRpulsators, the agreement between theory and observations in the B band is
slightly less accurate with respect to that in the V band, suggesting that we
should assume as acceptable a discrepancy theory-observation within 0.1 mag.
Similar results are obtained for the equivalent situation in the B band of Figs. 14 and 15.
On the basis of the results obtained so far for V128, V126 and V72, we
estimate a mean apparent distance modulus
mag.
At the end of this section we discuss problems
which arose in the analysis of the RR
variable V120, a fundamental
pulsator with a quite red color and a relatively long period, likely close to the red
boundary of the fundamental instability strip. For the very red RR Lyrae, the
modeling of the light curve morphology already appears problematic (see e.g.
MC05). This is probably due to the remaining uncertainties in the treatment
of the coupling between pulsation and convection in the hydrodynamical models.
In particular theoretical light curves for red pulsators can show a spurious
feature before the maximum (see e.g. the atlas reported in Bono et al.
1997a,b) which is not observed in real curves, even if the pulsation amplitude
is well reproduced.
For V120 we are not able to find acceptable solutions even allowing
the physical parameters and the
values to vary over wide ranges.
Figure 17 shows, as an
example, some attempts to find possible solutions. A detailed investigation of
this problem and of its dependence on variations of the parameters entering in
the turbulent convective models will be addressed in a forthcoming paper.
The light curves of the two pulsators, namely the RR
V6 and
the RR
V75, show morphological features similar to those of the
pulsators discussed above. For this reason we do not present the same detailed
analysis of the dependence of results on the model input parameters, but we
show one of the possible solutions for each object. The
value adopted to
obtain a good reproduction of the observed light curve is, in agreement
with the previous pulsation models,
for the RR
and
for the RR
.
These results are shown in Fig. 18. We note that the predicted values for the physical parameters and the distance modulus are
within the range shown in Table 3. Similar results are
obtained for the B band.
The pulsational results can be compared with evolutionary model predictions.
To this end, we adopted the Zero Age Horizontal Branch (ZAHB) properties
for Z=0.001, as reported in recent papers, as based on different assumptions
concerning the original helium abundance, the set of physical inputs
(radiative and conductive opacities, EOS tables, nuclear reactions rates,
neutrino energy losses etc.) and the efficiency of the external convection.
The adoption of different evolutionary predictions allows us to obtain a rough
estimate of the uncertainty on current stellar models. By comparing the
obtained ranges of pulsational mass and luminosities, corresponding to a given
effective temperature, with the same results for the evolutionary models, one
can roughly evaluate the consistency between the two theories. As already
discussed, pulsation models are very sensitive to the effective temperature,
with the consequence that the estimated effective temperature range is quite
small. Due to the significantly lower sensitivity of evolutionary predictions
on this parameter and to the unavoidable uncertainty in the evolutionary
predictions for this quantity, a mean value for the
theoretical effective temperature of all the variables taken into account in
our analysis, namely
K, can be used. Table 4 summarizes the
evolutionary values for masses and luminosities by the different quoted
authors, corresponding to the selected temperature value. All results are
within the pulsational ranges given in Table 3. It is also worth
noting that the pulsational calculations are less sensitive than the
evolutionary ones to masses and luminosities. However, the evidence that two
completely independent methods give consistent results is encouraging,
in spite of the several possible uncertainty sources in both the evolutionary
and pulsational calculations. Finally, we note that for some variables
off-ZAHB evolution effects can be not negligible, producing luminosities
higher by about 0.1 dex, for a fixed effective temperature. By taking into
account the evolutionary predictions reported in Table 4 and the
possibility of off-ZAHB evolution, we can restrict the ranges reported in
Table 3, discarding all masses lower than
and
higher than
.
This selection implies that the ranges of the
inferred individual distance moduli are significant reduced, giving a weighted
mean value of
mag.
Table 4: Evolutionary masses and luminosities predicted by different sets of ZAHB models for a mean effective temperature of 7000 K, obtained from our best fit pulsational models (see text). The selected evolutionary papers are: V00, VandenBerg et al. (2000); DC02/C05, D'Antona et al. (2002) and V. Caloi (2005, private communication quoted in MC05); P04, Pietrinferni et al. (2004); CDC04, Cariulo et al. (2004).
Until now, as discussed in the introduction, we have analyzed variables in a restricted range of luminosity, to deal with a sequence of pulsators spanning the temperature range of the predicted instability strip, at a given luminosity level. As already discussed, for each variable, we found a range of physical values (and thus distance moduli) which are in agreement within each other.
A further check of the reliability of the estimated cluster distance modulus is to analyze the properties of pulsators with a different luminosity: as these stars are also cluster members, the obtained distance modulus has to be in agreement with the previous results.
To this aim we selected the
RR
V152, which has a well-sampled light curve in
the V and B bands and is brighter by
0.1 to 0.2 mag than the
other selected variables (see Table 2). It takes into account
firstly the above-mentioned evidence that the temperature is the key parameter for
good agreement between the theory and observations, and secondly that the expected mass,
luminosity and distance modulus ranges are essentially obtained by investigating
the quality of the comparison between the theory and observations at a fixed
effective temperature. Figure 19 shows the results for models of
different mass (luminosity) with a suitable temperature value (
K). We selected as acceptable models those for which the difference with
the observations is within 0.05 mag at all the pulsation phases. We also
checked that even if the temperature is allowed to vary,
really represents the upper limit in mass (see Fig. 20). Similar to what was obtained for the other variables, the conclusions are the same when the comparison is performed in the B band.
The ranges obtained of mass, luminosity and distance modulus are shown in Table 3. We note that the distance modulus range for this brighter pulsator is consistent with results obtained for the other variables, even if the visual magnitude is higher. A similar agreement is obtained for the stellar mass and luminosity.
Including V152 in our sample and taking into account the evolutionary
constraints discussed in the previous section, we obtain a weighted mean
distance modulus of
mag.
As mentioned above, all calculations have been performed for a
metallicity value Z=0.001 and a helium abundance Y=0.24. We
discuss here how our results are affected by the assumed chemical composition.
In previous papers (e.g. Bono et al. 1996,
BCM00) we have shown that the
shape of light curves is affected by metallicity, with a minor dependence on
the Helium content (see Bono et al. 1997a). To check how this dependence
can affect our estimates of the stellar intrinsic parameters and distance,
we calculated some additional models for a significantly lower metallicity
value, which is outside the range of metallicity values indicated in the
literature for M3, namely Z=0.0004. Figure 21 shows that, at least for variable V128, in spite of a change in the best fit model intrinsic parameters, we
are able to find solutions with mass, luminosity (and thus distance modulus)
within the previous accepted ranges of Table 3. A slight
increase of the effective temperature is needed, confirming that the
temperature is a key parameter affecting the morphology and the
amplitude of RR Lyrae light curves. However, in the comparison with the evolutionary
prescriptions (see Sect. 2.5) we have to consider that
for Z=0.0004 the expected stellar masses on the ZAHB (around
K)
are about 0.7
.
This implies that the models fitting the observations within
in Fig. 21 should be excluded. From this perspective, our approach seems to confirm the most recent spectroscopic measurements, which exclude for M3 a
metallicity as low as Z=0.0004.
Regarding the helium abundance, within reasonable variations of the relative helium to metal enrichment ratio (
)
and assuming a primordial helium content no larger than 0.24, we do not find any significant effect on the predicted light curves.
Recently Hartmann et al. (2005) published three bands (BVI) light curves for several variables of different types in M3; in particular all the variables selected by us, except V126, are included in the Hartmann et al. (2005) sample. Moreover during the completion of this study another paper on the light curves of the M3 RR Lyrae in the V, B and I photometric bands was published (Benko et al. 2006) in which all the light curves selected in our paper are analyzed. Figure 22 shows the comparison of the V, B and I data by Hartmann et al. (2005) and Benko et al. (2006) with those by CC01 for all our analyzed variables.
Regarding the V band we note that the light curves by the different authors are almost coincident or vertically shifted by only a few hundredths of magnitude, an amount which is within the errors due to photometric uncertainties and the fitting procedure. A similar agreement is found in the B and I bands. We also note that the periods quoted by Hartmann et al. (2005) and Benko et al. (2006) for the selected variables are the same as those by CC01.
From Fig. 22 it is clear that the theoretical models which reproduce the CC01 B and V light curves are also able to match the Hartmann et al. and the Benko et al. data with, at most, a slight variation of the distance modulus, still consistent with the results of Table 3 within the errors.
To investigate if the data in the I band confirm the results obtained in the Vand B bands, Fig. 23 shows the comparison between the models corresponding to the bottom panels of Fig. 2 (for V128) and Fig. 13 (for V72) and the data by Hartmann et al. in the I band. We note that the agreement between theory and observation is good (within 0.05 mag), thus confirming the results obtained in the previous section. Moreover the apparent distance moduli obtained for the V and I bands are very similar, confirming the occurrence of a low reddening value. Similar results are obtained for the other selected variables for which I data are available.
In a recent paper, Cacciari et al. (2005) performed a detailed study of the pulsational and evolutionary characteristics of 133 RR Lyrae stars in M3. On this basis physical parameters were derived from B-V colors and accurate color-temperature calibrations, as well as from Fourier coefficients. The results obtained from the color-temperature calibration, which are considered by the authors themselves to be more precise than those based on the Fourier parameters, are reported in Table 5.
Table 5: Physical parameters of V128, V126 and V72 variables as derived by Cacciari et al. (2005) from color-temperature calibration.
By comparing these values with the ranges reported in Table 3 we see that an agreement for the intrinsic stellar parameters can be found, within the error bars, for all the variables, even if in some cases the overlap between the ranges proposed by Cacciari et al. and those obtained in the present work is marginal.
We have successfully modeled the light curves of six variables (four
RR
and two RR
)
of the globular cluster M3.
For four variables (V128, V126, V72, V152), spanning a significant period range and showing different light curve morphologies, we searched for multiple solutions analyzing the dependence of the theoretical light curves on the model input parameter. We remark that, due to the degeneracy inherent to the method, from the theoretical reproduction of the observed light curve of a given variable one can obtain only a range of stellar parameters; the analysis of several variables belonging to the same cluster allows us to provide sounder constraints and also mean results. To our knowledge, this is the first time that such an analysis has been done in a systematic way.
The obtained acceptable ranges of stellar parameters (mass, luminosity and effective temperature) and distance modulus are self-consistent and in agreement with stellar evolutionary predictions. Moreover a general agreement can be found with the physical parameters provided by Cacciari et al. (2005), on the basis of a multicolor study. For the other two pulsators (V75 and V6), that show morphological features similar to the previous ones, the obtained stellar parameters and distance moduli are consistent with the previously-identified ranges, confirming the previous estimates. These results represent a check for both evolutionary and pulsational theories.
The failure in the reproduction of the light curve morphology for the reddest pulsator V120, likely due to some remaining uncertainties in the adopted turbulent convective models, represents a challenge for future model developments.
Our final estimate for the distance modulus is
,
derived as a mean of the individual determinations, while taking also into account the
evolutionary constraints on the pulsator masses, obtains a weighted mean
value of
.
Our obtained distance modulus is consistent, within the uncertainties, with the results from other methods, for example with the determination by Ferraro et al. (1999), obtained from the comparison between the observed and predicted Zero Age Horizontal Branch luminosity, as well as with the results by M03, based on additional pulsational methods (see their Table 4). The precision of the technique could be further improved if accurate radial velocity curves are available together with photometric data covering both the optical and the near-infrared bands.
Acknowledgements
This paper is dedicated to the memory of V. Castellani who inspired this work and constantly supported us. We shall miss his personality, clearness and enthusiasm in approaching the different issues of stellar astrophysics. We are grateful to P. G. Prada Moroni and S. Shore for careful reading of the manuscript and to F. Caputo for useful suggestions. We are grateful to the anonymous referee for her/his suggestions and comments which greatly improved the paper. Financial support for this work was provided by MIUR under the scientific projects "Continuity and discontinuity in the Galaxy formation'' (P.I.: R. Gratton) and "On the evolution of stellar systems: a fundamental step toward the scientific exploitation of VST'' (P.I.: M. Capaccioli).
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Figure 1: Observational data in the V band for the selected light curves, from Corwin & Carney (2001). Periods (in days) are labeled. |
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Figure 2: Predicted light curves constrained to the observed period of V128 for the labeled fixed values of mass ( top panel), luminosity ( middle panel) and effective temperature ( bottom panel). The right panels show the differences between theory and observations, computed as described in the text. |
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Figure 3:
Predicted light curves constrained to the observed period of V128 for two different mass values, namely
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Figure 4: As in the bottom panel of Fig. 2 but for the B band. |
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Figure 5: The same as Fig. 3 but in the B band. |
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Figure 6:
The behavior of the light curve shape for the V128 fitting case, when the ![]() |
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Figure 7:
As in the bottom panel of Fig. 2 but for
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Figure 8:
As in Fig. 3 but for
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Figure 9: As in Fig. 2 but for the variable V126. |
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Figure 10:
Predicted light curves constrained to the observed period of V126 for
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Figure 11: As in the bottom panel of Fig. 9 but for the B band. |
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Figure 12:
The best match for the V72 light curve by adopting models with
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Figure 13: Dependence of the modeling of V72 light curve on the input parameters, by fixing them one at a time (see text). Upper panels: fixed mass; middle panels: fixed luminosity level; lower panels: fixed effective temperature. |
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Figure 14:
Comparison between the data and the predicted light curves
for
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Figure 15:
Comparison between the data and the predicted light curves
for V72 with fixed luminosity (
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Figure 16: As in the bottom panels of Fig. 13 but for the B band. |
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Figure 17:
Attempts of modeling the V120 light curve when ![]() |
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Figure 18: Observed light curves in the V band for the V75 and V6 variables from CC01, compared with one of the possible theoretical matches. The periods (in days), the physical parameters of the selected models and the corresponding distance moduli are labeled. |
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Figure 19: Models at fixed effective temperature constrained to reproduce the observed V152 period compared with the observed data. |
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Figure 20:
Models for V152 with
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Figure 21: As in the bottom panel of Fig. 2 but for Z=0.0004 and a suitable value of the effective temperature (see text). |
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Figure 22: Comparison (in the V, B and I bands) of the observed light curves analyzed in this paper by CC01 (black circles) with those of Hartmann et al. (2005) (red diamonds) and Benko et al. (2006) (green triangles). |
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Figure 23: Upper panel: the models of the bottom panel of Fig. 2 for V128 compared with the data by Hartmann et al. (2005) in the I band. Lower Panel: as in the upper panel but for the models in the bottom panel Fig. 13 for V72. |
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