G. Raimondo1,2 - M.-R. L. Cioni1 - M. Rejkuba1 - D. R. Silva1
European Southern Observatory, Karl-Schwarzschild-Str. 2, 85748 Garching bei München, Germany - INAF - Osservatorio Astronomico di Teramo, via M. Maggini, 64100 Teramo, Italy
Received 26 August 2004 / Accepted 18 March 2005
Abstract
A sample of carbon-rich stars (C-stars) in the Small
Magellanic Cloud (SMC) was selected from the combined 2MASS and
DENIS catalogues on the basis of their
colour. This
sample was extended to include confirmed C-stars from the
Rebeirot et al. (1993) spectroscopic atlas. In this combined
sample (N = 1149), a smaller number (N = 1079) were found to have
MACHO observations. For this sub-sample, light curves were
determined and 919 stars were found to have high quality
light-curves with amplitudes of at least 0.05 mag. Of these
stars, only 4% have a well-defined single period - most of
these have multiple well-defined periods, while 15% have highly
irregular light-curves. The distribution of the logarithm of the
period versus magnitude, colour, period ratio (if applicable), and
amplitude was analyzed and compared with previous works. Variable
C-stars are distributed in three sequences: B, C, and D from Wood
et al. (1999), and do not populate sequences with periods shorter
than
.
Stellar ages and masses were estimated
using stellar evolutionary models.
Key words: stars: AGB and post-AGB - stars: variables: general - Magellanic Clouds
Asymptotic Giant Branch (AGB) stars can be separated into two classes based on their spectra: oxygen-rich (O-rich or M-stars) and carbon-rich stars (C-rich or C-stars). M-stars have more
oxygen than carbon in their atmospheres (
), while C-stars
display
enrichment (
)
due to dredge-up caused by
thermal pulsation (Iben & Renzini 1983). These thermal pulses
also lead to mass loss, as well as to luminosity variations with
periods of
100 days or longer and peak-to-peak amplitude
variations up to a few magnitudes at visual wavelengths.
All stars are oxygen-rich when they enter the AGB phase. Whether
or not they become C-stars depends primarily on the efficiency of
the third dredge-up and the extent and time-variation of the
mass-loss (e.g. Marigo et al. 1999; Iben 1981). In metal-poor
stars, fewer
atoms are necessary to change the envelope
from oxygen to carbon dominated (
); therefore, fewer
thermal pulses are needed to convert an M-star into a C-star.
Conversely, mass loss is expected to be stronger in metal-rich
stars, leading to shorter AGB and C-star phases.
In the past, it was thought that both oxygen-rich and carbon-rich Mira variables follow a well-determined period-luminosity (PL) relation in the near-IR regardless of the host system mean metallicity or type, e.g. in a globular cluster (Feast et al. 2002), dwarf galaxy (Glass & Lloyd Evans 1981), spiral galaxy (Glass et al. 1995; van Leeuwen et al. 1997), or elliptical galaxy (Rejkuba 2004). However, the availability of long-term photometric monitoring data provided by the microlensing observing projects, e.g. MACHO (Alcock et al. 1992), OGLE (Zebrun et al. 2001; Udalski et al. 1997), and EROS (Aubourg et al. 1993), and large-scale near-infrared (NIR) photometric surveys, like the Two Micron All Sky Survey (2MASS, Skrutskie et al. 1997) and the Deep Near-Infrared Southern Sky Survey (DENIS, Epchtein et al. 1997) have opened a new window on this issue and revealed that AGB stars lie on multiple parallel sequences in the PL diagram (Cook et al. 1997; Wood et al. 1999). Furthermore, red giant branch (RGB) stars at the tip of the RGB were also found to vary (Ita et al. 2002; Kiss & Bedding 2003). These results have provided new and significant constraints for theoretical pulsation models.
Differences between O-rich and C-rich Long-Period Variables (LPVs) have also been found. Using a small sample of AGB stars in the SMC observed by the Infrared Space Observatory (ISO), 2MASS, DENIS and MACHO, Cioni et al. (2003) concluded that the period distribution of C-stars peaks at about 280 days. They also noted that C-stars have a larger amplitude with respect to M-stars, contrary to what was derived for the LMC AGB stars, where both types showed a similar amplitude distribution (Cioni et al. 2001). Studying a much larger sample of C and M LPVs in both Magellanic Clouds, Ita et al. (2004b) confirmed that O- and C-rich Miras follow different period vs. (J-K) colour relations (Feast et al. 1989), that C-rich Miras tend to have greater I-band amplitudes at redder J-K colour, and that the amplitudes of O-rich Miras are independent of colour. Groenewegen (2004, G04) reached similar conclusions.
However, the studies by Cioni et al. (2001) and Ita et al. (2002) were limited to LPVs with P < 1000 days, since the OGLE-II observations only span a time-baseline of about 1200 days. More recently, Fraser et al. (2005) presented an analysis of the eight year light-curve MACHO data for LPVs in the LMC and found that C-stars occupy only two of the sequences in the period-luminosity diagram. Furthermore, dust-enshrouded stars are located in the high-luminosity ends of the both sequences.
In this paper, we have extended the work of Cioni et al. (2003)
and complemented the work of Fraser et al. (2005) by investigating
the variability properties of all C-stars observed by MACHO in
the SMC. We used the Master Catalogue of stars toward the
Magellanic Clouds (
)
by Delmotte et al. (2002) and
Delmotte (2003) to identify C-stars in the SMC. This catalogue
provides a cross-correlation between the DENIS Catalogue towards
the Magellanic Clouds (DCMC -
)
and the 2nd Incremental
Release of the 2MASS point source catalogue (
) covering the
same region of the sky. C-stars were selected statistically on
the basis of their red
colours (
;
Cioni et al. 2003). This selection was checked through cross-correlation
with the Rebeirot et al. (1993, hereafter RAW93) catalogue of
spectroscopically confirmed C-stars
(Sect. 2.2). C-stars found
spectroscopically by RAW93 but with
were later
included in the sample. In Sect. 2, we
present the selection of C-stars and the extraction of the
corresponding light-curves from the MACHO database. Section 3 discusses the analysis and resulting light-curve parameters. The (
,
), (
,
)
and other diagrams are discussed in Sect. 4,
while Sect. 5 concludes this work.
Details about the method developed to determine periods and
amplitudes and the quality assessment of the data and of the
relevant parameters are given in Appendix A.
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Figure 1:
Near-IR CMD of SMC stars from 2MASS within the observed
MACHO fields (dots). The C-stars region is marked by two perpendicular
dashed lines: C-stars with
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The
catalogue, containing the cross-correlation
between DENIS and 2MASS surveys, as well as optical UCAC1 and
GSC2.2 catalogues toward the LMC, was published by Delmotte et al. (2002). Here, we use its extension to the SMC (Delmotte 2003) and
the near-IR information of the catalogue only. The region
confidently populated by C-stars (
mag and
mag) contains a total of 1657 stars. 805 stars within the
MACHO fields satisfy the photometric criterion. The MACHO project
observed 6 fields (each of 0.49 deg2), covering the densely
populated bar of the SMC of approximately 3 square degrees in total.
Figure 1 shows the near-IR vs.
colour-magnitude diagram (CMD) of stars from
and
within the MACHO fields. Evolved AGB stars occupy the region above
the RGB-tip at
mag (Cioni et al. 2000) and
mag. At
mag the split into two branches
is significant, though it starts already at
.
C-stars populate the well-extended tail toward red colours,
while M-stars lie along the almost vertical sequence at
mag. They reach a maximum luminosity of
mag, except for 3 stars that have
and are likely to be more massive O-rich stars (G04). Thus, these 3 have been excluded from the sample. Dust-enshrouded AGB stars are located
at redder colours (
mag). These obscured stars can
be either C-rich or O-rich, and spectra are needed to
distinguish between the two types. We include them in our
analysis, unless they are explicitly rejected by RAW93 (see
discussion of contamination by O-rich stars below). The small box
in the upper right corner shows the histogram of sources with
vs.
colour. Indeed, the branch of C-stars is
well separated from O-rich stars at about
.
Other
approximately vertical sequences in the main figure at
are populated by either foreground galactic stars or red
supergiants and upper main-sequence stars that belong to the SMC
(i.e. Nikolaev & Weinberg 2000).
Figure 2 displays the J-H vs.
colour-colour diagram of the SMC sources within the MACHO fields.
This diagram is useful for identifying stars with large infrared
excess (Bessell & Brett 1988; Nikolaev & Weinberg 2000). The main
feature in the diagram is an extended branch to the right side of
the dashed line that marks our selection of C-stars (
mag). It has been recognized as the locus of TP-AGB stars (e.g. Marigo et al. 2003). Circles and triangles, as in
Fig. 1, correspond to stars in our sample. The less
populated region at
and
mag
corresponds to obscured AGB stars. Along this branch a small
contamination of stars with
mag is present (small dots).
A handful of them might also be C-stars with very large
amplitudes caught at minimum light or faint extrinsic C-stars
(Westerlund et al. 1995).
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Figure 2:
J-H vs.
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Figure 3:
Panel a): absolute distance between 2MASS
and RAW93 stars in the
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Our photometric selection criteria of C-stars were checked against
the spectroscopically confirmed C-stars in the RAW93 catalogue.
Among 1707 stars listed by RAW93 we rejected 27 that were classified
by the authors as "doubtful'' (flag = 1 in Col. 8 of their
Table 4). Then, we cross-correlated the coordinates of the remaining
1680 C-stars with stars in the entire
catalogue
brighter than the RGB tip (i.e.
mag), by adopting a
searching radius of 3
.
The absolute distance between the 2MASS counterpart of the RAW93 sources is shown in
Fig. 3a. We found that 1275 C-stars in the RAW93 catalogue have a 2MASS counterpart within 3
,
and the peak of the distance distribution is at
.
The majority of the spectroscopically confirmed C-stars are well matched to stars in our sample when the same colour and
-magnitude criteria are
adopted (see also Cioni et al. 2000). Restricting the area to those
fields observed by MACHO, there are 931 C-stars in common between
RAW93 and
.
The location of these stars in the
near-IR CMD is presented in Fig. 3b.
We find that 73% of 802 C-stars photometrically selected have
confirmed C-type spectra. Our photometric selection identifies
more C-stars at
mag with respect to RAW93. This
might be because 1) some stars with these colours might be O-rich
AGB stars, and 2) some obscured stars were probably below the
detection limit in the RAW93 survey. Due to metallicity dependence
of the C-star life-time, in a population with the metallicity of
the SMC we expect only few of our non-RAW93 stars to belong to the
first category. Many spectroscopically confirmed C-stars have a
colour bluer than
mag and a magnitude fainter than
mag, overlapping the region where O-rich stars are also
present. These C-stars cannot be disentangled using only a
photometric selection criterion (see also G04).
Figure 3c shows the histogram of the number of
photometrically selected C-stars versus
colour, together
with the number of spectroscopically confirmed C-stars by RAW93
versus
.
There is a shift between the two distributions
suggesting that the spectroscopic identification of C-stars is
biased to bluer colours. This is not surprising - in a sample
that was spectroscopically selected using CN bands near 8000 Å,
Blanco et al. (1980) also found more C-stars with bluer colours
than
.
We find that within the MACHO fields there are 117 stars with
in the MC2 that are not present in the
RAW93 catalogue. They amount to 17% of the total number of MC2 stars with these colours. Because of the bias discussed above this is an upper limit to the number of O-rich stars contaminating this
region of CMD. From comparison with the spectroscopic sample of G04, we also expect the contamination to be low due to the fact
that there are only 3 confirmed O-rich stars in the SMC with
,
1.58, and 2.80, respectively.
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Figure 4:
Locations of C-stars in the observed SMC MACHO fields
(6 large squares each indicated by its MACHO field number).
Small black dots are all photometrically selected C-stars in the
SMC; blue crosses refer to C-stars with
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Figure 5:
Histogram of the absolute distance between 2MASS sources and
their MACHO counterparts. The dashed line is drawn at 3
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Considering the mis-identifications or missed cross-identifications as a result of an automatic association criteria, we estimate a contamination of significantly less than 2.7% and 7.9%, respectively, based on Loup et al. (2003).
The light-curves of stars in our sample were extracted from the
on-line MACHO catalogue.
The MACHO project observed the central body of the SMC
simultaneously in non-standard (i.e broad) blue (
)
and red (
)
bands for roughly 8 years (1992-2000). On average there are 800-900 observations per filter for most stars. Since we
were interested in the characteristics of the temporal behavior,
we used instrumental magnitudes and analyzed the photometric
variations in both bandpasses.
2MASS and MACHO coordinates were cross-correlated using a search
radius of 3
and the nearest, and reddest star from MACHO catalogue was chosen as the counterpart to the 2MASS source. Of the 802 photometrically selected C-stars lying within the MACHO fields (Fig. 4), 25 stars were detected twice with
different identification numbers because of overlap between
adjacent MACHO fields. However, since we chose the nearest star to
a 2MASS star as a MACHO counterpart, we avoid double or multiple
identifications of the same variable. Nevertheless we did check
that the quality of the light-curves is similar. The histogram in
Fig. 5 shows the distance in arcsec between a
2MASS source and the corresponding star in the MACHO database. The
distribution peaks at
6, and 65% of the stars are
within
1
.
For 51 stars the nearest MACHO counterpart
is at d>3
.
Of these, 20 are within 10
(6 within
4
)
and 31 within 20
.
The majority of these stars are
located at the edges of the MACHO fields and have poorer
astrometry perhaps due to distortions. Moreover, they have bluer
instrumental magnitudes than other stars in our sample. Therefore
they have been excluded from our analysis. Finally, the
photometrically selected sample contains 751 stars with MACHO
light-curves.
Table 1: Cross-identification between the MASTER and MACHO catalogs of our sample of C-stars.
By cross-correlating the RAW93 and
sample with the
MACHO data set, we found that 328 spectroscopically confirmed
C-stars are not included in our photometrically selected sample.
Thus, the total number of C-star light-curves analysed in the
next section is 1079 (751+328).
Table 1 is an extract of the full table,
available electronically at Centre de Données astronomiques de
Strasbourg (CDS), and
reports the first 15 lines of the cross-identified MACHO and
sources. It contains: MACHO, DCMC, and 2MASS
identifier (Cols. 1-3); right ascension and declination (in degrees) from the second incremental release of the 2MASS catalogue (Cols. 4 and 5); the positional difference between
2MASS and MACHO coordinates in arcsec (Col. 6) and the RAW93 identification number (if appropriate, otherwise 0) (Col. 7).
An independent Fourier analysis of the
and
light-curves was performed to search for periodicities in the
data. The MACHO time-baseline is about 2700 days, more than
twice that of OGLE-II database used, for example, by Ita et al. (2004a,b), Kiss & Bedding (2004) and G04. Resulting periods above 2600 days (
)
should be considered
less significant. Only photometric measurements that are more
accurate than 0.1 mag are used. The method that extracts
the light-curve parameters is based on the Lomb-Scargle
algorithm (Lomb 1976; Scargle 1982) as used by Rejkuba et al. (2003). It is described in Appendix A.1.
Initially 751 light-curves of photometrically selected C-stars
were analysed. All the light-curve fits and their parameters were
inspected visually. Based on this inspection, and on parameters
returned by the light-curve analysis programmes, three quality
flags were assigned to each light-curve: flag(1) describes the
data quality, flag(2) describes the light-curve fit quality,
and flag(3) describes the detected periodicities. Table 2 summarizes flag values and their meaning while more details are given in Appendix A.2 with examples of
light-curves associated to a given flag value. Light-curves with
produce reliable period determinations, as
confirmed by the fact that for most of them the Fourier analysis
has provided good results (
). Only for some stars
(
and
,
respectively, in
and
photometry)
classified as good light curves (
), uncertain
periodicity (flag(2)=3) or no periodicity (flag(3)=0) was
detected because of highly irregular light-curves.
Table 2: Description of the values of different flags for the complete sample of 1079 C-stars.
The same procedure was applied to the 328 stars common to MC2,
RAW93 and MACHO but missed by the photometric selection. Aliases
were identified from the diagram vs.
magnitude
as those periods that create clear vertical paths (see also
Appendix A.1). These correspond to periods equal to 1 and 2 years exactly and were removed from our analysis.
Table 3:
Light curves parameters: X is the mean magnitudes; first (second) row refers to (
)-light curve.
The adopted procedure allowed us to define a semi-automatic algorithm to obtain the light-curve parameters and access their quality. It is summarized as follows: the best fitting period(s) are obtained from Eq. (A.1); the quality of the observations is derived from flag(1); the quality of the period determination is evaluated using the spectral power that is closely related with a semi-automatic definition of flag(2)(Appendix A.2). Note that this procedure can also be applied to stars of a different type (i.e. M-type stars) in the MACHO catalogue or to other measurements of stellar variability in a comparable sampling.
Amplitudes related to the main periodicity of light-curve variations were determined both from the sinusoidal fit of each light-curve and from the peak-to-peak magnitude difference. A comparison of both determinations is given in Appendix A.3.
Table 3 lists the parameters derived from our
analysis for a total of 1079 stars. Only the first ten lines are shown in this paper, but the complete table is accessible electronically via CDS. The table contains: MACHO
identifier (Col. 1); the quantities of the period-amplitude
analysis for
(first row) and
(second row) light
curves: (Cols. 2-9): mean magnitude X0, A1, B1, A2,
B2, P1, and P2 in days, power strength of the first
and second period
;
(Col. 10); flag(1), flag(2), and flag(3) values (Cols. 11-13).
Among the 1079 stars the following have passed the data-quality
criterium
:
919
and 893
light-curves.
In the following discussion, only the good 919
light-curves
are used, unless explicitly stated otherwise. Within this sample
785 stars also have
and a minimum amplitude of
0.05 mag, thus are all variables. About 4% show a very regular
variation with only one periodicity. The others appear
multiperiodic. Some (7%) show a well-defined first period with
amplitude variations and a less clear second periodicity, while
others (59%) clearly show two periods (examples are given in
Fig. A.3).
The availability of accurate photometry and long-term observations has made the distinction between regular and semi-regular (SR) variables more and more difficult (Whitelock et al. 1997). The stellar light-curves can be as regular in SR as in Mira class, but on average SR variables show smaller amplitudes (Cioni et al. 2003). However, due to a difficult and rather subjective classification we only distinguish between two broad groups: sources which show a clear single periodicity and sources which are multiperiodic. About 15% appear to be irregulars (flag(2)=3), with no clear period.
Table 3 has 131 stars are in common with Cioni et al. (2003). Figure 6a shows the comparison
between the periods derived in the present paper and those in
Cioni et al. (2003). Amplitudes are compared in
Fig. 6b. We plotted stars with
in our analysis and stars with
Flag < 9 and
in the table by Cioni et al. (2003). There are 78 stars in
common. The mean period and amplitude differences are:
76 and
0.8. The periods agree
within the uncertainty in the period determination that is on the
order of 5%, except for few stars for which Cioni et al. (2003)
derive only one period and flag the LPV as multiperiodic, while we
find 2 periods that are typically shorter. It is possible that
additional longer periods are present as well. In contrast,
amplitudes are systematically different. Note that Cioni et al. (2003) define amplitude as the difference between the minimum and
maximum value of MACHO photometry, which is different from our
definition of
(see Appendix A.3). This could explain
why Cioni et al. amplitudes are systematically larger than ours.
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Figure 6: Comparison between the period and amplitude derived in this work and those from Cioni et al. (2003). |
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Figure 7 shows the comparison between periods and amplitudes as derived in the present paper and those by Ita et al. (2004b) and G04. In both panels, for all the three works only stars satisfying our photometric criteria are reported. In Fig. 7a in the case of multi-periodic variables the period of G04 corresponding to the largest amplitude and our first period are reported. Ita et al. (2004b) only give the predominant period and do not analyze multi-periodic light-curves. The present results are in good agreement with those by G04, predicting periods as long as 2400 d, while Ita et al. (2004b) do not find periods longer than about 690 d.
In Fig. 7b the OGLE I-band amplitudes of
Ita et al. (2004b) and G04 are compared with the present
-band amplitudes as derived from the light-curves
(
,
see Appendix A.1). In the case of G04 we plot
the largest amplitudes from his Table 2. Although a direct
comparison between
- and I-band amplitudes is difficult,
the shape of the histogram illustrating our results is similar to
that by Ita et al. (2004b), even if we have a larger number of
stars with amplitudes smaller than
0.4. G04 found that the
majority of C-stars in his sample have
,
while
Ita et al. (2004b) have almost no stars in the same amplitude range.
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Figure 7: Comparison between the period and amplitude derived in this work and those from Ita et al. (2004b) and G04. |
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Both Figs. 6b and 7b
indicate that the various methods for estimating pulsation
amplitudes can give systematically different results (see also
Appendix A.3). Ita et al. (2004b) and Cioni et al. (2003) derive
pulsation amplitudes as
,
where Xis the photometric band used, while G04 estimate amplitudes from
the sinusoidal fit.
In the past five years new results for the log(P) vs. K-band mag diagram have been obtained as a by-product of microlensing projects such as MACHO, OGLE, and EROS. Wood et al. (1999) found that the bright red giant variables in the LMC form four sequences: three are the result of different pulsation modes and one, at the longest periods (seq. D or the long secondary period sequence - LSP), remains unexplained. Ita et al. (2004b) showed that some sequences possibly split into sub-sequences at the discontinuity around the tip of the RGB. G04 analyzed spectroscopically confirmed M- and C-type stars and concluded that the LSP sequence is independent of evolutionary and chemical (C-rich or O-rich) effects. A comparison of the SMC and the LMC sequences can also be found in Cioni (2003). More recently, Schultheis et al. (2004) compared variable stars in the different sequences between the Magellanic Clouds and the NGC 6522 field in the Galaxy. These authors conclude that all three fields contain similar types of variables, but the proportion of stars that vary decreases at lower metallicities and the minimum period associated with a given amplitude gets longer.
Figure 8 shows the distribution of the mean
mag vs. the logarithm of the period obtained by
analyzing
light-curves. However, the discussion that
follows applies also to a similar analysis of
light-curves.
In fact as expected from the comparison shown in
Fig. A.6, there are no differences in the periods
obtained from the two channels.
In Fig. 8 all sources in the sample are indicated by
small black dots. In addition, stars for which only one reliable
period was detected (
and flag(3)=1) are plotted as
larger dots, while for stars with two reliable periodicities
detected (
and flag(3)=2 and 3), the first and the
second periods are plotted with blue open squares and red crosses,
respectively. The
-band photometry is the mean of DCMC and
2MASS measurements (
). DCMC magnitudes are
corrected for the shift in the absolute calibration according to
Delmotte et al. (2002). The
-magnitude is also dereddened
according to a SMC mean reddening of
E(B-V)=0.065
0.05obtained by averaging different measurements (Westerlund 1997).
By adopting the extinction law of Glass (1999) and RV=3.1 we
obtain
and AJ=0.05.
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Figure 8:
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All periods range in the interval 1.5
log(P)
3.5
(Fig. 8). There are 3 well-defined parallel
sequences in the PL diagram with a small number of stars lying
between the sequences. Part of the scatter is due to the fact that
the
-band is a mean of only 2 measurements and part is due,
probably, to depth effects in the SMC. The line of sight depth of
the SMC is estimated to range between 5-20 kpc by various authors
(Westerlund 1997). Hence, assuming the SMC distance modulus of
(m-M) = 19 mag and the full depth of 10 kpc, we expect a scatter on
the order of
0.4 mag around the average.
Sequences can be identified with B, C, and D from Wood et al. (1999). It is interesting to note that C-stars do not populate
shorter sequences (i.e. sequence A), a result already visible from
Fig. 4 of Ita et al. (2004b), and noted in the LMC by Fraser et al. (2005). Since the baseline explored here is longer than that
of previous works (except for Fraser et al. 2005 who analysed
MACHO data in the LMC), we find a well populated D sequence, about 34% of the variable stars in our sample have a first or second period longer than 630 days. This should be compared with previous
results which found (i) 25% of all variable AGB stars in the
LMC (Wood et al. 1999), but with a very small fraction belonging to
C-rich LPVs (Fraser et al. 2005); and (ii) 24.6% of all the
spectroscopically selected C-stars with periods from OGLE-II photometry in the SMC on the
sequence D (G04). The bias against detection of variables with periods in excess of 800 days
in the latter work may be why we find more C-stars on this
sequence. This shows that very long-term monitoring is essential.
Even in the case of our sample from MACHO with an 8-year
time-baseline, it is not clear if the drop in the period
distribution at
is real or an artifact due to
incompleteness at longest periods (see Fig. 9).
Sequence D is much broader than the others, and the nature of its
stars is still a matter of debate Wood et al. (2004). Only a few of
them, those with
(see Sect. 4.5)
are probably dust-enshrouded AGB stars that could have either
carbon or oxygen-dominated chemistry, but their number is very
small. Thus, as already concluded by Wood et al. (2004) from the
similarity of the colour variations associated with both the
primary and secondary periods, dust is unlikely to cause the LSPs.
Clearly, a long-term spectroscopic and photometric monitoring of
these stars is necessary to gain some insight into the nature of
their variability.
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Figure 9:
Histogram of all the periods found for ![]() ![]() |
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In Fig. 9 we compare the histogram of the
first detected periods, which represent the dominant periodicity
in a given star, with the second best period. First periods mainly
occupy sequences C and D with the peaks at
and
,
respectively. Second periods mainly populate
sequence B and peak at
.
A weak peak is also seen
at
,
which is clearly part of sequence D.
According to Lattanzio & Wood (2003) a star that evolves up the
AGB first pulsates at low amplitude on sequence A, then with
further evolution the sequence B mode will become unstable and
its amplitude increases, while the sequence A mode amplitude
decreases. The star will be a multi-mode pulsator having periods
in sequences A and B. The pulsation amplitude of each mode
increases with increasing stellar radius. Subsequently a
fundamental mode in sequence C will also become unstable. At
this moment up to three different periods of pulsation can be
detected in a given star. However, the competing growth rate
of the amplitude of each pulsating mode may shade the
detectability of a given periodicity. For example for a
star the fundamental period dominates at
,
while at
the first and second
overtone pulsation modes are stronger. The star will finally
end up as a dust-enshrouded, large amplitude fundamental mode
pulsator prior to ejection of all its envelope and the beginning
of the Planetary Nebulae phase.
For stars with detected multiperiodicity we plot the ratio between the longer to the shorter period as a function of the longer one in Fig. 10. A well-defined group of stars have ratios ranging from 1 to 2. These are shown in zoom-in in the upper left corner of the figure. They belong to the sequence C (longer) and B (shorter) periods. Their ratio distribution agrees with the scenario proposed by Lattanzio & Wood (2003) for a star pulsating in the fundamental, first and second overtone (see their Fig. 53). An extended tail up to a ratio of 20 is populated by stars with the longer period on sequence D and the shorter period along either sequence C or B. There are no theoretical models available at present to explain these high ratios which involve an intrinsic stellar pulsation; thus a different nature for long-term modulation needs to be invoked. It is interesting to note that most of the sources with P1<P2 have lower period ratios and slightly larger values of longer periods with respect to sources with P1>P2.
![]() |
Figure 10: The ratio between the longer to the shorter period versus the longer period. Black-filled circles represent stars with P1>P2, while red open squares sources with P1<P2. In the upper left corner we show an enlargement of the lowest ratios. |
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Figure 11 shows the period distribution (of only
the first periods P1) as a function of amplitude in the MACHO bands. Most variables have amplitudes in the optical bands below about 0.5 mag. Stars that occupy sequences C and D are
clearly separated. Sequence B is not present in these diagrams
because it is mostly populated by secondary periods. It should be
also noted that amplitudes belonging to secondary periods are
typically smaller, and thus this figure should be compared with
those of other authors (e.g. G04, Ita et al. 2004b) with
caution. In addition, the definition of amplitude is not always a
trivial issue with these highly variable stars that often have
variable amplitudes as well. We discuss this in Appendix A.3. Here
we use amplitudes determined directly from photometry (
)
as defined in Appendix A.3.
![]() |
Figure 11: Period distribution as a function of amplitude in the two MACHO photometric bands. |
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Both C and D sequences present distribution tails to large amplitude (from a few tenth up to 3 mag) values that probably correspond to variables of Mira type. While Miras typically occupy sequence C, a few of them can be found on sequence D because they have fainter magnitudes due to dust obscuration.
Figure 12 shows the period distribution for small
(
;
open histogram) and large
(
;
shaded histogram)
amplitude variables. The two distributions are different. Small
amplitude variables have two peaks that correspond to the period-magnitude
relations C and D. Large amplitude variables are more homogeneously
distributed between
and 3.1. Fewer of them have
longer periods, though at
the sample might be incomplete.
In Schultheis et al. (2004) the period of both small and
large amplitude variables defined as in Fig. 12
increases progressively with decreasing metallicity, even though
the general period distribution of the two classes is fairly
similar. In the present paper stars located in sequence C(
)
span the full range of periods if
they are either small or large amplitude variables. Therefore
there seems to be no indication of a difference in metallicity
between the small and large amplitude LPVs on this sequence. On
the other hand, the majority of large amplitude variables that
occupy sequence D (
)
have
.
A longer monitoring time-baseline is
necessary to discern whether this lack of longer period large
amplitude LPVs on sequence D is real or due to incompleteness,
and thus if it could represent shortage of lower metallicity stars
among the large amplitude variables.
![]() |
Figure 12:
Period distribution (![]() ![]() ![]() |
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Figure 13 shows that redder variables have a larger
amplitude. In fact Ita et al. (2004b) also found that the C-rich
regular pulsators (Miras) have larger amplitude the redder the
star, while the O-rich Miras seem to have arbitrary amplitudes as
a function of
colour. When the
colour gets redder,
periods increase for variables in the C sequence (see Fig. 13b). This is apparently not the case for stars populating the D sequence, which again indicates a different
mechanism is responsible for the light-curve variations.
![]() |
Figure 13:
![]() ![]() ![]() |
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We found no difference in the location of stars belonging to the
sequences in the
vs.
diagram. In Fig. 14 we investigated the
luminosity function and
colour distribution of stars in each sequence for both the first (empty histogram) and second (shaded histogram) detected
periods. Most of the stars with
belong to sequence C,
and they have only one detected and dominant periodicity. A
handful of them show a long second period in sequence D. The
faintest stars analyzed in this sample (
)
have their first period either in sequence B or D and
eventually a second period in sequence C, which is often the
case for stars in sequence D that are multi-periodic, while
stars in sequence B usually have only one period detected. The
bulk of the C-star population has a first period that occupies
sequence C or D and somewhat less sequence B, which is
instead largely populated by the second period of these same stars.
![]() |
Figure 14:
Distribution of the number of sources populating each ![]() ![]() ![]() |
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Figure 15 shows the PL-relation for all stars in
the sample. Overplotted are the theoretical models by Vassiliadis & Wood (1993) for different masses and a mean SMC metallicity of
Z=0.004. To transform magnitudes into
we used
the relations by Bergeat et al. (2002). Their Fig. 1
shows that the bolometric correction is well-defined for C-stars
with
,
while for increasing colours the
uncertainty becomes larger. In our sample of 1079 stars, only 30
have
and
,
these sources are
emphasized in Fig. 15 as green triangles. These
stars have large amplitudes (see Fig. 13) which is
likely to influence their location in Fig. 15.
![]() |
Figure 15:
![]() ![]() ![]() ![]() |
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A comparison of the PL diagram with the models in Vassiliadis &
Wood indicates that the bulk of C-rich LPVs in the SMC have lower
masses than 2.5-3 ,
which at the metallicity of the SMC
corresponds to ages of 0.5-0.3 Gyr (Pietrinferni et al. 2004). This
agrees with the SFH derived by Harris & Zaritsky (2004), who
found an active star formation during the past 3 Gyr with three
enhanced episodes at 2.5, 0.4, and 0.06 Gyr. Few carbon stars show
higher masses up to 5
corresponding to an age of
0.1 Gyr.
The spatial distribution of stars in each sequence is investigated. There seems to be no indication of a spatial correlation between stars that occupy one or the other sequence in the period-magnitude diagram with their location in the galaxy. We also checked if there is any difference in the spatial distribution of stars where different masses were expected. Again, we do not find clear indication of a different distribution of sources with a different initial mass. There are overall less sources with high mass (or young), and a lack of these sources in the Northeast, Northwest and Southwest regions compared to other locations in the galaxy. The distribution of sources with intermediate and low mass is fairly similar.
This work analyzes and discusses the MACHO light-curves of 1079 C-stars. The sample consists of 751 photometrically selected stars from the MC2 catalogue according to
mag
and
mag criteria, and 328 spectroscopically confirmed
C-stars from RAW93 catalogue. Many of the photometrically selected
sample also have C-type spectra (RAW93), meaning that for 18% of
all the sample we do not know their spectral type. However, given
the low metallicity of the SMC and a very low number of
spectroscopically known O-rich AGB stars in the SMC (G04), we
expect the contamination by O-rich AGB stars to be negligible. For
all the stars, we performed Fourier analysis of their
light-curves identifying up to 2 significant periodicities.
All the stars for which good quality light-curves exist were found
to vary with amplitudes of at least 0.05 mag in the two MACHO
photometric bands. The analysis was carried independently for
and
light-curves, and the derived periods are
identical (to within the errors). After a selection based on the
quality of the light-curve, significance of the derived periods,
and quality of the periodicity fits, a total of 919 C-stars were
used in further analysis.
Carbon stars occupy bright parts of sequences B, C, and D in
the
diagram. None of the stars in our sample have
shorter periods characteristic of sequence A, which is in
agreement with recent studies by Fraser et al. (2005) in the LMC,
and Ita et al. (2004a,b), and G04 who observed this in the SMC as
well. The large majority of the stars have their primary
(dominant) periodicity on sequence C or D, while more than 2/3
of sequence B is populated by secondary periods of those stars
whose primary period is on sequences C or D. The stars whose
primary period is on sequence B are preferentially fainter and
bluer. This is in agreement with the models that predict change of
pulsation mode for the LPVs from higher overtones towards a
fundamental mode as they evolve along the AGB (e.g. Lattanzio & Wood 2003).
Most of the stars with
belong to sequence C, and only
a handful of these reddest variables show a long second period in
sequence D. This is a clue that dust obscuration cannot be a
cause of the long secondary periods (see Wood et al. 2004, for a detailed
discussion). The luminosity functions of the three
sequences span a similar range of magnitudes, but the faint
distribution tail of sequence D is more populated. Again, we
cannot explain it with the dust obscuration, as there are very few
stars redder than
on this sequence. The width of this
sequence in the period-magnitude diagram is larger than that of
other sequences, especially at brighter magnitudes, but the lack
of stars with a period longer than (
)
may be due
to incomplete time coverage.
Stars belonging to different pulsational sequences are
homogenously mixed over the SMC area. Their masses were derived
from a comparison with the theoretical tracks of Vassiliadis & Wood (1993), and the majority of them are indicative of the major
star formation episode that took place 0.3-0.5 Gyr ago.
This is in excellent agreement with Harris & Zaritsky (2004), who
found enhanced star formation episodes at 2.5, 0.4, and 0.06 Gyr
in the SMC. However, our sample indicates a considerably weaker
star formation event at younger ages (
0.1 Gyr ago), which
is expected as these younger, and thus more massive, stars are
preferentially oxygen-rich, hence not in our sample. There is no
clear difference in the spatial distribution of the stars with
different masses.
The very long-time baseline of MACHO observations has allowed us
to confidently extract long periods up to
.
We
found that about 10% of the variables fall on sequence B, 30%
on C, and 34% on D. The latter percentage is higher than the
25% derived by Wood et al. (1999) and than the 21% derived by
G04 in the LMC. It is also important to note that in the LMC,
Fraser et al. (2005) find only a small fraction of probable
C-stars with periods along sequence D. Monitoring of these
variables over more than an 8-year period photometrically and
spectroscopically is desirable in order to discern their nature.
According to Wood et al. (2004) these stars belong to the only
class of bright large amplitude variables whose properties cannot
be explained with theoretical models at present.
Acknowledgements
We thank the anonymous referee for providing constructive and insightful comments that have greatly improved the paper. We thank N. Delmotte for giving us thecatalogue of SMC data before publication. Support for this project was provided by the ESO Director General Discretionary Fund. The support given by ASTROVIRTEL, a Project funded by the European Commission under FP5 Contract No. HPRI-CT-1999-00081, is acknowledged. This paper utilizes public domain data originally obtained by the MACHO Project, whose work was performed under the joint auspices of the US Department of Energy, National Nuclear Security Administration by the University of California, Lawrence Livermore National Laboratory under contract No. W-7405-Eng-48, the National Science Foundation through the Center for Particle Astrophysics of the University of California under cooperative agreement AST-8809616, and the Mount Stromlo and Siding Spring Observatory, part of the Australian National University. It also makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. The DENIS project is partially funded by European Commission through SCIENCE and Human Capital and Mobility plan grants. It is also supported, in France by the Institut National des Sciences de l'Univers, the Education Ministry and the Centre National de la Recherche Scientifique, in Germany by the State of Baden-Würtemberg, in Spain by the DG1CYT, in Italy by the Consiglio Nazionale delle Ricerche, in Austria by the Fonds zur Förderung der wissenschaftlichen Forschung und Bundesministerium fuer Wissenschaft und Forschung, in Brazil by the Foundation for the development of Scientific Research of the State of Sao Paulo (FAPESP), and in Hungary by an OTKA grant and an ESOC&EE grant.
The method used to determined the first period is already
discussed in Rejkuba et al. (2003). To detect the second period
we used the following fitting function:
We call the first period (P1) the best fitting period
obtained from the first term of Eq. (A.1) and
the associated value of the power. We call the second period (P2) the best fitting period obtained from
Eq. (A.1) and
the associated value of
the power. Amplitudes (
)
associated with the first period
are defined from the sinusoidal fit as:
Table 2 summarizes the values of the three
quality flags associated to each light-curve and the number of
light-curves with a given value. In particular, flag(1) is
related to the accuracy of the photometric measurements. We
defined n0.05 and n0.01 as the ratio between the
number of observations with, respectively, magnitude error
and <0.01 and the total number of
measurements in each light-curve (i.e. with magnitude errors
). Thus, values to flag(1) are assigned as follows:
flag(1) | = | ![]() |
|
flag(1) | = | ![]() |
|
![]() |
|||
flag(1) | = | ![]() |
|
![]() |
|||
flag(1) | = | ![]() |
|
![]() |
![]() |
Figure A.3: As in Fig. A.2, but for flag(3). |
The fit quality flag (flag(2)) and the periodicity flag (flag(3); see Table 2) are less objective and much more related to an inspection by eye than flag(1). In order to assign values to these flags, first we carefully eye-inspected the first period in the phase and time diagrams. Then, we looked at the combined periodicities (Eq. (A.1)). On the basis of these inspections we assigned values for flag(2) and flag(3) according to Table 2. In Fig. A.2 we plot light-curves with different values of flag(2) referring only to their first periodicity. Figure A.3 illustrates examples of light-curves with different values of flag(3). In both figures for each star we plot the corresponding power spectrum and the light curve in phase and time domain and overplot the best fitting sinusoidal function.
In order to provide an automatic classification of the fit
quality, we analyzed the correlation between flag(2) and the
strength of the power of the first period (
)
as follows. In Fig. A.4 the histogram of
is reported for different values of flag(2). It is clear that starting from flag(2)=0 to flag(2)=4 the peak of the distribution moves toward lower power values. In
particular, for
flag(2)= 0 and 1
(Fig. A.4a) the only region populated is that
with
;
the bulk of variables with
flag(2)=2 have
with only a small number
of stars with
.
The distribution of variables
with flag(2)=3 peaks in the range
,
and those for which no satisfactory period could be
derived mainly have
.
Figure A.5 emphasizes the good correlation
between these two quantities. In panels a, b we plot vs.
(see next section for a discussion) for stars with
(a) and stars with
(b).
Panels c, d show the same quantities but for stars with
flag(2)=3 (c), and
(d). We conclude
that the selection described in panels b, d outlines three
sequences and the distribution of stars in both panels are very
similar. These results
are related to
light-curves, however an inspection of
light-curves gives similar results. Figure A.6
shows an almost perfect correlation between the first period
obtained from
and
light-curves. Only stars with
are plotted. For 94% of the objects the residual
dispersion of the differences is lower than 4%. Only a few points
scatter away from the 1:1 line, implying that the primary period
is the same in both bands. Most of these points occur because the
first period (here P1) found from the
light-curve
corresponds to the second period (P2) found from the
light-curve, and confronting only P1 between
both light-curves produces a point that does not follow the
correlation. A few other points occur when the period P1derived from the
light-curve is twice that derived from the
light-curve.
Summarizing, an automatic selection criterion (i.e. by using
)
can be applied with a good level of confidence for
the analysis of the primary periods of a large sample of variables.
To define flag(3) in a similar automatic way as flag(2), we
considered the values of the reduced
of the sinusoidal
fit. In most cases the
value decreases when two periodicities are considered, even in cases we classified with flag(3)=1, i.e. when only one period could be reliably derived.
Perhaps this indicates the complexity of stellar pulsation. For
these stars flag(3)=1 does not mean that the light-curve is
clearly characterized by one periodicity only; rather it
implies that the second period is much less regular and could not
be fitted well with a set of sinusoid functions.
Figure A.7 shows the relation between the
pulsation amplitude derived from the sinusoidal fit (
)
and that from the peak-to-peak magnitude difference (
)
for
and
light-curves separately. The latter value is defined
as
,
where
and
are the maximum and the minimum values averaged over
a few data points in order to avoid spurious detections. For
sources of regular periodicity with excellent observational
data (i.e. low photometric errors and flag(1)=0) that clearly
show only one periodicity, we discarded the first 5 measurements
and took the mean over the brighter (fainter) ten measurements
(see for example the light-curve 212.15910.2115 in
Fig. A.2). Since the majority of sources show
less regular light-curves with bumps and multiperiodicity, the
first brighter (fainter) 150 measurements, which could be
disturbed by the secondary periodicity with
days, are
excluded. Then, we computed the "maximum'' ("minimum'') value
as the average over the next brighter (fainter) 40 photometric
measurements.
![]() |
Figure A.4:
Histogram of the power of the first period (![]() |
![]() |
Figure A.5:
Distributions of the first period (![]() |
The sinusoidal fit clearly predicts smaller amplitudes compared to
.
Quantitatively,
0.22 and
0.25. This is expected in case of deviations from pure sinusoidal
variability, due to the presence of irregularities in the intrinsic
light variation and photometric uncertainties caused by measurement
errors and variations in seeing. The latter may lead to blending and
spurious measurements due to the presence of cosmic rays and defects
on the CCD. As can be seen from Fig. A.2 this applies
to all but the most regular variables with flag(2)=0.
The histograms of amplitudes in the two MACHO photometric
bands are reported in Figs. A.7c,d. The bulk
of sources have
and
.
As expected, amplitudes in
are slightly higher than those in
(see Fig. A.7e). The
difference in the mean
amplitude of the two bands is
0.18.