5 Cepheid variables

Cepheid variables are post-main sequence stars occupying the instability strip in H-R diagram. Their light curves display the characteristic "sawtooth'' pattern, with periods ranging from a few days to more than 100 days. We have identified 26 Cepheid variables in the disk of M 31 (Fig. 3). The Cepheids were identified by looking for the stars with light curves similar in appearances to known Cepheids of M 31. The period of the Cepheids was determined as described in Sect. 4.2. The periods estimated independently in two filters agree very well for most of the Cepheids; however difference in the period exists for a few short period variables due to their low amplitude as well as comparatively larger photometric error in I-band. Therefore, the period calculated using R filter data was considered as the period of the star and phase in both R and I filters have been evaluated using this period.

Since most of the Cepheids are believed to be in their second crossing of the instability strip, their period of pulsation is directly related to the stellar mass, and hence, to the main sequence life time of the star. Hence a rough estimate of the age of the star can be obtained from the observed pulsation period of the Cepheids. We have used the following relation given by Magnier et al. (1997b) to estimate the age of Cepheids:

$$\log A = 8.4 - 0.6 \log P$$ (3)

where A is the age of the Cepheid in years and P is the pulsation period in days. As Cepheids detected in our study have period ranging from $\sim$7.5 to 56 days, the age of the Cepheids varies from $\sim$22 to 75 Myrs with a peak of 48 Myrs. Though there is an uncertainty of $\pm$0.15 in $\log A$ (Magnier et al. 1997b), this relation provides an indicative age of the Cepheids in M 31, and hence some idea of the recent star formation history in the field of our observations.

Figure 3: Location of the 26 Cepheids identified in our data juxtaposed over the entire observed field of M 31. Small dots indicates the position of 4400 stars observed in our field. Overlapping area of DIRECT M 31D field is also shown.

In Table 4, we present characteristics of these 26 Cepheid variables detected in our study. The identification number of the star, celestial coordinate for J2000, phase weighted mean magnitude in both R and I filters, amplitude of variability in R filter, period, age of the Cepheid (estimated using Eq. (3)) and total number of points in R filter used for the light curve study are given. The Cepheids are sorted in the increasing order of their periods. The reference to variability reported in the literature and corresponding period, if known, are listed in the last two columns. Out of the catalogued 26 Cepheids, the variability is reported for the first time for stars V22 and V24; moreover 13 of the variables are identified as Cepheids for the first time. The remaining 13 Cepheids have been known from earlier studies. The present data confirms their variability and our periods are generally in good agreement with the reported value.

Figure 4: Phase-magnitude diagram is plotted for 26 Cepheids detected in the present study. Filled and open circles represents R and Imagnitudes respectively.

Figure 4: continued.

Figure 4: continued.

 

 

Table 4: A list of 26 Cepheids observed in our study with their characteristic parameters. Star identification by KAL99, Tomaney & Crotts (1996), MAG97 and Berkhuijsen et al. (1988) are prefixed with K, TC, M and B respectively in Col. 10. The periods of the 13 Cepheids given in the catalogues are given in the last column. The Cepheids discovered and classified by us are marked as $\dag$ and * respectively in the first column.

Star	$\alpha$	$\delta$	$\overline R$	$\overline I$	$\Delta_R$	Period	Age	N	Other	Period
ID	(deg)	(deg)	(mag)	(mag)	(mag)	(days)	(Myrs)		Identification	(days)
V1	10.9321	41.1970	20.48	19.98	0.27	7.459$~\pm~$0.002	75	123	K V883	7.459
V2*	10.8469	41.1737	20.17	19.69	0.15	8.566$~\pm~$0.003	69	124	TC 170	-
V3*	10.8669	41.2320	20.61	20.28	0.22	8.836$~\pm~$0.004	68	120	TC 18	-
V4	10.9366	41.2503	20.28	19.54	0.11	9.160$~\pm~$0.008	67	86	K V1219	9.173
V5	10.9721	41.2128	20.56	20.04	0.19	9.790$~\pm~$0.005	64	92	K V2879	9.790
V6*	10.8770	41.0601	20.43	19.76	0.15	10.383$~\pm~$0.009	62	93	TC 76	-
V7*	10.8736	41.2367	20.42	20.27	0.28	10.500$~\pm~$0.004	61	123	TC 16	-
V8	10.7500	41.1426	19.89	19.55	0.17	11.17$~\pm~$0.01	59	95	M 65	25.0$~\pm~$5.0
V9*	10.8594	41.2004	20.21	19.60	0.26	13.773$~\pm~$0.006	52	126	TC 20	-
V10*	10.9290	41.1715	20.77	19.84	0.48	14.420$~\pm~$0.006	51	116	TC 85	-
V11	10.9255	41.2489	19.57	18.87	0.16	15.26$~\pm~$0.01	49	96	K V635	15.255
V12	10.9576	41.2227	20.84	20.08	0.32	15.46$~\pm~$0.01	49	89	K V2286	15.464
V13	10.8259	41.1386	19.82	19.46	0.40	15.76$~\pm~$0.01	48	94	M 68	14.0$~\pm~$2.8
V14*	10.9049	41.2419	19.93	19.58	0.22	15.90$~\pm~$0.01	48	121	TC 194	-
V15*	10.9115	41.2396	20.79	19.91	0.30	15.95$~\pm~$0.01	48	126	TC 196	-
V16	10.9775	41.2348	20.28	19.74	0.40	16.38$~\pm~$0.02	47	47	K V3198	16.345
V17*	10.9069	41.1868	20.12	19.60	0.39	16.60$~\pm~$0.01	47	124	B 4614	-
V18	10.9853	41.2176	19.47	19.09	0.21	17.73$~\pm~$0.01	45	91	K V3583	17.703
V19	10.9839	41.2374	19.83	19.60	0.32	17.83$~\pm~$0.03	45	18	K V3551	16.699
V20*	10.9526	41.1540	19.20	18.99	0.35	20.09$~\pm~$0.01	42	96	TC 207	-
V21	10.8379	41.1514	19.74	19.31	0.39	21.44$~\pm~$0.02	40	96	M 69	13.0$~\pm~$2.6
V22$^\dag$	10.8272	41.1071	20.01	19.19	0.29	26.99$~\pm~$0.04	35	92	-	-
V23*	10.9059	41.2379	19.78	18.92	0.34	28.78$~\pm~$0.02	33	127	TC 30	-
V24$^\dag$	10.9002	41.1823	20.55	19.57	0.23	35.1$~\pm~$0.05	30	121	-	-
V25	10.9293	41.2475	18.89	18.35	0.31	43.53$~\pm~$0.08	26	97	K V836	43.371
V26	10.9183	41.1856	19.36	18.82	0.22	56.02$~\pm~$0.08	22	124	K V164	56.116

In Fig. 4, we show the phase-magnitude diagram for all the 26 Cepheids in both R and I filters. The I-band light curves show more scattering particularly for short period and low amplitude Cepheids. This could arise due to the following reason: the amplitude of Cepheids decreases with increasing wavelength (e.g. Freedman et al. 1985) and Freedman (1988) found a ratio of 1.00:0.67:0.44:0.34 in amplitudes of B:V:R:I filters. As most of these Cepheids have a pulsation of $\sim$0.1-0.2 mag in Rfilter, the amplitude of pulsation in the I filter is even less and becomes comparable to the photometric errors.

5.1 Comparison with earlier studies

Our target field in M 31 has been observed only in a few earlier surveys. The V band photometric observations have been obtained by Magnier et al. (1997a, hereafter referred to as MAG97) for about 25 days. They have observed 9 fields of M 31 and each field covered a region of $\sim$11 $\times$ 11 arcmin² on the sky. They found 130 Cepheids in the survey. As we have observed smaller field, only 3 of them are located in our field and we have identified all of them. In Table 4, we have compared period of these stars with those found in our data set. One can easily notice a large discrepancy between the periods obtained in two surveys which we attribute to insufficient observations by MAG97.

BVI photometry has been carried out by Kaluzny et al. (1999, hereafter referred to as KAL99) under the DIRECT project. Out of 35 Cepheids detected in DIRECT project in the field M 31D, 11 are located in our field of observations. We have identified 10 of them and notice that except in the case of V19, the periods determined in the two surveys are in excellent agreement. The discrepancy in the period of V19 appears to be due to inadequate data in the present study (see Table 4). One of the Cepheid (K V952) reported in KAL99 could not be detected in our data due to blending effect as this star is highly blended by a neighbouring bright red star.

For Cepheids V7, V9, V14, V15, V17, V23 and V26, the parameters given in Table 4 supersedes our earlier results given in Paper I, as these are derived using data with much longer temporal coverage.

5.2 The Period-Luminosity (P-L) diagram

The Cepheid variables exhibit a stable periodic variability and there is an excellent correlation between their mean intrinsic brightness and pulsation period. Consequently, the Cepheid Period-Luminosity relation provide an important standard candles to measure distances to galaxies up to the Virgo Cluster, by comparison of their absolute magnitudes inferred from P-L relation with their observed apparent magnitudes. Based on the study of Cepheids located in LMC, SMC and IC 1613 galaxies, Udalski et al. (2001) found that the zero points of the P-L relations have no significant dependence on metallicity. In fact, they are constant to within $\sim$0.03 mag. However, the values could be affected due to other effects, e.g. blending, which can cause up to 10% uncertainty in the distance determination of M 31 (Mochejska et al. 2000). We have therefore used metallicity independent P-L relation in the present analysis.

Using a set of 32 LMC Cepheid variables, Madore & Freedman (1991) have obtained an equation of the ridge line in P-L diagram for Cousin Rfilter as:

$$M_{R} = -2.94(\pm 0.09)(\log P-1) - 4.52(\pm 0.04)$$ (4)

while using a large sample of 658 Cepheids, Udalski et al. (1999c) derived a P-L relation for Cousin I filter as:

$$M_{I} = -2.96(\pm 0.02)(\log P-1) - 4.90(\pm 0.01)$$ (5)

where we adopt a true distance modulus of 18.5 mag for the LMC (Freedman et al. 2001) instead of 18.2 mag used by Udalski et al. (1999c).

Figure 5: Period-Luminosity relation for 26 Cepheids observed in our field. Lower panel indicates P-L relation for R filter while upper panel indicates the same for I filter. Filled circle: fundamental mode Cepheids; asterisk: possible Population II Cepheids. No corrections for reddening have been applied to either data set. The slope of the fitted straight lines are fixed at dm/d $\log P = -2.94$ and -2.96 for R and I filters respectively. The dashed envelope lines have been drawn 0.5 mag from the straight line fit (see text for details).

The apparent mean magnitudes of 26 Cepheids are plotted as a function of $\log P$in Fig. 5. The slope of the straight line is fixed as -2.94 and -2.96 for R and I filters respectively. The Cepheids V24 and V26 are located by about 1.0-1.5 mag below the ridge line in the P-L diagram which makes them a possible Population II Cepheid variables. Therefore, we did not consider these two Cepheids for the zero points evaluation. Using remaining 24 Cepheids, we found a zero points of $23.54\pm0.09$ and $23.02\pm0.07$ mag in R and I filters respectively. The dashed envelope lines are drawn 0.5 mag from the fitted line, representing the expected intrinsic scatter of P-L relation due to the finite width of the instability strip in the H-R diagram (Sandage 1958; Sandage & Tammann 1968), blending of Cepheids as well as the range of interstellar extinction.

The above zero points give us a mean apparent distance modulus of $25.12\pm0.09$ and $24.96\pm0.07$ mag in R and I filters respectively. This yields a mean colour excess of

$$E(R-I) = (m-M)_{R} - (m-M)_{I} = 0.16~$mag$ .$$ (6)

Using this and the standard extinction law given by Cardelli et al. (1989), we found a total extinction of 0.63 and 0.47 mag in R and I filters respectively. Correcting the apparent distance modulus for the total extinction, we obtain a distance modulus of $24.49\pm0.11$ mag for the M 31 which corresponds to a distance of $790\pm45$ kpc. Stanek & Garnovich (1998) based on red clump method and Holland (1998) based on globular cluster CMDs estimated a distance of M 31 as 780 kpc while a recent determination of Freedman et al. (2001) based on Cepheid P-L relation suggest a distance of 750 kpc for M 31. Present estimate of M 31 distance is thus agrees well with them.

Figure 6: The colour-magnitude diagram for the Cepheids under discussion. Symbols are the same as in Fig. 7. The solar metallicity isochrones by Bertelli et al. (1994) for $\log A = 7.4$ and 7.9 are also plotted. Average magnitude of errors in observations are shown in the right bottom corner of the figure.

5.3 The Colour-Magnitude diagram

In the M_R, (R-I)₀ diagram, locations of the Cepheids under discussion are shown (Fig. 6). The isochrones by Bertelli et al. (1994) for solar metallicity drawn for $\log A$ as 7.4 and 7.9 are also plotted which supports the age estimated from Eq. (5). As most of the detected Cepheids ranges in magnitude from 19 to 21 in R filter and 18 to 20 in I filter, there can be an error of $\sim$0.2 mag in the colour determination.

5.4 Location of the Cepheids

The locations of the Cepheids (see Fig. 3) indicate that most of them are towards the inner side of M 31 (bulge direction), where, possibly, the star formation was active a few million years ago. This inference is further supported by the fact that, the early type supergiants detected in our field ($R \sim 16$ to 19 mag) also occupy the same strip where the Cepheids are found. So it is tempting to suggest that we are possibly tracing one of the spiral arms of M 31.

5.5 Brief description of the Cepheids

We detect 26 Cepheids in which V22 and V24 are newly discovered while V2, V3, V6, V7, V9, V10, V14, V15, V17, V20 and V23, as reported variables in earlier studies, are identified Cepheids for the first time. Out of the 26 Cepheids, only 12 are monitored for a period of four years while remaining are observed only for last three years. The period of 4 Cepheids were cross-examined using INT data (obtained with 2.5 m telescope at La Palma, Canary Islands) using Lafler-Kinman (1965) periodogram which agree quite well with our estimated period except V8 which have a low amplitude, highly scattered light curve and needs further investigation. One of the Cepheid, V4, could not be ascribed as a Cepheid variable purely on the basis of our data. However, V band light curve of the same star in KAL99 clearly indicates its Cepheid nature. Other interesting Cepheids are briefly described below:

V19 -- KAL99 obtained a period of 16.999 days for this Cepheid while we obtained a period of 17.83$~\pm~$0.03 days using 18 data points. The light curve for this Cepheid is extremely under-sampled and poorly distributed in phase in our data which could be the reason for discrepancy between two periods.

V22 -- It is discovered in the present study with a period of 26.96$~\pm~$0.04 days. A small bump in the falling branch can be seen in both filters.

V24 -- This is another Cepheid discovered by us. It has a period of 35.12$~\pm~$0.05 days and an exceptionally large value of (R-I) colour of 0.98 mag. In the P-L diagram, this Cepheid deviates maximum towards the fainter side in both filters and could be a possible Population II Cepheid.

V26 -- This is the longest period Cepheid in our sample. We obtained its period as 56.02$~\pm~$0.08 days in agreement with a period of 56.116 days derived by KAL99. In the P-L diagram, this Cepheid is also located downwards in both filters and it could also be a Population II type Cepheid.