3 Search for periodic stars

3.1 Reconstruction of the light curves

Since the EROS photometry is described in detail in Ansari (1996) only the main features of the PEIDA++ package are summarised below. For each field, a template image is first constructed using one exposure of very good quality. A reference star catalogue is set up with this template using the CORRFIND star finding algorithm (Palanque-Delabrouille et al. 1998). For each subsequent image, after geometrical alignment with the template, each identified star is fitted together with its neighbours, using a PSF determined on bright isolated stars and imposing the position from the reference catalogue. A relative photometric alignment is then performed, assuming that most stars do not vary. Photometric errors are computed for each measurement, assuming again that most stars are stable, and parameterised as a function of star brightness and image sequence number. Figure 2 shows the mean point-to-point relative dispersion of the measured fluxes along the light curves as a function of $R_{\rm E}$ and $V_{\rm E}$. The photometric accuracy is $\sim$15% at $R_{\rm E} \sim 18$, and about 2% for the brightest stars.

Figure 2: Average value of the relative frame to frame dispersion of the luminosity measurements versus $R_{\rm E}$ (upper panel) and $V_{\rm E}$ (lower panel), for stars with at least 30 reliable measurements in each colour. This dispersion is taken as an estimator of the mean photometric precision. The superimposed hatched histograms show the EROS magnitude distribution of monitored stars.

Finally, using the PEIDA++ photometric package, we reconstruct the light curves of 1 913 576 stars.

 

 

Table 2: Impact of each selection criterion on the data. For each cut, the number of remaining light curves is given.

Cut	Criterion
	Total analysed	1 913 576
1	$N_{\rm m}$ > 30	1 299 690
2	$R_{\rm E} < 17$	330 089
3	Pre-filtering	41 545
4	Period search	2553
5	Aliasing	2424
6	Visual inspection	1362

3.2 Pre-selection

Each one of the light curves is subjected to a series of selection criteria in order to isolate a small sub-sample on which we will apply the time consuming period search algorithms. These analysis cuts are briefly described hereafter (see Derue 1999 for more details) and their effect on the data is summarised in Table 2:

cut 1:

At least 30 measurements should be available in both passbands and the base flux must be positive;

cut 2:

The search is restricted to stars whose magnitude is $R_{\rm E} < 17$ which corresponds to a photometric accuracy in $R_{\rm E}$ better than $\sim$10%.

cut 3:

A non specific pre-filter is applied which retains most variable stars. It selects light curves satisfying one or both of the following criteria:

• the relative dispersion of the flux measurements is 25% larger than the average one for the set of stars having the same magnitude;
• the distribution of the deviations with respect to the base flux is incompatible with the one expected from a stable source with Gaussian errors during the observation period (Kolmogorov-Smirnov test).

These cuts are tuned to select $\sim$10% of the light curves. We have checked that this procedure allows one to retrieve the previously known Cepheids observed by EROS in the Magellanic Clouds. We also keep a randomly selected set of light curves ($\sim$2%) to produce unbiased colour-magnitude diagrams, for comparison purposes;

At this stage a set of 41 545 light curves remains which is then subjected to a periodicity search.

3.3 Light curve selection

We use three independent methods to extract periodic light curves. The first two are classical methods already described in the literature: method 1 is based on the Lomb-Scargle periodogram (see Scargle 1982) while method 2 makes use of the One Way Analysis of Variance algorithm (see Schwarzenberg-Czerny 1996). Both provide the probability for false periodicity detection. In method 1 one computes the Fourier power over a set of frequencies. It is therefore well adapted to identify sinusoidal light curves. It can be improved by incorporating higher harmonics in order to detect any kind of variability such as eclipsing binaries (Grison 1994; Grison et al. 1995).

We developed a new method, the third one, in order to extract periodic light curves in a way which is insensitive to the particular shape of the variation. This method also provides a probability for false periodicity detection.

Figure 3: Distributions of $\chi ^2$ (left panels) and variation of x with the value of the test-period in days (middle panels) obtained with the third method for a stable star and for a typical Cepheid candidate. The bold dots are pointing to the actual period of the star. The right panels show the light curve (in $R_{\rm E}$) obtained once the period has been folded in.

It consists in searching for a frequency such that the corresponding phase diagram, i.e. the series of fluxes F_i versus phases $\varphi_{\rm i}$ in increasing order of $\varphi$, displays a regular structure significantly less scattered than for other frequencies. Let $T_{\rm obs}$ be the observation duration ($\sim$100 days in this analysis). We span the frequency domain from $T_{\rm obs}^{-1}$ to $(0.2\ {\rm days})^{-1}$ with a constant step of $(4 \times T_{\rm obs})^{-1}$. The total number of test-frequencies is thus $N_{\rm test}~\sim~2000$. This sampling ensures that the total phase increment over $T_{\rm obs}$is $\pi/2$ for two adjacent test-frequencies.

For each value of the test-frequency we compute the corresponding phase diagram. We calculate a $\chi ^2$ from the weighted differences of F_i and the fluxes interpolated between F_i-1 and F_i+1:

 

$$\chi^2 = \sum_{i=1}^{N_{m}}\left(\frac{F_i-(1-R_i)\times F_{i-1}-R_i\times F_{i+1}}{\sigma_i}\right)^2$$ (2)

where $R_i = (\varphi_i - \varphi_{i-1})/(\varphi_{i+1}-\varphi_{i-1})$ and $N_{\rm m}$ is the number of measurements. The uncertainty $\sigma_i$ takes into account the errors $\epsilon_i$ on the flux F_i and on the interpolated flux: $\sigma_i^2 = \epsilon_i^2 + (1-R_i)^2 \epsilon_{i-1}^2 + R_i^2 \epsilon_{i+1}^2$.

Figure 4: Distribution of $-\log({\rm Prob})$ for method 1 (top), 2 (middle) and 3 (bottom). The histograms show the distribution for all 41 545 stars on which the periodicity search is done (in white) and for the 2553 selected light curves (in black).

Expression (2) can be interpreted as the $\chi ^2$ of the set of differences between the odd measurements with respect to the line joining even ones, added to the $\chi ^2$ of the set of differences between even measurements with respect to the line joining odd ones. If the star is measured in both colours, we add the $\chi ^2$ obtained in each colour ($N_{\rm m}$ is then twice as large). For a given stable star with Gaussian errors, each phase diagram can be considered as a random realisation of the light curve. When the test-frequency spans the search domain, the distribution of the $\chi ^2$ parameter defined by Eq. (2) is the one of the standard $\chi ^2$ with $N_{\rm m}$ degrees of freedom. Since $N_{\rm m}$ is large enough, this distribution is close to a Gaussian with average $N_{\rm m}$ and variance $2N_{\rm m}$ (see upper left panel of Fig. 3). For a periodic variable star, the $\chi ^2$ distribution displays a main cluster, when the test-frequency results in a phase diagram with non-correlated point to point variations, and a few lower values when the test-frequency corresponds to a phase diagram with a regular structure (see lower left and middle panels of Fig. 3). In practice, instead of using the parameter defined by Eq. (2), we use the reduced variable:

 

$$x = \frac{\chi^2-<\! \chi^2 \!>}{<\! \chi^2 \! >/N_{\rm m}} \times\frac{1}{\sqrt{2N_{\rm m}}}$$ (3)

where $<\!\chi^2\!>$ is the average of the realisations of $\chi ^2$for all test-frequencies. For a stable star, the distribution of this variable x is a Gaussian centred at zero, with unit variance. If the errors are correctly determined $<\! \chi^2 \! >/N_{\rm m}$ is close to unity; if the errors are all systematically overestimated (or underestimated), then including this term ensures a global renormalisation of the errors in Eq. (3), and the distribution of our reduced variable x is also a normal distribution. Let $x_{\rm min}$ be the smallest value of x calculated among all test-frequencies for a given star. Under the hypothesis that the light curve is produced by a stable star, the probability to obtain at least one value $x \leq x_{\rm min}$ in a series of $N_{\rm test}$ realisations is ${\rm Prob}(x < x_{\rm min} \vert N_{\rm test}) = 1 - \left[ 1 - \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{x_{\rm min}} e^{-x^{2}/2} {\rm d}x \right]^{N_{\rm test}}$. If this probability is small, then:

 

$${\rm Prob}(x < x_{\rm min} \vert N_{\rm test}) \simeq \frac{N_{\rm test}}{2} {\rm erfc} \left( \frac{- x_{\rm min}}{\sqrt{2}} \right).$$ (4)

If the light curve exhibits periodic variations, then there exist test-frequencies for which x is significantly smaller than typical values of this variable (see middle and right panels of Fig. 3), and the probability for false detection is then extremely small. Figure 4 displays the probability distribution obtained with the three methods for the set of filtered light curves.

We apply the three algorithms which all give a probability for no periodicity. A star is accepted only if selected by all three methods with the following thresholds: $P({\rm method \ 1}) < 10^{-15}$, $P({\rm method \ 2}) < 10^{-10}$, $P({\rm method \ 3}) < 10^{-5}$, tuned in order to allow one to retrieve the previously known Cepheids observed by EROS in the Magellanic Clouds (see Derue 1999 for more details). This procedure (cut 4) selects a sample of 2553 stars.

Ten times more stars would have been selected if we had used method 1 or method 2 only (with the same thresholds), most of them being spurious variables. The third method would have added far less candidates if used alone, but still a factor 2 more. Combining independent methods has thus the advantage of considerably reducing the noise background (mostly due to aliases of one day or noisy measurements) while giving redundant information about the period. To obtain individual periods we perform Fourier fits with five harmonics:

$${\rm flux} = {\rm flux}_{0} + \sum^{5}_{l=1} a_{l} \cos(\frac{2 \pi}{P} l (t-t_0) + \phi_{l}),$$ (5)

where P is the period, $\phi_{l}$ the phase and a_l the amplitude. We define the amplitude ratios R_kl = a_k/a_l, and the phase differences (defined modulo 2$\pi$) $\Phi_{kl} = \phi_{k} - k \phi_{l}$, with k>l. Objects with non-significant harmonic amplitudes (i.e. with almost sinusoidal light curves) have $R_{21} \sim 0$ and their $\Phi_{21}$ is ill defined.

The selection of periodic variable stars is complicated by aliases. Some of the stars with periods equal to a simple fraction or a low multiple of one day may be badly phased because of the nightly sequence of measurements. These aliased periods are seen in Fig. 5 as vertical groups of dots at 2/3, 2 and 3 days. To eliminate them we demand (cut 5) that the fitted periods are not within $\pm 1\%$ of these values. One can also notice some vertical groups of points around 25 days which correspond to data gaps in our sample (see Fig. 1). Once these objects are removed, 2424 stars remain. The flux values of the remaining stars are folded using each period obtained with the three methods. The resulting phase diagrams are visually inspected. Some of them display an obvious spurious periodic or quasi-periodic variability due to a low photometric quality. After this final visual selection (cut 6) the list of variable stars includes 1362 candidates which exhibit unambiguous periodic variability.

3.4 Type of variability

The classification of the selected stars among different types of variability cannot be based on the position of the objects in the colour-magnitude diagram since the spread in distance of these stars entails a spread in magnitude and colour. It is desirable however to classify the various light curves according to some physical parameters. In the following we mainly use criteria based on the period P of the luminosity variations and on the amplitude ratio $\Delta V_{\rm E}/\Delta R_{\rm E}$. For each selected type of variable star the phase diagram of a typical candidate is displayed in Fig. 6.

Figure 5: Period-colour diagram (P in days vs. $V_{\rm J}$-$I_{\rm C}$) of the 2553 selected candidates before cut 5.

Three groups are distinguished depending on their period:

1st:

stars with a period larger than 60 days (543 objects):
For these objects an entire period has not been observed. There is thus no warranty that these objects are periodic ones. 34 display a nearly linear light curve and are thus catalogued as Miras candidates. The other 509 objects are catalogued as Long Period Variable stars (LPVs).

2nd:

stars with a period between 30 and 60 days (387 objects):
264 Semi-Regular variable stars are selected by requiring: $\Delta V_{\rm E}$/ $\Delta R_{\rm E} >$ 1.2 to select pulsating stars (see below), $V_{\rm J}{-}I_{\rm C}$ > 2.5 to discriminate from bluer variable stars and $\Delta R_{\rm E} < 1.0$ to avoid possible Miras or LPVs wrongly phased. The long term stability of these stars is not known. Some of the reported periods may change from season to season, as a result of their semi-regular behaviour. The remaining 123 objects are catalogued as miscellaneous variable stars.

Figure 6: Phase diagrams (or light curves for the two lowest panels) for typical variable stars of our catalogue in $R_{\rm E}$ magnitude.

 

 

Table 3: Number of selected objects for each type of variability.

Period range	Type	Number of objects
P>60 d		543
	LPV	509
	Miras	34
60 > P>30 d		387
	Semi-Regular	264
	miscellaneous	123
P<30 d		432
	pulsating	60
	RRc	14
	RRab	5
	classical-Cepheids	6
	s-Cepheids	3
	miscellaneous	32
	non-pulsating	372
	EA	130
	EB	35
	EW	11
	miscellaneous	196

3rd:

stars with a period smaller than 30 days (432 objects):
The colour change for a Cepheid in standard passbands is $\Delta V_{\rm J}$/ $\Delta I_{\rm C}>$1.3 (Madore et al. 1991) which corresponds to $\Delta V_{\rm E}$/ $\Delta R_{\rm E} >$ 1.2 in the EROS system. Two sets are thus distinguished based on this criterion:

-

The pulsating variable stars (60 objects):
For stars with period P<1 day, two samples of RR Lyræ are identified: the RRc have R₂₁ < 0.4 (14 objects) and the RRab have R₂₁ > 0.4 (5 objects). We adopt the morphological classification proposed by Antonello et al. (1986) and classify as s-Cepheids the stars that lie in the lower part of the R₂₁-P plane, and as classical Cepheids the remaining stars. s-Cepheids pulsate in the first overtone and classical Cepheids in the fundamental mode (see e.g. Beaulieu et al. 1995; Beaulieu & Sasselov 1996). We use the empirical function $R_{21}^{\rm cut}(P)=0.4 - (P/30$ days) in order to separate these pulsation modes (see Fig. 7). Among the five objects that pass the s-Cepheid cut, only three belong to the R₂₁-P and $\Phi _{21}-P$ distributions of galactic s-Cepheids and their phase parameter $\Phi_{21}$ is poorly constrained. Besides most of classical Cepheids have amplitudes larger than $\Delta R_{\rm E} >0.4^{\rm mag}$ (see e.g. Afonso et al. (1999)). For stars with period P>1 day, three samples are then identified: the classical Cepheids have $R_{21} > R_{21}^{\rm cut}$ and $\Delta R_{\rm E} >0.4^{\rm mag}$ (6 objects); the s-Cepheids have $R_{21} < R_{21}^{\rm cut}$ and $3 \ {\rm rad} < \Phi_{21} < 6 \ {\rm rad}$ (3 objects); the remaining 32 objects are catalogued as miscellaneous pulsating stars.

-

The non-pulsating variable stars (372 objects):
The remaining objects have similar amplitudes in both passbands. We classify them according to the following criteria. Algol systems (130 stars, type EA) display well-defined eclipses whose secondary one has a depth lower than half the primary one, and possibly flat light curve between them. EB type objects (35 in total) show a secondary eclipse equal to half the primary one. The EW type (11 objects) is characterised by similar depths of the two eclipses. The members of a residual sample of 196 objects do not look like convincing eclipsing binaries and are catalogued as miscellaneous variable stars.

As emphasised by Udalski et al. (1999a) a large number of variable objects show small amplitude sinusoidal variations, such as ellipsoidal binary variable stars. A contamination of the sample of pulsating stars by eclipsing binaries is thus possible.

Figures 8 and 9 show the location of the selected variables in the colour-magnitude diagrams. Also plotted are 10 000 stars located in the central part of the two fields tm550 and gn450. Most of the Cepheids are much brighter than our magnitude threshold (cut 2); this is not so for RR Lyræ (see Fig. 9, lower panel). Our catalogue is thus not complete for this type of variable stars, as already mentionned.