Periodic variations

The simulation principle used for generating the short period light-curves is the fol-lowing one. First, I build empirical, phase-folded light-curve templates, from light-curves found in the literature, for which relevant period and amplitude information are available.

Those templates are obtained by smoothing the observational phase-folded light-curves retrieved from literature, either with a rolling mean algorithm (eventually clipping the most extreme points in each window), through linear interpolation or via Fourier fitting with a number of harmonics between 2 and 9. Additionally, the resulting smoothed phase-folded light-curves are normalized to 1 from peak to peak. That way, the periodic variable templates retain only the characteristic shape of variation for the considered variable type.

I retrieved 17β Cephei and 30δ Scuti star templates from the ASAS-3 light-curves in the V band (Pojmanski 2002). I obtained 32 RR Lyrae and 44 eclipsing binary templates from the LINEAR optical wide-band photometry (Palaversa et al. 2013). For ZZ Ceti, I use a set of 21 previously simulated light-curves, described in Varadi et al. (2009). Finally, I obtained two AM CVn templates, one from Campbell et al. (2015) and the other from Anderson et al. (2005), whose measurements where taken inr and g bands respectively.

3.2. Simulations of short timescale variable light-curves

Figure 3.2 shows examples of templates for each of the simulated variable types listed in Table 3.1.

The second ingredient of my recipe is the magnitude of the source to simulate. I draw it uniformly between8and20mag. Finally, I choose the periodP and amplitudeAof the simulated variable star. To make my simulations realistic, when possible I draw the(P, A) pair from empirical period-amplitude probability distributions, retrieved from existing variable star catalogues. Figure 3.3 represents the 2D probability distributions used in my simulations, for β Cephei stars, δ Scuti stars, RRab stars, RRc stars, ZZ Ceti stars, Algol-like eclipsing binaries and contact eclipsing binaries. If there is not enough information in the literature, as it has been the case for AM CVn stars, I uniformly draw the period and amplitude in the appropriate ranges given in Table 3.1.

Note that the drawn amplitudes from those distributions are the amplitude in the lit-erature photometric bands, e.g. V orr bands, namely in the optical but with passbands slightly different from theGpassband. Hence, to ensure that my simulations are as close as possible to the observedGaia G photometry, I decide to apply a scaling to the drawn amplitude, taking advantage of the preliminary Gaia EPSL photometry made available to CU7. When possible, i.e. when I can find crossmatches between the sources in the catalogue used to get the period-amplitude distribution and theGaiadata, I calculate the difference between the 10^thand the 90^thpercentiles in both literature andGaiatime series for the considered crossmatched object. Those differences are taken as a proxy for ampli-tude estimation. Then, the scaling factor between the literature ampliampli-tude and the Gaia amplitude is defined type-wise as the slope of a linear regression between the literature amplitude estimates and theGaiaamplitudes estimates for all the crossmatched sources of the considered type and literature catalogue. Such scaling factors have been retrieved for the ASAS δ Scuti sample, the LINEAR RRab sample and the LINEAR RRc sample, with values of0.801,1.09and1.06respectively. For the other periodic variable types simulated, the scaling factor is taken equal to 1.

Once I have all these elements, I scale the phase-folded template at theGaia simula-tion amplitudeA, I generate the set of observing times, according to the appropriate time sampling over the required timespan, convert them into phases depending on the period P and finally compute the corresponding magnitudes from the scaled phase-folded tem-plate.

For my analysis, I generate two different types of light-curves:

• Thecontinuous light-curves, noiseless, with a dense and perfectly regular time sam-pling, over a timespan of∆t∼5P whereP is the simulated period, with 1000 points per light-curve for AM CVn simulations and 500 points for the other variable types.

The continuous data set is used to assess which periodic variable should be flagged as short timescale in an ideal situation. It comprises 100 distinct simulations for each of the eight variable types listed in Table 3.1.

• TheGaia-like light-curves, comprising simulations similar to those of the continu-ous data set (same variables simulated with the same period, amplitude and mag-nitude), but this time with a time sampling following the Gaiascanning law, for a random position over the sky, over a timespan ∆t ≈ 5years (i.e. the nominal du-ration of theGaiamission). Besides, I add some noise to the simulated magnitudes, according the a magnitude-error distribution retrieved from theGaiaEPSL per-CCD data (see Figure 3.4).

48

0.0 0.5 1.0 1.5

LINEAR RRc 9001883

P=0.309825[d]

LINEAR Algol 19030478 clippedMean

P=0.305943[d]

LINEAR ContactBinary 629195 clippedMean

P=0.229956[d]

GaiaSimuPerCCD ZZCeti WD0939+5609

P=0.00289236111111111[d]

Figure 3.2: Examples of light-curve templates for the simulated short period variables. From left to right and from top to bottom: β Cephei star,δ Scuti star, RRab star, RRc star, Algol eclipsing binary, contact eclipsing binary, ZZ Ceti star and AM CVn star. The black circle with red errors bars correspond to the observed literature light-curve, and the blue points to the derived template.

In the case of ZZ Ceti and AM CVn templates, the magnitudes are relative magnitudes.

50CHAPTER 3. PREDICTIONS ON THE DETECTABILITY OF SHORT TIMESCALE VARIABILITY WITHGAIA

Figure 3.3: Probability distribution in the period-amplitude diagram for the simulated short pe-riod variables. From left to right, and from top to bottom: β Cephei stars (Pigulski & Pojma ´nski 2008a,b),δScuti stars (ASAS-3 catalogue of variable stars Pojmanski 2002), RRab stars (Palaversa et al. 2013), RRc stars (Palaversa et al. 2013), Algol-like eclipsing binaries (Palaversa et al. 2013), contact eclipsing binaries (Palaversa et al. 2013), ZZ Ceti stars (Mukadam et al. 2006).

5 10 15 20 25

−5−4−3−2−101

Gaia estimated errors in G band per CCD

G per CCD magnitude

Log(G per CCD error [mag])

Figure 3.4: Gaiamagnitude-error distribution from the preliminary EPSL data.

I remind that theGaiaGper-CCD time series comprises the whole AF measurements of all FoV transits associated to the considered source. On the other hand, the data points forming the Gaia G FoV time series are obtained by averaging the brightness measure-ments of the nine AF within the FoV transit. The short timescale analysis makes use of the per-CCD photometry.