2.4 Time series analysis
2.4.2 Time domain analysis
In thetime domain analysis, instead, a regression is performed on past values of the time series, and then some parameters will be tested against the null hypothesis of uniformity of data.
2.4. Time series analysis Z-transformed Discrete Correlation Function
This method is based on the autocorrelation function, but was specifically tailored to uneven samples of data. The autocorrelation function, or correlogram, is a function which relates correlation coefficients to their trial periods, also called lags. An example for the function f(t) = 2sin(2t) + cos(5t) is shown in Fig. 2.1. Given a lightcurve, which is a function of timef(t), first a lagτ needs to be chosen (τ = 2 in the current example), then the correlation coefficient rτ will be computed between all pairs(f(t),f(t+τ)) over the entire timescale: rτ = PN i=1[f(ti)−f(¯t)][f(ti+τ)−f(¯t)] PN i=1[f(ti)−f(¯t)]2 (2.16)
rτ is basically the correlation coefficient in the diagramf(t) versusf(t+τ), as shown in Fig. 2.2, and its value is between [-1,1], as defined by the Pearson correlation coefficients explained in Section 2.2.1.
Figure 2.2: The pairs (f(ti), f(ti+τ)) are used to derive the correlation coefficientrτ for the lag
τ. The example refers to the previous function, where τ = 2. Data are clearly anticorrelated in this case.
In order to find the best possible period, rτ needs to be computed for a set of lags. All correlation coefficients will be then plotted versus lagτ, hence yielding a correlogram. The lag which provides the correlation coefficient closest to unity will be the best period for the lightcurve. Fig. 2.3 shows the correlogram for the example function depicted in Fig. 2.1, where it can be seen that the higher values ofrτ are achieved whenτ is between 2.5 and 3.5 or at 6.28, which corresponds to an entire cycle 2π. The dashed lines show that rτ =−0.59 for τ = 2. The correlation coefficient is always 1 when τ = 0, because no time delay is applied in this case.
The same method can be applied to two different lightcurves, but it will be called Correlation function instead of Autocorrelation. The only difference is that the data pointsf(t) andf(t+τ) will belong to different functions. It is worth mentioning that the autocorrelation function is the Fourier transform of the periodogram: they provide the same information, but the former is in the time domain while the latter in the frequency domain.
As explained before, astronomical data often lack of uniformity in the sampling and this issue had been dealt with in several ways when applying the autocorrelation function. In case of time gaps some authors (Gaskell & Peterson, 1987) interpolate the data, others (Edelson & Krolik, 1988) prefer to bin the data and use the mean and variance computed in each bin. The former method assumes that the lightcurve varies smoothly, but might introduce spurious data; whereas the latter tends to yield correlation coefficients which
2.4. Time series analysis
Figure 2.3: The correlogram shows that the best correlation is achieved when τ is between 2.5 and 3.5 or after an inter cycle, i.e. at 6.28, equivalent to 2π. In the current example r=−0.59 whenτ= 2.
are not normally distributed, hence generating biassed results (Alexander, 1997).
Alexander (1997) addressed this issue by introducing the ZDCF method. The main differences consist in using Fisher’s z-transform to estimate the correlation coefficients and in binning the data in equal number of items instead of equal time lag. The Fisher’s z-transform is defined as follows (Fisher, 1921):
z= 1 2ln 1 +r 1−r (2.17)
and has the advantage of being normally distributed better than r (Alexander, 1997). Further, binning the data by number of data points instead of by time allows improving the results in case of time gaps (Alexander, 1997).
The other two techniques in thetime domainare ’Epoch Folding’ and ’Phase Dispersion Minimisation’. They both make use of the folding method, but the former tests the data for uniformity in each phase bin, while the latter analyses the variance within each phase bin. They will be now described in detail.
Epoch Folding Technique
The Epoch Folding Technique tests different trial periods and folds the data onto them. It then returns, for each trial period, theχ2 computed to test the null hypothesis that the
mean magnitude is the same over the entire trial period. A periodicity is given through a highχ2, because in this case the null hypothesis is not true and data points show instead
a well defined pattern once the lightcurve is folded onto a specific period (Larsson, 1996). After folding the data, the period is divided in nbins. Theχ2 statistic is given by:
χ2 = N X i=1 (xi−x¯)2 σ2i (2.18)
wherexi is the number of counts in thei-th bin, ¯xis the mean over the entire sample, and σ2
i is the variance in thei-th bin. If there is no periodicity on the chosen trial period, all data in each bin will be scattered around the same mean value, because the folded light curve could be approximated with a straight line. In theχ2 distribution the mean is equal
to the degrees of freedom N−1, N being the total number of data points, so ifχ2 ≈N the null hypothesis of absence of periodicity will be confirmed. On the contrary, a high value of χ2 will be an indication of periodicity.
Phase Dispersion Minimisation
The Phase Dispersion Minimisation is similar to the Epoch Folding, in that data are folded on a trial period, but in this case the variance instead of the mean will be tested. If the trial period is divided intoM bins having ni data points in each bin and the total number of observations isN, letting σ2 and s2i be the variance over the entire sample and in each bin respectively, the PDM test statistic as defined by Stellingwerf (1978) is:
Θ2 = s 2 σ2 ≡ 1 σ2 PM i=1(ni−1)s2i N −M (2.19) where σ2= 1 N−1 PN
1=1(xi−x¯)2 is the sample variance over the whole unfolded data. If the folded data are uniformly distributed, i.e. there is no periodicity, the variance in each bin will be roughly the same, and consequently Θ2 ≈1. If there is a periodicity, instead, the variance in each bin will be smaller than the entire sample data, and Θ2 ≈0. In general, the more Θ2 is close to zero, the more correct the trial period is.
3
Optical variability in classical T Tauri stars
T Tauri stars are young stellar objects (YSO), first characterised because of their optical variability (Joy, 1942, 1945), and only later on found to host protoplanetary discs where new planets form. Although it was discovered more than half a century ago, variability is a feature which is still extensively studied and whose causes are not completely known yet. In fact, there seems to be a multiplicity of mechanisms, either star or disc related, which give rise to the observed flux variations, as it will be explained in Section 3.1. This Chapter makes use of the optical photometric data collected during almost a decade for the SuperWASP survey, described in Section 3.2, and aims to analyse and quantify over that timescale the optical variability in a sample of 39 CTTS (Section 3.3). For the most variable stars it will be searched for a periodic pattern through time series analysis (Section 3.4), while for the entire sample the causes of variability will be analysed in relation to stellar and disc properties (Section 3.5).
3.1
Optical variability in CTTS
T Tauri stars have long been known to present optical irregular variations which can be as high as 3 magnitudes (Joy, 1945), but it was understood very soon that there was not a single cause of variability.
One of the first attempts of variability classification was proposed by Parenago (1954), who provided a qualitative grouping in four classes based on the prevalence of bright or dim phases, as described in Weaver & Frank (1980). Cohen & Schwartz (1976) tried also to delve into the causes of variability and proposed two mechanisms: obscuration by dusty material in the form of protoplanetesimals and changes in the gas emission from the star, like the level of ionisation in the outer part of the stellar surface. The former one, however, required not only that objects were viewed edge-on but also a large number of protoplanetesimals in order to explain the observed variations. Moreover, the Hα flux related to the infrared variations did not seem to support this idea. Schmelz (1984) considered other possible explanations: changes in the spectral type of the star, and hence in its temperature, and changes in the optical depth either in the chromosphere or in the “dust shell”, now known as protoplanetary disc.
More convincingly, Herbst et al. (1994) classified the variability of T Tauri stars defin- ing three types, and offered an explanation still accepted today. Type I variability involves strictly periodic variations on a timescale ranging from half a day to slightly less than three weeks, explained as due to cool spots on the stellar surface. The observed periodicity is hence the rotation period of the star. Type I variations are more easily found on WTTS rather than CTTS, because those stars are devoid of disc and consequently do not present other kinds of disc-related variability. Type I periodicity shows a bimodal distribution (Attridge & Herbst, 1992), with peaks at either 2 or 8 days, and it was found that faster rotators are stars without a disc, while the presence of a disc makes the stars slow down (Edwards et al., 1993; Bouvier et al., 1993; Cieza & Baliber, 2007). Type II variations, observed only in CTTS, are more irregular and are deemed to be caused by hot spots, i.e. the regions where material accreting from the disc hit the stellar surface. Type II variations are observed on hourly timescales and are about ten times wider than Type I, with maximum amplitudes of 2-3 mag depending on the band. Type III are irregular variations like Type II but longer, on timescales of days-weeks instead of hours. The am-
3.1. Optical variability in CTTS
plitude of variations is less than one magnitude. These variations are explained as due to circumstellar variations. Extensive studies for the last two decades have investigated the presence of a warped circumstellar disc as cause of variation (Carpenter et al., 2002; Bouvier et al., 2003), favoured also by new infrared photometry available from satellites likeSpitzer (Flaherty et al., 2013).
Ismailov (2005) proposed a five-type classification scheme, based on the analysis of 28 T Tauri stars, which combines the causes of variability like in Herbst et al. (1994) with the lightcurve morphology like in Parenago (1954). The five groups differentiate mainly on how the mean brightness and the amplitude of each lightcurve vary. Type I, where there are not changes in the amplitude and the mean brightness remain constant, is explained as due to cool spots. Type II, characterised by variations in the amplitude without changes in the mean brightness, would be caused by hot spot or cyclic stellar activity similar to our Sun. In Type III only the mean brightness changes and is explained as due to the presence of a companion. The more complex shape of Type IV lightcurves, which combine the variations of type I and II, would be caused by both variations in the chromospheric activity and eclipses by a companion. The last group, type V, is characterised by a high level of brightness with occasional and quick dips. The origin of these events would be twofold: obscuration by circumstellar gas and dust when there are frequent dips, and eclipses by a companion when the drop in flux happens at least every two years.
A very detailed morphological classification was recently presented by Cody et al. (2014). They grouped lightcurves on the basis of shape and periodicity. The former parameter describes the level of asymmetry in the flux and is divided into three categories: bursters, symmetric and dippers. Burster are lightcurve which show increases in luminosity over less than one day, and which are explained as due to accretion instabilities. Dippers, on the other hand, are characterised by sharp and quick drops in flux with respect to the continuum level. These events are explained as extinction caused by circumstellar material or, when strictly periodic, by circumbinary discs. Symmetric lightcurves, instead, present the same level of increase and decrease in their fluxes. Concerning periodicity, they identified again three groups: periodic, quasi-periodic and aperiodic lightcurves. Very few YSOs in their sample present a strictly period behaviour, while most of them are quasi- periodic or aperiodic. Quasi-periodic means either that the period changes from cycle to cycle or that there is a superposition of periodicity and aperiodicity, where aperiodic
lightcurves have no sign of regular pattern.
In order to find possible patterns in the luminosity changes, the analysis of variability requires observations over at least several days to detect Type I variations, based on Herbst et al. (1994) classification, but possibly weeks if not months or years for Type II and III. Consequently, short-term variations have been more extensively studied than long-term variations, which require timescales of years to be properly identified. Beck & Simon (2001) used archival photographic plates to analyse the variability of three T Tauri stars over 50 years during the beginning of the 20th century. Gahm et al. (1993) analysed lightcurves, colours and periodicity of 16 YSOs between 1981 and 1991, and they found periods between a few days and three weeks in most of the stars in their sample, although clearly pointed out that no period was observable for the entire decade. Percy et al. (2010) made use of observations of 22 T Tauri stars over three decades, and found that most of the variable stars had variability on a timescale of 0-100 days, even longer for some of them. Proper periods were found only for few stars instead. The most remarkable study on long term variability was performed by Grankin et al. (2007), who analysed the variability of 73 CTTS over 20 years from 1983 until 2003. They used some statistical parameters and colours to characterise the variability, and found four typical kinds of lightcurves. In the largest group they found a rather stable pattern, explained as a reflection of the short term variations due to hot or cold spots. A small group exhibited instead large changes in brightness over the years, and was interpreted as due to high variations in the accretion rate or to changes in the circumstellar extinction. Stars which presented seasonal variations with small drops in brightness were explained as binary stars or stars partially occulted by circumstellar disc material. F8 to K2 stars showing a blue change in colour at the minimum brightness were explained as surrounded by variable circumstellar extinction. The authors, however, stressed that in most cases there is a combination and superposition of effects when analysing long term variability. Other long-term studies were performed in infrared (Rebull et al., 2014; Flaherty et al., 2013), but they will be discussed in the next chapter.
The goal of the work presented in this Chapter is the analysis of the optical lightcurves of 39 CTTS, observed for seven years between 2004 and 2011. Only two objects in the sample are in common with Gahm et al. (1993) and Percy et al. (2010); there is a larger overalapping with the work of Grankin et al. (2007), but in all cases SuperWASP im-