Astron. Astrophys. 351, 607-618 (1999)

2. Observations and results

Spectroscopic observations in the spectral region around H were obtained from the Rozhen Observatory from 1996 to 1999. We used a CCD camera attached to the coudé spectrograph of the 2.0-m telescope. Spectral resolution was 0.2 Å/px. Details of the observations are given in Table 1. The signal-to-noise ratio of the final one-dimensional spectra in the continuum is 15-37.

Table 1. Spectral CCD observations of TT Ari

The photometric data consist of 45 observing runs in the U and B bands obtained from 1987 to 1998. The duration of the runs varies from 1 to 7 hours. Details of the observing runs are given in Table 2. There are more than 600 individual measurements in the standard UBV system also. Some of them have already been published by Kraicheva et al. (1987, 1989). All observations were carried out with the one-channel photometers attached to the 60-cm Carl Zeiss telescopes of the Rozhen and Belogradchik Observatories of BAS. The observations were processed with the reduction software described by Kirov et al. (1991). The star "c" of the sequence of Götz (1985) was used as a comparison star. The magnitudes of this star were taken from Shafter et al. (1985). The integration time was 10 sec.

Table 2. Flickering properties

2.1. UBV photometry

The UBV data shown in Fig. 1 cover the interval from 1985 to 1999. The 6 yr cycle found by Kraicheva et al. (1997b) is seen in all passbands and in the color indexes, as the star becomes bluer in the maximum of the cycle. The variations of the colors indicate changes in the spectrum of the star, which may be caused by variations of the temperature profile of the accretion disc. For stationary optically thick accretion discs it depends on the mass transfer rate and is roughly . An increase of the mass transfer rate leads to a higher luminosity and bluer colors of the star. The UBV photometry shows similar behaviour of TT Ari and consequently suggests that changes in the mass transfer rate may be responsible for the long-term brightness variations of the star in high state. According to current ideas (Warner 1988; Bianchini 1990), the mass transfer rate can be modulated by solar-like magnetic cycles of activity in the secondary. Such cycles could cause small changes of the radius of the secondary and change the mass transfer rate.

Fig. 1. Nightly averaged UBV estimations of TT Ari from 1985 to 1999.

In addition, we added our new B data to the long-term light curve of TT Ari compiled by Kraicheva et al. (1997b) and analysed it for brightness variations in high state. We should notice that Kraicheva et al. (1997b) used -statistics (Stellingwerf 1978) to search for brightness variations in high state. The applied smoothing procedure was running mean with a window length of 1000 days. As a result of the long smoothing window, the data were too smoothed and the information about the true amplitude of the modulations might have been lost. Additionally, information is lost at the beginning and the end of each high state data segment. Because of the long gaps in the high state data, many false peaks appear in the -statistics and may mask the real ones. That is why in this paper we used other data smoothing and period searching procedures. The light curve is based on more than 800 nightly mean B magnitudes obtained from 1928 to 1999 in Sonneberg (Hudec et al. 1984) and Rozhen (Kraicheva et al. 1987, 1989, this paper) mainly. Since the data were very noisy, we averaged them in boxes with length of 1 year. Because of the small number of points per bin (5-15), the estimated mean values may be strongly affected by some outlying points. To avoid this, in each bin the mean and the standard deviation were calculated. Those individual points lying outside were removed and data was averaged again to obtain the final curve. To estimate the level below which the mean points may be considered to be at intermediate or low state, we determined the mean star magnitude and standard deviation in high state mag by using tree parts of the light curve in which the star had been in high state predominantly (JD 2427000-2430000, JD 2437000-2443000 and after JD 2446500). Finally, we analysed only those mean points that were brighter than mag (shown with filled squares in Fig. 2). The error bars shown in the figure are the standard errors of the mean values. Since the data are unevenly spaced and contain large gaps, the light curve was analysed by the CLEAN algorithm (Roberts et al. 1987). Fourier transform of a sampled signal is a convolution of the true spectrum with the spectral window of the data. When processing data with large gaps, the spectral window has a very complex shape and introduces many false peaks in the computed PS. This makes the recognizing of the real peaks very difficult. Roberts et al. (1987) have shown that the CLEAN algorithm is a powerful tool for analysis of such data. In practice, CLEAN is an iterative method for deconvolution of the dirty spectrum with the spectral window. Rarely, a very complex spectral window and a presence of more than one periodic signal may cause overlapping of some false peaks and the strongest peak in the PS to be false also. In this case CLEAN may fail.

Fig. 2. Upper panel: The yearly averaged long-term light curve of TT Ari and the fit to the high state magnitudes (filled squares) with the found periods. Open squares mark intermediate state magnitudes; Lower panel: The dirty spectrum, spectral window and clean spectrum of TT Ari high state light curve.

In Fig. 2 are shown the dirty spectrum, spectral window and clean spectrum of TT Ari data. The clean spectrum is formed by convolution of the clean components with a Gaussian, whose width is equal to that of the central peak of the spectral window. The clean spectrum shows two main peaks corresponding to 6.25 yr and 27.5 yr. The sinusoidal fit to the data with these two periods is shown in Fig. 2 also. The lengths of the individual cycles of the first modulation vary from 4 yr to 8 yr, which is distinctive for solar like cycles of activity observed in late type main sequence stars (Wilson 1978) and in other CVs (Warner 1988; Bianchini 1990). The second modulation is uncertain because the light curve spans over 2.5 cycles only. But it is seen from Fig. 2 that intermediate and low states happen more often in the minima of the 27.5 or 6.25 yr high state cycles when the mass transfer rate is lower. If the secondary of TT Ari is not properly filling its Roche lobe, as proposed by Bianchini (1990), the low and intermediate states may be due to a very low mass transfer rate triggered in the minima of the activity cycles.

2.2. Spectroscopy

During the nights of spectroscopic observations, TT Ari was in high state with mag in 1996-1997 and mag in 1998-1999. The H profiles of TT Ari are shown in Fig. 3. The equivalent widths and relative intensities of H emission line are listed in Table 1. The orbital phases were calculated according to the ephemeris given by Thorstensen et al. (1985). It seems that the strength of H emission varies with the orbital phase reaching maximum intensity near quadrature, according to the behaviour of the other Balmer lines observed by Cowley et al. (1975). Our observations assume also that the strength of H varies from cycle to cycle. The small number of observations did not allow us to test the intensity of H in the two "supernump" regimes and/or in the extrema of the 6 yr cycle, but the typical values of the H EWs in TT Ari high state are low and averages about 4.9 Å. Patterson (1984) and Smak (1991) have shown that the EWs of H and H emission lines are correlated with inclination and inversely correlated with accretion disc absolute visual magnitudes. With (Ritter 1990) and yr^-1 in high state, TT Ari is expected to have H and H emissions with low EWs as is observed. It is also seen that the spectra taken in the "positive superhump" regime suggest two components in the H emission (despite the relatively low S/N ratio 15-20), while in the "negative superhump" regime they are rather Gaussian (Fig. 3). In addition, some spectra obtained in 1996 suggest a broad weak absorption base. However, it is clearly seen in one spectrum only. We notice that this spectrum is with the highest S/N 37. To investigate the "EW-stellar magnitude" relation, we collected all available data on the EWs of the H and H emission lines in TT Ari spectra (Cowley et al. 1975; Patterson 1984; Williams 1983; Shafter et al. 1985; Hutchings & Cote 1985; Hutchings et al. 1986). The B magnitudes were estimated from the long-term light curve of the star. The results given in Fig. 3 show an increase of the H and H EWs with decreasing system brightness, confirming the empirical relation based on spectroscopy of different types of CVs given by Patterson (1984). But in the case of TT Ari, the different magnitudes correspond to different photometric states and we should have in mind that in low state a significant part of the emission in the lines may be contributed by the chromosphere of the secondary (Shafter et al. 1985).

Fig. 3. Left panel: H profiles of TT Ari from Rozhen data; Right upper panel: H profile observed on 03 Aug 1996 (); Right lower panel: Equivalent widths of H (open circles - the data from the literature, open squares - mean from our data taken in 1996-1997 and 1998-1999) and H (filled circles) the star B magnitudes.

2.3. Long photoelectric runs: "3-hour" and "4-day" modulations

The results from an analysis of the runs obtained before 1994 have been already published by Kraicheva et al. (1997a). The new data consist of 19 series in B filter and 2 series in U filter obtained in the four seasons from 1995 to 1998 (Table 2). The distribution of the B runs over the seasons is 5, 6, 4 and 4 respectively. The observations in both seasons, 1996/97 and 1998/99, were obtained within 6 days, while those in 1995/96 and 1997/98 for 3 and 2 months, respectively. The small number of runs spread over a long time interval in the last two seasons made the analysis of the data difficult. As was already discussed in Sect. 2.1, a suitable method for analysis of such data is the CLEAN algorithm. Fourier spectra of the data in each season were calculated in the range 0-20 [c/d] and the CLEAN algorithm was applied. To remove the variations of the mean star brightness and a possible difference in the instrumental systems of the two telescopes, the runs in 1995/96 and 1996/97 seasons were normalized to zero mean. Since the duration of the runs is not equal to an integer number of periods, the mean values should be determined by a sinusoidal fitting of the data with an unknown mean rather than a simple mean. Unfortunately, the duration of some of the runs is shorter than expected period value and in some of the others, the shape of the wave is very complex. In these cases, the fitting might give a more inaccurate estimation of the mean. The observations in the other two seasons were obtained at Rozhen observatory only and they did not show any variations of the mean star brightness. The absence of mean brightness variations in 1997 was pointed out by Skillman et al. (1998) also. The periods found in the four seasons are , , and with corresponding amplitudes 0.054, 0.041, 0.068 and 0.065 mag. The clean spectra, the runs and the fits with the periods found are shown in Figs. 4 and 5. In spite of the large gaps in 1995/96 and 1997/98 data leading to a very complex spectral window, CLEAN succeeded to find the periods. However, the clean spectra of the data in 1995/96, 1996/97 and 1997/98 still show many minor peaks. Fig. 4 shows the complexity of the light curves: some maxima of the "3-hour" wave are double peaked and the modulation is strongly affected by the flickering and "20 min" QPOs. Thus, both the complex light curves and spectral window are the reason for the minor peaks remaining in the clean spectra. In two of the seasons (1996/97 and 1997/98) some of the minor peaks may correspond to the secondary photometric period in the range 5-7 hours suggested by Wenzel et al. (1986) and Tremko et al. (1992). However, in 1997/98 the period of 7 hours exceeds the duration of the runs and it is uncertain. Moreover, Skillman et al. (1998) did not see this modulation in their long, dense observations. In the clean spectrum of 1996/97 data, two equally strong peaks are seen. The period of was chosen by direct fitting of the data and it is uncertain too. The presence of these peaks in the PS is most probably due to the complexity of the light curves rather than any real modulation. In spite of the uncertainty, both periods were included in the fits shown in Figs. 4 and 5. The amplitudes and phases of all detected periods were obtained by sinusoidal fitting to the equation:

where t is the time, are the found periods, n is the number of the periods, and the unknown parameters are the mean magnitude , the amplitudes and the phases .

Fig. 4. The runs obtained in 1995 and 1996, the best fits to them and clean spectra.

Fig. 5. The runs obtained in 1997 and 1998, the best fits to them and clean spectra.

Andronov et al. (1999) found period of the "negative superhumps" in 1994/95 season. They suggested that the increase from to could be a result of small secular period variations or may indicate the beginning of a gradual period change, led finally to the modulation observed by Skillman et al. (1998). The periods we found in 1995/96 and 1996/97 were significantly bigger than the previous determinations of the "negative superhump" periods. This may support the idea for a gradual change of the "3-hour" period, but we should keep in mind that the factors giving the minor peaks in the clean spectra increase the uncertainty in the period determination.

Skillman et al. (1998) found that modulation in 1997 was very stable without any period or amplitude changes during the whole interval of their observations (115 days). It seems that our data in both 1997/98 and 1998/99 seasons, confirm this finding. Keeping in mind that the periods and their amplitudes differ by only 0.5 min and 0.003 mag, respectively, and that we have a few runs only, we think that within the limit of the errors the values for the two seasons coincide. We note also the hump-like features in the minima of the wave seen in the runs taken on 17 and 19 Oct 1998 (Fig. 5).

Possible detections of a "4-day" period during "negative superhump" state were reported by Semeniuk et al. (1987) and Kraicheva et al. (1997a). This period was interpreted as a beat period between the photometric and the orbital period of the system. With photometric period of in 1996, the expected beat period is . The mean values and standard deviations of the runs taken in Nov 1996 are drawn in Fig. 6. It is seen that a cyclic change of the mean star brightness could be suspected, but we have only four consecutive observations from one telescope and could not say if its period exceeds .

Fig. 6. The mean values and standard deviations of the consecutive B runs carried out in Nov 1996. Open squares - Rozhen and filled squares - Belogradchik observations.

2.4. Long photoelectric runs: 20-min QPOs and flickering

To investigate the flickering and the QPOs we analysed our 45 photoelectric light curves obtained from 1987 to 1998 (Table 2) by using the method of Scargle (1982). As a result of the very complex light curves of the star, many (often equally strong) power peaks were detected in the range 40-120 [c/d]. The presence of too many peaks makes the recognizing of those corresponding to real periodicities difficult. An additional problem is the estimation of the statistical significance of the peaks as the PS of TT Ari light curves show a strong red noise. As far as we know, there is no theoretically justified procedure to evaluate statistical significance of the peaks when a red noise is present. Van der Klis (1989) suggested dividing the PS by some mean red noise shape and then to evaluate "false alarm probability". Kraicheva et al. (1999) tried to apply this procedure in their analysis of the light curves of MV Lyr and discussed the difficulties arising when one deals with low frequency QPOs plus a red noise. In particular, they showed that red noise PS are too noisy to allow precise fitting of their low frequency flat parts and it is hard to determine the smooth red noise shape by this way. However, if we search for QPOs in the low frequency flat part of the PS, "false alarm probability" may be used without any normalisation. In the case of TT Ari, the power in the QPOs is near the turnover of the PS (Fig. 8) and a normalisation is needed, thus, we did not try to estimate statistical significances of the peaks. In this case a more suitable procedure would be to perform sine fits to the data with all detected periods. Since the light curves are sawtooth rather than sinusoidal, an exact fit is not possible, but a careful inspection of the fits could give an idea about the real periodicities. In addition, multi-frequency fits with combinations between all detected periods could be useful.

The main result of the frequency analysis and data fitting is that the "20-min" QPOs in TT Ari are highly unstable. In spite of the many power peaks detected in the range 40-120 [c/d], there are no single oscillations that can describe a given run completely. Instead, in most of the runs, distinct parts could be fitted well by sinusoids with quite different periods ranging from 4 min to 26 min. The fits showed that these oscillations remained coherent for about 3-8 cycles and in most runs the distinct parts did not overlap. As examples, the series obtained on 25 Oct 1989 and 13 Oct 1993 are given in Fig. 7.

Fig. 7. The light curves of TT Ari obtained on Oct 13, 1993 and Oct 25, 1989 and their power spectra.

Light curves showing transient oscillations with varying periods, phases and amplitudes can be modelled by autoregressive processes (Robinson & Nather 1979). An autoregressive process of order p (AR(p)) is defined by the equation:

where are the filter coefficients and denotes an uncorrelated white noise process with zero mean and variance . If the coefficients are chosen properly, Eq. 2 describes the sum of dumped oscillators. Of particular interest for astronomy are two special cases: AR(1) and AR(2). The former may be recognised as a usual "shot noise" process which assumes the light curve as a sum of randomly occurring exponentially decaying shots. If the two parameters of an AR(2) process fulfil the condition , it corresponds to a dumped oscillator with characteristic period and dumping time determined by the particular values of the coefficients. Since the process is driven by a random noise, it is completely unpredictable and the resulting light curves are the sum of dumped oscillations with randomly varying period and phase. The PS of such processes show a single broad peak (Robinson & Nather 1979). If the coefficients do not fulfil the above condition, Eq. 2 describes a "shot noise" process but the shots are no longer simple exponents. Having the AR coefficients estimated, the PS of the process can be calculated.

The calculation of AR model coefficients is a linear problem and many efficient numerical algorithms have been developed. We used that published by Andersen (1974). Since the runs contained gaps, we adapted the algorithm to adjust AR model coefficients using only available data. The most difficult problem is the choice of the AR model order. For that purpose the Final Prediction Error (FPE) criterion (Akaike 1970) was used, namely, the correct model order is that minimising the quantity:

where N is the number of data points, m is the AR model order, k is the number of data segments and is the output error of the filter. Eq. 3 differs from the original FPE criterion by the factor k, which takes into account the fact that AR coefficients are computed from k data segments. The procedure was tested with simulated AR series and it was found that it determines both the AR coefficients and the model order correctly. AR models have the property that, if the model order is correctly chosen, the calculated PS are optimally smooth. Before the calculation of the AR coefficients, the "3-hour" waves were removed from the data. This was done by subtraction of a cubic spline through the mean points in non-overlapping bins of length 20 min. The spline interpolation was calculated by using a subroutine based on the tension spline algorithms given by Cline (1974). The tension splines are less likely to introduce spurious inflection points. The algorithm allows the tension in the interpolating curve to be a free parameter. If the tension is close to zero (), the interpolating curve is almost natural cubic spline and if it is large ( 10), the curve is almost polygonal line. We used tension of 1.

The calculations showed that most of TT Ari light curves could be modelled as AR(1) and AR(2) processes and only a few as AR(3). There are at least two factors that can influence AR coefficient estimation and FPE: observational noise and trend removal. In Eq. 2, is a random variable that only drives the AR process and any other noise processes are not taken into account. The real light curves, however, are always covered by observational noise. The numerical algorithm used cannot separate these two noise processes and the observational noise is treated as an addition to . Besides, any trend removal introduces residual noise. This noise was discussed by Kraicheva et al. (1999) and they concluded that if a cubic spline through mean points in long enough bins is subtracted, the residual noise should be small compared with characteristic amplitude of the flickering and it should weakly affect the results. Thus, the fact that we found different AR model orders does not mean that the light curves have different properties. This is most probably due to the influence of the noise. The most important result is that all PS of the estimated AR models follow roughly the shape:

without any peaks seen. Here f is the frequency and is the slope of the PS in the high frequencies. If =2, Eq. 4 exactly describes the PS of the usual "shot noise" process and is the e-folding constant of the exponential shots.

In general, any AR(p) process can be converted into an infinitely long moving average (MA) process defined by the equation:

where is again an uncorrelated white noise process and the filter is the inverse of (Scargle 1981). The MA representation has a more clear meaning than the AR one: Eq. 5 describes the reaction of the system to a random impulse sequence. Thus in the case of stochastic processes, MA representation is more reasonable. As can be seen from Eq. 5, the usual "shot noise" processes are a special case of MA when is a Poisson sequence, and where is the time step. In the common case, the MA filter coefficients may be arbitrary (the only restriction is the filter to be stable, i.e. ). The fact that the PS of the calculated AR models do not show peaks suggests that fast variability of TT Ari can be modelled as a general "shot noise" process. The strong low frequency variability in such a process is a result of the shots overlapping (Terebizh 1989). If the overlapping parameter, i.e. the number of the shots per one shot duration, is greater than 3-4, the "shot noise" process can easily produce transient QPOs. Then, in some parts of finite length series generated by such a "shot noise" process, transient QPOs may be detected. Therefore it is possible the QPOs observed in TT Ari light curves to be generated by the same instability mechanism as the flickering, i.e. the QPOs and the flickering could be the same thing. One may try to find the exact shape of the individual shots by an inversion of the estimated AR parameters. As Scargle (1981) pointed out, the procedure of Andersen (1974) presumes that the AR filter is minimum delay. In astronomy this is rarely the case and the shape obtained by the inversion may not be the real one. That is why we have not tried to invert the AR coefficients and only emphasise the ability of the "shot noise" processes to generate transient QPOs as those observed in TT Ari. However, the low frequency flat part and the values of the power slope (Table 2) suggest that the shots are not far from a single exponent.

The flickering is often modelled in term of "shot noise" process and many authors (Williams & Hiltner 1984; Elsworth & James 1982; Panek 1980; Bruch 1992) suppose that it is responsible for the red noise observed in CVs. The flickering may be characterised by some empirical quantities such as the power law index , the typical time scale , amplitude and the emitted energy. They have been determined and published for many stars, and the theoretical or numerical models have to be consistent with them. The determination of all these quantities and the difficulties related to it have been discussed by Kraicheva et al. (1999) and here we only mention some of the key points.

All estimated quantities are given in Table 2. As estimations of the flickering activity, the total amplitude (i.e. the difference between the brightest and the faintest points in the light curve) and the standard deviation around the mean of the light curves are listed in the table. If these quantities are taken as activity indications, it seems that there is no difference of the flickering activity in B and U bands.

The contribution of the flickering light source to the total light of the star can be estimated following the conception of Bruch (1992). The brightness of the star can be considered as a sum of the flickering light source and all other sources supposed to be constant on the flickering time scale. An upper limit of the constant light source magnitude can be defined as , where m is the mean star magnitude and is the total amplitude of the flickering. Then, if the amplitude of the flickering is assumed to be independent of the passband, the ratio of the flux of the flickering light source to that of the constant one, over the whole optical range, is given by

where and is the magnitude of some point of the light curve. In Table 2 the ratios of the fluxes calculated for equal to the mean and the maximal star brightness are given. As is an upper limit for the constant light source, the calculated ratios have to be considered as a lower limit. The estimated values suggest that a significant part of the visual light of TT Ari is emitted by the flickering light source.

The power law index was determined by least-squares linear fitting of the high frequency linear parts of the PS in log-log scale. The interval in which the fits were performed varied from 100-120 [c/d] to 1700-2000 [c/d] depending on the quality of the light curves. Small changes in this interval caused changes of the values of reaching 0.2-0.3 and this is the real error in the determination of . The values of vary, but the average is about 2. The fact that all PS have a shape similar to that defined by Eq. 4 and 's are 2, suggests that the underlying physical process could be mathematically described by a "shot noise" process with shot differing from exponents a little.

Since strong shot overlapping is expected, the duration of the shots cannot be directly measured. One way to estimate it is by autocorrelation function (ACF). may be defined as the time shift at which the ACF, , first reaches value . The ACF of an infinitely long "shot noise" process has a shape , where t is the shift time and is the e-folding constant of the exponential shots. Because of the finite length of the observational data, the ACF is strongly biased by the shots overlapping and it does not follow this theoretical shape. Instead of this, it crosses the zero level at some lag and after that oscillates around it. Additionally, the ACF is biased by trend removal and observational noise. A precise analytical theory of bias of ACF owed to all these factors can be found in Andronov (1994). In spite of the fact that the common shape of the ACF is strongly biased, using simulated "shot noise" series, Kraicheva et al. (1999) showed that a mean of several values might be a correct estimation of . For shots with arbitrary shape, the 's calculated by ACF may be considered as an effective duration or in case of varying duration, as a mean. In general, could be determined by direct fitting of the PS to the Eq. 4. But here, we have the same problem as that in the determination of the mean red noise shape of the PS, discussed in the beginning of this section. Thus, we give preference to the the 's determined by the ACF.

In Fig. 8 are shown the mean PS for the six seasons best covered by observations in log-log scale and the fits obtained by Eq. 4. The fits were performed with and fixed to values previously determined and the only unknown parameter was a constant multiplying Eq. 4. 's were determined from the mean spectra as was discussed above and 's are the mean of the individual determinations. To avoid the influence of the power excess due to the "3-hour" modulations, the PS were fitted for frequencies 1.25 only. In spite of some deviations from the fits near the turnover, it is seen that the shape of the PS is described by Eq. 4.

Fig. 8. The mean seasonal power spectra of TT Ari fitted by Eq. 4.

© European Southern Observatory (ESO) 1999

Online publication: November 3, 1999
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