To study the long-term photometric behavior of TT Ari, we used our
UBV measurements, obtained in the period 1985 - 1994.
The UBV light curve in this
time interval is shown in Fig. 1 (click here). The interval 
covers the ascending branch top end of
the long-term light
Figure 1: TT Ari long-term light curve for the 1985-1994 time interval
curve after the star had left its 1979 - 1985 low state.
Kraicheva et al. (1989) showed that the star reached its
maximal brightness gradually: in a time interval of two years, the
brightness increased by about 
in the B and V bands and
by 
in the U band. Since 1987, TT Ari
has been mainly in a high state and its brightness
fluctuated about the mean values of almost 
in U,
in B and 
in V. The fluctuations around these mean
values were considerable, and in two nights (JD 2448151 and
2448176) the brightness of TT Ari reached values typical for
the intermediate state: 
in U, 
in B and 
in
V. Unfortunately, the time interval when TT Ari was supposed
to be in an intermediate state is not very well covered by our
observations. There is a gap in the observations between JD
2448283 and 2448650 and it makes it impossible to describe how
and when the star did return to its high state. It is
interesting to note that the star returned to its previous
high state mean brightness in the B and V bands,
while in the U band, TT Ari stayed at an average of 
which is by 
fainter as compared with the previous high state value.
Our light curve continues the curves given in Hudec et al. (1984) and Wenzel et al. (1992). Their sum gives an exhaustive description of the photometric behavior of TT Ari from 1928 to the present time.
We used two different methods to determine the 20-min oscillations. They are the maximum entropy method (MEM) described, e.g., by Burg (1972) and Laccos (1971), and the period dispersion minimization method (PDM) described by Stellingwerf (1978).
By using these methods, we computed the MEM power spectra
and 
-statistics for the individual measurement series (from
Nos. 1 to 29) listed in Table 1 (click here). To increase the accuracy in the
determination of the 20-min oscillations periods,
the influence of the 3-hour modulation was removed.
The peak corresponding to the 20-min oscillations is
well seen in 21 nights both in the power spectrum and 
-statistics.
On four other nights it is seen either in the 
-statistics
or in the power spectrum.
In most of the power spectra and 
-statistics, this peak is
accompanied by peaks that are comparable or even bigger,
than that of the 20-min oscillations. Their positions
are highly variable and different in nights during a given
observing season. We accepted the recurrence of the peak
position as a criterion for the presence of quasi-periodic
oscillations. The MEM power spectrum and 
-statistics
for the observing run No. 16, showing the longest period,
are shown in Fig. 2 (click here).
Figure 2: Frequency analysis of observing run of Oct. 29, 1989
The values of the period
obtained for the 20-min oscillations by the two methods
are in good agreement. Using the PDM method we computed also the
-statistics for several nights in close succession of a given
season. The results of the observations in the U band are
presented in Fig. 3 (click here). The mean amplitude of quasi-periodic
brightness variations, obtained by a sinusoidal
approximation, varied in the interval 
.
Figure 3: The averaged seasonal behaviour of the TT Ari 20-min period
Semeniuk et al. (1987) found that the period of 20-min oscillations in TT Ari decreased from 27 min in 1961 to 17 min in 1985. Udalski (1988) found values for the 1987/88 season in the 18-22 min interval. We extended the curve of long-term changes of the 20-min oscillations given in Semeniuk et al. (1987) with our period values for the individual nights found by using the PDM and MEM methods. In our opinion, they give a more realistic picture of the behaviour of the 20-min period changes. The data are shown in Fig. 4 (click here). In the figure are also plotted the period values obtained by Volpi et al. (1988) and Hollander & van Paradijs (1992), as well as the range in which the period varied according to Udalski (1988). Fortunately, we had observing series in the time interval close to the observing runs carried out by Volpi et al. (1988) and Hollander & van Paradijs (1992). The coincidence with their results is excellent. In Figs. 3 (click here) and 4 (click here) it can be seen that the period of quasi-periodic oscillations did not decrease continuously. From 1987 to 1990 there was a noticeable increase of the mean period, but we have not enough observations for a detailed study of its long-term behaviour.
An interesting feature in the power spectra
and 
-statistics corresponds to a period of about 13
min. We discovered it on 4 of the nights,
where the 20-min oscillations were not clearly established.
An inspection of the results for other observing nights shows
that in 14 of the remaining observing nights
the peak representing a 13-min periodicity can be either
clearly seen or there are indications of it.
Figure 4: The long-term behaviour of TT Ari 20-min
period: 
- our values found by MEM;
- our values found
by PDM; 
- the data of Semeniuk et al. (1987); + - the
value of Volpi et al. (1988); 
- the value of Hollander &
van Paradijs (1992); | - interval in which the period varied
according to Udalski (1988)
In all observing runs of sufficient length 3-hour modulations are clearly seen. However, their amplitude and shape changed from night to night. The star also showed variations in its mean brightness. We applied the method of extrema for the 3-hour period determination, for observations in the seasons 1987/88, 1989/90 and 1990/91.
Table 2: Moments of extrema of 3.2 hour wave
For TT Ari, as is the case with the other novalike stars, the determination of moments of extremum of the brightness was embarrassed by the strong flickering activity and the 20-min quasi-periodic oscillations. In some of the cases the flickering amplitude was bigger than the amplitude of the 3-hour modulation. Therefore we had to smooth the light curves in order to prepare them for a period analysis. For the purpose we used the method of moving average. The smoothing windows were equal to the characteristic time of 20-min quasi-periodic oscillations for the studied season, the step was 1 min. The exact moments of extrema were determined by polynomial approximation.
Since the observations were carried out with the one-channel photometers, there were time gaps in the observing runs during the time of background and comparison star measurements. In order to perform the smoothing procedure, we filled the gaps by linearly interpolating the data. Our observing material, (Table 1 (click here)), enabled us to determine 21 moments of maximum and 14 moments of minimum brightness, which can be found in Table 2 (click here).
To determine the photometric period in the 1987/88 season, we
used the moments of maxima of brightness published by Udalski
(1988) for the same interval plus moments determined by our
series 3, 4, 5 and 7 (see Table 1 (click here)). The cycle numbers E,
corresponding to the four maxima calculated from observational
runs, were determined by the photometric period found by
Udalski (1988) 
. Moments of maximum
covered relatively regularly the 165-day observation time interval.
This gave the opportunity to determine
accurately the cycle numbers. The following ephemeris were
obtained by the least square method:
The (O-C) values obtained according to this ephemeris are shown
in Fig. 5 (click here). Generally they are less than 0.11. The two exceptions, 0.15
and 0.17, are probably due to the high flickering activity
during the nights. The data of Udalski (1988) are also compatible
with a period of 
. We examined periods of 
, 
and 
. It was found that the values of (O-C) for the last
three periods were much higher than for the first
period. Therefore, in our further discussion we concentrated on the period
of 
.
Figure 5: (O-C) diagram for the 1987/88 season computed by using Eq. (1)
Our observing material for the 1989/90 season enabled us
to determine five moments of maximum and five moments of
minimum of the three-hour modulations. This was an
interesting possibility to compare periods obtained from two
extremum systems of the same observing data. All maxima were found from
the observing series 13, 14 and 16. Four moments
of minimum were obtained from the observing runs 12, 13, 14 and 16
and one by the UBV measurements of October 26, 1989 (Table 1 (click here)). Four of them
are within a 5-day time interval
which gives a certain confidence that the cycles were
counted accurately.
The following ephemeris were obtained:
The (O-C) residuals with these elements are less than 0.1
which is in good agreement with the accuracy of other
already published determinations. The same procedure,
applied to the moments of maxima, gave best results with
. The (O-C) residuals of this period are also
less than 0.1. We should mention that two moments of maximal
brightness (numbers 10, 12 in Table 2 (click here)) occurred at the beggining
or at the end of the observing runs and for the exact
determination of the moments we had to perform some
extrapolation of the data. This could result in a certain
enhanced error in these determinations. Thus we consider the period
based on the moments of minima more reliable. In a sense, the
difference between the results of the analysis of moments of minimal
and maximal brightness gives an estimation of the accuracy
when we work with a small number of points.
The 1990/91 season is best covered by our observations.
Seven observation runs were performed during the season,
five of them in consecutive nights. Six moments of
maximal brightness in the observing runs 18, 20, 21, 22, 23
and 24 (Table 1 (click here)) were found. They are presented in Table 2 (click here) under
numbers 13-18. The least squares method determination gives the elements
The (O-C) residuals calculated by these elements were less than 0.11. As for the 1989/90 season, however, the moments of extremal brightness were few. Unfortunately, we have not found any published moments of extrema for this observing season which would enable us to determine the period more accurately.
The values of the 3.2-hour photometrical period we
obtained for the 1987/88, 1989/90 and 1990/91 seasons are
not significantly different from the values obtained by
other authors for other seasons. At first glance there is
an impression that the period decreased from 
in the
1987/88 season to 
in 1990/91. Although the (O-C)
residuals for respective seasons (except for two
in 1987/88) were less than 0.11, the small number of moments
of extremum used in 1987/88 and 1990/91 was crucial. This
decreases the reliability of the obtained results.
We plotted the period
vs. the season (Fig. 6 (click here)) using
values obtained by us and published by different
authors (Smak & Stepien 1975; Udalski 1988;
Semeniuk et al. 1987; Volpi et al. 1988). It is seen
that there was no tendency toward an increase or decrease of the
period. Despite the large number of
well-distributed moments of maximum determined in the 1985
and 1987/88 seasons, the periods obtained were different 
and
Figure 6: The three-hour periods found for the 1961-1991 interval.
- the values obtained by us; 
- the values published by
Smak & Stepien (1975), Udalski (1988), Semeniuk et al. (1987),
Volpi et al. (1988)
, respectively.
We tried to associate the moments of
maxima published by Wenzel et al. (1986) and Udalski
(1988) with the moments obtained by us.
The calculated (O-C) values in the 1961-1994 interval were minimal for
the period of 
.
The (O-C) residuals from the ephemeris:
reached to 0.36. The period obtained by us and the (O-C) values with
additional 21 moments of maximum are close to that obtained
by Wenzel et al. (1992).
There are a few reasons to doubt the correctness of the period derived from the observations spanning over 33 years. Because of the short period and the rather long time interval of the investigation in which the observations occur in groups which are occasionaly separated by several years, it may have been easy to count wrong the total number of the cycles. In this case even the ``good-looking" behaviour of the (O-C) values could be caused by accidental coincidence.
We used our observing material to check the presence of a four-day brightness modulation of TT Ari. In both 1989/90 and 1990/91, we had intervals of four consecutive nights of observations that clearly showed the existence of a four-day modulation.
For a period determination, we applied the method of period
dispersion minimization. Because of the large gaps in the time
distribution, it was impossible to apply the maximum entropy
method. For analysis the observing data sets were smoothed with a 20 min
window and 5-min step. The 4-day modulation period was
determined for the 1987/88, 1989/90 and 1990/91 seasons
because they were best covered by our observations. Observing sets 2-7, 11-14
and 18-22, as well as UBV measurements 32 and 33 (Table 1 (click here)) were used for
analysis. These observations were distributed irregularly over the seasons.
Thus we can
expect that false frequencies might appear in our analysis and therefore
we limited the interval of period search around
the value of 
found by Semeniuk et al. (1987) and the
value determined by the equation
The time interval searched for was between 
and 
for
all the three seasons.
The results from analysis gave periods 
, 
and
for 1987/88, 1989/90 and 1990/91 seasons respectively.
The mean light curves based on this periods are shown in Fig. 7 (click here).
The amplitudes of 4-day modulations in U varied from 
in
1989/90 to 
in 1990/91. It was 
for 1987/88.
Figure 7: The nightly average U magnitudes for the 1987/88, 1989/90, 1990/91
runs versus the phase of the corresponding beat period
With a mean photometric period 
and an orbital period
of 
Eq. (5) gave the value 
. Accounting the
factors influencing the accuracy of the 3.2 hour period determination discussed
in Sect. 3.3, this value is in agreement with the abovementioned values.
These results confirm the
existence of 4-day modulations found by Semeniuk et al.
(1987). Answers to the questions associated with the long-term
behaviour of the photometric periods of TT Ari can only be
expected from a future intensive patrol observing programme.
Performing long photoelectric series of the star in
consecutive nights and a proper coverage of the observing season are
highly desirable for the development of a model of this binary
system.
Acknowledgements
This work was partially supported by NFSR under project No. 346/93.
