3 Analysis of the data

3.1 General description  

The whole light curve, constructed from the AAVSO one-day means (about 6070) of observations, is displayed on a condensed time scale in Fig. 2. The respective parts, plotted on an expanded time scale, can be seen in Fig. 3 and Fig. 4. The seasonal gaps are usually short and the dense coverage allows to follow the course of the brightness variations almost day by day. One can immediately resolve that the amplitude significantly varies on the time scale of years. There are definitely groups of large variations with an amplitude about 2 mag$_{\rm vis}$. These changes have a repeating character and occur on a typical time scale of hundreds of days. On the other hand, extended intervals (years) of a flat light curve with just minor fluctuations can be seen in other years.

  

Figure 3: The same AAVSO light curve as in Fig. 2 but plotted with high temporal resolution (the first half)

  

Figure 4: The same AAVSO light curve as in Fig. 2 but plotted with high temporal resolution (the second half)

The AAVSO data set begins with an epoch of large-amplitude variations (from 10 mag$_{\rm vis}$ to 13 mag$_{\rm vis}$). The extrema of brightness are sharp, especially the maxima have a character of distinct peaks ("outbursts"). Both gradual and abrupt transitions between extrema can be resolved. Notice that after 3.8 years this season abruptly ends around JD = 2438970 with a transition into an intermediate level of brightness (about 11 mag$_{\rm vis}$); a brief dip may have occurred here. This transition can be established with an accuracy of several days.

V Sge spends the next 3.3 years in this intermediate level. Just minor variations can be seen till JD = 2440190 when a rapid decline of brightness introduced another series of large-scale variations. In this season which lasted for 8 years (till JD = 2443000) the brightness varied from about 10.5-11 mag$_{\rm vis}$ to 12.5 mag$_{\rm vis}$ (several brief excursions up to about 10 mag$_{\rm vis}$). Most extrema of brightness are flatter and more rounded now. The light curve is gaining a character of alternating high and low states in this season (hereafter HS and LS). Part of this segment was already plotted with smaller time resolution in Fig. 5 of Paper I, but owing to weaker coverage the AFOEV data were averaged over several days. Despite the heavier smoothing, which left only the long-term changes in the AFOEV data, the mean course of the HS/LS variations is in agreement with the AAVSO observations.

Another extended interval of an almost flat light curve, lasting for about 8 years, spans $\rm JD = 2\,443\,000 - 2\,445\,900$. The interval is interrupted near JD = 2444400 by one event, having the character of an isolated outburst. Its analysis was given in Paper I. The duration of this outburst is much shorter than the surrounding interval of the flat light curve.

The next season of large-scale variations, which was introduced by a well defined rapid decline into a low state, set in near JD = 2445900. Most variations with an amplitude larger than about 1 mag$_{\rm vis}$ can be described as alternating HSs and LSs again. However, they occur on a shorter time scale (up to 100 days) than in the previous season of the HS/LS transitions. Besides the episodes of well defined HSs and LSs one can resolve short-term variations in the high state; they take place on the time scale of days and have an amplitude of several tenths of mag$_{\rm vis}$. The alternating HSs and LSs vanished around JD = 2447300 and the brightness remained near the level of HS. Although some fluctuations of brightness are present after this date (till about 2447500) they do not resemble the episodes of LSs anymore. Brightness of V Sge showed no episodes of low state till JD = 2448300 (for about 3 years).

Although some data are missing near JD = 2448300 it is evident that an episode of LS occurred here and introduced a new extended series of alternating HSs and LSs. These states are very pronounced now and repeat on the time scale of 200-300 days. Most transitions between the states are relatively abrupt (i.e. much shorter than the duration of the state) and can be easily resolved. The high states usually display some structure. Brightness of a HS immediately after recovery from a LS is often higher than that before the decline into LS. Also some shallow minimum near the middle of a HS can be resolved. Detail of the brightness fluctuations in HS is shown in Fig. 8 in Paper I, again averaged over several nights.

In summary, this examination of the AAVSO light curve revealed that the seasons of the suppressed brightness variations (hereafter called flat segments) interchange with intervals of pronounced changes (hereafter active segments). The borderlines of the segments are usually well defined and the light curve can be divided into seven seasons. They are marked in Fig. 2 and abbreviated as S1-S7. Changes of brightness in the active segments often have a character of the HS/LS transitions. The flat segments display just minor fluctuations of brightness and although some rare larger excursions (a few days) from the main level may be present, they can better be compared to the brief dips, documented by [33, Robertson et al. (1997)]. Our data do not allow to carry out a reasonable assessment of such short rare events.

One more segment, formed from the AFOEV data and representing the light curve in the years 1934-1944, is included. Its large part is displayed in Fig. 2 in Paper I and can be characterized as a long-lasting epoch of low brightness (near 12 mag$_{\rm vis}$) with occasional relatively narrow outbursts (see Paper I for details). In the following analysis we will treat it as segment S0. We note that there are just very few scattered data within the years 1944-1961 and do not enable any reasonable analysis in this time period.

  

Figure 5: a) Two-sided moving averages of the brightness for the AAVSO data. b) Standard deviations $\sigma_{\rm mag}$ of the moving average of brightness. Result for the parameter Q=400 days is shown. See Sect. 3.2 for details

3.2 Moving averages  

Moving averages is one method which enables us to suppress the high frequency variations and obtain thus a better insight into the long-term trends. We will present the results of this method first because it allows to analyse variations through the whole data set without further presumptions. Two-sided moving averages of the AAVSO data were calculated for several values of Q. The parameter Q refers to the interval of days, within which the data were averaged. We also introduce the standard deviation $\sigma_{\rm mag}$of every calculated mean point. This quantity gives information about the current amplitude of the variations.

We calculated the moving averages for several values of Q. Figure 5 displays the results for Q=400 days. This filter half-width washed out the variations inside the active segments and emphasized the general levels of brightness. 200 to 570 observations are inherent in each mean. It can be resolved in Fig. 5a that the lowest mean brightness occurs in S1 but increases towards the transition from S1 to S2 (from active to flat segment) near JD = 2439000. Also initially large $\sigma_{\rm mag}$ rapidly decreases towards S2. The average level of brightness in S3 (active segment with HSs and LSs) is clearly lower than in the neighbouring flat segments S2 and S4. $\sigma_{\rm mag}$ slightly grows through segment S3 but then rapidly falls as the border with S4 approaches.

The active segment S5 is clearly defined by a large bump in $\sigma_{\rm mag}$(Fig. 5b). However, the light curve in Fig. 5a shows that the mean brightness level through this segment stayed almost unchanged with respect to adjoining S4 and S6.

HSs and LSs in S7 are very pronounced and have a long cycle-length. They therefore tend to be more sensitive to the seasonal gaps. The course of the moving averages for this segment may be then distorted to some extent. Nevertheless, the mean light curve shows that the highest brightness did occur in this segment.

In summary, the moving averages reveal the long-term changes of the mean level of brightness of V Sge. Mainly the respective seasons of the large-amplitude variations (manifested by enhanced $\sigma_{\rm mag}$ and corresponding to the active segments) tend to brighten with time; this is clear especially for the sequence S1-S3-S5. The course of $\sigma_{\rm mag}$confirms the obvious fact that amplitude of the brightness changes varies in agreement with the division into segments. We note that the transitions between the neighbouring segments are inevitably rounded in Fig. 5a,b due to the large value of Q used.

  

Figure 6: The statistical distributions of brightness constructed from the one-day means of observations in the respective segments. The intervals in JD and parameters of the segments are given in Table 1

3.3 Statistical distributions of brightness in the segments

We have argued above that the light curve of V Sge can be divided into segments with the well defined limits. The dense coverage ensures that this division is not oversimplified. When the data set displays seasons of such clearly divergent behaviour then it is a reasonable approach to analyse each segment separately. It also gives an opportunity to search for the overall characteristics of the activity in a given interval. The segment can be characterized by its net parameters (mean brightness, median, skewness, excess). These net parameters can be used for an assessment of the evolution of the long-term activity.

The histograms of the one-day means, constructed for the brightness in segments S0-S7, are displayed in Fig. 6. The width of each bin is 0.25 mag$_{\rm vis}$ in all cases. Parameters of the distributions can be found in Table 1. The distribution of the whole AAVSO data set (S1-S7) is clearly unimodal without any other features and is roughly symmetrical. We preferred not to include segment S0 in this histogram since it represents a season separated from the rest of the data by a gap of about 17 years.

Large changes of the distribution from segment to segment can be seen when the respective histograms are constructed for S1 to S7. They confirm the division presented above. Notice three similar segments S2, S4, S6 which are clearly unimodal, relatively narrow, roughly symmetrical and all have almost identical mean levels (near 11.3 mag$_{\rm vis}$). On the other hand, very broad distributions are observed in active S1, S3, S5, S7. The histogram for S1 is flat and has an asymmetrically placed peak near 11.9 mag$_{\rm vis}$. The distributions of S3 and S7 are bimodal. The histogram for S5 is slightly more narrow than S3 and S7 and its bimodality is at most marginal.

  

Table 1: Parameters of the respective segments of the light curve of V Sge. Intervals of JD-2400000, duration (in days) and coverage by the one-day means (in %) of the respective segments (S) are given along with the mean visual magnitude, its median, skewness and excess. S1-S7 are identified in Fig. 2 (AAVSO data) while S0 is formed from the AFOEV data (see the text). Standard deviation $\sigma$$_{\rm mag}$ of the brightness in the whole segment is included. Number of the night means in the segment is given in the column N(obs). $M_{\rm d}$ refers to the average number of observations in the one-day mean

3.3.1 Evolution of the brightness changes  

The divergent distributions, apparent in the histograms for the individual segments, suggest changes on a very long time scale (years to decades) in the activity of V Sge. The net parameters of each segment (mean brightness, skewness, excess), listed in Table 1 and plotted versus JD in Fig. 7, allow for a better insight into these trends.

The evolution of the net parameters of the active segments S0, S1, S3, S5, S7 deserves special attention. The most striking variation is a gradual change of asymmetry of the histograms (skewness) in the sequence S0-S1-S3-S5-S7 (Fig. 7a). This change through the four latter segments can be considered linear; the fit is denoted by dashed line in Fig. 7a. We admit that the limits of segment S0 are not so clearly defined as the rest of the segments. Nevertheless, skewness of S0 follows the trend of the later active segments.

A gradual increase of the mean brightness in the sequence S1-S3-S5-S7, apparent in Fig. 7c, amounts about 0.4 mag$_{\rm vis}$. The points are connected by a line for clarity. Mean levels of HSs and LSs are included; they allow to resolve their contribution to the resulting trend. It turns out that the mean levels of HSs in active segments are always brighter than the levels of the adjacent flat segments S2, S4, S6. Brightening of the active segments is in accordance with the moving averages (compare Fig. 5a and Fig. 7c). Two levels of LS in S7 are displayed, with the last episode and without it (see below).

  

Figure: 7 Net parameters of the respective segments from Table 1 plotted versus Julian Date. Each point is centered on the middle of the appropriate segment. See Sect. 3.3.1 for details

Comparison of Fig. 7 with Figs. 2, 3 and 4 allows to relate evolution of the net parameters with the changes of the character of the light curve in the respective active segments. The large negative skewness and low mean brightness of S0 are caused by the relatively narrow outbursts from the quiescence level (Fig. 2 in Paper I). The type of variations in segment S1 is intermediate between the separated outbursts in S0 and HS/LS transitions in S3, S5, S7. The well defined transitions between HS and LS, analysed in Sect. 3.4, began in S3. With increasing mean brightness in the next active segments V Sge spends more time in HS than in LS, causing thus an increase of skewness.

3.4 Characteristics of the HS/LS transitions  

The alternating high and low states are a characteristic photometric activity of several kinds of interacting binaries: cataclysmic variables (CVs) of the VY Scl type (e.g. [36, Shafter et al. 1985),] polars (e.g. [14, Hudec & Meinunger 1977),] X-ray binaries (e.g. [15, Hudec & Wenzel 1986).] However, due to its short duration, only rarely is the course of the transition between the states covered by the observations. V Sge displayed the typical HS/LS transitions in active segments S3, S5, S7 (see Sect. 3.1). Dense coverage of the AAVSO data allowed to resolve the course of these transitions in 36 cases. The typical appearance of the HS/LS variations in V Sge can be described as follows. The onset of the episode of LS begins with a rapid fall of brightness from HS by about 1 mag$_{\rm vis}$. The phase of LS is not quite flat in some cases; it displays a very slow brightening (e.g. LSs centered on JD = 2449300 and 2449520). The final return to HS is approximately as rapid as the decline. Brightness after return to HS is sometimes (mainly in S7) slightly higher than immediately before onset of LS. Interchanging HSs and LSs in segment S5 occurred on a short time scale (Sect. 3.5) and the stable level of LSs was not always fully developed.

  

Figure 8: a) Duration of the respective HS/LS transitions in three active segments (S3, S5, S7) of typical HS/LS behaviour of V Sge, plotted versus Julian Date. Logarithmic scale is used for the y-axis. b) Rate of the brightness change during the transitions. Those transitions which belong to the same episode of the low state (fall and rise of brightness) are connected by lines. See Sect. 3.4 for details

The declining and rising parts of the light curve of the well covered transitions were approximated by straight lines. Linear least squares fits were used inside the intervals, adjusted interactively, for determination of the parameters of the transitions. The results for S3, S5 and S7 are summarized in Table 2.

Not all transitions have equal amplitude; many states, especially HSs, display their own structure (fluctuations on the time scale of days or trends-often brightening through LS and decline through HS). The transition therefore does not always begin and finish in the mean level of the state. In order to describe the transitions in S3, S5, S7 fully, both duration and the rate of change $\Delta {\rm mag}/\Delta t$ are needed (Fig. 8ab). The lower limit of the duration is comparable for all three segments (typically several days, the shortest transition only 4 days). The respective segments largely differ mainly in the range over which the durations are scattered. Transitions in S3 are generally slow with a large scatter (7 to 80 days), which also is reflected in their low rate of change. On the contrary, S5 represents a relatively tight group of very rapid transitions (4 to 11 days). The rate of change for the respective transitions in S5 varies by one to four, which is larger than the ratio of the durations and confirms scatter of the amplitudes. Parameters of the transitions in S7 are intermediate to S3 and S5.

Statistical distribution of durations of all measured transitions (Fig. 9) shows a clustering between 4 and 20 days and a weak tail towards prolonged transitions (up to 80 days).

  

Figure 9: Histogram of durations of HS/LS transitions in S3, S5, S7

  

Table 2: Parameters of the well covered HS/LS transitions in segments S3, S5, S7. The upper two rows give the numbers of the respective measured H-L (from the high to the low state) and L-H (from the low to the high state) transitions analysed. Durations and amplitudes (in mag$_{\rm vis}$) of the transitions are listed along with their standard deviations

We further analysed a possibility of gradual variations of the brightness levels of the respective HSs and LSs through the whole active segment. We mainly checked the working hypothesis that the brightness difference between HS and LS is largest near the middle of the segment and decreases towards its borders (possible gradual development and vanishing of the instability). The mean brightness of each state with the removed transitions was calculated, received a unit weight and was centered on the middle of the appropriate state. Inspection of the data confirmed that the level of the given state was achieved also in several cases of weaker coverage. The means were fitted by linear and quadratic polynomials. Upward curvature of the quadratic fit of HSs and downward curvature for LSs would be expected if our working hypothesis were true.

The linear fits both to HSs and LSs satisfy the course well in segment S3; the quadratic terms are very small. However, the brightness difference between HS and LS linearly increases through S3. Notice that this behaviour is also apparent in the evolution of $\sigma_{\rm mag}$ in Fig. 5b. Linear fits are justified also in S5. They reveal small decrease of brightness of both HSs and LSs through this segment while the brightness difference between HS and LS stays almost constant. Brightness of HSs increases linearly through S7. At the same time, levels of the respective LSs in S7 display more complicated changes with a prominent increase of brightness during the last two episodes. It supports the interpretation of the medium state after JD = 2449940 as an episode of an exceptionally shallow LS.

We can conclude that with the exception of LSs in S7 the changes of the levels of the respective states are small and their courses can be considered linear. It strengthens the above given arguments that the transitions between the adjoining segments are rapid and that the characteristics of HSs and LSs are usually kept for the whole duration of the given segment.

3.5 Periodicities in the HS/LS transitions  

Already visual inspection revealed that most HS/LS transitions occur in at least a semi-regular manner. A search for periodicities was therefore undertaken using the PDM program and the autocorrelation method. The PDM program ("phase dispersion minimization"), based on the method of [40, Stellingwerf (1978)] and written by Dr. J. Horn at the Ondrejov Observatory, is suitable for nonsinusoidal time variations covered by irregularly spaced observations. Significance of a given period is evaluated by the parameter $\Theta$, lying in the range 0-1. Insignificant periods have $\Theta\approx1$ while highly significant periods are expected to have $\Theta\approx0.4$ or lower. Horn's program enables not only an automatic searching for the best period inside a given interval but also an interactive examination of the resulting data foldings. It allows to assess whether the course of the particular folded light curve is reasonable.

The autocorrelation method, described by [29, Percy et al. (1981),] allows to search for characteristic time scales or quasiperiods which extend just for several cycles. This method makes use of the brightness difference $\Delta$mag versus the time difference $\Delta$t of each observation, divided into equal bins. (Quasi)periodic behaviour then gives rise to the minima in the resulting $\Delta$mag versus $\Delta$t curve.

Segment S7 was analysed first because it contains the most pronounced HS/LS transitions. We restricted ourselves to $\rm JD~= 2\,448\,314 - 2\,449\,930$and excluded the last LS which brightness is roughly intermediate to HS and LS. The period search revealed two possible periods: 286 days ($\Theta=0.684$)and 238 days ($\Theta=0.807$). Notice that the latter value is very close to the period of 240 days, found by [33, Robertson et al. (1997)] in their CCD data. However, it gives too many points in antiphase. Our data folded with P=286 days can be seen in Fig. 10. $\Theta=0.684$ implies that the HS/LS transitions cannot be considered strictly periodic; 286 days is just a typical time scale. The autocorrelation diagram displays a prominent deep minimum at 270 days (Fig. 11) which is in good agreement with the result of the PDM method. We therefore can conclude that the quasiperiod near 280 days has its meaning and is not just accidental because the folded light curve bears much of the course of the original data.

  

Figure 10: The data inside the epoch of the HS/LS transitions (segment S7; JD = 2448314-2449930) folded with the period of 286 days. The peak at JD = 2449080, which occurred after recovery from LS, was taken as the initial moment of the folding. See Sect. 3.5 for details

  

Figure 11: Autocorrelation diagram for the segments S3, S5, S7. Notice the deep minimum at 270 days for the segment S7 which is in good agreement with the period of 286 days, found by the PDM program (Fig. 10). See Sect. 3.5 for details

The PDM program did not reveal any period with $\Theta<0.75$ in S3. The autocorrelation diagram displays a minimum at 260 days but it is not as deep as in S7 (Fig. 11).

Inspection of segment S5 revealed that the cycle-length is shorter than in S3 and S7. However, no periods with $\Theta<0.85$ were found. Even the autocorrelation diagram does not display any prominent feature. Part of this segment was also covered by the photographic data, analysed by [23, Marsakova (1998).] The dominant period of 61 days, found by Marsakova, may represent a characteristic time scale of the HS/LS transitions, persisting just for several consecutive transitions; it only confirms a shorter cycle of the transitions in this segment.

In summary, only typical cycle-lengths, not strict periods, can be traced in the active segments. We can state that while the HSs and LSs in segments S3 and S7 tend to occur on the time scale of about 270 days, a significantly shorter time scale (<100 days) prevails in S5.