SV Cep: The results of a search of the literature for photometric data were published by Friedemann et al. (1992). In the meantime the results of visual brightness estimates collected by the AAVSO were made available to us (Mattei 1996, private communication). A small number of photoelectric uvby observations were reported by Reimann & Friedemann (1991). Further UBV data were published by Kardopolov & Filip'ev (1981).

WW Vul: Visual brightness estimates on sky patrol plates from the archives of Sonneberg, Bamberg, and Harvard observatories were published by Friedemann et al. (1994b, 1996). The results of a literature search are given by Friedemann et al. (1993). In the present study we include new UBV observations by Zaytseva & Lyutyi (1997) and visual estimates provided by the AAVSO (Mattei 1996, private communication).

The lightcurves of the six stars appear quite different. While the lightcurves of SV Cep and WW Vul show conspicuous wave-like patterns with amplitudes of up to 1 mag, for the rest of the stars (BH Cep, BO Cep, VX Cas, RZ Psc) the "normal light'' is represented by a broad band of data points. In most regions of the lightcurves of these stars the standard deviation of the average brightness is smaller than $\pm 0.2$ mag. Comparing this value with the maximum error of the eye estimates discussed in Sect. 3.2, we have to conclude that the uncertainty of the estimated magnitudes can account for a large percentage of the scatter of the data points, but short-scale activity certainly also contributes to it. Our data suggest that all of the stars, with the possible exception of RZ Psc, exhibit occasional short-lived brightenings. The existence of such brightenings has already been known from photoelectric observations so that they may not generally be attributed to plate defects.

$\begin{figure}\includegraphics[width=18cm]{fig02.ps}\medskip \end{figure}$

Figure 2: Lightcurves for VX Cas, RZ Psc, WW Vul, BH Cep, BO Cep, and SV Cep based on the brightness estimates on plates of the archives at Harvard College Observatory and Sonneberg Observatory covering the period 1898-1997 and incorporating additional photometric data from the literature as described in the text. B magnitudes are plotted as ordinates

We searched the data of the six stars for possible periodicities by means of the Discrete Fourier Analysis (DFA). The results are displayed in Fig. 3 as amplitude spectra.

$\begin{figure}\includegraphics[width=10cm]{fig03.ps} \par\end{figure}$

Figure 3: Amplitude spectra for the data displayed in Fig. 2. The positions of the annual and monthly periods are indicated

$\begin{figure}\includegraphics[width=8.8cm,clip]{fig04a.ps}\includegraphics[width=8.8cm,clip]{fig04b.ps}\end{figure}$

Figure 4: Mean lightcurve of BO Cep adopting a period of 10.658289 d. The lower diagram is based on normal points using a bin width of 0.05. Please note that the ordinates have different scales in the diagrams

Earlier observers of SV Cep have differentiated between medium-term wave-like light variations with durations from 100 to 1000 days and stochastic fluctuations. From a very fragmentary lightcurve we conjectured a secular decline of the stellar brightness (Friedemann et al. 1992 and references therein). The more complete data correct that picture and show that the long-term changes have a wave-like character, too. The fictitious secular decline was pretended by the highly fragmentary data. This experience underscores once again how important long and coherent time series are for the study of irregular variables. The DFA did not reveal any single dominant period. The component with the largest amplitude ( $\approx 0.18$ mag) has a period of 7599 d, but several additional components with only slightly lower amplitudes are also present. Therefore, we have to conclude that the wave-like light variations are largely irregular. We attribute them to intrinsic variations of the star.

The light variations of WW Vul are very similar to those of SV Cep, but the periodicities are far less marked. The component with the largest amplitude ( $\approx 0.11$ mag) has a period of 5714 d, which agrees with our earlier result (Friedemann et al. 1993) satisfactorily. Again we attribute the wave-like variation to the star itself.

Our amplitude spectrum of BO Cep shows a single prominent peak corresponding to a period of 10.658289 d. This value confirms earlier results by Grankin et al. (1991) and Wenzel (1991). Based on photoelectric photometry, these authors had also found that the amplitude of the mean lightcurve is about 0.04 mag, but minima which have amplitudes up to 0.4 mag occasionally occur. Figure 4a shows the mean lightcurve based on our brightness estimates, where we computed the phase from the epoch derived by Wenzel (1991). Of course we may not expect that the regular minima with their extremely small amplitudes can be recognized in our photographic data, but obviously the occasionally occurring deeper minima have been detected in sufficient numbers to produce an accumulation of data points around the epoch of the minimum. In order to show the periodic nature of the light variations more clearly, we calculated normal points and show the resulting mean lightcurve in Fig. 4b. Our amplitude of $A \approx 0.13$ mag is larger than that found by Grankin et al. (1991) and Wenzel (1991). The reason may be that the number of deep minima in our sample is larger than in their data since the distribution of the Algol-like minima shows smaller numbers during their period of observation. Moreover, our diagram reveals that sometimes very deep minima were observed. It seems remarkable that even these very deep minima occur at phase zero.

We wish to emphasize that the period of the light variations as well as the epoch hold for the full time interval covered by our data set and no glitches seem to have occurred. Because of the large standard deviation of the data points, it appears premature to discuss whether the slight asymmetry in the mean lightcurve is real or not.

The modelling of the light variations seems to call for at least two different mechanisms. The strictly periodic low-amplitude variation is very reminiscent of the lightcurve of an eclipsing binary and may be caused by an unseen secondary component in a binary system, which partially eclipses the primary every 10.658289th day. The deeper minima have to be classified as Algol-like and are presumably due to occultations by circumstellar dust clouds. To synchronize these two physically distinct mechanisms, Grankin et al. (1991) proposed a model in which the circumstellar clouds move around the secondary but can occasionally leave it through the inner Lagrangian point onto the primary. When near the Lagrangian point, the cloud produces the observed minimum. A major objection against this model is that the dimensions of the binary are such that dust residing in a shell around the secondary is heated by the primary to temperatures which are too high for the grains to survive. Here we wish to offer an alternative mechanism. The mean lightcurve shows no indication of a secondary eclipse. We assume, therefore, that both components of the binary have identical properties and the orbital period is twice the observed period of the light variations. Adopting the mass of each star as 1.5 $M_{\hbox{$\odot$ }}$ , the distance between the stars would be 0.2 AU. It follows from the discussion of the duration of the Algol-like minima that the clouds have distances of 2.5 AU and more from the centre of gravity of the system. Let their orbits be inclined to the orbital plane of the binary. Then an occultation will become visible from the earth whenever a cloud is near the orbital plane of the binary and in conjunction with either star. A conjunction with either star is, however, possible only if both stars are situated on, or very near the line of sight, i.e., during an eclipse. Thus a configuration where clouds which move in the orbital plane of the binary are extremely rare can account for the observed synchronization. To test this model, it is important to prove the binary nature of BO Cep and to check (e.g., by multicolour observations) whether the origin of the Algol-like minima is actually circumstellar dust clouds.

The normal light of VX Cas, BH Cep, and RZ Psc is represented by a broad band of data points without any distinct wave-like patterns. Our DFA confirms this qualitative impression. It revealed no dominant periodic component in the lightcurve of any of these three stars. This contradicts results by Shevchenko et al. (1993a,b), who claimed a period of 4.46 a for VX Cas from their photoelectric data, and by Grankin et al. (1991), who reported a period of 13.78 d for RZ Psc. For the latter star Zaytseva (1985) claimed a period of 12.67 d for minima fainter than $13^{\rm m}$ . This period is also absent in our data. One has to keep in mind that the time intervals considered by these authors were much shorter than the interval covered in this study. It is therefore possible that the periods found by them are held only temporarily and cannot be recognized in the long-term data set.

To summarize the foregoing discussion of the long-term behaviour of the light variations, it must be stated that only in the case of BO Cep the existence of a stable periodic component of the lightcurve could be confirmed. The variations of the other stars are more or less irregular or cyclical at best.

4.3 The Algol-like minima

Because of the compressed timescale of Fig. 2, only the deepest Algol-like minima are clearly discernible as single observations well below the broad data bands representing the normal light of the stars. Additional minima with smaller amplitudes are present within this band as is indicated by the asymmetric distribution of the data points. Due to gaps in the time series of the photographic observations, the course of the minima is only partly covered in most cases. The photographic data also cannot give any hints on the cause of the minima as multicolour observation can do. Therefore, in the context of this paper the term Algol-like minimum is applied to any decrease of the stellar brightness on the timescale of days or weeks and we assume that it is caused by an occultation of the star by a circumstellar dust cloud.

Because the uncertainty of any particular data point is relatively large and the intervals between successive observations differ widely, the visual inspection of the lightcurves gives an only incomplete result and the hazard of subjective misjudgement is large. To make the search for short-lived minima as objective as possible, we applied the following procedure. First we divided the lightcurve in time intervals of equal length $\Delta t$ . The mean brightness $\langle B \rangle$ and the standard deviation $\sigma$ were then calculated from the combined data of any three adjacent intervals and then taken as representative for the normal light in the middle interval. As the consequence of this procedure, the normal light of the stars was approximated by something similar to a running mean. We excluded the data points below a critical limit $B_{\rm crit}$ in the averaging process to minimize the influence of the deep minima on the average brightness. Test runs were performed with different interval lengths $\Delta t$ to find the most appropriate parameters for each star. In the second step we selected as possible minima all observations, for which $A = B -\langle B \rangle \ge 2.5 \sigma$ . If there were several observations per night, the average of them must satisfy this condition, otherwise the possible minimum was rejected. The number of minima initially selected but then rejected, may serve as one indication of how reliable the identification of a minimum really is if it is based on a single observation only. We found that the fraction of rejected minima is between 0.06 for VX Cas and 0.457 for SV Cep. While the fraction of rejected minima increases slightly with the total number of observations (because the number of multiple observations per night increases), the high rejection rate makes SV Cep very special. However, this is also the star with the lowest detection rate. As for Algol-like minima, this star very likely shows a very low level of activity.

$\begin{figure}\includegraphics[width=8.8cm]{fig05a.ps}\includegraphics[width=8.8cm]{fig05b.ps}\end{figure}$

Figure 5: Examples of a deep peaked Algol-like minimum a) and a trough-like minimum b). Photoelectric data for VX Cas by courtesy of Shevchenko (private communication)

Bad weather, seasonal visibility, and bright moonlight are the main causes for gaps in the time series of the observations. One problem that arises from such gaps is that we possibly cannot decide whether two successive fadings which are several days apart belong to a single minimum or represent two minima, but the star was unfortunately not observed while in its normal light. Therefore, we have defined that successive fadings belong to a common minimum when they occurred not more than 10 d apart. Lack of observations is also the reason why only upper limits for the duration of the minima could often be determined or why the duration could not be derived at all. A third consequence of the gaps is that a certain number of Algol-like minima remains undetected because the most are of short duration only.

Table 4: Characteristics of Algol-like minima
VX Cas RZ Psc WW Vul BH Cep BO Cep SV Cep

Total number of observations $n_{\rm obs}$ 3907 2971 6958 3976 3679 6000

Total number of minima $n_{\rm min}$ 107 93 213 183 95 44

Number of minima

with the duration estimates 56 46 141 73 53 25

Mean amplitude $\bar{A}$ of the

5 deepest minima (mag) 1.28 1.65 1.33 1.29 1.15 1.16

Mean duration of minima

with $\tau \le 4$ d (d) 3.1 2.4 2.9 2.9 3.5 3.2

Bin width $\Delta t$ in calculating

the running mean $\langle B \rangle$ (d) 240 240 120 240 240 60; 80; 120

$n_{\rm tot}/n_{\rm min}$ 36 32 33 22 39 136

**Table 4:** Characteristics of Algol-like minima
	VX Cas	RZ Psc	WW Vul	BH Cep	BO Cep	SV Cep
Total number of observations $n_{\rm obs}$	3907	2971	6958	3976	3679	6000
Total number of minima $n_{\rm min}$	107	93	213	183	95	44
Number of minima
with the duration estimates	56	46	141	73	53	25
Mean amplitude $\bar{A}$ of the
5 deepest minima (mag)	1.28	1.65	1.33	1.29	1.15	1.16
Mean duration of minima
with $\tau \le 4$ d (d)	3.1	2.4	2.9	2.9	3.5	3.2
Bin width $\Delta t$ in calculating
the running mean $\langle B \rangle$ (d)	240	240	120	240	240	60; 80; 120
$n_{\rm tot}/n_{\rm min}$	36	32	33	22	39	136

Table 4 summarizes some general properties of the Algol-like minima accepted as such by our standards. In addition to the total number of minima found for each star, we listed the number of minima, for which durations $\tau$ could be derived, the mean amplitude of the 5 deepest minima, the mean duration of the minima with durations $\le 4$ d, the parameters of the minimum search procedure, and the mean number of observations per minimum.

To get some idea of the reliability of the numbers, we first look for evidence of incompleteness. As already mentioned above and also discussed in same detail by Friedemann et al. (1995), gaps in the observational data have prevented us from detecting all minima. We would expect, however, that the gaps become shorter and the number of detections increases with an increasing number of observations. If the stars are compared with each other, no close correlation between the total number of observations and the total number of minima detected exists. This means that the frequency of the minima varies among our stars. A trend is, however, visible if for each star the number of minima per year is plotted against the number of observations per year. It appears that a large number of observations are a necessary condition for a large detection rate of minima, but it is not sufficient. On the contrary, a low number of observations always resulted in the detection of few minima. With other words, for a certain number of observations, there is an upper limit for the number of minima detected. It is clear that a large scatter of the data points of the normal light may mask some minima, especially those with relatively low amplitudes. The relatively large error produces spurious fadings. It follows from our selection criterion for Algol-like fadings ( $A \ge 2.5 \sigma$ ) that the probability of a selected fading to be due to the estimation error is about 0.0062. For all stars but SV Cep the number of minima found is much larger than predicted from the scatter of the data points alone. Therefore, most of these minima should be real and suited for further analysis. Because of the strong wave-like variability of SV Cep, the portions of the lightcurve where $\sigma$ is relatively small are much shorter than for the lightcurves of the other stars. The recognition of Algol-like minima, especially minima with small amplitudes, is hindered considerably. The number of fadings initially detected (81) is only twice the number expected based on the scatter of the data (37). From this and the high rejection rate we conclude that the fraction of the genuine minima among the total number of accepted minima is smaller for SV Cep than for the other stars.

The standard deviation $\sigma$ computed in the averaging procedure is not constant over the whole lightcurve. Consequently we could not select the minima everywhere from the same minimum amplitude upward and the numbers of small-amplitude minima are systematically too low for $A < 2.5\,{\rm max}(\sigma)$ . We made an attempt to correct crudely for this systematic effect. We assume that both the number of minima and the relative distribution of their amplitudes are constant over the whole time interval. Then the number of minima with $A \le 2.5\,\sigma$ is too low by the factor $\int\limits_0^\sigma n(\sigma')\,{\rm d}\sigma'/n_{\rm min}$ , where $n(\sigma)$ and $n_{\rm min}$ are the number of minima detected in parts where the observations have a standard deviation of $\sigma$ and the total number of minima, resp. Figure 6 shows the normalized distributions of the amplitudes of the Algol-like minima for our six stars. Both the uncorrected and corrected distributions were plotted. While the uncorrected distributions apparently signalize significant differences among the stars, the serious influence of the different values of $\sigma$ becomes evident through our correction procedure. Apart from RZ Psc the amplitude distributions seem to be quite similar with about 50 per cent of the minima having amplitudes smaller than 0.5 mag. In the case of RZ Psc the low number of shallow minima appears to be not an effect of observational errors alone but a property of the circumstellar shell.

In Fig. 7 we present the relative distributions of the duration of the minima. The number of minima is not corrected for any incompleteness due to the amplitude-to- $\sigma$ ratio. Different distributions occur among stars with similar time series. A comparison with the data in Table 3 shows that a large number of observations does not necessarily mean a higher fraction of shortest minima. Therefore, we feel that the low rate of short minima in the lightcurve of VX Cas may not be attributed to a lack of observations alone. On the contrary, SV Cep and RZ Psc have exceptionally large fractions of short minima. However, SV Cep is not a fully convincing case. Because for this star the fraction of spurious minima may be larger than for the other stars, the distribution function could be biased. More interesting is the preponderance of short minima in the case of RZ Psc. Since this star is physically different (it is of relatively late spectral type), the different distribution functions for amplitude and duration may well point to differences in the shell properties between young stars of early and late spectral types.

$\begin{figure}\includegraphics[height=21cm]{fig06.ps}\end{figure}$

Figure 6: Relative frequency distribution of the amplitudes of the Algol-like minima. Both the observed distribution (broken line) and the distribution corrected for the effects of the varying standard deviation over the lightcurves (solid line) are shown

4.4 Photometric properties of the Algol-like minima - R values and colour excess ratios

No photoelectric observations of deep Algol-like minima are available for BH and BO Cep. However, Pugach (1988) concluded from his UBVR observations of BH Cep that the small-amplitude variations are compatible with R = 3.2and the circumstellar and interstellar dust may, therefore, be similar.

$\begin{figure}\includegraphics[width=8.8cm]{fig08.ps}\end{figure}$

Figure 8: Relative frequency distribution of the R values of 41 Algol-like minima observed for WW Vul, SV Cep, VX Cas, and RZ Psc

To check whether the R values derived by us are compatible with the common assumption that larger R values are the consequence of larger grain sizes, we compared our observational results with model calculations by Steenman & Thé (1991). They assumed the dust model of Mathis et al. (1977) for the general interstellar medium and studied the effect of changing the lower and/or upper limits of the particle size distribution on the reddening law. The solid line in Fig. 9 shows the predicted effect of increasing the minimum particle size. The dashed line is our extension of this model to higher R values. Within the observational errors the observed colour excess ratios and R values agree well with the model predictions. This seems to be convincing evidence for the assumption that the individual clouds responsible for the various Algol-like minima may be characterized by different depletion of small grains. The small systematic difference between the theoretical curve and the majority of the observations may indicate that the circumstellar reddening law is not altogether identical to the interstellar one.

$\begin{figure}\includegraphics[width=8.8cm]{fig09.ps}\end{figure}$

Figure 9: Relation between reddening parameter R and colour excess ratio E(V-R)/E(B-V) for WW Vul (crosses) SV Cep (open diamonds) and VX Cas (filled circles). The filled square is the result by Pugach (1988) for BH Cep. The grey triangle marks the standard value of the interstellar mean. The solid line (and the dashed part of it as an extrapolation from us) represent model calculations by Steenman & Thé (1991) discussing anomalous extinction curves. The error bars in the lower right corner indicate one typical standard deviation

4 Discussion of the light variations

4.1 The available photometric data

4.2 The long-term lightcurves

4.3 The Algol-like minima

4.4 Photometric properties of the Algol-like minima - R values and colour excess ratios