CHAPTER 6
RESULTS AND INTERPRETATIONS

This chapter describes the results for the individual systems as they relate to the information in the corresponding sections of Chapter 5 and the Figures and Tables of Appendix B. The phase of minima are determined from the observational information from Table 6-1 and the ephemeris information in Table 6-2. The periods and phases cited in the text are the average of the values from the three colors which are summarized in Table 6-3.

The errors for the periods are either the geometric average of the statistical errors, from equation 2-6, or the standard deviation from the three colors, whichever is larger. The errors for the phases are the standard deviation of the three colors. No errors are quoted for the specific amplitudes of the signals since there is no simple way of defining them, but analysis of synthetic data sets indicates an average error of 0.005^m. Only the V-band amplitudes are cited in the text for the direct comparison of values taken from the literature. The amplitudes for the U and B-band data can be found in Table 6-4, along with the corresponding filter phases, f in radians, and false alarm probabilities, Á. The difference between the filter phases, Df, is used as an absolute measure of phase changes in the photometric wave.

The first problem that must be addressed is to define the minimum signal than can be extracted from a data set given the noise level of 0.01^m and the data precision of three decimal places. The time sequence of 33 Psc was arbitrarily used for the test case. Three groups of signals were generated, with ~0.01^m random noise, and truncated at three decimal places. These data were then averaged into a single file as was done with the actual APT data. The full amplitude of the artificial signal was reduced until it could no longer be detected. The results of this analysis indicate that the full amplitude of 0.007^m is the smallest signal than can be readily detected, but an amplitude of 0.006^m can be seen at the expected noise level (equation 2-8) if the presence of the signal is know a priori.

The two primary sources of variability in non-eclipsing systems are the differential reflection effect and ellipticity. The reflection effect is a result of the light from one star being absorbed and re-emitted by the other. The theoretical constructs and characteristics have been described by several authors (Hopf 1934; Russel 1948; Kopal 1959; Napier 1968; Chen and Rein 1969; Wood 1969; Hill and Hutchings 1970; Wilson and Devinney 1971). The resulting photometric period will be equal to the orbital period, or half of that value, and the phase of minimum will be at the time of conjunction.

The ellipticity effect results from the fact that the intensity at any point on a star is directly related to its gravitational potential (von Zeipel 1924; Chandrasekhar 1933; Kopal 1959; Wood 1971b). The period is always one half the orbital value and the phase of minimum is at the time of conjunction. If the observed variations of a system can possibly be attributed to either reflection or ellipticity then they are discussed, otherwise no mention is made.

6.1 33 Psc

No structure is visible in the periodogram for this system (Figure 6-1). If the ellipticity effect cited by Barksdale et al. (1985a) is present, at the cited level of DV = 0.007^m, it would be detectable. This conjecture is confirmed by replacing the data with a synthetic signal to simulate the variability observed by Barksdale et al. (1985a) as it would be observed by the APT. However, if their error was ~0.002^m, this could put the signal below the detection level for this data. The limitation for the analysis for this system is not the signal to noise ratio, but rather the limit of three decimal places for the APT data. Higher significance data is needed to resolve the question of the existence of the photometric period. The low activity level is consistent with the observed weak X-ray emission and the lack of detectable radio emission.

6.2 13 Cet (A)

No period is visible in the periodograms for this system (Figure 6-2). The data set is limited, 61 observations over 385.9466 days, but analysis of synthetic data indicates that periodic structure would be visible down to the 0.007^m full amplitude limit. The lack of any detectable periodicity is consistent with previous null detections.

6.3 z And

Two clear periods are found for this system (Figure 6-3). The 8.888 ± 0.008 day period is half the orbital period (P_orb/2 = 8.8846^d) and implies the ellipticity effect. The phase of minimum at 0.964^p ± 0.003 would seem to be inconsistent with this effect. If the orbital period of 17.7692 days is in error by only 0.000045 days then the phase of minimum is within the time of conjunction for the ephemeris used. The presence of the ellipticity effect is consistent with the observations of Herbst (1973); however, my full amplitude of DV = 0.069^m is half of his reported value.

The secondary period of 18.25 ± 0.09 days is 2.7% longer than the orbital period. The periodograms for this signal are noisy, but the structures are clearly visible and are a factor of two above the predicted noise level. There is also strong agreement between colors. It is not surprising that the secondary period has not been previously observed considering its amplitude (DV = 0.018^m) as compared to that of the primary period (DV = 0.069^m).

The data for this system is marginally separable into two groups. The analysis of the groups clearly shows the previously determined periods. The periods and amplitudes show no variation between the groups and there are no apparent phase changes for the primary period. The phases for the secondary period are poorly characterized and cannot be compared.

From the analysis, I must conclude that if the primary periodicity is a result the ellipticity effect, then I cannot account for the large discrepancy in the amplitudes between my results and those of Herbst (1973). The secondary period has not been previously detected and represents the first clear detection of a distortion wave for this system.

6.4 AY Cet

This is an extremely active system, and a look at the light curve of Figure 4-B clearly shows the variability. There is a trend in increasing mean brightness in the system, as also reported by Poretti et al. (1986), and decreasing amplitude. Unfortunately, the length of the period makes it impossible to analyze the data in smaller groups, so only a qualitative observation of the amplitude variations can be made. The mean magnitude has risen by ~0.04^m in all three colors.

The periodogram analysis yields four frequencies, the last three of which show dependence on the primary signal and cannot be considered real. This additional structure is a result of the dramatic changes visible in the light curve and indicates that a single sinusoid is not completely representative of the data. The primary period (Figure 6-4) of 77.7 ± 0.6 days is the same, within the errors, as that found by Eaton et al. (1983a), although my amplitude of DV = 0.069^m is about half of theirs. This amplitude is similar to the value obtained by Poretti et al. (1986) for their 77.22 day period. Unfortunately the time span of the observations, 471.8256 days, implies a resolution limit of 11.3 days for a period of 77.7 days, as described in section 2.3, so the possibility of a 79 day period, as described by Poretti et al. (1986), cannot be investigated. A longer term data set is needed to search for multiple periodic structure.

6.5 AR Psc

The data set for this system contains only 62 observations over 403.9235 days, in which two periods are identified. The second period is dependent on the first, so it cannot be considered real. The primary period (Figure 6-5) of 12.23 ± 0.02 days is the same as that found by Hall, Henry and Louth (1982) except that my amplitude of DV = 0.105^m is a little larger and my phase of minimum is 0.77^p ± 0.02.

AR Psc seems to have undergone a significant amplitude change while the mean value has been steady. Unfortunately, the data cannot be separated into two sets since there are only 12 observations in the first group. Analysis of the second group results in a single period of 12.16 ± 0.06 days with a phase of minimum at 0.91^p ± 0.04. This period agrees with that found for the entire data set, but there has been a significant change of Df = 1.1 radians in the filtered phase values. This phase shift would account for the residual signal found near the primary period.

6.6 TZ Tri

Two weak periods are found for this system (Figure 6-6). The light curve is unusual in that the primary period is not visible at U-band, but the secondary is present at a low confidence level (Á = 0.5). The periods of 7.371 ± 0.008 days and 14.66 ± 0.06 days are at the error limits for the values expected for the ellipticity and reflection effects, respectively. The phase of minimum at 0.955^p ± 0.005 for the primary period would seem inconsistent with ellipticity, but an error of 0.004 days in the cited orbital period of 14.732 days of would place my value within the time of conjunction. My amplitude of DV = 0.041^m for this signal agrees with that found by Hall et al. (1980). The phase of minimum at 0.45^p ± 0.05 for the secondary period is consistent with the reflection effect. There is no evidence for a 100 day period as suggested by Hall et al. (1980).

This system is severely underobserved with only 75 observations over 411.9296 days. It is barely separable into two data groups that show no indications of variation. The results of my analysis are just within the errors for the ellipticity and reflection effects so further observations are needed to reduce the errors and confirm the source of the variability.

6.7 UX Ari

The analysis for this system yields two strong periods at 6.425 ± 0.002 days and 3.217 ± 0.002 days (Figure 6-7). Two residual periods are also found, at very low confidence levels, and indicate changes in the photometric properties of the light curve over the period of the observations.

The primary period is very close to the orbital value, differing by only 0.2%, but it is well outside the error ranges for the reflection effect. The secondary period of 3.217 ± 0.001 days is equal to half the orbital period (3.218955 days) with a phase of minimum at 0.919^p ± 0.003. Based on this period alone, I might associate this signal with the ellipticity effect, but the phase is not close enough to the time of conjunction given the confidence level of the signal. The orbital period of 6.43791 days would have to be in error by 0.00015 days to account for the variation.

The data are separable into two groups, and I find two periods in each group with no trace of the residual signals that are found for the entire data set. The analysis indicates that the primary period changed from 6.42 ± 0.01 days to 6.441 ± 0.007 days with a corresponding change in the filtered phase values of Df = 1.2 radians, and a decrease in amplitude from 0.263^m to 0.229^m at V-band. The secondary period does not show any significant change, but there does appear to be an increase in the amplitude of the signal in all three colors with the greatest variation appearing at U-band.

My primary phase of minimum is the same as that found by both Guinan et al. (1981) and Zeilik et al. (1981)– an indication that the system has been basically stable from 1980 to 1984 which is interesting since it is known to have been highly variable in the past. There has been no direct mention of the secondary period found in this analysis except for the necessity of two components used in starspot modeling.

6.8 V711 Tau

Three periods are found for this system, with the second signal showing a clear dependency on the primary signal and resulting in a period at the Nyquist harmonic of one day. The two remaining periods observed are 2.8370 ± 0.0003 days and 1.4186 ± 0.0005 days (Figure 6-8). The primary period is only 0.026% shorter than the orbital value, but with a phase of minimum at 0.391^p ± 0.006 and a full amplitude of DV = 0.119^m, the signal is clearly not a result of the differential reflection effect. The secondary period is only 0.018% lower than half the orbital value, but the phases show too great a variation to be used in the characterization of the signal. The primary signal does not filter well and results in strong, dependent residual signals that indicate changes in its photometric properties.

This data set is separable into two groups. Both groups show three periods with the second signal at the Nyquist harmonic. The primary period shows a 0.14% increase from 2.836 ± 0.001 days to 2.840 ± 0.002 days with a corresponding increase in amplitude from DV = 0.094^m to DV = 0.142^m. The secondary period shows a possible decrease from 1.43 ± 0.01 days to 1.417 ± 0.001 days, but the difference is very close to the error limits of the values. The amplitude, on the other hand, shows a clear increase from DV = 0.014^m to DV = 0.024^m. The phases for both data groups are too poorly defined to be compared.

It is clear from the period and amplitude variations that these signals are not the result of ellipticity or the differential reflection effect. My primary period and amplitude agree with previous findings as does the presence of the secondary signal which has been observed as a second maximum in the light curve.

This system is known to be highly variable and with its relatively short photometric period of three days it would be an ideal candidate for intensive observation. With a high time-resolved data set, the time scales for the variations could be determined.

6.9 HD 22403

One period at 1.895 ± 0.002 days is found for this system. The periodograms are extremely noisy and only the combined use of the UBV data sets reveal the structure (Figure 6-9).

My period agrees with that found by Raveendran, Mohin and Mekkaden (1985). My amplitude of DV = 0.03^m is only one-third their reported value and indicates the presence of large amplitude variations. The data set for this system is severely undersampled and further observations are needed to identify the presence of any multiple periodic structure.

6.10 EI Eri

The analysis of the data yields four periods with the second showing up at the Nyquist harmonic and the last two being dependent on the primary signal. The presence of these residuals indicates changes in the photometric properties of the light curve over the period of the observations. One aspect of these variations can been seen as in the dramatic amplitude change in the light curve of Figure B-10.

The periodograms show two distinct, but illformed peaks (Figure 6-10), one of which is an alias. The two peaks reflect periods of 2.0479 days and 1.994 days, which have also been observed by Bopp et al. (1983) and Hall et al. (1987) respectively. Since the orbital period is 1.94722 days, I suspect that Bopp et al. (1983) reported an alias, or the photometric period is highly variable. The irregular structure observed in the peaks is caused by the dramatic amplitude change in the light curve and an associated decrease in the photometric period and changes in phase. Fortunately, the data set can be separated and analyzed.

The separated periodograms, as expected, still show the aliased signal. The peak nearest the orbital period is taken as real. Only two periods can be extracted from each group, so the residual signals found in the entire data set have been eliminated. The second period in the data groups is still at the Nyquist harmonic value of one day.

The analysis indicates that the period increased from 1.920 ± 0.002 days to 1.949 ± 0.001 days, with a corresponding change in the filtered phase value of Df = 2.3 radians. The amplitudes also increased dramatically with DV changing from 0.070^m to 0.156^m. There is an apparent increase in mean brightness on the order of 0.03^m in all three colors. These two periods are 1.4% shorter and 0.08% longer than the orbital period and are nearest the value cited by Hall et al. (1987) but my amplitudes are consistently lower, which is not too surprising considering the variations observed in this data set.

6.11 RZ Eri

This source has an insufficient number of data points to perform any useful analysis (20 data points over 55.8521 days for a system with a 39.2826 day period). The periodogram (Figure 6-11) does indicate periodicity near half the orbital period, caused by the eclipse, but it is very weak with only a 60% confidence level. A phase plot of the data indicates that most of the observations were made outside of eclipse, so the low signal level is not surprising.

6.12 BM Cam

Four periods can be identified by the periodograms, with the last two appearing at the Nyquist harmonic. The periods for the primary and secondary signals are 40.5 ± 0.10 days and 80.3 ± 0.9 days with the phase of minima at 0.592^p ± 0.008 and 0.41^p ± 0.07 (Figure 6-12). Hall and Osborn (1986) noted that from 1979 to 1985 the photometric period changed from ~80 days to ~40 days accompanied by changes in amplitude and mean brightness. I find that the photometric period from 1982 to 1985 is actually a composite of these two periods with the 40 day period being the more pronounced.

I attempted to separate the data into two groups and analyze them independently in order to investigate the possibility of period, phase and amplitude variability. Unfortunately, the length of the periods precluded the possibility of making any meaningful comparison between the two groups.

6.13 HD 37824

The analysis of the entire data set reveals the presence of a 53 and 26 day period (Figure 6-13) along with a large collection of residual signals. These residuals indicate the presence of variability in the period, amplitude or phase of the two primary signals. Fortunately, the observations can be separated into two data groups. The analysis of the two groups also shows the 53 and 26 day periods along with a signal at the Nyquist harmonic period of one day. The initially observed residual structures do not appear.

The primary period shows a clear change from 51 ± 1 days to 53.5 ± 0.3 days with a corresponding change in the filtered phase of Df = 1.3 radians. There is also a dramatic change in amplitude by about a factor of two (0.12^m to 0.24^m at V-band) in all three colors. These periods are in agreement with that found by Hall et al. (1983) as is the smaller of the two amplitudes.

The secondary period has not been previously identified. It does not show any significant variations and is found, for the entire data set, to be 26.83 ± 0.05 days. The phase of minimum also appears stable at 0.719^p ± 0.003, although the errors are large in the separated groups. The amplitude is also seen to decrease by a factor of two in all three colors (0.080^m to 0.044^m in V-band). The period and phase values just cited were obtained after filtering the primary signals from the separated data sets and then recombining. The difference between these values and those obtained after filtering the primary signal of the entire data set is within the errors.

6.14 s Gem

This system reveals five periods, with the last three showing dependency. The residual structure is apparent in the periodograms for the two strongest signals (Figure 6-14). This large amount of residual structure indicates changes in the photometric properties of the light curve during the observing period. Fortunately, the data can be separated into two groups. Each group shows only three periods, with the last one being the Nyquist harmonic value.

The primary period of 19.32 ± 0.02 days does not show any sign of variation. The phase of minimum is stable at 0.986^p ± 0.009 as are the amplitudes (DV = 0.066^m). This period is in rough agreement with the characteristic period near 19.4 days as reported by Fried et al. (1983) and Strassmeier et al. (1988), although my V-band amplitude is about half that reported by the latter.

The secondary period is a different story. It shows a clear variation from 9.76 ± 0.02 days to 9.46 ± 0.04 days with a phase shift of Df = 1.8 radians. The amplitudes of all three colors decreased by about a factor of two (0.045^m to 0.027^m in V-band). The presence of this secondary period was reported by Fried et al. (1983) who noticed two distinct minima in the light curve, but the characterization of this periodic structure has never been established.

6.15 AE Lyn

Only one poorly defined period, at the Nyquist harmonic value of one day, could be identified (Figure 6-15). Along with being illspecified (meaning that the results were not consistent between colors) the confidence levels are low (89% at V-band) and the periodograms are exceptionally noisy. If the 10 day period reported by Eaton et al. (1981) were still present, it would be easily identified with the large amount of coverage for this system (222 observations over 502.7864 days), thus the amplitude of the signal has dropped below the sensitivity limits of my analysis.

6.16 53 UMa

The analysis of this system shows some unusual behavior. When the data set is processed in its entirety, a strong period of 0.998 ± 0.003 days appears, which is the Nyquist harmonic value. If the observations are separated into two data groups, this period vanishes (Figure 6-16). This behavior indicates a period that is slightly longer than the fundamental value of 515.7867 days. These results are consistent with the suggestion of Percy and Welch (1982) of a possible period near 500 days. With the large amount of data for this system (376 observations over 515.7867 days), any existing periodic structure would be identified.

Visual inspection of the light curve (Figure B-16) hints at a long term periodicity, and is clearest a U-band. The optical companion of this system needs to be studied for variability since it is a known 670 day spectroscopic binary.

6.17 DQ Leo

The analysis of this system demonstrates an unexpected level of variability (Figure 6-17). A single period of 56.0 ± 0.7 days is found which is substantially different from the orbital period of 71.69 days and the photometric period of Hall et al. (1980), which was near the orbital value. The phase values for this system are poorly defined, which is a surprise considering the coverage of 358 observations over 515.7641 days, and indicates the presence of variability.

Separating the observations into two data groups reveals that there is no periodic structure in the fist data set. The second data set yields a period of 59 ± 3 days with a large variation in the phase values, which indicates the presence of even additional structure. Unfortunately, the length of the photometric period precludes any further separation of the data.

6.18 DK Dra

Three periods are found for this system, with the second showing a dependency on the first. The periodograms (Figure 6-18) show a great deal of structure and indicate changes in the properties of the light curve during the observing season. The increase in amplitude and the 0.04^m decrease in mean magnitude are clearly visible in the light curve of Figure 18-B. Unfortunately, insufficient data were taken in the second observing season to allow for separation, but the data from the first season can be evaluated independently.

The two periods isolated are 65.2 ± 0.5 days, with a phase of minimum at 0.49^p ± 0.03, and 30.6 ± 0.1 days, with a phase of minimum at 0.497^p ± 0.004. Neither of these two periods is a candidate for either the ellipticity or reflection effects. The primary period is consistent with the values found by other authors which range from 63.75 to 64.4 days, while the secondary period has never been reported.

Independent analysis of the data from the first observing season is difficult owing to the long period and the short time interval giving only 2.4 orbital cycles. The periods found are within the errors of the period for the entire data set. The residual signal found for the entire data set in not present in this analysis.

The data does not filter well, resulting in periodograms with a great deal of ragged structure. From the form of the periodograms, I suspect that the two cited periods are real, but there are changes that are either long compared to the observing window or nonperiodic in nature. More observations are clearly needed of this system.

6.19 BH CVn

The results for this system are similar to those found for 53 UMa. When the data set is process in its entirety, a strong period of 1.000 ± 0.003 days appears, which is the Nyquist harmonic value. If the observations are separated into two data groups, this period vanishes (Figure 6-19). This behavior indicates a period that is slightly longer than the fundamental value of 513.7563 days.

No evidence for the reflection effect, reported by Hall et al. (1978), Burke et al. (1980), Dorren and Guinan (1980) and Barbour and Kemp (1981), is observed. The coverage for this system is very good (207 observations over 513.7563 days) so the variation of DV = 0.01^m found by both Hall et al. (1978) and Burke et al. (1980) would be easily identified. Thus, it is entirely possible that previously observed effects attributed to reflection were actually a distortion wave as concluded by Shore and Adelman (1984).

6.20 HD 136901

Four periods are identified for this system (Figure 6-20), however the last value appears at the Nyquist harmonic period of one day. The primary period of 9.335 ± 0.002 days is exactly half the orbital period and the phase of minimum (0.5034^p ± 0.0008) is at the time of conjunction. The secondary period of 18.67 ± 0.03 days is equal to the orbital period and the phase of minimum (0.49^p ± 0.01) is also at the time of conjunction. These two signals are most likely a result of the ellipticity and differential reflection effects, respectively. The value of my primary period is in agreement with that reported by Boyd, Genet and Hall (1984). Their V-band full amplitude is also in agreement with my value of DV = 0.16^m. The third signal has a period of 6.225 ± 0.003 days, a phase of minimum (0.509^p ± 0.001) near the time of conjunction and a full amplitude of DV = 0.029^m. The secondary and tertiary periods have not been previously reported.

The periodograms are extremely stable and show no signs of residual structure. The data set is separable and further analysis of the data groups confirms the lack of variation.

6.21 TZ CrB

Two periods are found for this system, with the primary signal appearing at the Nyquist harmonic value of one day. When the data are separated; however, this signal vanishes. This behavior indicates a period that is slightly longer than the fundamental value of 506.6880 days.

The secondary period of 1.165 ± 0.002 days (Figure 6-21) is 2.2% longer than the orbital period and is in general agreement with the values cited by other authors. The phases for this signal are poorly defined, which is surprising considering the coverage of 168 observations over 506.6880 days. The poorly described phases are a result of the clear variation in the mean magnitude of the system and an increase in the amplitude of the signal by nearly a factor of two (0.015^m to 0.029^m in V-band) between the observing seasons. This amplitude variation was found from the analysis of the separated data groups.

As a result of the sampling rate, the d Scuti-type variations, suggested by Skillman and Hall (1978), cannot be investigated with the current observing program of the APT. If the variations were real, then the harmonic nearest one day would be selected. Replacing the data with a signal to simulate the d Scuti-type variations results in a low, but detectable, signal. From this analysis, I do not find supporting evidence for the proposed short term variations.

6.22 HR 6469

This newly discovered eclipsing system was analyzed in two ways. First, the raw data were processed in order to test the methods for this extreme case. The resultant periodograms are full of structure and are nearly unreadable (Figure 6-22). Applying the selection criteria, I found a period of 1.1147 ± 0.0002 days with a phase of minimum at 0.51^p ± 0.01. This value is half the orbital period, with a minimum at the time of conjunction, as would be expected for an eclipsing system. Thus the methods can be applied even in the case of a severally non-sinusoidal signal. Unfortunately, filtering the signal is not feasible because of the strong deviation from a sinusoidal structure.

In order to analyze the outside-of-eclipse variations, I visually cropped the eclipses and reprocessed. I found a single period of 0.9859 ± 0.0001 days, with a minimum at 0.76^p ± 0.3 and DV = 0.026^m. It is possible that this period is a product of the spectral window, but it is far enough from the one day Nyquist harmonic to be considered real. There is no evidence for the 83.2 day period as reported by Boyd et al. (1985).

6.23 DR Dra

Two periods are found for this system, with the primary being equal to the Nyquist harmonic value of one day (Figure 6-23). The secondary period of 31.6 ± 0.1 days (DV = 0.069^m) is the same, within the errors, as that found by Hall et al. (1982) but at about half the amplitude. This system shows a clear increase in mean amplitude, on the order of 0.1^m in V-band, during the observing season and a suggestion of a final decline. This large scale variation results in the poorly defined phases for the secondary period. Unfortunately, very little data are available for this system (59 observations over 615.2923 days), which precludes the possibility of analyzing the observations in separate groups.

6.24 Z Her

The periodograms for this system show two clear peaks, one of which is an alias (Figure 6-24). The value nearest the orbital period is considered real. Analysis of this system results in a single period of 3.97 ± 0.01 days (DV = 0.038^m) with a minimum near the time of conjunction. These results are in agreement with distortion wave observed by Evren et al. (1982).

The distortion wave, rather than the eclipse, is detected because the phase coverage of the orbital period is so poor. A phase plot of the data indicates that only four observations lie within the time of eclipse.

6.25 V815 Her

The periodograms for this system show two strong peaks, one of which is an alias (Figure 6-25). Two periods are isolated for this system. The primary of 1.825 ± 0.001 days (DV = 0.078^m), with a minimum at 0.70^p ± 0.2, is in agreement with that found by Mekkaden, Raveendran and Mohin (1980) except that my amplitude is only 78% of their reported value. The secondary period of 0.8930 ± 0.004 days (DV = 0.025^m) has a minimum at 0.92^p ± 0.02 and has not been previously identified. The data set is barely separable into two groups and shows no signs of variation in the periods, phases or amplitudes.

6.26 47 Dra

This system suffers from the electronic dead time correction problem addressed in Chapter 4. It is clearly visible in the light curve of Figure 26-B with the change in mean magnitude observable in B and V but not in U. The data were processed before and after adjusting the mean amplitudes in order to investigate the sensitivity of the solution to such problems.

For the uncorrected data, three signals are identified, although the primary period was not visible in U-band and the third value is dependent. The resulting periods are 26.1 ± 0.1 days (DV = 0.074^m) with a phase of minimum at 0.442^p ± 0.002, and 54.8 ± 0.5 days (DV = 0.023^m) with a phase of minimum at 0.39^p ± 0.02. The secondary period is consistent with the results of Hall and Persinger (1986), except that my amplitude is only 68% of their value. The 26 day period has not been previously reported.

The data are normalized to a constant mean magnitude level for all three observational segments. Only two periods are found (Figure 6-26) which correspond to the first two periods in the uncorrected data. The order of their appearance is reversed and both periods are visible in all three colors. The periods are 54.9 ± 0.4 days (DV = 0.035^m) with a minimum at 0.38^p ± 0.01 and 26.3 ± 0.1 days (DV = 0.019^m) with a minimum at 0.568^p ± 0.005. The periods are the same, within the errors, as the unprocessed data. The phases and amplitudes show sizable differences. The amplitude of the 55 day period for the corrected data set agrees with the value cited by Hall and Persinger (1986).

6.27 V478 Lyr

The periodograms show two peaks, one of which is an alias (Figure 6-27). The peak nearest the orbital period is considered to be real. Two periods are found for this system, with the second showing dependency and indicating variations in the photometric properties of the light curve.

The primary period of 2.162 ± 0.002 days (DV = 0.043^m) is 1.5% longer than the orbital value and has a phase of minimum at 0.70^p ± 0.03. My value is about 1% lower than the photometric period found by Henry (1981) and has an amplitude which is larger by 0.01^m.

The data set is separable and a single period can be found in each of the two data groups. There is no apparent change in the orbital period, but the amplitude of the signal shows a sizable decrease (0.076^m to 0.028^m at V-band) between observing seasons. The phases are poorly defined in these data sets because of the limited number of observations (25 observations over 52.8639 days and 49 observations over 90.7640 days), so they cannot be compared.

6.28 HR 7428

Three signals are found for this system, with the second appearing at the Nyquist harmonic. The two periods identified of 54.12 ± 0.2 days and 111 ± 3 days (Figure 2-28) are near those cited by Barksdale et al. (1985b). The first period is within the range for the ellipticity effect with the phase of minimum at 0.03^m ± 0.03. The second period is just at the error limits for differential reflection, but has a phase of minimum at 0.83^p ± 0.05. I am inclined to associate the primary period with ellipticity, but the phase of the secondary indicates that it is not a result of the reflection effect. I find no evidence for any other distortions as was suggested by Barksdale et al. (1985b). The long period of this system makes analysis especially difficult, and a longer set of observations are needed to remove the ambiguity in the results.

6.29 V1764 Cyg

Two periods are identified for this system (Figure 6-29). The primary (20.2^d ± 0.5) is within the errors of half the orbital period and the phase of minimum at 0.50^p ± 0.01 indicates that this signal is most likely a result of the ellipticity effect as reported by Lines et al. (1987). The secondary (40d ± 1) is within the orbital period range but the phase of minimum at 0.37^p ± 0.01 precludes the possibility of the differential reflection effect. It is interesting to note that the secondary period is not visible at U-band. My amplitude of DV = 0.101^m for the 20 day period is 0.024^m lower than that of Lines et al. (1987) while the amplitude of DV = 0.047^m for the 40 day period is within the range they quoted for its variable value. Unfortunately, there are too few observations to separate the data set.

6.30 ER Vul

Two periods can be identified for this system. Unfortunately, with only 30 observations over 51.8596 there is severe aliasing for this 0.69809510 day system.

For the primary signal, the orbital period is used to distinguish it from the aliased values. The period of 0.6959 ± 0.0008 days is 0.2% shorter than the orbital period and has a minimum at 0.17^p ± 0.03, which is consistent with the observations of Akan et al. (1987). The distortion wave, rather than the eclipse, is identified because all but five observations are outside the eclipse.

The secondary period is difficult to specify. Because of the severe undersampling and small number of datum, aliasing has occurred over a wide range of plausible values (Figure 6-30), although the signal at 0.3488 ± 0.002 days is enticing because it is half the primary photometric period. I can safely state, however, that there is a second period present, somewhere. The presence of a second period is consistent with the two spot requirement in the models of Zeilik and Budding (1986).

6.31 HK Lac

Three signals are identified for this system, with the second showing dependence on the primary signal and indicating variations in the photometric properties of the light curve over the term of the observations. The primary and secondary periods for the entire data set are 24.53 ± 0.02 days and 12.16 ± 0.03 days (Figure 6-31).

After separating the data into two groups, I find that the primary period changed from 25.08 ± 0.07 days to 24.49 ± 0.03 days with a corresponding phase shift of Df = 0.4 radians. There is also a decrease in amplitude from 0.248^m to 0.190^m at V-band. The dependent period is not visible in the separated data sets. The secondary period has large errors and so its variability cannot be confirmed, although there is a clear decrease in amplitude.

My primary photometric period is consistent with other works and my V-band amplitude is in rough agreement with Olah et al. (1985). The dependent secondary frequency and the poorly shaped periodograms are consistent with the findings of Olah (1983), that no single period can be identified that can fit a broad band of observations. The presence of my secondary period is consistent with the two spot model of Olah et al. (1985). They proposed differential spot motions of 1.025% and 0.08% of the orbital period which are beyond the resolution limits of the periodograms for this data set.

6.32 AR Lac

Three signals are identified for this system (Figure 6-32), with the last one being dependent on the secondary signal. The primary period of 0.9962^p ± 0.0002 (DV = 0.045^m) is only 0.47% longer than half the orbital period and has a minimum at 0.024^p ± 0.005. The secondary period of 2.0053 ± 0.003 days (DV = 0.046^m) is 1.1% longer than the orbital period and has its phase of minimum at 0.22^p ± 0.02. Neither of these signals has a period, or phase of minimum appropriate to the eclipse of the system.

A phase plot of the data indicates that there are only a few observations during the time of eclipse, so I must attribute the signals to a double peaked distortion wave as reported by Ertan et al. (1982). The quality of the period near one day must be considered with care, since it is so close to the value expected for the Nyquist harmonic.

The light curve of Figure 32-B indicates an apparent variation in mean magnitude, however a synthetic data set for the two signals isolated from the observational set shows the same apparent increase in mean magnitude. The change is clearly a selection effect caused by the proximity of the periods to integer days.

6.33 V350 Lac

Three signals are isolated for this system with the second one appearing at the Nyquist harmonic value of one day (Figure 6-33). This signal is a result of the clear decrease in mean magnitude by 0.03^m. Unfortunately, this system has only 62 observations over 510.3231 days and so it cannot be separated.

The primary period of 8.873 ± 0.008 days (DV = 0.089^m) has a phase of minimum at 0.453^p ± 0.006. This period is equal to half the orbital value within the errors. The phase of minimum is within the errors of the time of conjunction if an error as small as 0.0009 days in the orbital period of 17.755 days is folded into the ephemeris. These results are consistent with the ellipticity effect as concluded by Herbst (1973), but my amplitude is only 89% of his reported value. The secondary period of 17.62 ± 0.05 days (DV = 0.035^m) is 0.77% longer than the orbital period and has not been previously reported.

6.34 IM Peg

Three signals are found for this system, with the last one appearing at the Nyquist harmonic value of one day (Figure 6-34). The primary period of 24.43 ± 0.02 days is 0.88% shorter than the orbital period. The secondary period of 12.22 ± 0.02 days has not been previously reported and is just at the error limits for the period expected for the ellipticity effect. The phase of minimum at 0.589^p ± 0.007 is appropriate if I allow for an error of 0.0015 days in the orbital period of 24.649 days, however, further analysis shows this period to be variable.

Analyzing the two large data sections independently indicates no change in period or amplitude for the primary signal, but there was a noticeable change in the filter phase of Df = 0.9 radians. The secondary period shows a clear increase from 12.18 ± 0.08 days to 12.7 ± 0.1 days, which rules out the possibility of the ellipticity effect. There is also a clear decrease in amplitude from 0.050^m to 0.026^m at V-band, which is consistent with the observations of Eaton et al. (1983b). The errors in the phases are too large for any change to be evaluated.

6.35 l And

Four signals are found for this system. The third signal is at the Nyquist harmonic value, and I suspect the last signal is a residual as a result of changes in the photometric properties of the light curve over the period of observation (Figure 6-35).

The primary period of 53.4 ± 0.2 days is in agreement with the general 54 day period that characterizes this system. The secondary period of 26.18 ± 0.08 days, which is 27.6% longer than the orbital value, has never been explicitly reported. The presence of the secondary is consistent with the canonical, two-component starspot model, which is known to describe the light curve.

This data set is separable, and I find that the primary period changed from 59 ± 5 days to 49 ± 3 days with an increase in full amplitude from 0.036^m to 0.074^m at V-band. The secondary period shows no apparent variation, but there was a visible increase in the full V-band amplitude from 0.025^m to 0.036^m. The fourth period found for the entire data set is not visible in the separated data groups, which confirms my suspicion that it is indeed a residual. The Nyquist value is still present in the separated data sets. The poor phases and large errors are a result of the long photometric periods and short observing intervals.

6.36 II Peg

Two clear periods are identified for this system (Figure 6-36) with only 43 observations over 113.7577 days, although aliasing forced the use of the orbital period for the selection of the signals. The primary period of 6.75 ± 0.02 days (DV = 0.148^m) is 0.36% longer than the orbital period and is consistent with the results of other authors. The secondary period of 3.351 ± 0.005 days (DV = 0.072^m) is 0.32% shorter than half the orbital period and has a phase of minimum at 0.38^p ± 0.01. The presence of this period is consistent with the double peak light curve observed by Nations and Ramsey (1981), Wacker and Guinan (1986) and Boyd et al. (1987).

Figure 6-1
V-band Periodogram for 33 Psc

This periodogram shows a clear lack of structure. The periodograms for U and B-bands are similar in appearance as are the DCDFT convolution functions.

Figure 6-2
V-band Periodogram for 13 Cet (A)

This periodogram shows a clear lack of structure. The periodograms for U and B-bands are similar in appearance as are the DCDFT convolution functions.

Figure 6-3
V-band Periodograms for z And

The two periodograms are plotted on the same time scale so the lack of visibility of the secondary signal in the primary periodogram can be seen. The secondary signal is noisy, and at a considerably lower level than the primary signal, but it undeniably present.

Figure 6-4
V-band Periodogram for AY Cet

This poorly formed periodogram shows a clear indication of the signal. The malformation is a result of the large variation in amplitude and mean magnitude between the observing seasons (see Figure 4-B).

Figure 6-5
V-band Periodogram for AR Psc

This poorly formed periodogram shows a clear indication of the signal. The malformation is a result of the variation in amplitude between the observing seasons and the poor coverage (see Figure 5-B).

Figure 6-6
V-band Periodograms for TZ Tri

The two low level signals are just visible in the two V-band periodograms. The presence of the secondary signal is barely visible in the unfiltered data. The low power levels are most likely a result of the poor observational coverage (Figure 6-B). The periodograms share the same time scale so the presence of the secondary signal is more easily identified.

Figure 6-7
V-band Periodograms for UX Ari vThe two V-band periodograms are text book examples of the analysis methods used in this dissertation. The two signals are clearly visible, with the secondary signal becoming visible only after filtering. The periodograms share the same time scale so the dramatic appearance of the secondary signal is more easily seen.

Figure 6-8
V-band Periodograms for V711 Tau

The primary signal of V711 Tau is unmistakable, but the secondary only appears after filtering. The secondary signal is at a considerably lower level than the primary and can only be positively identified with the use of all three colors.

Figure 6-9
V-band Periodogram and DCDFT Convolution for HD 22403

The uppermost figure is a plot of the V-band periodogram for HD 22403. The periodic structure is not immediately obvious. The lower figure is the convolution of the data-compensated discrete Fourier transforms for the U, B and V-band data. The plot is scaled to a full amplitude of one. The periodic structure, which is shared by the three colors, is now clearly visible, along with an alias.

Figure 6-10
V-band Periodogram for EI Eri

This periodogram shows two distinct, but illformed peaks, one of which is an alias. The malformations are a result of period variations between the observing seasons which result in a collection of dependent subsidiary frequencies after filtering.

Figure 6-11
V-band Periodogram for RZ Eri

This periodogram shows the lack of structure visible in this system as a result of severe undersampling. The eclipse is barely detectable in the peak near 0.3.

Figure 6-12
V-band Periodograms for BM Cam

Two periods are clearly identified for this system. The periodograms show a noticeable lack of symmetry which is most likely a result of variations in the period, amplitude or phase of the signals. Unfortunately, the data are not separable.

Figure 6-13
V-band Periodograms for HD 37824

The periodograms clearly identify the secondary signal after filtering, although a large amount of residual power remains from the primary. The poor shape of the curves is a result of period, amplitude, and phase variations in the signals.

Figure 6-14
V-band Periodograms for s Gem

The secondary signal is visible in the periodogram of the unfiltered data, although it would most likely have been labeled an alias. Filtering clearly indicates the independence of the signals. The figures are plotted on the same time scale so the correlations can be more easily observed.

Figure 6-15
V-band Periodogram for AE Lyn

This broad band periodogram shows the lack of visible structure for this system. The U and B-band periodograms show a similar lack of structure. A convolution of the DCDFT results indicate a weak signal at the Nyquist harmonic.

Figure 6-16
V-band Periodograms for 53 UMa

This pair of periodograms illustrates the loss of the visible signal at the Nyquist harmonic when only the first half of the data is processed. The analysis of the second data group gives similar results. This dramatic, time independent, change in structure is most likely a result of the presence of a period slightly longer than the fundamental value for this data set.

Figure 6-17
V-band Periodograms for DQ Leo

This pair of periodograms illustrates the dramatic variation of the periodic structure of DQ Leo from the first to the second observing season. The signal is clearly present in the second data set and not present in the first.

Figure 6-18
V-band Periodograms for DK Dra

The periodograms show the presence of the primary and secondary signals as well as the high noise levels. No clear changes in the periods are observed, but the errors in the analysis of the separated data sets is large.

Figure 6-19
V-band Periodograms for BH CVn

This pair of periodograms illustrates the loss of the visible signal at the Nyquist harmonic when only the first half of the data are processed, as is also the case for 53 UMa. The analysis of the second data group gives similar results. This dramatic, time independent, change in structure is most likely a result of the presence of a period slightly longer than the fundamental value for this data set.

Figure 6-20
V-band Periodograms for HD 136901

The periodograms clearly identify the three signals found for this system. The first two periods are most likely a result of the ellipticity and reflection effects. The secondary and tertiary periods have not been previously identified.

Figure 6-21
V-band Periodogram for TZ CrB

This periodogram shows the single distinct signal found for this system. The strong ragged structure is the result of amplitude variations of the signal.

Figure 6-22
V-band Periodogram for HR 6469

This periodogram illustrates the large amount of structure that appears when analyzing eclipsing systems. The primary period is not readily identified, but the combined information of all three colors unambiguously locates the signal at half the orbital period with a minimum at the time of conjunction.

Figure 6-23
V-band Periodogram for DR Dra

The periodogram shows the high level of sidelobe structure in the signal for this system. Such structure is usually associated with changes in the photometric characteristics of the system, unfortunately, the data set cannot be separated.

Figure 6-24
V-band Periodogram for Z Her

This periodogram clearly identifies the primary signal and its alias. The peak near the orbital period (w » 1.57) is taken to be real.

Figure 6-25
V-band Periodograms for V815 Her

The periodograms clearly show the presence of the primary and secondary signals, but the noise level is seen to be high.

Figure 6-26
V-band Periodograms for 47 Dra

The periodograms are for the corrected data set. The figures clearly show the poor structure of the periodograms and the poor filtering of the primary signal. It is only with a combination of the three colors can a distinction be made between the aliases. The poor quality of the periodogram for the filtered data set indicates that the values for this signal must be viewed with caution.

Figure 6-27
V-band Periodogram for V478 Lyr

This periodogram clearly identifies the primary signal and its alias. The peak near the orbital period (w » 2.7) is taken to be real.

Figure 6-28
V-band Periodograms for HR 7428

The primary and secondary signals are clearly visible in the periodograms, but the shapes of the peaks are poor as a result of the low number of datum as a function of the orbital period.

Figure 6-29
V-band Periodograms for V1764 Cyg

The signals for this system are both well defined, but they are very close the fundamental frequency of the data set. Considering the strength of the signals, a longer term set of observations would greatly reduce the errors in the period determinations.

Figure 6-30
V-band Periodograms for ER Vul

The periodograms illustrate the severe aliasing problem for this system. The aliased signals are a result of sampling this system (P_orb» 0.7^d) at intervals near one day. The primary signal is identified by its proximity to the orbital period (w » 9), but the large class of plausible aliased signals for the filtered data makes any absolute identification impossible.

Figure 6-31
V-band Periodograms for HK Lac

The two signals for HK Lac are clearly identified, but the secondary signal has a lot of sidelobe structure as a result of variations in the photometric properties of the light curve.

Figure 6-32
V-band Periodograms for AR Lac

Three periodograms are shown for AR Lac. The topmost figure shows the broad band characteristics of the periodogram with the secondary signal already visible prior to filtering. The remaining two figures are the periodograms for the primary and secondary signals, respectively.

Figure 6-33
V-band Periodograms for V350 Lac

Two signals are clearly present in the periodograms. The ragged structure would indicate changes in the photometric properties of the light curve, but the data cannot be separated.

Figure 6-34
V-band Periodograms for IM Peg

Two signals are clearly present in the periodograms. It is clear that the primary signal filters well and that the secondary signal would normally have gone undetected.

Figure 6-35
V-band Periodograms for l And

The periodogram for this system shows a large amount of sidelobe structure, but the presence of the two periods is clear. The high sidelobe levels are a result of the dramatic changes is the period of the primary signal.

Figure 6-36
V-band Periodograms for II Peg

Two periods are clearly visible in the periodograms. It is obvious that the secondary signal can not be identified in the presence of the primary.

Table 6-1
Tabulation of Data Characteristics

This table summarizes the general information of the data base. Column two is the number of points in the data file. Columns three and four are the starting Julian Date of the observations and the total time span, in days, of the data. Column five is the expected noise peak level in the periodogram as determined from equation 2-8 and the last column is twice the pseudo-Nyquist angular frequency as determined from the separation of the peaks in the periodograms when the data are replaced with a high frequency sine wave [sin(20t)].

Table 6-2
Source Ephemeris Table

This table lists the times of conjunction and orbital periods for each of the 36 sources. With the exceptions of HR 6469, DR Dra and V478 Lyr, which were discussed in Chapter 5, all values are taken from the Catalog of Chromospherically Active Binary Stars (Strassmeier et al. 1988). These ephemerides are used in determining the phase of minima for the extracted periodic signals.

Table 6-3
Photometric Periods and Phase of Minima

This table summarizes the photometric periods and phase of minima for each of the sources averaged over the three colors. Only those signals that are considered real are listed here. The second and third columns are the list of photometric periods and their associated errors. The primary, secondary and tertiary periods are listed in order. If the data set is separable, then the periods found for the individual data groups are listed. The phase of minimum, for each period, and the associated error are listed in the next two columns. The last column indicates what part of the data were analyzed and any other special notes in reference to the primary text of this chapter.

Table 6-4
UBV Full Amplitudes, Phases and False Alarm Probabilities

This table summarizes the UBV full amplitudes A, filter phases f and false alarm probabilities Á. The full amplitudes and phases correspond to a filtered signal of the form (A/2)cos(wt - f) where zero time is the first element of the full data set. False alarm probabilities <10^-13 are listed as zero. Only those signals that are considered real are listed here. The information for the primary, secondary and tertiary periods are listed in order. If the data set is separable, then the values found for the individual data groups are listed. The structure of this table is the same as Table 6-3, which should be referenced for the periods, phase of minima and associated comments.