2.1. Data Acquisition and Reduction

The observations were done in the summer of 2016, along with those of other variable stars and clusters. The procedure to determine the physical parameters has been reported elsewhere (^{Peña et al., 2016}). The photometric system, if well-defined and calibrated, provides an efficient method to investigate physical conditions, such as effective temperature and surface gravity, using a direct comparison of the unreddened indexes with those obtained from theoretical models. These calibrations have been described and used in previous analyses (^{Peña & Peniche; 1994}; ^{Peña & Sareyan, 2006}).

The observations consisted of one long season with two different observers (ARL and CVR), one in June (22 to 30 by ARL) and the other in July, 2016 (1 to 8 by CVR) with different target objects in each one, although two were obtained in both sea-sons (NGC 6633 (Paper I) and V2455 Cyg).

The reduction was done considering both seasons together to make a longer season in order to increase the accuracy provided by the standard stars. Over the five nights of observation of V 2455 Cyg, the following procedure was used: for each measurement we took at least five ten-second integrations of each star and one ten-second integration of the sky for the uvby filters and the narrow and wide filters that define Hβ.

We also observed a series of standard stars nightly to transform the data into the standard system. The chosen system was that defined by the reported values of ^{Olsen (1983)} although some standard bright stars were chosen from the ^{Astronomical Almanac (2006)}. The transformation equations are those defined by Crawford & Barnes (1970) and by ^{Crawford & Mander (1966)}.

The coefficients defined by the following equations and which adjusted the data to the standard system were:

In these equations the coefficients D, F, H and are the slope coefficients for (b − y), m ₁, c ₁ and β, respectively. The coefficients B, J and I are the color terms of V, m ₁, and c ₁. The averaged transformation coefficients of the season are listed in Table 2 along with their standard deviations. Season errors were evaluated with the eighteen standard stars observed for a total of 272 observed points. These un-certainties were calculated through the differences in magnitude and colors for all nights, for (V , b − y, m ₁, c ₁ and β) as (0.024, 0.010, 0.011, 0.015, 0.015) respectively, which provide a numerical evaluation of our uncertainties for the season. Emphasis is made on the large range of standard stars in the magnitude and color ranges: V: (5.0, 8.8); (b − y):(-0.06, 0.80); m ₁:(0.10, 0.68); c ₁:(0.11, 1.18) and β:(2.60, 2.82).

Table 3 lists a sample of the photometric values of the observed star. The complete data set will be published elsewhere. In this table Columns 1 to 4 list the Strömgren values V, (b − y), m₁ and c₁, respectively; Column 5, Hβ, whereas Column 6 reports the time of the observation in HJD. The photometry is presented in Figure 1.

Fig. 1 Phase light curves of V 2455 Cyg obtained in uvby − β absolute photometry in 2016. The magnitude and the color indexes are shown in each frame.  

3.1. Differences of Consecutive Times of Maximum Light, DCTM

To determine the period from scratch, as a first guess we considered the period calculated through the differences of two or three times of maxima that were observed on the same night. Since they are separated by only one cycle they are, by definition, one period apart. The sample of periods determined in this fashion is constituted of 44 times of maxima difference and was compiled from both those listed in the literature and the few that we measured two years later (Table 5). There were fourteen consecutive times of maxima; the mean value was 0.0944 ± 0.0006 (d). The uncertainty is merely the standard deviation of the mean.

This period served as seed and was utilized with T ₀, the first time of maximum of ^{Wils et al. (2003)}, and the epochs in the list of times of maximum light, Table 5, to calculate new ephemerides through a linear regression of epoch vs. HJD, thus determining refined values of T ₀ and P . The period we established in this fashion (0.0944 d) is in agreement with the assumed period reported in the literature. The results of this method are presented in the second row of Table 4.

3.2. Time Series

As a second method to determine the period we used a time series method favored by the δ Scuti star community: Period04 (^{Lenz & Breger, 2005}). The V magnitude of the uvby − β set was analyzed with this code.

The analysis of these data gave the results listed in Table 6 with a zero point of 8.8184 mag, residuals of 0.0178 mag and 5 iterations. The analysis of Period04 is presented in Figure 2. Beginning at the top is the power spectra of the original data; followed by the consecutive set of residuals. The scale of the axis is the same to show the relative importance of the residuals. The frequencies obtained from this analysis are presented in Table 6. Three different frequency values are reported, but one must realize that F 2 = 2 F 1 and F 3 = 3 F 1, that the second and third frequencies are multiples of the first.

Fig. 2 Power spectra of V 2455 Cyg with the SPM data. Top to bottom: first is the power spectra of the original data; then, the sets of residuals. The scale of the Y axis is the same to show the relative importance of each frequency. 

3.3. O − C differences

Before calculating the coefficients of the ephemeris equation, we searched the literature for previous papers related to V 2455 Cyg. Only one source conducted studies of the O − C behavior of this particular object (^{Wils et al., 2003}). The reported ephemerides are listed in Table 4; the newly observed ephemerides are also presented.

3.4. Period Determination Through Minimization of the Standard Deviation of the O − C Residuals (MSDR)

The well-known O − C diagram method is a tool utilized to compare and analyze the difference between the observations and the calculated value obtained with the model; in this case, the ephemerides equation. A good reference for this method can be found in ^{Sterken (2005)}.

To find the ephemerides equation of the variability of V2455 Cyg we implemented a method based on the minimization of the standard deviation of multiple O − C diagrams for V2455 Cyg (see ^{Peña et al., 2016} for details). This method is based on the minimization of the standard deviation of the O−C residuals by considering that the model must be fairly close to the data. Therefore, its differences tend to zero and the standard deviation has to be close to zero. The first step was to calculate the mean of the differences of all the consecutive times of maxima available in the literature (first method) and its standard deviation. All these differences of consecutive times of maxima have the approximate length of the period of the star. Then, with each of the periods given in the range provided by the mean of the consecutive, plus and minus its standard deviation, the number of cycles E was calculated for these multiple O − C diagrams. In all cases, the T ₀ utilized was the oldest time of maxima in the literature. The step precision in this range was of 1 × 10^-9. After obtaining the number of cycles for each period, a linear fit was performed to every set of times of maximum with the cycle number E (HJD vs. E). With the new fitted parameters, the diagrams were calculated as well as their standard deviation. Finally, the diagram with the smallest standard deviation of the residuals was selected and the parameters with which it was calculated are taken to be the parameters of the ephemerides equation.

Sweeping around the average of the differences of consecutive times of maxima within a short period range, taking as limits the standard deviation of the differences around 0.0943052 d, and utilizing the minimization of the standard deviation of the O − C values as criterion of goodness, the best period is 0.094205989 d (Figure 3). The resulting ephemerides equation is

Even with the maximum resolution (1 × 10^-9) in the MSDR method, there was some scatter around the minimum in the Standard Deviation vs. Period plot (Figure 3). To clarify this, a binning procedure was implemented. For this binning the size of the intervals was 1 × 10^-8. After this operation the minimum value of the standard deviation was checked. This was 0.094205989, which is the same as the original before the binning procedure. Continuing with the analysis, two new binning procedures were implemented, but this time with intervals of 1 × 10^-7 and 1 × 10^-6. In the first new case, the period for the minimum standard deviation remained the same, but in the final case the period with the minimum standard deviation was 0.094206015. The value of the period changes only after the size of the intervals for the binning is three orders of magnitude larger than the original precision.

Fig. 3 Standard deviation vs. period. This diagram served to determine the best period. In the upper panel the original data that served to determine the best period are shown. In the lower panel the data have been binned with a window of 1 × 10−⁶. 

With these elements we calculated the O − C residuals which are shown schematically in Figure 4. They are clearly fitted by straight line, which implies a constant period (at least at this stage) with a time basis of 12.8 yrs.

Fig. 4 Behavior of the O − C values with the proposed ephemerides determined utilizing the method of Minimization of the Standard Deviation of the O − C Residuals (MSDR) 

3.5. Period Determination Through an O − C Differences Minimization (PDDM)

We implemented a method based on the idea of searching for the period which minimizes the chord length which links all the points in the O−C diagram for different values of the periods. Being a classical O−C diagram, we plotted the time in the x-axis and the O − C values in the y-axis. Since in the x-axis distances remain constant, we just concentrated on the change in the distance in the y-axis generated by each period. Once the difference was calculated for each period, the minimum one indicated, at this stage, the best period (period determination through an O − C differences minimization PDDM).

We considered the set of T_MAX listed in Table 5. Given the mean period determined from the consecutive times of maxima and the associated standard deviation, (0.0938 days and 0.0944 days), we calcuated values of epoch and O − C by sweeping the period in the range provided by the standard deviation limits, calculating 595,956 steps, a number fixed by the the difference of the deviation limits and the desired precision of one billionth. This provided the new period for the minimum difference (Figure 5). The T₀ time used for the present analysis was the one of the last observation run, 2457575.9157, be-cause we are certain of its precision. As a result we determined the linear ephemerides equation as:

Fig. 5 Period Determination Through an O − C Differences Minimization (PDDM). 

Figure 6 shows the O − C diagram for the ephemerides equation found by the above method (PDDM).

Fig. 6 (O − C) diagram obtained with the period of the PDDM method. 

Assuming the wave behavior is caused by a light travel time effect (LTT) as a part of the mechanics of the system, we adjusted a sinusoidal function to the O − C, performing a fit with the Levenberg-Marquardt algorithm for the best 1,000 O − C lengths. This would be, in this particular case, an-other way of finding the best period (if the LTT effect is present) and, at the same time, the sinusoidal function that gave us a first approach to the orbital period of a two body system. The parameters which best represent the system are listed in Table 7. The parameter used to test the goodness of the fit is the residual sum of squares RSS. Then, we plotted the periods of the best O − C lengths vs the RSS value of every fit (Figure 7).

Fig. 7 Period vs RSS. The best value is indicated by a triangle. 

After this, the ephemeris equation was set as:

As can be seen, we do not have enough data to say more about the O − C diagram. The parameters presented in Table 7 are shown schematically in Figure 8.

Fig. 8 Diagram of the O − C adjusted to a sinusoidal function. The time span of the observations is of 4691 d or 12.8 years which is 49795 cycles. 

3.6. Period Determination Conclusions

V 2455 Cyg has been little observed. Since the first report more information has been gathered, but no period analysis has been done. In the present paper, different approaches were utilized to determine the stability of the pulsation. First, differences of consecutive times of maximum light were evaluated to determine a coarse period. The second method utilized time series analyses. The set employed was that of the V magnitude of the uvby −β photometry of the present paper. However, we had a very limited time coverage, only 10 d, merely 107 cycles. The third and fourth methods utilized the entire coverage of this star since its discovery, with the times of maxi-mum light, which in this case are 4691 d or 12.8 years (49795 cycles). The four methods used the available data up to now. The goodness of each method and that proposed previously were determined using the mean and the standard deviation of the residuals of the O − C values, the minimum chord length, and RSS (residual sum of squares) for the sinusoidal fit-ting of the O − C diagrams. They were obtained for each ephemeris listed in Table 4 and numerically they are listed in Columns 4, 5 and 6 of this table. It is quite evident that the MSDR method yields the smallest standard deviations for the O − C residuals, but if we consider the sinusoidal behavior as part of the mechanics of the system, the PDDM gives us the best ephemerides equation.

However, this star was discovered to be a variable very recently and has been observed too little to be able to make a solid conclusion since a large spread can be seen in a section of the O − C diagram (Figures 4 and 6). Those points belong to the observational group of ^{Hubscher (2015}, ^2015b). There are two possible explanations for the large spreads: an inaccurate determination of the times of maxima for those seasons or the possibility that this group is a set of observations acquired during one of the changes in the period inflicted by the presence of a second body which corresponds to the LTT effect mentioned in the PDDM method. We feel that more time must pass, with a more frequent sampling, before we can definitively arrive at conclusions about the pulsational nature of V 2455 Cyg, the LTT effect, and a possible secular variation.

4.1. Physical Parameters: Conclusions

New uvby −β photoelectric photometry observations were carried out for the HADS star V 2455 Cyg. From these observations we first determined its spectral type, which varies between A3V and F0V, different from that previously found. From ^{Nissen’s (1988)} calibrations the reddening was determined, as well as the unreddened indexes. This served to obtain the physical characteristics of the star: effective temperature in a range from 7200 K to 7900 K and log g from 3.6 to 3.9 with two methods: (1) from the location of the unreddened indexes in the LGK86 grids and (2) through the theoretical relation (P&J72).