2.1. Data Acquisition and Reduction at Tonantzintla

Three 10-inch Meade telescopes were utilized at the TNT Observatory and are denoted by m1, m2 and me. These telescopes were equipped with CCD cameras and the observations were done using V and G filters. During all the observational nights this procedure was followed: the integration time was 1 min and there were around 11,000 counts, enough to secure high precision. The reduction work was done with AstroImageJ (^{Collins & Kielkopf (2012)}). This software is relatively easy to use and, besides being free, it works satisfactorily on the most common computing platforms. With the CCD photometry two reference stars were utilized whenever possible in a differential photometry mode. The results were obtained from the difference Vvariable −Vreference and the scatter calculated from the difference V _reference1− V _reference2. Light curves were also obtained. In order to calculate the times of maximum light a program that considers the derivative function was used. A fifth polynomial was adjusted between the selected points in order to enclose the point of maximum light. On applying the derivative, as a criterion, the roots were determined in such a way that an f adjustment function was found, so that in the open selected interval, the derivative equal to zero was a minimum (if it was convex) or maximum (if it was concave) relative to f. The new times of maximum light are listed in Table 1. In this table, in Column 1 the date of observation is listed, in Column 2 the observers/reducers; in Column 3 the obtained number of points during the night, and in Column 4 the time span of the observations. Column 5 specifies the number of maxima for that night; Column 6 the time of maximum in HJD-2400000; and Column 7 the telescope utilized (m1, m2 and me stand for small 10” Meade telescopes, whereas 84 describes the telescope at the SPM observatory); Column 8 describes the filter utilized (V in the UBV system and G for the GRB filter system, whereas in the uvby−β we utilized only the y filter transformed into ^{Olsen’s (1983)} photoelectric photometry system. Finally, the last two Columns 9 and 10 specify the CCD camera and the observatories, Tonantzintla (TNT) and San Pedro Mártir (SPM).

2.2. Data Acquisition and Reduction at SPM

As was stated in Peña et al. (2016) reporting on BO Lyn, the observational pattern as well as the reduction procedure have been employed at the SPM Observatory since 1986 and, hence, have been described many times. A detailed description of the methodology can be found in ^{Peña et al. (2007)}.

The star AD CMi was observed in uvby −β photometry over various seasons: November, 2013; January, December, 2016 and 2017. The first one covered six nights, whereas the second one three nights in January and seven in December, 2016. AD CMi was observed on only one night in the 2017 December season.

Over the nights of observation of the AD CMi star the following procedure was used: Each measurement consisted of at least five ten-second integrations of the star and three ten-second integrations of the sky simultaneously for the uvby filters, and almost simultaneously for the narrow and wide filters that define Hβ.

The details of the second season are described in ^{Peña et al. (2019)}, whereas for the 2013 season the transformation coefficients of that night are listed in Table 2. In the 2016 season three nights were lost due to bad weather.

What must be emphasized here are the transformation coefficients for the observed seasons (Table 2) and the season errors which were evaluated using the observed standard stars. These uncertainties were calculated through the differences in magnitude and colors for (V , b−y, m ₁, c ₁ and Hβ) which are (0.0870,

0.0065, 0.0085, 0.0212, 0.0179); (0.054, 0.012, 0.019, 0.025, 0.012) and (0.032, 0.010,0.011,0.013, 0.012) for the 2013 and both the 2016 and 2017 seasons, respectively, for a total of 94 points in uvby and 68 points in Hβ, which provide a numerical evaluation of our uncertainties. We must call attention to the bad quality of the December 2016 season. However, since the effects of the clouds compensate each other, the color indexes gave reasonable values, whereas the V showed significant larger scatter and, will not be taken into account. The obtained data will be sent to the archives and are available upon request.

3.1. Time Series

As a first method to determine the period, we used a time series method which is commonly utilized in the δ Scuti star community: Period04 (^{Lenz & Breger, 2005}). There are several data sets which can be analyzed with this code:

The analysis of Period04 is presented in Figure 1. At the top is the periodogram of the original data and at the bottom, that of the residuals after the produced frequency was subtracted. Table 4 summarizes these results (set 3).

Fig. 1 The analysis of Period04 is presented. Descending, the periodogram of the window function and subsequently each maxima is presented. At the bottom, the residuals after the three frequencies have been subtracted. Emphasis should be made on the Y-axis scale.  

The results of utilizing Period04 gave as the first frequency 8.13830066c/d with an amplitude of 0.1409 and a phase of 0.1759. Evaluating the residuals we obtained the following numerical values: 16.254, 0.02898 and 0.4234 respectively. Fitting the data to these values gave as residuals a frequency of 0.0025c/d with an amplitude of 0.2881 and a phase of 0.1689. These values correspond to a period of 0.12287577d for the first frequency and to 0.061523d. Given the numerical frequency values the results of Period04 gave f and 2f, or a value close to it (16.276c/d). When a phasevs.magnitude plot was constructed, the phasing shows the correctness of the determined period (Figure 2).

Fig. 2 Top: Phase diagram after the analysis of Period04 using frequency F1 set 3 of Table 4. Bottom: The residuals after the three frequencies have been subtracted.  

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

We employed another method based on the concept of the minimization of the chord length linking all the points in the O−C diagram for different values of changing periods, looking for the best period which corresponds to the minimum chord length (see the case of KZ Hya, ^{Peña et al. 2018}).

A set of 94 times of maxima was considered for this analysis. Taking this into consideration, we used the determined period value in § 3.1 and the standard deviation of those consecutive differences to keep an interval span in which the period is located; those values are 0.1230±0.0002d. Maintaining a period precision of a billionth and taking the interval span period into consideration, 500,000 periods were used to perform this method. The T ₀ used to calculate the O−C diagrams was 2458559.7805. Then the best period is the one with the smallest chord length and it is shown in Figure 3.

Fig. 3 Period determination through an O−C difference minimization. 

The ephemerides equation is then:

As we can see in Figure 4, the plot shows a slope and marginally a sinusoidal behavior which can be related to the light-travel time effect (LTTE), revealing the possible presence of a second body. This visualization is later confirmed (Figure 5).

Fig. 4 O−C residuals after the adjustment to the obtained period. 

Fig. 5 O−C diagram with the sinusoidal function fit obtained by the PDDM method.  

Then, we adjusted a sinusoidal function to the O−C diagram performing a fit using LevenbergMarquardt Algorithm. We then used a combination of the chord length minimization and the non-linear fit for a sinusoidal function, taking the periods with the minimum chord length and, as a second criterion, the RMS error of the non-linear fitting.

The fitting equation for the O−C diagram is

and the continuous line in Figure 6 shows the sinusoidal fit over the O−C diagram with a new period.

The sinusoidal function parameters are listed in Table 5. Accordingly, the period of the sinusoidal behavior is 10,247.8759 days or 28.1 years.

Fig. 6 Upper: O-C diagram and fit to the equation 4. Lower: residual of the fit. 

The ephemerides equation determined by this method is:

The goodness of the model is determined by the RSS residuals listed in Table 3.

3.3. O−C and Mass Determination

Before calculating the coefficients of the ephemerides equation, we reviewed the existing literature related to AD CMi. Several authors have conducted studies on the O−C behavior of these particular objects and, in this preliminary stage, we took the existing updated list of times of maxima from the literature, plus the data that we acquired. We built the O−C diagram with the obtained period values from Period04 (set 3).

The O−C diagram (Figure 6) shows a variation and a cyclic pattern with no quadratic component. Hence our solution is of the form:

where τ is the equation given by Irvin (1952)

The fit (Figure 6) was done by least squares with the Levenberg-Marquard algorithm, using the relation provided by Li & Qian (2010) in their equation 3:

Where A'=Asin⁡ic is the projection of the semi-major axis and c is the speed of light. Kepler’s equation provides the relationship between the mean anomaly (M) and the anomalous eccentricity (E ⁺)

With

where e, ω, P _orb , T, P and E are: the eccentricity, the periastron longitude, the orbital period, the time of periastron passage, the period of pulsation of AD CMi (P = 0.122974518) and the epoch, respectively.

Since equation 5 is trascendental, the solutions for E ^∗ are obtained through Bessel series

where

The obtained orbital parameters with the fit are presented in Table 6 and have been used to build the plot of Figure 6.

The mass function is given by Fu et al. (2008) with P _orb = 32.33 (yr).

To calculate the mass of the companion, the solution of the following function was evaluated for m ₂:

for a mass of m1=1.93M⊙ obtained by ^{McNamara (1997)}. A mass was calculated for several inclination angles as shown in Table 7, where the minimum mass that can be obtained for the companion star is 0. 0.12M⊙ (Figure 7).

Fig. 7 Mass as a function of angle.  

Pongsak ^{Khokhuntod et al. (2007)} stated that the binary orbit has a period of 27.2 ± 0.5 years, with an eccentricity of 0.8 ± 0.1. However, since there is a big gap between the first 12 data points and the rest, and the bulk of the data spans only one cycle, this result should not be considered to be an accurate solution of the orbital parameters.

However, this is a plausible solution because it diminishes the dispersion, as can be see in the residuals in Figure 6. From the data shown in Table 7 and represented in Figure 7 we can consider the mass determination of the companion as a function of this angle since we cannot determine the inclination angle. If we do so, the probability that the companion is an M type star is 75%. The minimum mass is discarded because in our light curves there is no evidence of eclipses. Of course, more observations are needed to corroborate this result

3.4. Period Determination Conclusions

Despite having been determined to be a variable star many years ago, the nature of this star has been variously described, from a binary star to a pulsating star, as well as an RR Lyrae and, finally a δ Scuti star. Since then, more information has been gathered but no period analysis was done. In the present paper, different approaches were utilized to determine the stability of the pulsation. With respect to its period determination and interpretation, it has varied from a periodic star to a quadratic fit which should be interpreted as a changing period.

The first method used to determine the periodic content utilized a time series analysis of different data sets. The first set is that of the V filter of the photometry of ^{Rodriguez et al. (1988)} and that of the V filter of the uvby − β photometry of the present paper. The time span covered for all the data is 17084d or 47 years, which in cycles is 138926. The second method we used was PDDM which gives a first approximation to the period of a binary system. Finally, in the third method we utilized O−C analyses via the LTTE.

The amazing result is that the uvby−β photoelectric photometry adjusts to a phase both in magnitude and color indexes with the properly determined frequency.

In conclusion, there was a systematic improvement of the period, but not only that: Yang et al. (1992) postulated that the quadratic fit was much better, so the period was considered to be increasing at a rate of (1.3 ± 0.07 × 10⁻⁸ days/year) although they could not fit the curve well with the observations from 1985 and 1986.

They suggested that more observations were urgently needed to clarify the true situation of the period variation. It is just now that these observations and analysis have been done with congruent results.

Data from other independent sources, such as the Kepler field or TESS, would have been very useful but, unfortunatelly, the star was not observed by either.

4.1. Calibration Utilizing all the Available uvby − β Observations

To diminish the large scatter in the magnitude and photometric indexes, we averaged the entire set with available uvby−β photometry: SPM over three seasons: 2013, 2016 and 2017 (the data will be sent upon request), along with those of ^{Epstein et al. (1973)} and of ^{Rodriguez et al. (1988,)} covering a time span of 12314 days, or 100135 cycles in phase bins of 0.05 intervals, utilizing the epoch of ^{Kilambi & Rahman (1993)} and the frequency determined in this paper. Given the large number of points in the whole sample (509) we cleaned the data deleting points that were notoriously out of the trend. The reduced sample includes 333 data points.

The averaged points are presented in Table (8), (Column 1 phase; Column 2 magnitude; subsequent columns list (b − y), m ₁, c ₁, next column presents values of Hβ). The standard deviation for each bin is presented in the subsequent columns. The last column presents the number of entries on average.

We applied Nissen’s prescription to the averaged values and determined the reddening, the unreddened indexes (b − y)₀, m ₀,c ₀, as well as the absolute magnitude, and distance for each phase point. Mean values were calculated for E(b − y) for two cases: (i) the whole data sample and (ii) in phase limits between 0.3 and 0.8, which is customary for pulsating stars to avoid the maximum. It gave, for the whole cycle, values of 0.010 ± 0.010; whereas for the previously mentioned phase limits we obtained, 0.010 ± 0.005. The distance, for the same limits, was 370 ± 38 pc and 369 ± 36 pc, respectively. The uncertainty is merely the standard deviation. The procedure to determine the physical parameters has been reported elsewhere (Peña et al., 2016). The metallicity values are the same in both cases [Fe/H] of 0.13 ± 0.013.

The physical parameters determination was performed by ^{Rodriguez et al. (1988)} through uvby −β photoelectric photometry. We have utilized the same procedure (see for example ^{Peña & Martinez; 2014}; ^{Peña & Sareyan, 2006}).

They derived an E(b-y) of 0.017 ± 0.007 and T _e (K) of 7550 ± 190 and logg (dex) of 3.83 ± 0.06. The values we obtained are presented below and the results are equal, since the same uvby − β photometry was considered and the same analysis was done, though thirty years later with an extended time basis. A comparison between the photometric, the unreddened indexes c ₀ and (b−y)₀ obtained for the star with the models allowed us to determine the effective temperature T _e and the surface gravity logg (Figure 10). Table 9 lists these values. Column 1 shows the phase, Columns 2 the reddening; Columns 3 to 5 the unreddened indexes and β. Figure 10 shows the position of the unreddened color indexes of AD CMi in the theoretical models of LGK86. Phase is indicated as integers. Subsequent columns present the unreddened magnitude, the absolute magnitude, the distance modulus, the distance in parsecs and the effective temperature from the LGK86 plot. Column 12 lists the effective temperature obtained from the theoretical relation reported by Rodriguez (1989) based on a relation from ^{Petersen & Jorgensen (1972}, hereinafter P&J72) Te = 6850+1250×(β−2.684)/0.144 for each value and averaged in the corresponding phase bin. In the final column the surface gravity logg from the plot is presented.

Fig. 10 Position of the unreddened color indexes of AD CMi in the theoretical models of LGK86. Phase is indicated in integers. 

4.2. Physical Parameters. Conclusions

One of the major contributions of the present paper is the analysis of uvby −β photoelectric photometry with an extended time basis. From the uvby−β photometry we have obtained basically the same results as ^{Rodriguez et al. (1988)} but with a time span extended thirty years. This has proven the stability of the star.