3.1. O−C

Before calculating the coefficients of the ephemeris equation, we studied the existing literature related to BL Cam. Several authors have carried out studies of the O−C behavior of this particular object and developed models. When these models were constructed, they were built with the data available to them based on the length of their observation time. However, they often differ in the interpretation of the data.

^{Hintz et al. (1997)} presented a thorough study of this star concluding that BL Cam is a double-mode variable with a primary period of 0.0391 day, with evidence that the fundamental period had increased by 0.009 s in the previous 20 years. Their determined variation differs from that of ^{McNamara and Feltz (1978)} who proposed a linear variation. They found that the best fit to the data was given by equation (1), and concluded that the period had changed by 0.009 s in the last 20 years:

It was ^{Hintz et al. (1997)} who decided that this star had an increasing period and ^{Blake et al. (2000)} corroborated this assertion. Using their observations those authors proposed that the amplitude of the star’s light curve is modulated and that the physical cause may be tied to the fact that the star is known to exhibit the features of double-mode pulsation. However, works such as ^{Wolf et al. (2002)} could not confirm the results of Hintz et al. (2000) implying that the star had a constant increasing pulsational period. Their conclusion, as expected, was that this star deserved continuous monitoring. One year later ^{Kim et al. (2003)} found that the parabolic period variation had recently reversed.

In view of the discrepancies among the different authors, we did a follow-up of the literature that contained ephemerides equations with the data available at that epoch. Most of the ephemerides equations have been calculated utilizing equation 2, in which P is the period, in days; β is the rate of the variation; Z, is the zero point; B is the amplitude; Ω, the frequency and α, the phase. These ephemerides elements are presented in Table 2.

with A=12β.

In Table 3, we list the author and the year of publication in Column one, Columns two and three show the initial and final dates of the data they analyzed, Column four shows the time span they considered, in years; the next column shows the number of points in their analysis consisting of the points they obtained combined with those of the literature at that time; the last column lists the number of points each author observed. The behavior/ of their O−C analysis is shown schematically in Figure 1.

For our analysis we collected a total of 520 times of maximum light from the literature and our observations. To these data we added those of VizieR to get 1985 data points, some of which were duplicated. Removing these repeated values, we got a total of 1606 data points. The elapsed time of observations is from 2443125.8046 to 2458904.7320 for a time span of 15778 days, or forty-three years, a considerable length of time.

The starting point for our O−C analysis was that provided by ^{Zhou et al. (1999)}, who utilized the ephemerides listed in equation 1 proposed by ^{McNamara et al.(1978)}. As a first stage, we reproduced what ^{Zhou et al. (1999)} obtained in their O−C residuals with their 136 times of maximum light, in which we can see that the parabolic behavior interpretation was logically inferred (see Figure 2). It is important to mention that they report a change in rate of 7.1457×10-13day/cycle.

We have extended the time basis with the new values of Tmax determined in January and February, 2020 for a total of 1606 times of maximum light and a time basis of twenty three years elapsed since the ^{Zhou et al. (1999)} study, increasing the accuracy of the ephemerides elements, T0 and P given by ^{Zhou et al. (1999)}. We followed the prescription provided by ^{Zong et al. (2019)} who calculated the number of elapsed cycles adjusting a linear fit to the times of maximum number of cycles presented in Figure 3 in the XY plane with X, the number of cycles and Y the times of maximum light. The newly determined values for T0 and P are given in equation 3:

With these new determined values we calculated the new O−C values represented in Figure 4, analogous to those presented by ^{Zong et al. (2019)} in their Figure 5.

However, they arbitrarily discarded all O−C data points with E less than 150000 cycles because they showed an unexplained behaviour. Given the analogous results we obtained from ^{Zong et al. (2019)}, we could have proceeded in the same fashion but, we preferred to consider all the O−C elements in the ephemerides procedure in ^{Zhou et al. (1999)}.

^{Zong et al. (2019)} carried out an exhaustive analysis of the O−C residuals. Their origin was based on the work of ^{Fu et al. (2008)} who suggested that the main period of BL Cam might have undergone an abrupt change. This change was studied by ^{Fauvaud et al. (2010)} and by ^{Conidis & Delaney (2013)} who suggested that the change of the main period might be caused by a third body explained by a cubic curve adjustement shown in their Figure 5, which is presented here in Figure 4.

However, the model proposed by ^{Zhou et al. (1999)} shows that the main frequency derived in the O−C analysis is quite consistent with the fundamental frequency decomposed by Fourier transforms which explains all the times of maxima. Following this approach, we utilized ^{Zhou et al.’s (1999)} equation and considered the number of cycles and the O−C for the extended time basis all the elements, including those below an E of 150000 which ^{Zong et al. (2019)} had discarded. The smooth variation presented in Figure 5 was obtained if a sinusoid was assumed. If we only take the time span employed in ^{Zhou et al (1999)} into account, the logical deduction would be a parabolic behavior. This parabolic assumption was also assumed by ^{Fauvaud et al. (2006)} although their time basis was not long enough to reach a different conclusion, but when the elapsed time was extended, a sinusoidal variation can be seen.

Due to the fact that a sinusoid can be considered, in Figure 5 we calculated the variation parameters through a fit obtained with Period04, a canonical procedure utilized for short period variable stars (^{Lenz & Breger, 2005}). Period04 is a computer program especially dedicated to the statistical analysis of large astronomical time series containing gaps. The program offers tools to extract the individual frequencies from the multiperiodic content of time series and provides a flexible interface to perform multiple-frequency fits. The Fourier analysis in Period04 is based on a discrete Fourier transform algorithm.

This fit to a sinusoid is shown in the Figure 5 by the continuous line, which explains those points discarded by ^{Zong et al. (2019)} and the newly observed points in 2020. The output of Period04 is presented by the following equation 4, with the numerical values presented in Table 4. It is shown schematically in Figure 6.

With an extended time span it becomes evident that the conclusion reached previously was partial because it was obtained based on the limited time span at that time. Now, we postulate a sinusoidal behavior that could be explained by the presence of another unseen companion star and the light travel time effect (LTTE). Due to the above, we adjusted the equation:

to determine the optimized parameters for the equation to get the best fit to the data, we utilized the algorithm of Levenberg-Marquardt. As initial parameters we considered a ₀ equal to 2Z and w = 2πΩ₁. To the remaining coefficients we assigned zero values. The obtained optimized parameters and their standard deviations are presented in Table 5 for a frequency of w = (122 ± 2) × 10⁻⁷ (1/d). This new analysis provides an orbital period of the star with a result of P _orb = 55.2 years. This postulation will be tested with continuous monitoring of this star over the next 12 years when the 55.2 years period that we predict will be completed. The results are shown in Figure 7.

Subtracting the calculated fit, we obtained the results presented in Figure 8. No clear evidence of another frequency exists as ^{Fauvaud et al. (2010)} and ^{Conidis et al. (2013)} proposed, but from our analysis it cannot be discarded due to the lack of observations in the cycle intervals between [0, 142000] and [330000, 382000] which makes the analysis difficult. Calculating the standard deviation of the residuals we get σ = 0.0012, less than that obtained by ^{Zhou et al. (1999)}, ^{Conidis et al. (2013)} and ^{Zong et al. (2019)}. This supports the sinusoidal behavior shown in the O−C diagram of Figure 7.

3.2. Determination of Orbital Parameters and Mass of the Companion Star

Carrying out an analysis of a binary system such as that developed by ^{Zong et al. (2019)}, it is possible to determine the parameters of the companion star utilizing the formulae 6, 7 and 8 proposed by ^{Borkovits et al. (1996)}, which are function of the Fourier coefficients.

where a ₀ is shown in equation 5; a _1,2 ; b _1,2 are the Fourier coefficients. These are listed in Table 5. A ^’ denotes the semi-major axis. i ,e y w are the elements of the orbit of the companion, c is the speed of light in astronomical units per day.

Then, we followed ^{Borkovits & Heged¨us (1996)} for the determination of the mass of the companion star.

Considering a three body system, ^{Borkovits & Heged¨us (1996)} present equation 11 in their paper as a solution which we used, adapting it to a binary system in which BL Cam orbits around another unseen star and the encountered variations in the O−C are provoked by the LTTE. This follows a prescription first employed by ^{Fu et al. (2008)}. Combining them, equation 9 provides the mass function in terms of the orbital parameters.

Substituting the above-mentioned values listed in Table 6 calculated in this work in the mass function equation given in equation 9, it is possible to calculate the mass of the companion. Considering the mass of BL Cam as m ₁ = 0.99 from ^{McNamara (1997)}, we calculated the mass m ₂ for different angles i , shown in Table 7, determining the roots of the polynomial given in the equations:

Being a cubic equation, it has three solutions. In each case we considered as a solution the real root. The other two are imaginary roots without physical interpretation. The mass of the companion star as a function of the inclination angle is shown in Table 7.

The behavior of mass as a function of angle is shown in Figure 9. Given the obtained values we can say that we are dealing with a M-type star.

3.3. Period Determination Conclusions

Despite having been determined to be a variable star many years ago, the variability of the SX Phe star BL Cam has been described with several interpretations as more information has been gathered. Considering the interpretation of this century only, we highlight the following studies.

^{Kim et al. (2003)} found that the parabolic period variation had recently reversed. They also determined five frequencies for BL Cam using Fourier analysis. They stated that in order to confirm the periodic variation of the O−C values, BL Cam should be observed for at least 10 more years.

^{Rodriguez et al. (2007)} reported that their results confirmed the existence of multi-periodicity in this star. In addition to the main frequency f ₀=25.5769 c/d and its harmonics 2f ₀ and 3f ₀, with stable amplitude, a secondary frequency f ₁ exists in the region 31-32 c/d with variable amplitude.

A more recent article, that of ^{Fauvaud et al. (2010)}, states that the short-term O−C variation, if interpreted as a light travel-time effect, is indicative of a massive stellar component (0.46 to 1 M⊙with a short period orbit (144.2 d), within a distance of 0.7 AU from the primary and encourage more observations to confirm the long-term O−C variations. If they were also caused by a light travel-time effect, they could be interpreted in terms of a third component, in this case probably a brown dwarf star (≥0.03M⊙), orbiting in 3400 d at a distance of 4.5 AU from the primary.

^{Conidis and Delaney (2013)} found a more accurate period of 0.039097912(1) days, and presented an updated linear ephemeris. This newly presented linear ephemeris was used to calculate revised O−C values, which were fitted with a parabolic curve to measure the rate of change of the pulsation period.

Although the parabolic fit has a physical interpretation, it is noted that a cubic fit more appropriately describes the behavior of the O−C diagram, but they concluded that this assumes the O−C diagram is best represented by a quadratic fit. This has been shown to be a poor assumption, since a cubic polynomial is a better representation of the O−C diagram. This, according to them, is problematic since the physical meaning of the third order term cannot be explained physically.

The most recent paper, that of ^{Zong et al. (2019)} did not find evidence of a triple system as stated by ^{Fu et al. (2008)}. ^{Fauvaud et al. (2010)} performed a triple system analysis of the O−C diagram. However, the determination of the second companion’s parameters was not successful. The residuals of fitting the O−C curve implied that BL Cam might be a binary system in an eccentric orbit with a period of 14.01 (9) yr. The companion might be a brown dwarf.

In this study, with an extended time basis of 15779 days or 403578.6 cycles, we performed an O−C analysis and found that the residuals do not conform to the parabolic variation proposed by other authors but rather present a sinusoidal variation with a period of 60 years.

In our analysis we do not need to invoke multi-periodicity as was previously proposed, nor a parabolic behavior. Instead, with only one stable period, using Period04, we adjusted the 1606 times of maximum light shown in Figure 5. Our final fit is obtained using equation 5 and is presented in Figure 7; its residuals are shown in Figure 8.

Considering the obtained results of the present work, and taking into account the residuals in the σ interval presented in Figure 8, this paper has shed light on the nature of BL Cam. Before we consider that it might be a triple system, as ^{Fauvaud1 (2010)} proposed, continuous monitoring of the star is mandatory.

4.1. Data Acquisition and Reduction at SPM

At the SPM Observatory the observational, as well as the reduction procedures, have been employed since 1986 and have been described many times. A recent detailed description of the methodology can be found in ^{Peña and Martinez (2014)}.

The star BL Cam was observed in uvby−β photometry in February, 2020. The season covered five nights during which BL Cam was observed on two. Over all the nights of observation the following procedure was used: each measurement consisted of five ten-second integrations of the star and three tensecond integrations of the sky simultaneously for the uvby filters and almost simultaneous for the narrow and wide filters that define Hβ.

The accuracy of our observation was evaluated by three numerical procedures: (i) the accuracy of each point; (ii) a comparison of the observed standard stars with those values in the literature, and (iii) a comparison of the observed standard stars obtained in each night throughout the whole season.

In the first case, the accuracy is a direct result of the star counts. Although BL Cam is a faint star for our telescope-spectrophotometer system we obtained in the five ten-seconds integrations, enough counts to calculate the errors for the u, b, v, and y filters that define the uvby−β photoelectric photometry, equal to (18491, 53040, 68053, 24413), respectively, which translated into uncertainties of (0.0074, 0.0043, 0.0038, 0.0064) magnitudes.

For the second criterion we reduced each night separately calculating the transformation coefficients and the difference in magnitude between the obtained values and the values reported in the literature for each night. To calculate the final photometric values we averaged the nightly coefficients and the differences with the literature. With the obtained photometric values we reduced the data as described in ^{Peña & Martínez (2014)}. What must be noted here are the transformation coefficients for the observed season (Table 9). This led to the obtainment of the final photometric values. The season errors were evaluated using the observed standard stars with those employed from the literature. These uncertainties were calculated through the differences in magnitude and colors for (V , (b − y), m ₁, c ₁ and Hβ) which are (0.035, 0.008, 0.011, 0.011, 0.012) for the 2020 season.

For the final numerical evaluation of the accuracy we considered all the standard stars on each night in the whole season, for a total of 118 points in uvby and 106 points in Hβ, of the standard stars. For each star we calculated its mean value as well as the standard deviation. This provides a numerical evaluation of our uncertainties considering the whole season, as well as the goodness of the season. The obtained results are presented in Table 8 with the ID of the star in Column 1. Columns two to six present the magnitude and color indexes (V , (b−y), m ₁, c ₁ and Hβ) and Columns seven to eleven, their corresponding standard deviations; the last column lists N, the number of data points of each star in the season. The mean values of the season with the standard deviations are shown in the last two lines. With the exception of the star HD122563 the standard deviation of all the standard stars is on the order of thousandths in the V magnitude. For this reason, HD122563 may be a new variable star since it is similar to HD 115520, which was discovered to be a new δ Scuti variable by ^{Peña et al. (2007)}.

4.2. Metallicity ([Fe/H] ) Determination Through uvby − β Observations

^{Nissen (1988)} also proposed equations to determine metal abundance (for stars of F spectral type). The metal abundance [Fe/H] is given by Nissen (1988) through the equation:

We applied ^{Nissen’s (1988)} prescription (described in detail in ^{Peña & Martínez 2014)} to the uvby − β photometric values presented in Table 10 arranged by decreasing β, and we determined the reddening, the unreddened indexes (b − y)₀, m ₀, c ₀, as well as the metallicity when the star passes through an F type stage. The metallicity values distribution is presented in Figure 10 which shows the histogram of all the values. As can be seen the mean value is −1.2±0.3. The continuous line is a Gaussian fit.

A comparison between the unreddened indexes c ₀ and (b − y)₀ was obtained for the star with the ^{Castelli & Kurucz (2006)} models which are based on ATLAS9 model atmospheres. This allowed us to determine the effective temperature T _e and the surface gravity logg (Figure 11). The effective temperature varies between 7250 K and 8000 K, whereas the surface gravity is around logg = 3.5. Table 10 lists these values. Column 1 shows the HJD, Column 2 to 5 the unreddened indexesandβ.

Subsequent columns present metallicity, effective temperature 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, and suface gravity. The averaged temperature along the phase interval of 0.3 to 0.8 is 7682 ± 503 K.

4.3. Physical Parameters Conclusions

Most of the papers devoted to BL Cam have emphasised its pulsational nature and few have tried to determine its physical parameters. In this paper, we were able to unredden the color indexes and determine the reddening and the metallicity with uvby − β photoelectric photometry and the well-calibrated equations of ^{Nissen (1988)}. Locating these indexes on the grids of ^{Castelli & Kurucz (2006)} we determined the effective temperature and surface gravity of the star along its cycle of pulsation.