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Revista mexicana de astronomía y astrofísica

versión impresa ISSN 0185-1101

Rev. mex. astron. astrofis vol.52 no.2 México oct. 2016

 

Articles

A study of BO Lyn, a neglected hads star1

J. H. Peña1  2  3 

C. Villarreal1  3 

D. S. Piña1  3 

A. Rentería1  3 

J. Guillén3 

A. A. Soni3 

H. Huepa2 

1 Instituto de Astronomía, Universidad Nacional Autónoma de México, Ciudad de México, México.

2 Observatorio Astronómico Nacional, Tonantzintla, México.

3 Facultad de Ciencias, Universidad Nacional Autónoma de México, México.

Abstract:

uvby − β photometry of the high amplitude δ Scuti (HADS) star BO Lyn allowed us to determine its physical characteristics. A secular period variation was established through the O-C of all the available times of maximum light and those newly acquired through CCD photometry. In the present study we have demonstrated that BO Lyn is pulsating with one stable varying period whose O-C residuals show a sinusoidal pattern compatible with a light-travel time effect.

Key words: stars: variables: delta Scuti; techniques: photometric

Resumen:

Fotometría uvby−β de la estrella tipo HADS BO Lyn nos permitió determinar sus parámetros físicos. La variación secular se estableció mediante el análisis de los tiempos de máximo con el método de O-C recabados de la literatura y con los nuevos adquiridos con fotometría CCD. En este trabajo hemos demostrado que la estrella BO Lyn se encuentra pulsando con un periodo estable, cuyos residuos muestran un patrón sinusoidal compatible con el efecto del tiempo de viaje de la luz.

1. Introduction

Very little has been done on BO Lyn since its discovery by Kinman et al. (1994) (as reported by Kazarovets & Samus, 1997, star 73243 in their Table 1). Kinman (1998) analyzed the star and assigned to it a period of 0.0933584 days and a V -magnitude range of 11.85-12.05 with a slightly variable light curve. He further stated that “the location, space motion, and other properties of this star indicate that it is a higher amplitude δ Scuti star (or ‘dwarf Cepheid’) that is a member of the old disk population”. The most recent study was done by Hintz et al. (2005) who examined the star both photometrically and spectroscopically and proposed that the period is decreasing at a constant rate.

Table 1 Log of observing seasons. 

Date Telescope Npoints ΔT (d) ΔV Observers*
16/01/1112 84 cm 37 0.079 0.054 aas, jg
16/01/1314 84 cm 41 0.086 0.054 aas, jg
16/01/2122 14 inch 297 0.228 0.020 ESAOBELA16
16/01/2324 14 inch 266 0.123 0.018 ESAOBELA16
16/01/2425 14 inch 356 0.165 0.020 ESAOBELA16
16/02/0607 10 inch 469 0.227 0.031 jg, ap

*aas, A.A. Soni; jg, J. Guillén; ap, A. Pani; ESAOBELA16: A. Rodríguez; V. Valera; A. Escobar; M. Agudelo; A. Osorto; J. Aguilar; R. Arango; C. Rojas; J. Gomez; J. Osorio; M. Chacon.

2. Observations

This star was observed at both the Observatorio Astronómico Nacional of San Pedro Mártir, México with the 0.84 m telescope and a uvby−β spectrophotometer and at Tonantzintla, México with 14-inch and 10-inch telescopes provided with SBIG ST 8300 and ST1001 CCD cameras, respectively. The log of the observations is given in Table 1. Column 1 reports the date (year month day), Column 2 the tele-scope and implicitly, the observatory; Columns 3 and 4 the number of points and the time span of the observations; Column 5 the uncertainty in each night; it should be kept in mind that in the first two rows we present the uncertainty in the absolute transformation to the standard system and in the remaining rows, the error of differential magnitudes; finally, the last column lists the observers.

2.1. Data Acquisition and Reduction at Tonantzintla

During all the observational nights the following procedure was utilized. Sequence strings were obtained in the V filter with an integration time of 30 sec. There were 551,111 counts for BO Lyn, 1,555,128 for the comparison star C1 and 2,774,289 counts for the check star C2, enough counts to secure high accuracy. The error in each measurement is, of course, a function of both the spectral type and the brightness of each star but they were observed long enough to secure sufficient photons to get a good S/N ratio and an observational error of 0.001 mag in all cases. The reduction work was done with AstroImageJ (Collins, 2012). This software is relatively easy to use and has the advantage that it is free and works satisfactorily on the most common computing platforms.

For the CCD photometry of BO Lyn (08:43:01.224, +40:59:51.79) two reference stars were utilized. A bright star TYC 2985-290-1 identified in the present paper as C1 (08:42:39.883, +40:59:48.30, V = 10.91, SpT= N/A) was used as comparison star, and a brighter star, BD +41 1869, identified as C2 (08:42:33.428, +41:05:59.97, V = 10.30, SpT=F8) as a check star to obtain the light curves in a differential photometry mode. The reason for using the less bright star C1 as a comparison object instead of C2 is that, during the observation, C2 was out of the observed field near the time of maximum due to the rotation of the telescope caused by the alt-azimuthal mounting. The results were obtained from the difference Vvariable - Vcomparison and the scatter calculated from the difference VcomparisonVcheck. Each of these times of maxima has an accuracy of 3 × 10-4 day. Figure 1 presents the light curves of BO Lyn.

Fig. 1 Light curves of BO Lyn. Top: those obtained by uvby − β absolute photometry, bottom: those observed at Tonatzintla with CCD detectors by differential photometry. 

3. O-C analysis

To investigate the secular behavior of the period of BO Lyn we studied the literature related to it. Besides Kinman (1998) only two groups of researchers have observed BO Lyn: Klingenberg et al. (2006) with few observations and, previously, Hintz et al. (2005) who performed studies of the O-C behavior of this star. A summary of their findings is presented in Table 2. In Column 1 the author is presented, Column 2, 3 and 4 the ephemerides determined by each author; T0, P and β respectively. Columns 5 and 6 list the mean value of the (O-C) for all the times of maximum light and the standard deviation, respectively. Hubscher et al. (2013) published only one time of maximum light. All are presented schematically in Figure 2.

Table 2 BO Lyn ephemerides equations. 

Author T 0 P β (O-C)mean (O-C)std dev
Kinman (1998) 2438788.0355 0.0933584 −0.021 0.028
Hintz et al. (2005) linear 2447933.8183 0.09335724 −0.007 0.014
Hintz et al. (2005) parabolic 2447933.7988 0.09335800 −7.2(1.0) ×10−12 −0.001 0.009

Fig. 2 (O-C) diagrams for the ephemerides proposed in the literature (Table 2). 

Table 3 lists the observed times of maximum light of BO Lyn and includes the new observations. In this table, Column 1 reports the time of maximum light (in HJD), Column 2, the source and Column 3 the epoch with the ephemerides parameters determined in this paper.

Table 3 Times of máxima of BO Lyn. 

Time of Maximum Reference Epoch
2447933.7964 Hintz-Kinman 0
2447938.7478 Hintz-Kinman 53
2448274.9308 Kinman 3654
2448577.9691 Hintz-Kinman 6900
2449010.7845 Kinman 11536
2449685.9520 Kinman 18768
2452252.8044 Hintz 46263
2452252.8993 Hintz 46264
2452264.7545 Hintz 46391
2452264.8459 Hintz 46392
2452266.7141 Hintz 46412
2452266.8091 Hintz 46413
2452288.6538 Hintz 46647
2452288.7455 Hintz 46648
2452288.8399 Hintz 46649
2452310.6849 Hintz 46883
2452331.7861 Hintz 47109
2453075.7484 Hintz 55078
2453075.8406 Hintz 55079
2453083.7767 Hintz 55164
2453427.7978 Hintz 58849
2453429.7592 Hintz 58870
2453487.7356 Hintz 59491
2453795.5356 Klingenberg 62788
2453795.6334 Klingenberg 62789
2455654.4061 Hubscher 82699
2457409.7380 Esaobela16 101501
2457409.8276 Esaobela16 101502
2457409.9249 Esaobela16 101503
2457411.7932 Esaobela16 101523
2457412.7228 Esaobela16 101533
2457412.8180 Esaobela16 101534
2457425.7953 jg, aas 101673
2457425.8890 jg, aas 101674

Since the study of Hintz et al. (2005), more observations have been carried out, some of them recently, and they are presented in this paper in Table 3. We tested the old proposed ephemerides equations with the complete set of times of maximum light which is constituted of a set of only 34 times of maximum light including those observed in 2016 (Figure 2). At the time of the Hintz et al. (2005) study, the time basis was only 5554 days or 15 years. In 2016 the time basis has been extended to 9492 days or 26 years, almost double that used for the calculations of Hintz et al (2005).

4. Period determination

To determine the period behavior of BO Lyn we followed three methods.

4.1. Period04

In this method all detailed photometry was used: Kinman (1998) which is presented in apparent magnitudes, and that of the present paper which includes that observed at the Tonantzintla and San Pedro Martir Observartories, Mexico and, as has been explained, consisted of two samples: absolute photometry in San Pedro Martir for two nights and differential photometry in Tonantzintla for four nights. Because of the amplitude difference between the instrumental and the apparent magnitudes and for a consistent analysis, all the light curves were normalized subtracting the average of each night from itself; in this way the amplitudes became similar and capable of giving more accurate results.

The whole time series was analyzed with Period04 (Lenz & Breger, 2005) to determine a representative period of the whole sample. Three data sets were created. The first one, the Kinman (1998) data, consisted of 142 data points over 17 nights separated by a time span of 2187 days. The second was the one currently presented, with 76 points over 2 nights for the absolute photometry and 1364 points over 4 nights for a time span of 27 days for the differential photometry; the third data set was the whole data set, for which, as has been said, amplitudes were normalized.

For the first data set (Kinman, 1998) Period04 gave the following frequency: 10.71140620 c/d, with an error of 5.12 × 10−6. The 2016 data season gave a frequency of 10.710874 c/d with an error of 2.43 × 10−4. Finally, the whole data set gave 10.711447500 c/d with an error of 7.42×10−7. These final results are presented in Figure 3. The corresponding periods were 0.093358423, 0.093363062 and 0.093358064 d, respectively.

Fig. 3 Periodogram of all the available light curves of BO Lyn. Top: the window function. Bottom: the obtained periodogram. 

Once the whole time string was analyzed the period was utilized as a seed period, which was taken to calculate the number of cycles, E. A least squares fit of TMAX vs. E was implemented giving as output the refined period and the corrected initial epoch T0 of the ephemerides equation, as well as the error parameters. The resultant equation is the following:

Tmax=2447933.7845 ±4.7 × 10-3+0.093358109±7.4×10-8×E

4.2. Minimization of the Standard Deviation of the O-C Residuals (MSDR)

The second method utilizes as criteria of good-ness, the minimization of the standard deviation of the O-C residuals (MSDR).

We implemented a method based on the O-C standard deviation minimization analogous to the idea proposed by Stellingwerf (1978) for period determination by phase dispersion minimization. We considered the set of TMAX listed in Table 3 in our analysis. The mean period was determined through the differences of two or three times of maxima that were observed on the same night and the associated standard deviation. Given the standard deviation and the period we determined, we swept between these limits calculating 5000 steps which gives the sufficient accuracy provided by the time span of the observations. The obtained precision of one millionth provides the new period and the limits for the iteration (Figure 4). In each iteration, the O-C standard deviation was calculated. We chose as the best period that which showed the minimum standard deviation. The resulting equation is:

Tmax=2447933.7845±4.7×10-3+0.093358109±7.4×10-8×E

Fig. 4 Standard deviation vs. period. This diagram served to determine the best period. 

The previous methods gave basically the same result. Therefore, it can be seen that the O-C residuals show a sinusoidal behavior. This pattern in the O-C diagram is usually related with the light-travel time effect (LTT). In view of this, the O-C residuals were analyzed in Period04 and fitted with an equation of the following type:

O-C=Z+A×sin2πΩE+Φ (1)

The result is shown in Figure 5 and the elements of the fit are listed in Table 4.

Fig. 5 O.C residuals by the minimization of the standard deviation. 

Table 4 Equation parameters for the sinusoidal fit of the O-C residuals. 

Value Period04 & MSDR PDDM
Z 1.17 × 10−3 −1.67 × 10−2
Ω 9.85 × 10−6 1.18 × 10−5
A 1.76 × 10−2 1.68 × 10−2
Φ 1.14 × 10−1 2.39 × 10−1
ZErr 4.25 × 10−4 3.62 × 10−4
ΩErr 2.43 × 10−7 2.31 × 10−7
AErr 5.82 × 10−4 4.89 × 10−4
ΦErr 7.81 × 10−3 1.65 × 10−2
Residuals 2.00 × 10−3 1.9 × 10−3

4.3. Period determination through an O-C differences minimization (PDDM)

In the third procedure, we implemented a method based on the idea of searching the minimization of the chord length which links all the points in the O-C diagram for different values of the periods, looking for the best period which corresponds to the minimum chord length. With this idea in mind, we tried to obtain the smoothest curve. Since we were dealing with the 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 are constant, we just concentrated on the change in the distance in the y-axis in each diagram, generated by one 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 Tmax listed in Table 3 in our analysis. Given the mean period determined from the consecutive times of maxima and the associated standard deviation, (0.0936 days and 0.0027 days), we calculated values of epoch and O-C by sweeping the period in the range provided by the standard deviation limits, 0.091 to 0.096 days, calculating 5 × 106 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 6). The T0 time used for the present analysis was the one of the 2016 observation run, 2457412.8196, because we are certain of its precision.

Fig. 6 Period determination through an O-C differences minimization, PDDM. 

As a result we determined the linear ephemerides equation as:

Tmax=2457412.8196+0.093358338×E

Figure 7 shows the O-C diagram for the ephemerides equation found by the above method (PDDM).

Fig. 7 (O-C) residuals after the adjustment to the obtained period. 

Assuming the wave behavior as a part of the physics in the system, we adjusted a sinusoidal function to the O-C, performing a fitting using Levenberg-Marquardt Algorithm for the best 1,000 O-C lengths. This would be, in this particular case, another way of finding the best period and, at the same time, the sinusoidal function which is similar to the one assumed in the second method. The parameters which best represents the system are listed in Table 4. The parameter used to prove the goodness of the adjustment 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 8.

Fig. 8 Zoom RSS vs. Period. 

After this, the ephemeris equation was set as:

Tmax=2457412.8196+0.093357995×E

As we can see, only one frequency explains this sinusoidal behavior. The parameters are presented in Table 4 and are shown schematically in Figure 9. This frequency corresponds to a period of 7,779.83 days or 21.49 years.

Fig. 9 Diagram of the O-C adjusted to a sinusoidal function. 

4.4. (O-C) Discussion

BO Lyn was discovered relatively recently and has been scarcely observed. The whole sample of times of maximum light contains only 34 unevenly distributed entries.

As can be seen, the parameters of the equations obtained with the light curves analyses and the standard deviation minimization for the times of maxima give the same results. This confirms the results among themselves since both solutions converge to the same period. Then in both cases a sinusoidal behavior can be seen. The third method, PDDM, with a completely different approach, gives basically the same results.

After a quick review of the most conspicuous and well-studied HADS stars, the majority (nine) have increasing period changes and only a few (four), the opposite. What we have found in BO Lyn is that it is a star in a stable evolutionary stage.

With respect to its amplitude, a variation in the light curves was suggested by Kinman (1998), and later Hintz et al. (2005) suggested that this could be due to two probable secondary frequencies in the pulsation modes. In our analysis we can explain the behavior of the star with only one pulsational period and an orbital period.

Continued monitoring of times of maximum will be crucial, and such observations are encouraged.

5. Physical parameters

Physical parameters can be obtained using the advantages given by Strömgren photometry with calibrations made by Nissen (1988) for the A and F stars or by Shobbrook (1984), for earlier spectral types. These calibrations are described in detail in Peña et al. (2002).

The evaluation of the reddening was done by first establishing to which spectral class the stars belong. As a primary criterion the location of the stars in the [m1] − [c1] diagram of the classical textbook of Golay (1974) (Figure 10) or the results derived for the open cluster α Per (Peña & Sareyan, 2006) were employed. As can be seen in this figure, the spectral type of BO Lyn varies between A5 and A8. There has been until now, no assignment of a spectral type for BO Lyn. Once a spectral class is assigned, we can choose the prescription for unreddening which, for the spectral type of BO Lyn, is that of Nissen (1988).

Fig. 10 Position of BO Lyn in the [m1] − [c1] diagram. 

5.1. Data Acquisition and Reduction at SPM

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 method-ology can be found in Peña et al. (2007). During the three nights of observations the following procedure was utilized: each measurement consisted of 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. It is important to emphasize here that the transformation coefficients for the observed season (Table 5) and the season errors were evaluated using the ninety-one observed standard stars. These uncertainties were calculated through the differences in magnitude and colors for (V, by, m 1, c 1 and β) which were (0.054, 0.012, 0.019, 0.025, 0.012), respectively. We emphasize the large range of the standard stars in the magnitude and color values: V: (5.62, 8.0); (by):(-0.09, 0.88); m 1:(-0.09, 0.67); c 1:(-0.024, 1.32) and β:(2.495, 2.90).

Table 5 Transformation coefficients obtained for the 2016 season. 

Coefficient B D F J H I L
value 0.031 1.008 1.031 −0.004 1.015 0.159 −1.362
σ 0.028 0.003 0.015 0.017 0.005 0.004 0.060

Table 6 lists the photometric values of the observed star. In this table Column 1 contains the time of the observation in HJD, Columns 2 to 5 the Strömgren values V, (by), m 1 and c 1, respectively; Column 6 lists β, whereas Columns 7 to 9 list the unreddened indexes [m1], [c1] and [u-b] derived from the observations. Unfortunately in none of the SPM observations (two nights) a time of maximum light was reached, although the observations were almost long enough to completely cover the whole pulsation cycle.

Table 6 uvby-β photoelectric photometry of BO Lyn. 

HJD V (b − y) m 1 c 1 β [m1] [c1] [u − b]
-2457000.00
399.8477 11.875 0.130 0.195 0.928 2.706 0.237 0.902 1.375
399.8506 11.865 0.160 0.154 0.952 2.754 0.205 0.920 1.330
399.8546 11.897 0.149 0.186 0.872 2.743 0.234 0.842 1.310
399.8567 11.894 0.162 0.180 0.865 2.791 0.232 0.833 1.296
399.8591 11.901 0.172 0.169 0.881 2.774 0.224 0.847 1.295
399.8617 11.937 0.160 0.167 0.893 2.804 0.218 0.861 1.297
399.8638 11.944 0.166 0.177 0.881 2.751 0.230 0.848 1.308
399.8662 11.974 0.164 0.157 0.882 2.683 0.209 0.849 1.268
399.8706 11.980 0.193 0.139 0.877 2.730 0.201 0.838 1.240
399.8728 11.974 0.210 0.123 0.877 2.695 0.190 0.835 1.215
399.8747 11.999 0.195 0.157 0.833 2.682 0.219 0.794 1.233
399.8768 12.012 0.192 0.167 0.824 2.664 0.228 0.786 1.242
399.8790 12.015 0.206 0.135 0.856 2.731 0.201 0.815 1.217
399.8809 12.056 0.155 0.222 0.802 2.691 0.272 0.771 1.314
399.8846 12.076 0.157 0.215 0.784 2.675 0.265 0.753 1.283
399.8864 12.064 0.174 0.190 0.812 2.713 0.246 0.777 1.269
399.8895 12.057 0.188 0.178 0.791 2.698 0.238 0.753 1.230
399.8926 12.059 0.196 0.155 0.830 2.694 0.218 0.791 1.226
399.8946 12.061 0.181 0.171 0.823 2.715 0.229 0.787 1.245
399.8966 11.964 0.253 0.121 0.717 2.739 0.202 0.666 1.070
399.9008 11.956 0.247 0.118 0.739 2.730 0.197 0.690 1.084
399.9027 11.962 0.232 0.115 0.791 2.708 0.189 0.745 1.123
399.9049 11.950 0.224 0.139 0.751 2.719 0.211 0.706 1.128
399.9073 11.939 0.205 0.148 0.795 2.715 0.214 0.754 1.181
399.9093 11.907 0.221 0.124 0.792 2.717 0.195 0.748 1.137
399.9114 11.955 0.173 0.162 0.869 2.777 0.217 0.834 1.269
399.9157 11.916 0.167 0.159 0.878 2.731 0.212 0.845 1.269
399.9176 11.918 0.161 0.173 0.907 2.811 0.225 0.875 1.324
399.9193 11.878 0.168 0.151 0.913 2.792 0.205 0.879 1.289
399.9208 11.911 0.135 0.190 0.912 2.743 0.233 0.885 1.351
399.9224 11.857 0.163 0.163 0.891 2.757 0.215 0.858 1.289
399.9240 11.840 0.153 0.160 0.950 2.742 0.209 0.919 1.337
399.9257 11.828 0.151 0.163 0.925 2.777 0.211 0.895 1.317
399.9276 11.833 0.142 0.176 0.911 2.770 0.221 0.883 1.325
401.9013 11.842 0.155 0.171 0.918 2.764 0.221 0.887 1.328
401.9033 11.838 0.159 0.164 0.923 2.796 0.215 0.891 1.321
401.9052 11.845 0.180 0.146 0.939 2.799 0.204 0.903 1.310
401.9070 11.863 0.167 0.166 0.901 2.773 0.219 0.868 1.306
401.9086 11.879 0.157 0.177 0.913 2.789 0.227 0.882 1.336
401.9103 11.888 0.169 0.151 0.934 2.779 0.205 0.900 1.310
401.9135 11.909 0.152 0.195 0.878 2.730 0.244 0.848 1.335
401.9152 11.897 0.171 0.173 0.895 2.794 0.228 0.861 1.316
401.9168 11.914 0.187 0.144 0.909 2.774 0.204 0.872 1.279
401.9187 11.927 0.173 0.173 0.895 2.794 0.228 0.860 1.317
401.9207 11.921 0.189 0.164 0.866 2.783 0.224 0.828 1.277
401.9227 11.951 0.182 0.164 0.875 2.788 0.222 0.839 1.283
401.9245 11.964 0.185 0.153 0.893 2.806 0.212 0.856 1.280
401.9264 11.961 0.202 0.133 0.881 2.766 0.198 0.841 1.236
401.9301 11.969 0.195 0.174 0.855 2.746 0.236 0.816 1.289
401.9319 11.983 0.187 0.184 0.825 2.753 0.244 0.788 1.275
401.9337 11.992 0.199 0.169 0.829 2.802 0.233 0.789 1.255
401.9362 12.000 0.204 0.162 0.814 2.719 0.227 0.773 1.228
401.9378 12.005 0.217 0.146 0.822 2.753 0.215 0.779 1.209
401.9395 12.015 0.208 0.155 0.834 2.743 0.222 0.792 1.236
401.9411 12.039 0.195 0.159 0.827 2.728 0.221 0.788 1.231
401.9430 12.033 0.218 0.147 0.790 2.769 0.217 0.746 1.180
401.9450 12.030 0.231 0.121 0.825 2.769 0.195 0.779 1.169
401.9469 12.039 0.212 0.133 0.852 2.734 0.201 0.810 1.211
401.9509 12.033 0.231 0.130 0.837 2.672 0.204 0.791 1.199
401.9526 12.046 0.219 0.141 0.812 2.670 0.211 0.768 1.190
401.9547 12.058 0.193 0.159 0.829 2.784 0.221 0.790 1.232
401.9567 12.031 0.210 0.146 0.811 2.770 0.213 0.769 1.195
401.9586 12.025 0.203 0.151 0.830 2.773 0.216 0.789 1.221
401.9610 12.010 0.212 0.147 0.810 2.758 0.215 0.768 1.197
401.9631 12.015 0.183 0.185 0.791 2.770 0.244 0.754 1.242
401.9652 11.990 0.187 0.176 0.818 2.791 0.236 0.781 1.252
401.9677 11.959 0.213 0.137 0.845 2.779 0.205 0.802 1.213
401.9697 11.971 0.170 0.169 0.836 2.753 0.223 0.802 1.249
401.9714 11.921 0.190 0.156 0.863 2.747 0.217 0.825 1.259
401.9735 11.913 0.188 0.147 0.868 2.828 0.207 0.830 1.245
401.9772 11.891 0.185 0.148 0.885 2.779 0.207 0.848 1.262
401.9791 11.885 0.166 0.176 0.866 2.751 0.229 0.833 1.291
401.9812 11.846 0.166 0.154 0.920 2.781 0.207 0.887 1.301
401.9832 11.828 0.171 0.150 0.919 2.817 0.205 0.885 1.294
401.9852 11.814 0.179 0.132 0.948 2.810 0.189 0.912 1.291
401.9869 11.813 0.166 0.144 0.940 2.833 0.197 0.907 1.301

5.2. Physical Parameter Determination

The application of the above mentioned numerical unreddening packages gave the results listed in Table 7 for BO Lyn. This table lists, in the first column, the HJD. Subsequent columns present the reddening, the unreddened indexes, the unreddened magnitude, the absolute magnitude, the distance modulus, and the distance. Mean values were calculated for E(b − y), the distance modulus (DM) and the distance 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. We obtained, for the whole cycle, values of 0.020 ± 0.021; 10.7 ± 0.9 and 1497 ± 756 for E(b − y), DM and distance (in pc), respectively, whereas for the mentioned phase limits we obtained, 0.022 ±0.022; 10.5 ±0.8 and 1383 ±702 respectively. The uncertainty is merely the standard deviation. In the case of the reddening, most of the values for the spectral type F of BO Lyn produced negative values which is unphysical. In those cases we forced the reddening to be zero in which case the (b − y) index is the same. If the negative values are included, the mean E(b − y) is 0.009 ± 0.038.

Table 7 Reddening and unreddened parameters of BO Lyn. 

HJD E(b − y) (b − y) 0 m 0 c 0 Hβ V 0 M V DM d(pc)
-2457000.00
401.9832 .052 .119 .166 .909 2.817 11.60 1.76 9.84 928
399.9176 .036 .125 .184 .900 2.811 11.76 1.78 9.98 990
401.9852 .058 .121 .149 .936 2.810 11.57 1.40 10.17 1080
401.9245 .055 .130 .170 .882 2.806 11.73 1.83 9.89 952
399.8617 .028 .132 .176 .887 2.804 11.81 1.80 10.01 1005
401.9337 .059 .140 .187 .817 2.802 11.74 2.35 9.39 754
401.9052 .049 .131 .161 .929 2.799 11.63 1.31 10.32 1159
401.9033 .024 .135 .171 .918 2.796 11.73 1.41 10.32 1160
401.9152 .032 .139 .183 .889 2.794 11.76 1.63 10.13 1060
401.9187 .034 .139 .183 .888 2.794 11.78 1.63 10.15 1070
399.9193 .029 .139 .160 .907 2.792 11.75 1.44 10.31 1154
399.8567 .017 .145 .185 .862 2.791 11.82 1.86 9.96 982
401.9652 .038 .149 .187 .810 2.791 11.83 2.28 9.55 811
401.9086 .015 .142 .182 .910 2.789 11.81 1.40 10.41 1210
401.9227 .036 .146 .175 .868 2.788 11.80 1.73 10.07 1034
401.9547 .039 .154 .171 .821 2.784 11.89 2.08 9.81 917
401.9207 .038 .151 .175 .858 2.783 11.76 1.73 10.03 1012
401.9812 .019 .147 .160 .916 2.781 11.77 1.21 10.55 1288
401.9103 .022 .147 .157 .930 2.779 11.80 1.06 10.74 1404
401.9677 .057 .156 .154 .834 2.779 11.72 1.86 9.86 935
401.9772 .033 .152 .158 .878 2.779 11.75 1.50 10.25 1122
399.9114 .017 .156 .167 .866 2.777 11.88 1.61 10.27 1130
399.9257 .001 .150 .163 .925 2.777 11.82 1.11 10.71 1389
399.8591 .015 .157 .174 .878 2.774 11.84 1.46 10.37 1188
401.9168 .033 .154 .154 .902 2.774 11.77 1.21 10.56 1296
401.9070 .011 .156 .169 .899 2.773 11.81 1.27 10.55 1287
401.9586 .040 .163 .163 .822 2.773 11.85 1.91 9.95 976
399.9276 .000 .142 .176 .911 2.770 11.83 1.13 10.70 1382
401.9567 .043 .167 .159 .802 2.770 11.85 2.03 9.81 918
401.9631 .014 .169 .189 .788 2.770 11.95 2.21 9.74 889
401.9430 .048 .170 .161 .780 2.769 11.83 2.21 9.62 840
401.9450 .065 .166 .140 .812 2.769 11.75 1.89 9.86 938
401.9264 .039 .163 .145 .873 2.766 11.79 1.34 10.45 1232
401.9013 .000 .155 .171 .918 2.764 11.84 .98 10.86 1489
401.9610 .035 .177 .158 .803 2.758 11.86 1.86 10.00 998
399.9224 .000 .163 .163 .891 2.757 11.86 1.12 10.74 1405
399.8506 .000 .160 .154 .952 2.754 11.86 .53 11.34 1851
401.9319 .008 .179 .186 .823 2.753 11.95 1.66 10.29 1144
401.9378 .038 .179 .157 .814 2.753 11.84 1.68 10.16 1076
401.9697 .000 .170 .169 .836 2.753 11.97 1.56 10.41 1209
399.8638 .000 .166 .177 .881 2.751 11.94 1.12 10.82 1458
401.9791 .000 .166 .176 .866 2.751 11.89 1.26 10.63 1334
401.9714 .010 .180 .159 .861 2.747 11.88 1.23 10.65 1347
401.9301 .013 .182 .178 .852 2.746 11.91 1.29 10.62 1332
399.8546 .000 .149 .186 .872 2.743 11.90 1.09 10.80 1447
399.9208 .000 .135 .190 .912 2.743 11.91 .73 11.18 1719
401.9395 .022 .186 .162 .830 2.743 11.92 1.44 10.49 1250
399.9240 .000 .153 .160 .950 2.742 11.84 .38 11.46 1961
399.8966 .052 .201 .137 .707 2.739 11.74 2.42 9.32 730
401.9469 .021 .191 .139 .848 2.734 11.95 1.09 10.86 1483
399.8790 .014 .192 .139 .853 2.731 11.96 .98 10.97 1564
399.9157 .000 .167 .159 .878 2.731 11.92 .79 11.13 1682
399.8706 .002 .191 .140 .877 2.730 11.97 .77 11.20 1737
399.9008 .042 .205 .131 .731 2.730 11.77 2.01 9.76 895
401.9135 .000 .152 .195 .878 2.730 11.91 .76 11.15 1695
401.9411 .000 .195 .159 .827 2.728 12.04 1.17 10.86 1489
399.9049 .016 .208 .144 .748 2.719 11.88 1.64 10.23 1114
401.9362 .000 .204 .162 .814 2.719 12.00 1.07 10.93 1531
399.9093 .015 .206 .129 .789 2.717 11.84 1.22 10.62 1333
399.8946 .000 .181 .171 .823 2.715 12.06 .87 11.19 1730
399.9073 .000 .205 .148 .795 2.715 11.94 1.13 10.81 1451
399.8864 .000 .174 .190 .812 2.713 12.06 .91 11.15 1701
399.9027 .020 .212 .121 .787 2.708 11.88 .95 10.93 1532
399.8477 .000 .130 .195 .928 2.706 11.88 -.44 12.32 2909
399.8895 .000 .188 .178 .791 2.698 12.06 .60 11.46 1955
399.8728 .000 .210 .123 .877 2.695 11.97 -.40 12.38 2986
399.8926 .000 .196 .155 .830 2.694 12.06 .04 12.02 2537
399.8809 .000 .155 .222 .802 2.691 12.06 .20 11.86 2351
399.8662 .000 .164 .157 .882 2.683 11.97 -.98 12.95 3892
399.8747 .000 .195 .157 .833 2.682 12.00 -.49 12.48 3140
399.8846 .000 .157 .215 .784 2.675 12.08 -.19 12.27 2838
401.9509 .000 .231 .130 .837 2.672 12.03 -.89 12.93 3847
401.9526 .000 .219 .141 .812 2.670 12.05 -.68 12.72 3506
399.8768 .000 .192 .167 .824 2.664 12.01 -1.02 13.04 4048

If the photometric system is well-defined and calibrated, it provides an efficient way to investigate physical conditions such as effective temperature and surface gravity via a direct comparison of the unreddened indexes with the theoretical models. These calibrations have already been described and used in previous analyses (Peña & Peniche; 1994; Peña & Sareyan, 2006).

A comparison between theoretical models, such as those of Lester, Gray & Kurucz (1986), hereinafter LGK86 and intermediate or wide band photometry obtained for the stars allows a direct comparison. LGK86 calculated grids for stellar atmospheres for G, F, A, B and O stars with different values of [Fe/H] in a temperature range from 5500 up to 50 000 K. The surface gravities vary approximately from the main sequence values to the limit of the radiation pressure in 0.5 intervals in log g. A comparison be-tween the photometric unreddened indexes (b − y)0 and c0 obtained for each star with the models al-lowed us to determine the effective temperature Te and surface gravity log g.

In order to locate our unreddened points in the theoretical grids of LGK86, a metallicity had to be assumed. LGK86 calculated their outputs for several metallicities. Particularly in the case of BO Lyn, for which we determined a mean metallicity of [Fe/H] = −0.39 ± 0.31, there are two applicable models, either [Fe/H] = 0.0 or −0.5. We tested both since our determined mean metallicity of [Fe/H] = −0.39 ± 0.31 lies in between. To diminish the noise and to see the variation of the star in phase, mean values of the unreddened colors were calculated in phase bins of 0.1 starting at 0.05. As can be seen in Figure 11, for the case of [Fe/H] = −0.5, the effective temperature varies between 7000 K and 7700 K; the surface gravity varies between 3.4 and 3.9. Table 8 lists these values. Column 1 shows the phase, Columns 2 and 3 list the temperature obtained from the plot for each [Fe/H] value; Column 4, the mean value and Column 5, the standard deviation for a [Fe/H] = −0.5 metallicity. Column 6 lists the effective temperature obtained from the theoretical relation reported by Rodriguez (1989) based on a relation of Petersen & Jorgensen (1972, hereinafter P&J) Te = 6850 + 1250 × (β − 2.684)/0.144 for each value and averaged in the corresponding phase bin. The last column lists the surface gravity log g from the plot.

Fig. 11 Location of the unreddened points of BO Lyn (dots) in the LGK86 grids. The numbers indicate the phase. 

Table 8 Effective temperature of BO Lyn. 

Phase T e T e Mean OP&J log g
[Fe/H] 0.0 -0.5 -0.5 -0.5
0.05 7300 7100 7200 7251 3.4
0.15 7400 7200 7300 7235 3.8
0.25 7300 7200 7250 7259 3.8
0.35 7200 7000 7100 7234 3.6
0.45 7400 7500 7450 7494 3.7
0.55 7800 7500 7650 7582 3.8
0.65 8000 7700 7850 7781 3.9
0.75 7800 7500 7650 7469 3.7
0.85 7700 7400 7550 7642 3.7
0.95 7700 7400 7500 7569 3.5

5.3. Physical Parameter Discussion

New observations in uvby −β photoelectric photometry were carried out on the HADS star BO Lyn. From this uvby −β photoelectric photometry we determined first its spectral type, varying between A5V and A8V. From Nissen’s (1988) calibrations the reddening was determined as well as the unreddened indexes. These served to obtain the physical characteristics of this star, log Te, in the range from 7000 K to 7700 K and log g from 3.2 to 3.6, using two methods: (1) from the location of the unreddened indexes in the LGK86 grids and (2) through the theoretical relation (Petersen et al., 1972). They are similar within the error bars, and give a good idea of the star’s behavior. Furthermore, when mean values are obtained from the two closest metallicity values, the result is closer to the obtained theoretical value.

6. Discussion

According to Rodriguez & Breger (2001) “only 14 % of the known δ Scuti stars are part of binary or multiple stellar systems Only five variables are fainter than V = 10.0.... Hence, multiplicity is catalogued for 22 of all the δ Scuti known up 10. 0. This percentage is very low because more than 50% of the stars are expected to be members of multiple systems”. They later state that “pulsating stars in eclipsing binaries are important for accurate determinations of fundamental stellar parameters and the study of tidal effects on the pulsations.... During the last two decades, unusual changes in the light curves have been detected, leading to a number of different interpretations...”

They later say that “pulsation provides an additional method to detect multiplicity through a study of the light-time effects in a binary system. This method generally favors high-amplitude variables with only one or two pulsation periods (which tend to be radial). Several decades of measurements are usually required to study these (O-C) residuals in the times of maxima”.

At that time they listed, in their Table 4, only six stars with known orbital periods. Since then, with a longer time basis for those stars, and for an increased number of measured times of maximum, a better definition of their orbital elements is available. There have been numerous studies of HADS stars with this purpose. For example, Boonyarak et al. (2011) carried out a study devoted to the analysis of the stability of fourteen stars of this type. Many other authors carried out analyses on a star-by-star basis. Some of the HADS stars show a behavior of the O-C residuals compatible with the light-travel time effect that is expected for the binaries AD CMi, KZ Hya, AN Lyn, BE Lyn, SZ Lyn, BP Peg, BS Aqr, CY Aqr, among others; whereas there are some stars that, on the contrary, are varying with one period and its harmonics and do not show a light-travel time effect. To this category, according to Boonyarak et al., (2011) belong GP And, AZ CMi, AE UMa, RV Ari, DY Her, DH Peg.

In the present study we have demonstrated that BO Lyn is pulsating with one stable varying period whose O-C residuals show a sinusoidal pattern compatible with a light-travel time effect. In relation to this topic, it is interesting to mention that in the excellent discussion of Templeton (2005), he states that: “In all cases except SZ Lyn, the period of the purported binarity is close to that of the duration of the (O-C) measurements, making it difficult to prove that the signal is truly sinusoidal. A sinusoidal interpretation is only reliable when multiple cycles are recorded, as in SZ Lyn. While the binary hypothesis is certainly possible in most of these cases, conclusive proof will not be available for years or even decades to come. Continued monitoring of times of maximum will be crucial, and such observations are encouraged. In the meantime, however, other possible interpretations of their behavior must also be explored”.

We feel that the results presented in this paper fulfill Templeton’s (2005) requirement that “a sinusoidal interpretation is only reliable when multiple cycles are recorded”.

Acknowledgements

We thank the staff of the OAN for their assistance in securing the observations. We thank an anonymous referee whose comments and suggestions improved this paper. This work was partially sup-ported by the OAD of the IAU (ESAOBEL), Papiit IN106615 and Papime PE113016. Proofreading and typing were done by J. Miller and J. Orta, respectively. C. Guzm´an, F. Ruiz and A. Diaz assisted us in the computing. We thank B. Juarez and G. Perez for bibliographic assistance. All the students thank the IA for allotting telescope time. Special thanks to O. Trejo, J. Diaz, K. Vargas, A. Rodriguez, V. Valera, A. Escobar, M. Agudelo, A. Osorto, J. Aguilar and A. Pani for observation assistance and discussions. We have made use of the Sim-bad databases operated at CDS, Strasbourg, France; NASA ADS Astronomy Query Form.

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1Based on observations collected at the San Pedro Mártir and Tonantzintla Observatories, México.

Recibido: 07 de Marzo de 2016; Aprobado: 07 de Julio de 2016

J. Guillén and A. A. Soni: Observatorio Astronómico Nacional, Universidad Nacional Autónoma de México, Tonantzintla, Puebla, México.

H. Huepa: Facultad de Ciencias, Universidad Nacional Autónoma de México, Ciudad de México, México.

J. H. Peña, D. S. Piña, A. Rentería, and C. Villarreal: Instituto de Astronomía, Universidad Nacional Autónoma de México, Apdo. Postal 70-264, Ciudad de México, M´éxico (jhpena@astroscu.unam.mx).

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