Near-infrared and UBVRI photometric study of open cluster : IC 361

Serb. Astron. J. } 183 (2011), 1 - 5 Editorial REVISITING OF THE VARIABLE STARS OF OC: NGC 6866 G. Chandra1 1 Department of Physics, DSB KU, 263002 Nainital, INDIA E–mail: gshunyansh@gmail.com (Received: October 2, 2018; Accepted: October 2, 2018) SUMMARY: The search of secondary pulsations is carried out to understand the possible relations among the known parameters of variables of the cluster, NGC 6866. These pulsations arise due to the various ongoing physical phenomena of the variables. Moreover, pulsations of the variables are identified through the visual inspection of their frequency-amplitude distributions. A total of 18 variables among the 28 known variables is showing the secondary pulsation modes. Furthermore, these pulsation modes do not occur in PV, EA, EB, Elliptical and semi-regular. In addition, the field variables seem to be the red-component-stars (RCS) for the studied cluster. The smoothness of the frequency-amplitude curves, signal-to-noise ratio and the significant limits are play a major role in deciding the real peak or frequency values. We are not rejected those amplitude peaks of parabolic patterns, for which, the amplitude is greater than the significant limit of variables. The weight of pulsation frequencies is given to be 0.5 for non full cyclic variation, but the amplitude is greater than significant limit. Similarly, our present analysis does not support the HADS characteristics of ID 1077 and also indicates its position beyond of the stellar cluster NGC 6866. We are also proposing to new correlation between the secondary pulsation modes (depending on frequencies) and the absolute magnitude of known variables. Key words. (Galaxy): Open star cluster – individual: NGC 6866 – variable: pulsation – method-data analysis 1. Introduction Open star clusters (OSC) are the host of various variable stars such as, δ-Scuti, γ-Doradus, rotational, elliptical, etc., and their detailed study need to impose the constraint of stellar pulsation modes. Since, stars have been changed their brightness with the time; and the variation of stellar brightness shows the various stages of stellar evolution in the form of instability strips. A star may show various pulsating cycles either on a stage of stellar evolution or in the time scale of its evolution. Since, the time scale of the pulsations and that of the stellar evolution are very different to each other. For an instance, Delta Sct Stars or Classical Cepheids etc., do not show their various stages of evolution during their pulsation cycles. Moreover, the pulsation variability is arisen due to those certain conditions, which translate into the various instability strips in the Hertzsprung-Russell (HR) or colour-magnitude (CM) diagram (Eyer & Mowlavi 2008). In addition, these pulsations are not found in those stars, which are lying on a fixed instability strip (Briquet et al. 1 G. Chandra 2007); although, the study of stellar variability of members of the clusters is most effectively used to constrain the model of stellar evolution processes. These constraints depend on the basic properties and the evolutionary status of each stellar members. In this connection, the basic properties of variables can derive from the properties of their associated cluster (Mowlavi et al. 2013). The simultaneous photometric analysis of the observations of variables of a cluster would be required more precise time-series data, which get by the observations of the stars of cluster under the same weather and equal instrumental conditions (Kim et al. 2001). In the present work, we have used the time series data of Joshi et al. (2012) (available in the VIZIER service) for the detailed frequency-amplitude analysis of the variables. The stellar dynamics and evolution study of the Eclipsing binaries are more attractive in the comparison of a single star. In addition, their period limit is still an open issue, and may need revolutionary revision for documentation of very short period systems (Liu et al. 2015). Detection of the stellar pulsation modes of a variable and analysis of their associated frequencies may offer a unique opportunity to understand the internal structure of that variable (Mowlavi et al. 2013). These prescribed pulsation modes are providing important clues for probing the physical properties of variables, including their masses, luminosities, temperatures and metallicities (Salmanzadeh et al. 2015). Furthermore, these pulsating variables cover a broad range of the stellar-parameters and associated evolutionary stages (Derekas et al. 2009). The present manuscript describes about the pulsation search procedure and results within the known variable stars of Joshi et al. (2012). The importance of search of secondary pulsations of variables is briefly described in the Section 2. The identification procedure of these secondary pulsations and smoothing procedure of their associated light curves prescribe in the Sections 3 and 4 respectively. The multi-periodic analysis of HADS stars is carried out in the Section 5. The specific discussion and analysis of the identified WUMA stars of the open cluster, NGC 6866 have discussed in the Section 6. In the Section 7, we are summarizing our important results and prescribed their important features/uses for constraining the model of stellar evolution processes. 2 2. Pulsation search in known variable stars The high variation of reddening may change the measurement accuracy of the distance modulus. The nature of stellar variability does not change through the small variation in stellar magnitude. Since, all science frames of interested target gathered in the similar environmental conditions, therefore, an analytic view/method needs to cut an effect of the positional variation of the target. In addition, the constant environmental conditions are not possible for a 6-8 hours observation of the target in a particular night and these conditions are varying nightto-night. Such type effect may withdraw through either the simple differential photometric method or the secondary standard method (STM). The linear relation of common hundred stars of various frames is a fundamental property of the STM and this procedure may create the least scattering due to the many data points. In this paper, we pretend to a frequency analysis of known variables in the open cluster NGC 6866. For this purpose, we use published data (Joshi et al. (2012)), already transformed to the standard V magnitudes through STM. They were observed the cluster for 29 nights between 2008 September 26 and 2011 January 10 (over two observing sessions). According to them, a total of 768 frames in the V-band have accumulated using the 1.04-m Sampurnanand telescope at Manora peak, Nainital and a brief log of their observations are given in Table 1 of Joshi et al. (2012). Since, we have revisited the variable work of Joshi et al. (2012), therefore, the data of light curves of their studied variables extracted from VIZIER services. These light curves occur due to the changing behaviour of the stellar magnitudes. Such magnitude variation of stars arisen due to the various ongoing physical and the stellar evolution phenomenon. Since, each particular phenomena may produce a periodic variation of the stellar magnitudes of studied variable, therefore, the light curves may contain the various periodic cycles of different amplitude. In the earlier study of these variables, the prominent period (having the largest amplitude) have determined. As a result, it is further needed to find out other possible pulsation frequencies/periods. The frequency analysis is a valid intention for knowing the frequency components of a multiperiodic variable certainly contributes to the understanding of its nature and physics. In this connection, we have reconstructed the frequency distribution of each identified variable by using the PERIOD-4.0 program (Lenz & Breger 2005). This computer program allows us to fit all the frequencies in the given magnitude-domain. The frequency spectrum and primary peak of all prescribed variables shown in the figure 1. Revisiting of the variable Stars of OC: NGC 6866 Fig. 1. The frequency distribution of variable stars in range of 0 to 79.8 d−1 . The X-axis and Y-axis represent the frequency (d−1 ) and Amplitude (mmag) in V-band respectively. The frequency spectrum of these variables contain the various asymptotically parabolic pattern of the amplitude within the entire frequency domain. These Gaussian profiles of amplitudes can consider to be the possible profiles of the ongoing stellar physical phenomena. The highest peak of amplitude of any asymptotically parabolic pattern defines as the amplitude of a periodic event of the variable and its corresponding frequency used to estimate the period of said physical phenomena. The major/prominent highest peak of all these asymptotically parabolic pattern may lead as the primary frequency (f0 ) of variables, where other amplitudes of remains asymptotically parabolic pattern may use to find the other possible periodic phenomenon. A new prominent frequency of 28 variables and values of their corresponding amplitudes have listed in Table 3. Moreover, the known literature values Joshi et al. (2012) of their periods and amplitudes are also prescribed in the same table. 3 G. Chandra Table 1.The new frequencies and amplitude values of the variables have listed here with their earlier values. Furthermore, they also classify as RCS and BCS. Star ID 0016 0020 0027 0036 0039 0047 0058 0074 0081 0094 0158 0191 0221 0231 0239 0248 0253 0332 0349 0444 0487 0494 1077 1088 1274 1292 1421 1583 Period (d) 0.465333 0.380373 0.143616 0.214684 0.836120 0.275710 0.911577 0.321750 1.239157 0.740741 2.285714 1.090513 0.476872 0.327547 37.037037 0.437446 7.407407 11.494253 12.048193 16.260162 0.415110 0.366704 0.033559 0.184775 0.462428 0.295247 0.041263 0.082055 Joshi et al. (2012) < V > Amplitude Membership (mag) (V-mag) status 11.747 0.064 Likely 12.003 0.032 Unlikely 12.222 0.039 Member 12.623 0.035 Likely 12.677 0.035 Likely 12.995 0.041 Unlikely 13.246 0.052 Field 13.469 0.055 Likely 13.548 0.058 Likely 13.878 0.087 Member 14.902 0.052 Member 15.285 0.085 Likely 15.564 0.079 Likely 15.615 0.052 Member 15.630 0.260 Unlikely 15.660 0.210 Likely 15.730 0.060 Field 16.190 0.140 Field 16.310 0.100 Unlikely 16.820 0.310 Field 17.210 0.340 Member 17.260 0.440 Unlikely 18.580 0.340 Unlikely 18.670 0.280 Field 19.050 0.460 Unlikely 18.950 0.370 Unlikely 19.150 0.300 Unlikely 19.260 0.320 Likely Joshi G.C. (2016) is used following magnitudecolour relation to separate main sequence (MS) from the stellar distribution of the observed field of the cluster, Vo = 7.66×(B − V )o + 7.32, (1) where Vo and (B − V )o are the observed magnitude and colour of stars. If, Vo value of star satisfy the expression Vo >7.66×(B − V )o + 7.32; then, it will be bluer member of studied cluster. The group of bluer member defines as the blue component stars (BCS) of a cluster, which are lying on the left side of main sequence (MS) of the (B − V ) vs V colourmagnitude diagram of study cluster. The other hand, the red component stars (RCS) is a group of redder members of stellar cluster and these stars are laying on the right side of the MS of the above prescribed CMD. The redder members of stellar clusters are satisfied a condition Vo ≤7.66×(B − V )o + 7.32. Our investigate indicates that all members and likely 4 Variable type Binary? Binary? δ Scuti δ Scuti γ Doradus ? γ Doradus Elliptical γ Doradus γ Doradus EA Binary? PV PV Semi-regular Elliptical Rotational Semi-regular Semi-regular EB W UMa W UMa HADS HADS W UMa HADS HADS HADS Period (d) 0.465462 0.380264 0.143638 0.515320 0.835128 0.275606 0.911635 0.160855 1.236996 0.739284 1.141031 20.907380 0.937075 0.478590 37.778617 0.218657 7.554011 11.500862 11.870845 8.115565 0.207608 0.183312 0.033559 0.184864 0.229491 1.318235 0.041262 0.297459 Present Study V-Amplitude Category (mag) 0.082 Blue 0.042 Blue 0.064 Blue 0.066 Blue 0.085 Blue 0.079 Blue 0.182 Red 0.117 Blue 0.225 Blue 0.168 Blue 0.089 Blue 0.173 Blue 0.153 0.078 Blue 0.283 Red 0.166 Blue 0.114 Red 0.237 Red 0.163 Red 0.177 Red 0.298 Blue 0.285 0.100 0.024 0.020 0.022 0.015 0.012 - known variables are found to BCS while all field variables will be RCS. Unlikely members may either BCS or RCS. Joshi et al. (2012) were assigned various membership status of the variables according to their spatial Psp ), photometric (Pph ) and kinematic probabilities (Ppm ) to assign the stellar membership probabilities of detected stars within observed cluster field. They have chosen these stars as the cluster members either due to photometric probability Pph = 1 or due to proper motion criteria, i.e. kinematic probability (Ppm > 0.6). According to them, Unlikely members are those stars which satisfy only one criterion and belong to outside from the coreregion (Psp < 0.71) of the cluster. Those unlikely variables are BCS, selected through the photometric criteria, whereas others are RCS. Since, the proper motion distribution of the BCS and RCS does not separate them from each other, therefore, field stars do not identify by kinematic probabilities. The period of a variable is the reciprocal of the pulsa- Revisiting of the variable Stars of OC: NGC 6866 tion frequency (P = 1/f ). By keeping in the view of giving values of variables by Joshi et al. (2012), this physical definition does not found true for the primary/prominent period of thirteen variables, whom IDs numbers are given as, 0074, 0158, 0191, 0221, 0231, 0248, 0494, 0487, 0494, 1274, 1292 and 1583 respectively. These IDs numbers have extracted from the list of variables of table 4 of the manuscript of Joshi et al. (2012). Furthermore, variable star of ID 191 has prescribed as the possible binary star by Joshi et al. (2012), but it is an unusual result of this star. Similarly, variable stars of ID 1292 and ID 1583 have identified as the HADS type variables by Joshi et al. (2012), which are not obeying the above prescribed fact, whereas other HADS variable (ID 1077, ID 1088 and ID 1421) are showing good agreement with this fact. The estimated frequency of the contact binary stars is the sum of prominent frequencies of both stars. Both stars so much near to each other that they are not resolved by the present astronomical telescopes and detected frequency occurs due to their combined physical evolution processes. Let, each star shares equal proportion of mass phenomena of binary system, then the real frequency of both stars will be same. Thus, actual frequency will consider to be the half of the resultant f0 of the contact binary stars. On the behalf of the present analysis, the known literature values of f0 of the PV and EB variables have not justified through the above prescribed statement. These results/facts are also motivated to use for searching the other pulsation frequencies in these known variables. 3. Technique of identification of new pulsation in variables The patterns of frequency spectrum of the variables are asymptotically parabolic (anyGaussian, Shannon’s function etc.), in which, the highest peak of 3σ level known as amplitude and corresponding frequency is prominent/pulsation frequency (f0 ). We have run Period 04 on a given data file and take the frequency spectrum as it comes, but it is not possible to take any peak apparently above the noise level as a secondary frequency, without realizing that some may be aliases of each other and the low frequency range the window spectrum may play an important role in producing spurious log periods. For example, we are discussing the nature of amplitude versus frequency pattern of the variable star ID 016, which is a possible Binary. Its frequency spectrum has depicted in the Figure 2. Fig. 2.1. The frequency spectrum of variable star having ID 0016 (Range:0-360 c/d−1 ). The X-axis and Y-axis represent the frequency (d−1 ) and Amplitude in V-band (mmag), respectively. 5 G. Chandra 10 90 8 72 6 4 54 2 0 36 24 30 36 42 48 54 18 0 0 6 12 18 24 30 36 Fig. 2.2. In the upper panel, we have shown the frequency distribution of variable star, having ID 0016. The lower panels depicted the phase diagrams of variable ID 0016 in two frequencies such as, 2.15 d−1 and 25.88 d−1 . In this figure, the amplitude is continuously decreasing within in a frequency domain from 2.15 d−1 to 20 d−1 and an asymptotic parabola appears. The highest peak of said parabola finds to 25.88 d−1 through visual inspection. The various asymptotic parabola patterns also find without prominent peaks compare to their surrounding. Such patterns occur either due to the instrumental error in the magnitude estimation or due to the stellar amplitude variation by the atmospheric conditions. In addition, a peak of 7mmag is comparatively higher of 3σ limits of the remaining peaks of the order of 3 − 4 mmag, therefore, peak of 7mmag may additional pulsation frequency of variable ID 16; and peaks with amplitude of 3 − 4 mmag are representing aliases frequencies (pseudo asymptotically parabolic peaks) due to noise. Both pulsation frequencies (2.1484 d−1 and 25.8787 d−1 ) with their corresponding phase diagrams are depicted in the Figure 2. At the amplitude spectrum for f = 2.15d−1 (upper panel of fig 2), the corresponding amplitude is 0.082 mag. However, the peak to peak complete amplitude from the smoothed phase plots (left lower panel of fig 2)) computed as 11.792-11.724=0.068 mag i.e. less than 0.082 mag. This result indicates decrement of the amplitude with the data smooth6 ness. The smoothed semi-amplitude of the sinusoid, with no harmonics, will be 0.02, so the complete amplitude will be 2 times of semi-amplitude i.e. 0.04 with no self-consistency. Obviously, the light curve at 2.15 c/d shows an eclipse-like narrow trough, which occur due to the presence of harmonics of the main frequency, the main at 4.3 c/d and its amplitudes seems larger than that of the ”fast variations”. Similarly, the amplitude spectrum shows one more peak at f = 28.88d−1 . Here, data point strength for f = 2.15d−1 is comparatively higher than data points f = 28.88d−1 and characteristics of the former has contained more data points compare to later. In this connection, the scattering of the phase curve of f = 2.15d−1 occurs low due to presence of the least characteristic data points of f = 28.88d−1. The other hand, the scattering of the phase curve of f = 28.88d−1 is high due to effect of dominant characteristic of f = 2.15d−1 . Since, smoothness has been found in the cost of decrement amplitude, therefore, the completed amplitude has found to 0.026 mag(11.760 − 11.734) from the phase plot at f = 25.88d−1 (right lower panel of Fig 2). Thus, we have identified a new pulsed frequency of the variable ID 0016 through this procedure. In the present work, we are identifying new frequencies of variables. Generally, frequencies of variables could be identified by keeping consideration of prewhitened procedure of data-points of light curves. In pre-whitened process, the data points are turned out from the theoretical sine curve of a known amplitude and known pulsation frequency. This process is effective to compute independent pulsation frequencies of the variables, whereas this process reduces the data points to identify more pulsation frequencies. Martinazzi et al. (2015) have been found pulsation frequencies through the pre-whitened process at the same place of peaks as appears in frequency spectrum of studied variables. Since, aliases is also present in the frequency spectrum, therefore; we have considered only highest peak frequency as pulsation frequency of asymptotic pattern of resultant spectrum. In this connection, we are proposing to an alternative procedure of pre-whitened to identify the pulsation frequencies. Revisiting of the variable Stars of OC: NGC 6866 4. Phase diagram and its terminology A phase of variable is the measuring scale in the terms of cyclic variation and depends on the period of cyclic variation. Thus, the phase diagrams represent the picture of full cyclic variation of stellar magnitudes. The phase diagrams of variable ID 0016 (as shown in the lower panels of Figure 2 have constructed through the moving average procedure. The phase of each data point of light curve computes through the following relation, P hase = (T − T0 )×f, (2) where T , T0 and f are the JD time of observation epoch, initial JD time, i.e. JD0 (The JD time of the first observed science frame of V-band of NGC 6866 by Joshi et al. (2012) on the date 26 September 2008 i.e. 4736.088461) and corresponding frequency of the variable. The integral part of these computed phase has removed and remaining decimal part have arranged in the increasing order. As a result, the cyclic variation are overlapped with each other in the phase-folded diagram. Here, we have used movingaverage-algorithm to overcome the effect of scattering of data points in resultant phase diagram. The moving average procedure improves the smoothness of phase diagram and decreases the amplitude of pulsation frequency. 4.1. Estimation of specific ratio, significant limits and S/N values The specific ratio, signal-to-noise (S/N) values and significant limits of variables will need to decide the real peak or frequency values. Specific ratios are proposing to co-relate the amplitude of frequency peak (A(mmag)) to the mean error of estimation errors of all data points (ErrMV ) of time series photometric data of variables. These specific ratios of variables have computed as, s A(mmag) , (3) specef ic ratio(s) = ErrMV The higher value of s shows higher smoothness of the phase diagram and it comes to be zero for the perfect non-variable stars. Since, an amplitude level of significant limit will require to find the true pulsation frequency, therefore, we are proposing to compute the values of said limit as equal to or more than 1.5 times that of the noise level. The noise level is the above limit of 3σ of background frequencies. These background frequencies are never showing the pulsation characteristics i.e. full cyclic variation. The S/N value is proposed to the ratio of the pulsation amplitude of the noise level of FFP following as, S 2×A(mmag) value ≥ . N Signif icant limit (4) The computed values of specific ratio, S/N and significant limits of each variable are given in the table 4. A deep analysis of Table 4 indicates that the value of the S/N ratio of pulsation frequencies finds more than 3.2 for variables, having a specific ratio greater than 1.4. This prescribed value of S/N is near to the value of Breger criterion. For example, the 3σ values of frequency spectrum of ID 494 find near of 2.5 mmag. A peak of 4.76mmag (frequencies range 24.5d−1 to 49.5d−1 ) comes to be more than 1.5 times of 2.5mmag. The scattering of data-points leads the deformation in the shape of the light curve of the variables. Such deformation reduces the S/N ratio of the pulsation frequencies. Since, the fainter stars show the higher uncertainty in their computed magnitudes, therefore, identification of pulsation frequencies of these stars becomes more complex and less precise. Due to the less precision of photometric data, the S/N ratio of secondary pulsation is found too low. A deeper and more precise photometric data will need to confirm new identified pulsation frequencies, having lower S/N . We have found secondary pulsations for 18 variables of a sample of 28 variables as listed in the Table 4. Table 2. In this table, we have listed the estimated values of the specific S/N and significant limits of the known variables of the cluster NGC 6866. The errors of pulsation amplitudes assumes same as mean values of photometric magnitudes as found in the light curve of the variables ID Error (mag) Amplitude (mag) Specific ratio 0016 0020 0027 0036 0039 0047 0058 0074 0081 0094 0158 0191 0221 0231 0239 0248 0253 0332 0349 0444 0487 0494 1077 1088 1274 1292 1421 1583 0.006 0.007 0.007 0.008 0.008 0.009 0.011 0.010 0.011 0.011 0.011 0.013 0.013 0.013 0.013 0.015 0.014 0.016 0.017 0.025 0.026 0.027 0.047 0.074 0.062 0.071 0.060 0.065 0.082 0.042 0.064 0.066 0.085 0.079 0.182 0.117 0.225 0.168 0.089 0.173 0.153 0.078 0.283 0.167 0.114 0.237 0.163 0.177 0.298 0.285 0.100 0.024 0.019 0.022 0.015 0.012 3.566 2.467 2.969 2.896 3.195 2.984 4.014 3.407 4.428 3.957 2.821 3.649 3.469 2.432 4.689 3.359 2.873 3.804 3.120 2.646 3.399 3.215 1.455 0.570 0.567 0.551 0.495 0.429 Significant limits (mmag) 3.10 6.84 3.48 5.45 6.21 6.89 5.45 4.31 5.14 4.48 4.38 5.02 5.78 7.19 3.73 2.96 5.89 6.16 5.87 3.78 3.57 2.51 6.18 6.65 5.99 6.77 6.53 6.58 Signal to noise ratio (S/N) 52.91 12.19 36.57 24.35 27.55 22.95 66.91 54.31 87.68 74.95 40.65 68.81 52.99 21.82 152.01 112.83 38.62 77.08 55.58 93.83 167.09 227.06 32.53 7.23 6.63 6.41 4.53 3.67 A ratio of amplitude signal/noise ≥ 4.0 provides a useful criterion for judging reality of a peak (Breger et al. 2002). According to S/N values in Table 2, all prominent and computed frequencies values are statistically meaningful and significant according to Breger criterion. The phase diagrams of fainter stars, having S/N ≥ 3.2, show smooth and full cyclic variation. In addition, peak of their corresponding pulsation frequencies is greater than significant limit. As a result, the significant limit is more useful to identify pulsation frequency compare than 7 G. Chandra Breger criterion. The new identified pulsation frequencies of variables have been listed in Table 4 and their corresponding S/N is given in the parenthesis of each frequency. 15 12 9 6 5. Revisiting towards the Known HADS stars 3 0 0 High amplitude Delta Scuti (HADS) variables are those late A and early F type pulsating stars which are changing their absolute magnitude (light) and radial velocity in the periods of one to six hours and amplitude of these variables is greater than 0.2 mag (Breger & Montgomery 2000). The values of the period and absolute magnitude of variable stars are satisfied by the following equations (McNamara 2011), MV = (−2.89±0.13)log P − (1.31±0.10, 6 12 18 24 30 36 19.1 19.2 19.3 19.4 0 0.4 0.8 1.2 1.6 20 0.4 0.8 1.2 1.6 2 (5) where MV and P are the absolute magnitude and period of the Delta-Scuti stars, which is used by Salmanzadeh et al. (2015) for the HADS stars. Similarly, (McNamara 2011), have also given relation between (B − V )o and period as given below, (B −V )o = (0.105±0.004)log P +(0.336±0.005, (6) where (B − V )o is the colour-index. In the core field of OSC NGC 6866, a total of five HADS (high amplitude- delta scuti) variables have identified by Joshi et al. (2012). The IDs of these variables are given as 1077, 1088, 1292, 1421 and 1583. All of these given stars have observational (B − V )0 > 0.80mag, three of them even more than 1 (Table 3). Since, HADS are A-F, main sequence (or slightly more evolved) stars, all of them should have (B − V )0 < 0.50mag; therefore, classification of HADS stars by Joshi et al. (2012). is very suspicious. Here, we are checking their characteristics by establishing relationships among the various parameters such as (B − V )o , period, absolute magnitude, etc. In addition, we are also analyzing the multi-periodicity of these stars as described in the subsection below. 5.1 ID 1583 It is fainter variable in the field of NGC 6866 and its study is beneficial to understand the nature of cut-off frequency for fainter variables. Its frequency distribution diagram shows four frequencies having Amplitude greater than 7 mmag. There are two frequencies (3.36 d−1 and 12.18 d−1 ) are found in a single Gaussian distribution and such frequency distribution may classify as twins frequencies of the variables. The first one is the highest amplitude, whereas later matches with the periodic value of this variable as given by Joshi et al. (2012). These twins frequencies with their phase diagram have shown in Figure 3. 8 Fig. 3. The frequency distribution of variable star having ID 1583. The X-axis and Y-axis represent the frequency(d−1 ) and Amplitude in V-band respectively. The frequency distribution of variable star (ID 1583) in higher frequency range. The X-axis and Yaxis represent the frequency (d−1 ) and Amplitude in V-band respectively. A higher frequency not shown any evidence of the full cycle of variability, therefore, it is declining from the possible pulsation frequency of the variable. Other two frequencies, find at 222.367 days−1 and 288.641 days−1 , respectively. The positions of both frequencies have shown in the upper panel of Figure 4 and corresponding phase diagrams have depicted in the lower panels of said figure. It is noticeable fact that the phase diagram of above prescribed first frequency shows a full cyclic variation of the magnitude, while such pattern does not occur in the later frequency. Since, HADS stars are fainter members of the cluster, therefore, their magnitude estimation error is prominent, which may cause of arisen of the later frequency. From all fig- Revisiting of the variable Stars of OC: NGC 6866 ures of the sample stars showing phase diagram, it is clear that all frequencies have similar amplitudes regardless the particular frequency value. This is physically impossible and makes very suspicious to find the higher frequencies. In addition, frequencies above 100cd−1 are hardly detectable in the best quality space data or in very fast ground based phototmetry. This suggests that the detected frequencies may instrumental nature and leads to overestimated results. Other hand, we are not ignoring that these frequencies are found for standardized data points of Joshi et al. (2012). In this connection, we give half weight to this frequency for constraining its relationship with various known parameters. 5.2 Pulsation Search in other stars We have applied above prescribed approach to find the pulsation modes in other HADS stars with ID 1274. We have shown these identified frequencies and their corresponding phase diagrams in the Figure 5. Moreover, the phase diagrams of pulsation frequencies of variable ID 1274 have depicted in the Figure 6. Fig. 4. The frequency distributions of HADS stars in the upper panels. The lower panels represent the corresponding phase diagrams through the moving average procedure. 9 G. Chandra The deep investigation of these diagrams does not show evidence of variable nature for identifying frequencies 15.66 d−1 and 192.90 d−1 . The amplitude of these frequencies of ID 1088 are 10.088 mmag and 7.963 mmag, respectively. The continuous decrement amplitude does not seem in the both sides of these peak values. As a result, such type peaks did not consider for further search of pulsations in other variables. The 182.58 d−1 frequency of ID 1421 shows clear variability, which leads our idea about the cut-amplitude of pulsations, which arises through the background effects. In addition, the data points of the fainter stars are strongly influenced by the daily aliases and necessarily scattered. The STM has used for reducing these influence by Joshi et. al. (2012). Therefore, the pulsation weight of these frequencies is also considered as 0.5 in the present case. of prominent frequency and Ps1 ...Psn are the periodic values of the secondary pulsations. The value of weight takes to 1 for the sinusoidal curve of phase, whereas its value taken to 0.5 for the half sinusoidal curve of phase. Since, the value of MV of open cluster depends on the value of colour-index, (B − V )o , therefore, we are proposing to add a new term k (B − V )o on the McNamara (2011) relation, in which k is the proportional coefficient of linearity. Since, Joshi et al. (2012) have been categorized some stars as a HADS star, therefore, we are revisiting these stars by assuming that these stars belong to the same group of variability. In this connection, the value of k is selected as 3n/2 for present sample stars through the error and trial method, where n is the number of pulsation modes of prescribed stars. As a result, the new proposed relation becomes as, 5.3 New modeling for the period and absolute magnitude MV = (−2.89±0.13)log Pef +ni (B−V )o −1.31±0.10, (8) here ni is defined as a sum of weights of detecting frequencies i.e., We have found that the fundamental parameters of variation through the cluster studies do not match with the computed parameters through the well-established relation like MV − P , (B − V )0 − P , Age-Period etc. Since, the emitted photons/light of the variable may be the result of various ongoing physical phenomena, therefore, the captured data provide the light curve of dependent pulsation frequencies. On this background, we are adding a new term in the well-established relation to establishing the similarities between the parameters through the cluster studies and parameters through known relation for independent frequencies. Since, relations like MV − P , (B − V )0 − P , Age-Period set up for the independent and prominent frequencies of the pulsation variables, therefore, the modified version of this relation need for the dependent pulsation frequencies. This change may possibly through the effective weights and effective number of pulsation frequencies. As a result, we have assumed that every pulsation mode can effectively give to constrain the co-relationship among the various parameters of the variable. The real frequency of contact binaries is low (half for system of identifying stars) compare than the estimated frequency leads to an increment in the period. The interfered frequencies (estimated/obtained) is always higher than the prominent frequency. The extra frequencies affect the prominent frequency on the power from leading the change in the period values. Thus, the modified term log P of dependent frequencies would be linked to various pulsations as follows, log Pef = w1 log Pp + w2 log Ps1 + .... +wn log Psn , (7) where w1 , w2 , ...wn are the weight of pulsation modes Pp , Ps1 , ..Psn , in which Pp is the period 1 www.real-statistics.com/correlation/multiple-correlation 10 ni = w1 + w2 + ... + wn . (9) Since, above prescribed relation is a statistical relation, therefore, the correlation coefficient of this relation is computed by following relation1 , s 2 + r2 − 2r r r rxz xz yz xy yz Rz,xy = , (10) 2 1 − rxy where x = log Pef , y = (B − V )0 and z = MV . The correlation between two variable (rxy ) has been computed through the following relation (Taylor 1997), Σ(xi − x̄)(yi − ȳ) rxy = p , Σ(xi − x̄)2 Σ(yi − ȳ)2 (11) where xi and yi are the mean values of the variable x and y, respectively. The correlation coefficient for absolute magnitude, colour excess and pulsation period finds to 0.59 through relation of Rz,xy . The said value shows strong linear relationship among MV , (B − V )0 and log P . Furthermore, the corresponding p-value of 5 HADS stars sample comes to 0.15 (one-tail test) through the online calculator (Soper 2016; Cohen et al. 2003). Similarly, the modified relation to computing the colour-excess of HADS as given below, (B−V )o = (0.105±0.004)log Pef f +ni (0.336±0.005. (12) The correlation coefficient [rx−y ] of the above relation comes to 0.28, which does not suggest strong linear dependence between the (B − V )0 and log P , but very weak. Thus, the present correlation is statistically incoherent and this analysis cannot give relevant results. However, the p-value of this relation Revisiting of the variable Stars of OC: NGC 6866 is 0.32 through a one-tail test (Soper 2016). A total of 5 sample stars use to find out the correlations of above prescribed equations. Due to the sample of lack stars (only 5), the statistical test and estimated p-values do not so meaningful to constrain any conclusion for present sample. Present sample stars are also contained very high frequency modes and such modes would definitely be of non-radial nature. The empirical relations for HADS defines for radial modes and arise from the pulsation equation. These informative facts show that HADS categorization by Joshi et al. (2012) is falsified. Thus, we can not make any decision about new proposed empirical relations due to fact that these stars may completely different stars. Since, their classification seems to uncertain by statistical algorithm, therefore, we are further investigating their characteristics in the view of values of colour-excess, absolute magnitude and membership probabilities. The resultant model MV and (B − V )o of variables with their observed values are given in the Table 4. Table 3.The model values of absolute magnitude and colour-excess for various HADS stars have been listed here. I st column gives the ID of HADS star. The ni and n of second column represent the total and effective number of pulsation mode of variables. The abbreviations such as, Mo. and Ob. are the short form of model values and observed values. Star ID 1088 1274 1292 1421 1583 ni , n 7, 5, 7, 3, 4, 5.5 4.0 5.5 3.0 3.5 log Pef (days) -5.4184 -5.6254 -7.3576 -4.2033 -5.4184 MV (Mo.) 8.739 9.267 8.009 8.227 8.913 (B − V )o (Mo.) 1.783 1.089 1.579 0.556 0.775 MV (Ob.) 7.92 8.30 8.20 8.40 8.51 (B − V )o (Ob.) 1.02 1.42 1.72 0.87 0.99 After a deep analysis of Table 4, it seems that the model absolute magnitudes of HADS (having kinematic probability less than 0.73 or 0.73) are greater than their observed absolute magnitude, whereas for others HADS members, this fact becomes vice versa. We have found that the model magnitudes seem too close to each other for last three candidates of Table 4, which indicates that they are still cluster members. Other hand, first two candidates show very high deviation between their model and observed values, therefore, they consider as field stars by us and these variables are laid within the cluster boundary. The colour excess values through observation and model are close to each other for cluster, whereas these described values are far away to each other for field stars. These all HADS stars are fainter member of the cluster, therefore, the photometric procedure/technique of estimation of their apparent stellar magnitude may show larger scattering in their light/phase curve. The colour excess values fainter members of the cluster seams to close to each other compare to field stars. Since, these field stars are brighter of the corresponding cluster members, therefore these field stars are either evolved from the cluster region or born different interstellar environment of the cluster region. 5.4 ID 1077: is a SX Phoenicis variable ? Due to the spatial position in CMD, it is an interesting object for understanding the cluster dynamics. It is classified as HADS star by Joshi et al. (2012). δ-Scuti stars (including HADS) are also known as dwarf-Cepheid. Cepheid and Cepheid like variables show the co-relation between age and period, which is given by Joshi & Joshi (2014) as follow, log(Age) = 8.60±0.07 − (0.77±0.08) log P, (13) where P is the period of Cepheid. Most Classical Cepheids (CCs) are monoperiodic, although stars pulsating in an overtone are more luminous and larger than a fundamental mode pulsator with the same period Bono et al. (2001). The amplitude of frequency spectrum of HADS stars is more than 0.3 mag at V-band i.e. δmV > 0.3 mag (Breger & Montgomery, 2000), but we find the amplitude of frequency spectrum approximately 0.1 mag. In addition, it is a blue straggler stars according to the position of variable ID 1077 in the CMD plane. Since, ID 1077 is a low mass star and it does not satisfy relation of Joshi & Joshi (2014). Due to above prescribed reasons, ID 1077 is not a HADS star, therefore, more analytic discussion need to understand its variability nature. Since, it does not fulfilled all criteria of a HADS star; therefore, we search nearest familiar variability nature for it due to its similar characteristics, that of δ-Scuti variables. It is noted that most SX Phe stars are also HADS, but not vice versa. Furthermore, all known SX Phoenicis variables in globular clusters are blue straggler stars (Jeon et al., 2004). In the light of its former variability type and above mentioned facts, we have examined its variable nature to find its similarity with SX Phe stars. In the frequency spectrum of ID 1077, we have found two peaks of amplitude 11.07 mmag and 100.51 mmag at frequencies 1.36 d−1 (f1 ) and 29.79 d−1 (f2 ) respectively. The value of (f2 ) is 21.90 time that of (f1 ) leads an approximate overtone value, but ratio of values of frequencies does not define by any known pulsation model. The peaks of both prescribed frequencies is shown in the frequency spectrum of ID 1077, and their corresponding phase diagrams are depicted in the first two lower left panels of Figure 4. The peak of first frequency (f1 ) is low compere to that of second one (f2 ). Pulsations in an overtone higher than first are rare but interesting (Soszynski et al., 2008). As a result, the peak of later frequency becomes a crucial result to understand the stellar dynamics of evolution of the variable. As the record of PPMXL catalogue (Roeser et al., 2010), the proper motion values of this variable are −1.2±5.6 mas/yr and −2.3±5.6 mas/yr in RA and DEC directions, respectively. These values gives the radial velocity of the variable ID 1077 in the order of 5.33±11 Km/s. The log(age) value of this variables comes to be 9.43 yr through modified Joshi & Joshi (2014) relation, which is greater than the log(age) value of cluster NGC 6866 i.e. 8.85 yr. The difference of both log(age) values comes to be 0.58. 11 G. Chandra tems of ID 487 and ID 494, they had considered twice of obtaining period values. They have argued that EBs are better represented by two sine waves as first author of that paper has accepted in his new paper (Joshi et al. 2015). It is noticeable fact that they did not apply the same technique for other possible binary systems ID 16 and ID 20 (Joshi et al. 2012). As a result, they have created a confusion to decide the so-called true period of binary systems and does not make a clear scientific reason for obtaining the log(Age) = 8.60±0.07−(0.77±0.08) Ps log P −(B−V )o . true period of EBs as a twice of obtaining period. (14) The cause of better representation may not a good where Ps is the period of first pulsation mode scientific reason due to fact that the scattering may of the variable ID 1077. The Amplitude of each pul- also occur by the environmental effect. Similarly, the sation mode of ID 1077 has shown in the correspond- position of both stars seems to differ in the Figure ing phase diagrams. The above prescribed relation 9 of Joshi et al. 2012 and Figure 6 of Joshi et al. provides the value of log(age) to be 8.89±0.15, which (2015). It is highly ethic due to fact that both figslightly high compare to the log(age) of NGC 6866. ures have constructed by the same author through Since, the present age-period relation indi- same results from the same dataset. cates that the variable may old compare to the clusWe have noticed that first one depends on the ter age, therefore; ID 1077 does not have a member of observed stellar magnitude by observation, whereas the cluster. If, the variable follows the membership the latter one depends on the so-called true period criteria of the cluster, then it must show agreement of both variables. Joshi et al. (2015) have been with the ZAMS of the cluster. Unfortunately, the computed the true period of ID 487 and ID 494 (B − V )o ) finds unsatisfactory in this regards. as 0.415110±0.000001 d and 0.366709±0.000004 d, Joshi et al. (2012) are reports that its periodic respectively. Furthermore, the absolute magnitude is 0.033559 d, which is further supplemented by our MV of these variables estimate to 4.32 mag and 4.86 present prominent frequency (29.79d−1 ) for ID 1077. mag, respectively, through the following Rucinski & Duerbeck (1997) relation, This periodic value satisfies a short period pulsation behavior of SX Phoenicis variables (i.e. their periMV = −4.44 log(P ) + 3.02(B − V )o + 0.12, (15) odic values vary on, time scales of 0.03-0.08 days). The masses of SX Phe variables are in the range 1.0- where (B − V )o and P are the intrinsic colour and 1.1 M⊙ (Fiorentino et al. 2014). Moreover, such orbital period respectively. The shifted values of absolute magnitudes of both variables must occur due type variables appear bluer (having a higher tem- to the true period through their assumption. perature) compared to similar luminous stars of the Since, the members of the cluster does not main sequence of studied cluster (Santolamazza et change their position in observed CMD, therefore, al. 2001). All above prescribed properties of ID 1077 we are rejecting Joshi et al. (2015) procedure for are showing similarities with the SX Phoenicis vari- estimating of periods of these variables. Our power ables, therefore it may an SX Phoenicis type variable. spectrum analysis indicates that the prominent period values of ID 487 and ID 494 are found in Since, SX Phe stars belong to the old Galactic disk 0.207608508 d and 0.183312602 d, respectively, which population and its presence in an open cluster does leads the absolute magnitude values of these varinot possible. It is also an Unlikely member of the ables as 5.58 mag and 6.14 mag respectively. Other cluster NGC 6866 as listed in Table 2, and it does hand, we get these values from solution of best fitnot lies in the main sequence of studied cluster, which ted isochrone as 6.45 mag and 6.51 mag respectively, which are close to our new finding results through suggest that ID 1077 does not having confirm mem- Rucinski & Duerbeck (1997) relation. However, new ber of the open cluster NGC 6866. As a result, we periods offer close values of absolute magnitude that conclude that ID 1077 is situated in Galactic Hollow of comes through CMD but still less. We are analysis frequency distribution of these variables to know the regions rather than cluster. cause of less absolute magnitudes through their primary or fundamental period value. For this purpose, we have carried out a search procedure for identify6. WUMA Stars ing the pulsation modes in these variables. On the basis of visual inspection, we are found 5, 2 and 2 A W Ursae Majoris (WUMA) variables are pulsation modes in ID 1274, ID 487 and ID 494 relow mass contact binaries, which are a subclass of spectively. The number of pulsation modes of ID the eclipsing binary variable stars. The light curves 1274 seems to more than that of WUMA. Variable of these variables contain the continuous brightness ID 1274 studied here due to fact that Joshi et al. variations with the strongly curved maxima and min- (2012) were classified it as a WUMA type variable. ima of nearly equal depths. Joshi et al. (2012) iden- Other hand, present analysis indicates that it does tified three WUMS stars in the cluster NGC 6866 not a WUMa. The pulsation modes of ID 1274 have and their estimated period has listed in the Table 3. shown in Figure 6 and results of its detail study have For obtaining the true period values of binary sys- shown in the Table 3. 12 The reddened value of colour (B − V ) of this variable is 0.66 mag (Joshi et al. 2012), which leads the value of (B − V )o as 0.54 mag within the periphery of cluster. Since, the difference of above prescribed values of log(age) and value of (B − V )o of ID 1077 indicates to add an addition term (B − V )o ) in modified relation. As a result, we are proposing a new relation for ID 1077, which is given as below, Revisiting of the variable Stars of OC: NGC 6866 Fig. 5. The frequency distribution and five pulsation mode of ID 1274. 13 G. Chandra Fig. 6. The frequency distribution and identified pulsation modes for ID 20, ID 36, ID 27, ID 74, ID 47, ID 81, ID 94 and ID 248. Fig. 7. The frequency distribution and pulsation mode of ID 487 and ID 494. 14 Fig. 8. The frequency distribution and three pulsation mode in ID 0191. Revisiting of the variable Stars of OC: NGC 6866 The pulsation modes of ID 487 and ID 494 are shown in Figure 7. Table 4.The identified secondary pulsation peaks for variables in the field of view of NGC 6866. Joshi et al. (2012) & Pulsation frequencies/ Sub-frequencies (in d−1 ) Star RA DEC f1 f2 f3 f4 f5 f6 f7 (S/N) (S/N) (S/N) (S/N) (S/N) (S/N) (S/N) ID (J2000) (J2000) 0016 20:03:26.12 44:10:05.3 2.148 25.879 – – – – – (52.91) (4.52) – – – – – 0020 20:04:25.52 44:10:16.2 4.693 20.689 – – – – – (12.19) (3.98) – – – – – 0027 20:03:47.13 44:09:25.7 6.962 41.216 66.840 – – – – (36.57) (3.71) (3.36) – – – – 2.629 26.940 – – – – – 0036 20:03:42.47 44:10:06.4 (24.35) (4.40) – – – – – 3.628 18.742 – – – – – 0047 20:04:11.20 44:05:33.3 (22.95) (3.33) – – – – – 0074 20:03:34.93 44:14:50.1 6.216 14.492 21.367 – – – – (54.31) (9.15) (5.77) – – – – 0081 20:03:27.93 44:09:19.1 0.808 12.784 – – – – – (87.68) (4.21) – – – – – 1.352 15.014 – – – – – 0094 20:03:59.34 44:10:25.8 (74.95) (8.39) – – – – – 0191 20:03:33.48 44:13:53.4 0.049 15.828 21.718 – – – – (68.81) (3.27) (3.45) – – – – 0248 20:03:38.79 44:04:53.0 4.573 20.059 – – – – – (112.83) (6.24) – – – – – 0487 20:03:49.82 44:11:08.5 4.187 19.812 – – – – – (167.09) (6.53) – – – – – 5.455 28.174 – – – – – 0494 20:04:00.17 44:14:03.2 (227.06) (3.79) – – – – – 29.797 1.366 – – – – – 1077 20:04:13.87 44:03:45.8 (32.53) (3.58) – – – – – 1088 20:03:56.20 44:12:49.9 5.409 0.534 15.667 33.018 51.303 60.128 192.909 (7.23) (6.19) (3.03) (2.87) (2.66) (2.51) (2.39) 1274 20:04:26.66 44:05:35.9 4.358 14.198 57.708 82.633 118.726 – – (6.63) (4.02) (3.17) (3.35) (2.87) 1292 20:03:41.23 44:12:17.9 0.758 3.351 22.877 41.456 52.343 75.111 245.455 (6.41) (6.30) (2.39) (2.50) (2.27) (2.94) (2.56) 3.613 24.235 182.588 – – – – 1421 20:03:41.08 44:08:47.4 (3.30) (4.53) (2.16) – – – – 1583 20:03:58.70 44:11:33.5 3.362 12.179 222.367 288.641 – – – (3.67) (3.35) (2.31) (2.22) – – – Since, the power of the secondary pulsation mode of ID 487 and ID 494 is very low to compare to its primary pulsation mode but it may use for correction in the Rucinski & Duerbeck (1997) relation. The said secondary pulsation mode are interfering with the primary mode, but not changed the nature of the phase curve due to their low strength. Since, secondary pulsation of these variable superimposed in its primary mode, therefore, we are adding a new term 2PS log(P ) in the value of log(P ) of Rucinski & Duerbeck (1997) relation, in which PS is the period of secondary pulsation mode. The corrected relation is given as below, MV = −4.44 (1 + 2PS ) log(P ) + 3.02(B − V )o + 0.12. (16) This relation provides the values of absolute magnitude as 5.89 mag and 6.37 mag for ID 487 and ID 494 respectively, which are close to the absolute magnitude as comes through CMD. 6.1 Other Results Joshi et al. (2012) were mentioned in their manuscript that they have identified seven such variables which having better representing with twice the 15 G. Chandra period given by the Lomb-Scargle periodogram. On the behalf of Section 4. 2 of their manuscript, there are also some binary stars, for which, the best fitted period has considered to half that of the real ones. In both cases, they have argued that the corresponding phase diagram is visually good. The scientific statement is totally absent about the half/twice of the real period as obtained through their data points. It is noticeable in Table 3 that they might be taken the half of periods for PV variables, whereas twice for the WUMA, EA, EB and Elliptical type variables. In our present analysis, the variable of ID 191 shows deviated frequency from their given value and not satisfied their argument like for deviate period of other variables. 7. Conclusion The stellar dynamics and the stellar evolution history of the clusters may constrain through the detailed analysis of the identified variables within the cluster. The ongoing physical and evolutionary phenomena of the stars are producing various types of variation in their stellar magnitudes. These variations are leading the secondary pulsations of variables. These stellar variations would be resolved through the temporal analysis of the frequency spectrum of variables. In the present work, we are revisiting the frequency spectrum of variables through the available photometric data. The peak of the greatest amplitude of FFP of a variable is defined as the prominent/principal peak, whereas, other remaining peaks may secondary pulsation modes. We have not found secondary pulsation modes for EA, EB, PV, rotational and semi-regular type variables. For other variables, the high scattering of data points find in the phase-diagram of each secondary pulse. This scattering may produce either due to superimposed of the wave nature of other pulsation or due to estimation error. Since, the scattering decreases the sharpness of the characteristics of the variables, therefore, an iterative-moving-average-procedure has adopted to remove the effect of scattering. The continuous decrements of amplitude of peak occur with each cycle of this procedure and makes major disadvantage/drawback of this procedure. These secondary pulsation modes are utilized for modifying the relations among various parameters of the HADS and W UMA type of variable stars. The ID 1274 does not appear WUma, whereas it claimed as the W Uma by Joshi et al. (2012). The difference in model and observed values of colour [(B − V )o ] and absolute magnitude for ID 1088 and ID 1274 indicates that both are field stars. We are re-classified ID 1077 as an SX Phoenicis variable instead of HADS type variable. Since, SX Phoenicis variable are cousins of the Delta Scuti variables (i.e. dwarf-Cepheid). The variable ID 1077 also satisfies the age and period relation of Cepheid variables, the result of said relation shows close agreement with the log (age) of the cluster. The number of secondary pulsation modes in the WUMS stars used to change the earlier relation, which provides an opportunity to 16 estimate the model absolute magnitude of the variables. Furthermore, the present finding of W Uma variables have confirmed by the photometric results of observations. As a result, we conclude that investigation of secondary pulsations is open an opportunity to develop their identification techniques and to constrain the models of their arisen. Since, we have proposed to signify limit for searching the pulsations, therefore, we have examined each pulsation of variables, having an amplitude peak of asymptotically parabolic pattern is greater than this value. We have been finding clear evidence of pulsation for those variables, having specific S/N is greater than 1. 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