Photometric and statistical comparisons of the old open cluster M67 (NGC 2682) using KFISP and Gaia EDR3 astrometry

ABSTRACT We present the BV photometric observations of the open cluster M67 using the Kottamia Faint Imaging Spectro-Polarimeter (KFISP), mounted at the 74-inch telescope. We build three colour-magnitude diagrams of the Gaia EDR3 and the BV. We derived the photometric and astrometric parameters of the cluster. The mean parallax is 1.167 mas (distance is 857 ± 22 pc). The age and the reddening E(B-V) of the open cluster are 4.00 Gyr and 0.03 mag, respectively. We identified 434 members from the Gaia EDR3, with a probability >70%. The mean proper motion is obtained as mas/yr and mas/yr. We used statistical comparisons to test the quality of the observations taken from the KFISP. The statistical comparisons in the BV bands showed that there are no significant differences between observations and published catalogues.

This work aims to test the quality of the observations from the Kottamia Faint Imaging Spectro-Polarimeter (KFISP) of the 74-inch telescope, Egypt. Determination of physical parameters of the standard open cluster M67 using the high-quality of the astrometry from the latest version of the Gaia telescope, Gaia EDR3, and the BV observations taken from the KFISP. Then compare with previous studies. We used the statistical comparisons between our observations and the available data from the literature to test the quality and deviations from the published literature.
We used three CMDs of the Gaia EDR3 and the BV CCD observations of the KFISP. We determined the distance (astrometry) of the cluster using high-quality parallax from the Gaia EDR3 and three CMDs (photometry). Observations and CCD calibration data are presented in Section 2. The membership, photometric, and physical parameters are explained in Section 3. The statistical comparisons methods and discussions between our BV magnitudes and the published photometric data are described in Section 4. The conclusions are presented in Section 5.

Observations and CCD calibration data
The charge coupled device (CCD) photometric observations of the open cluster M67 in the B and V bands were obtained using the KFISP. The KFISP is mounted at the Cassegrain focus of the 74-inch telescope of the Kottamia Astronomical Observatory (KAO), Egypt (Azzam et al. 2020& Azzam et al. 2021. The observations were taken on 28 January 2020 with an exposure time of 3 × 60 and 3 × 15 s in the B and V bands, respectively. The CCD (2k x 2k) of the direct imaging mode has a field of view of 8 × 8 arcmins and the pixel scale is 0.25 arcsec/pixel. The mean seeing is 1.4 and 1.3 arc seconds in the B and V bands, respectively, while the mean air mass is 1.08 in B & V ( Table 1).
The CCD data were observed with one amplifier, the basic reduction of CCD frames was corrected by using the IRAF's code (Hendy and Bisht 2021). The sky map for M67 in V-band is shown in Figure 1.  To obtain instrumental magnitude, we used the Point Spread Function (PSF) of the DAOPHOT package on the IRAF (Stetson 1987&1992). We used several isolated stars not saturated and distributed over the entire CCD to obtain good photometric results. We used aperture corrections of isolated and unsaturated bright stars in the CCD frame.
To convert instrumental magnitude into standard magnitude, we cross-identified stars in BV observed data with the APASS catalogue (Henden et al. 2016), we obtained 32 stars, see the yellow circles in Figure 1. We used the calibration curves to convert the instrumental magnitude to the standard magnitude V APASS , see Figure 2. We obtained the linear fitting B APASS = 0.9998 B inst -1.3369 and V APASS = 1.0075 V inst -1.8147 with a correlation coefficient are 0.99.
We obtained a standard magnitude of 245 stars in the B and V bands. The mean error in the V is 0.1 at V ~ 19.5 mag, Figure

The membership, photometric, and physical parameters
We searched in the Gaia EDR3 (Gaia Collaboration et al. 2021) using a large radius within our criteria, the maximum radius is 3 r 50 ~ 30 arcmins. The r 50 is the radius containing half the members (Cantat-Gaudin et al. 2018). We present the photometric errors in the G-mag in Figure 5, the maximum error is 0.015 mag. The distributions of proper motions and their errors with G-mag are displayed in Figure 6.
Proper motions have an important role to remove field stars from the main sequence of an open cluster (Hendy and Bisht 2021). To obtain a good quality from the astrometric data of Gaia EDR3, we remove the stars with the error of the parallax and proper motions >0.5. We used the Vector Point Diagram (VPD) of the proper motions (μ α cosδ & μ δ ) to select the over-density region of the open cluster M67. We selected 591 candidate member stars inside the red box. These candidates show a good trend with parallax and parallax error Figure 7.
To obtain membership probabilities of stars in the M67, we remove the stars with proper motions and parallax larger than 3 Median Absolute Deviation (MAD) from the median proper motions and median parallax of the M67. We obtained 434 member stars from Gaia EDR3, with membership probability >70% in Cantat-Gaudin et al. (2018). The CMD of the Gaia EDR3 and the combined CMD of V-mag and the colour (G BP -G RP ) show a good trace of the main sequence. The co-moving member stars move in the same direction in the space (Figure 8).
We used the three CMDs of the Gaia EDR3, the combined CMD of the V-mag and the colour (G BP -G RP ), and the CMD V (B-V) to obtain the best fitting of the age, distance modulus, and the reddening,  The photometric distance of the M67 d photometric = 802 ±36 pc. The correcting of mean parallax is 1.167 mas with the offset 0.017 of Lindegren et al. (2021). This led to the distance from inverting parallax d astrometric = 857 ±22 pc. This is in good agreement with our fitting. Our measurements of the centre, mean parallax, mean proper motion in RA and DEC, distance, age, and reddening are in very good agreement with the literature, see Table 2.

Statistical comparisons methods and discussions between our BV magnitudes and the published photometric data
We compared our BV bands observations of the same stars in the open cluster M67 with the same stars in published catalogues as listed in Table 3. The comparison is based on the correlation coefficients and statistical testing hypothesis by independent Samples T-Test.

The correlation coefficients
First, we examined the correlation coefficients between the V-band observations and the previous studies for M67. We calculated these correlations by the Pearson's correlation coefficient (Bobko 2001) from the next formula as follows: where r is the correlation coefficient between the populations x and y. cov(x,y) is the covariance between populations. σ x and σ y are the standard deviations for the populations    Table 3 contains the sample sizes of stars combined with us and Pearson's linear correlation coefficients (r) between our observations of B and V bands and the previous studies. These correlation coefficients are calculated by applying equation (1). The correlation coefficient between our observations of BV bands and the previous studies is very strong, which is equal to 99.80%, 100%, 98.70%, and 99.90% in B-mag and 99.30%, 99.60%, 99.20%, and 99.60% in V-mag as listed in Table 3.
Second, by an independent statistical T-Test for comparing equal population means. This method depends on basic assumptions such as the samples being independent of each other. The population of observations from which the sample was taken follows a Gaussian (normal) distribution and the sampled populations have homogeneous variances.
The next step is to confirm the comparison using the testing hypothesis. To apply the independent statistical T-Test, it is necessary to verify the previous assumptions and see whether they are fulfiled or not given that the first of the three conditions are fulfiled.

Test of normality
There are different methods to test the normality of the data. The common methods are the two tests Kolmogorov-Smirnov (K-S) and the Shapiro-Wilk (W).

• First: The (K-S) test
Null hypothesis, H 0 : The observational data follow a normal (Gaussian) distribution.
Alternative hypothesis, H 1 : The observational data do not follow a normal (Gaussian) distribution.
Test Statistic: The relation of (K-S) test statistic is defined as (Chakravarty et al. 1967) Where F is the cumulative normal (Gaussian) distribution function. The normality hypothesis is accepted H 0 if the test statistic, K s is smaller than the critical value obtained from the K-S table.  The statistical programs have used the probability value (p-value) to determine the critical region for rejection or acceptance of the null hypothesis, where the p-value is the probability of getting results close to the real ones, assuming that the null hypothesis is true. We will use this value in all our results.

Second: The W test
The Shapiro-Wilk test is denoted by W, The W statistic is calculated from the next formula (3) (Chakravarty et al. 1967): where the x (i) is the ordered sample values and the a i are constants generated from the central tendency and dispersion measures of sample size (n) taken from a Gaussian distribution. We compared with the tabulated value that was reproduced by (Pearson and Hartley 1972) or by using the p-value. The first condition for applying the Independent Sample T-Test is to test the normality between the different populations under test. We apply the two Equations (2) and (3), we get Table 4 for our observations and different authors.
The significant level (α = 0.05) is smaller than the p-value in column 2 (Kolmogorov-Smirnov) and column 3 (Shapiro-Wilk) as shown in Table 4. Then, we cannot reject H 0 that all populations follow a normal (Gaussian) distribution.

Test of homogeneity
Homogeneity means variance between the groups as equals, Levene's test is used to test the homogeneity between populations (Levene1960). If the resulting  probability value (p-value) of this test is less than some significant level (0.05), the null hypothesis of equal variances will be rejected and there is a difference between the variances in these observations.
The steps of the test are the following: The Statistical hypothesis H 0 : The variances are homogenous in two populations.

Statistic test
If the size of the observations is N concerning the Y variable and divided into K subgroups and the size of the i-th subgroup is N i , the Levene test statistic is described as follows: where � Y i is the mean of the i-th subgroup. and � Z i are the means group of the Z ij and � Z is the overall mean of the Z i.

Critical region
The Levene's test rejects the null hypothesis that the variances are equal if where F α, k−1, N-k is the F distribution with degrees of freedom k-1 and N-k at a significant level of α. We used equation (4) to get Table 5. We find that the p-value of the Levene test is greater than the significant level (α = 0.05), so we cannot reject the null hypothesis. Then, the variances are equal between these populations.

The independent sample T-Test
After confirming the validity of the data test fulfiling the normal distribution and homogeneity, we compared between BV observations and four published catalogues in different observatories for cluster M67 by independent Samples T-Test. We   compared the mean of observations of BV bands with each of the other four observatories to determine whether there is a significant difference between the means in two independent groups.
The T-Test is the test statistic of the statistical hypothesis test and follows the student T-distribution under the hypothesis of the study. We explain the steps of testing the hypothesis for means μ.

I-Hypothesis of study
The t statistic is calculated from the following equation: where � X 1 and � X 2 are the means of two samples. And S 2 p is the pooled variance of equal size for the two samples (n 1 = n 2 ) and S 2 2 , S 2 2 are the variances of the two samples.

Iv-Critical region
we reject the null hypothesis, if the statistical test is located in the rejection or acceptance region, or the P-value is smaller than the significant level α = 0.05.

V-The decision
We can reject or accept H 0 according to the last step (iv).
We apply Equations (7) and (8) to calculate the statistical T-Test. Table 6 reveals the comparison between our observations and other studies.
The most important column in Table 6 is the third because it is the decisive factor to determine the acceptance or rejection of the hypothesis. The p-value is greater than the significant level (0.05) with all the previous studies. Thus, we cannot reject the null hypothesis that there is no significant difference between our study and the other studies.

Conclusions
We presented the physical parametric study of the old open cluster M67 using the CCD BV-bands of the Kottamia Faint Imaging Spectro-Polarimeter (KFISP) and the Gaia EDR3. We estimated the 434 members using the Gaia EDR3, with a membership probability higher than 70%. We used those members to obtain the photometric and astrometric parameters. These measurements of the parameters are in very good agreement with published results, Table 2. We determined the distance using the inverting parallax (d astrometric = 857 ± 22 pc) from Gaia EDR3 and using three CMDs of Gaia EDR3 and BV CCD observations (d photometric = 802 ± 36 pc). The correlation coefficients between our measurements of the BV bands and the others are very strong correlation. The normality and homogeneity for the KFISP observations of all populations show the normal (Gaussian) distribution and equal variances (homogeneity). The T-Test showed that there are no significant differences between BV observations and published results. This means that the observations of KFISP are consistent with the published catalogues of photometric data in BV bands.

Disclosure statement
No potential conflict of interest was reported by the author(s).