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JOURNAL OF SCIENCE OF HNUE Mathematical and Physical Sci., 2012, Vol. 57, No. 7, pp. 65-70 This paper is available online at http://stdb.hnue.edu.vn USING APERTURE PHOTOMETRY TO STUDY MESSIER 67 Doan Duc Lam and Nguyen Anh Vinh Hanoi National University of Education Abstract. Studying star clusters is a way to test theories of stellar evolution in astronomy. Using current theories, the mass, age and components of a star are affected and decide the position of star in the Hertzsprung-Russell color-magnitude diagram. To study the role of mass in stellar evolution, we need to look at star clusters of which all stars were formed at the same time and are located at the same distance from planet Earth. In our study we have done photometry for open cluster M67. Photometric results are plotted in the color-magnitude diagram. The number of stars in each evolutionary phase has been estimated. The existence of blue stragglers in our result requires further research about stellar evolution and interacting binary stars. Keywords: Photometry, Messier 67, astrophysics. 1. Introduction A star is a physical entity which evolves over time and changes in phase. The stellar phase is indicated by parameters such as the temperature, pressure, generation and transportation of energy within a star. Stars, having different initial masses, evolve through different stages over time. A cluster is a system of stars which were formed from a single gas cloud. There are two types of clusters: open cluster and global cluster. All stars in a cluster are presumed to have been born at the same time so the current phase of the stars is decided by their initial mass. Messier 67 (M67) or NGC 2682 is an open cluster that is located in the Crab constellation. This is one of the oldest clusters and is of importance in the study of stellar evolution. Within it are at least 500 stars [1], and among them are about 150 white dwarfs, 100 solar-type stars, some blue stragglers and some giant stars. The age of M67 is about Received December 27, 2011. Accepted September 10, 2012. Physics Subject Classification: 60 44 19 . Contact Nguyen Anh Vinh, e-mail address: vinhastro@yahoo.com 65 Doan Duc Lam and Nguyen Anh Vinh 4 to 5 billion years. It is located at a distance of about 800 to 900 pc from the Earth and the distance modulus of the V filter is (m − MV ) = 9,59 [2]. 2. Content 2.1. Aperture photometry Astronomical photometry aims to measure energy of celestial objects. The information obtained helps determine the magnitude, luminosity and temperature of stars. Aperture photometry is one of the easiest of the methods available. Suppose that images which are captured by CCD camera are calibrated using bias and darkness, they are on a flat field. The following are steps to be taken to carry out aperture photometry (Figure 1): Step 1. Identify the star one wishes to measure Step 2. Determine the center (xc , yc ) and radius of the aperture. Brightness distribution is Gaussian. The radius of the aperture must be large enough to hold entire star but it not so large that it will reduce background noise. The radius of the annulus used to measure brightness of background is about three to five time that of the aperture. Figure 1. Illustration of aperture photometry xc = i=L P i=−L i=L P i=−L where
Ii − I xi Ii − I
; i=L X 1 I= Ii , 2.L + 1 i=−L j=L X 1 Jj , J= 2.L + 1 j=−L yc = j=L P j=−L j=L P
Ji − J y j j=−L Ii = Ji − J j=L X Ii,j j=−L Ji = i=L X Ii,j i=−L and i, j are indexes to indicate row number and column of pixel. 66
(2.1) Using aperture photometry to study Messier 67 Step 3. Add the counts (Nap ) within measurement aperture (area of aperture Aap ). It is equal the number of electrons multiplied by Gain. Step 4. Estimate the value of background per pixel Aap using an annulus aperture. Step 5. Compute instrumental magnitude:   Nap − Aap Ssky mI = −2.5 log (2.2) texp where texp is the time of exposure. Step 6. Correct for Earth’s atmospheric extinction and interstellar extinction to determine apparent magnitude m0 : mI − m0 = −2.5 log(F/F0 ) = −2.5 log(e−K sec Z ) (2.3) where K is the extinction coefficient for one air mass and Z is zenith distance. Step 7. Convert to standard filter photometry with transformation equations. Because all stars in the clusters are located at the same distance, so mI - Mcata is the same for all stars in cluster where Mcata is the magnitude of the reference star. We have computed mI - Mcata for reference stars and applied this value to other stars in the cluster. 2.2. Calculation and discussion All frames of M67 were taken by a system of telescopes and CCD cameras at an observatory in Marseille, France. We obtained two frames of M67 using a B and R filter in a Johnson-Kron-Cousins standard system. Exposure time of the CCD camera with the B and R filter was 10.04 seconds and 5.03 seconds, respectively. Data is processed with darkness, bias and flat field. The positions of reference stars are marked in a frame in Figure 2. Because of limited exposure time, each frame of M67 contains only 60 of the brightest stars. We have calculated instrument magnitude of all stars, “zero points” of reference stars and applied these values for all stars in the cluster: (mI − Mcata )B = 26.32 ± 0.05 (2.4) The color index for the B and R filter is (B − R) = 0.068. Extinction coefficients are K(B) = 0.154 for B filter and K(R) = 0.088 for R filter. After obtaining absolute magnitude of all stars in the cluster we have plotted them in the color-magnitude diagram (Figure 3). The horizontal axis is the color index for B − R and the vertical axis is the B magnitude. Our result shows that 40% of the stars in the main sequence are low to intermediate mass stars of A or B spectral type while 30% of the stars are leaving the main sequence and are concentrated near the ‘turn-off point’. So 67 Doan Duc Lam and Nguyen Anh Vinh Figure 2. Image of Messier M67. The reference stars are marked Figure 3. Color-magnitude diagram of open cluster M67 we can say that about one third of stars in the cluster have approximately the same mass. The ‘turn-off point’ is at 13 of B magnitude and 0.9 of color index B − R. 68 Using aperture photometry to study Messier 67 Table 1. Magnitude of stars in M67, MB and MR are the magnitude of stars using the B and R filter Star # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 MB 16.16 13.86 11.22 14.24 16.01 14.65 14.73 12.54 15.13 11.62 13.24 16.81 15.09 13.3 13.38 14.07 11.08 13.87 13.77 15.97 13.28 16.59 15.15 13.4 14.89 15.23 11.52 12.61 13.37 13.8 MR 14.96 12.94 10.84 13.39 14.79 13.73 13.77 12.12 13.86 9.988 12.32 15.15 14.02 12.36 12.62 13.13 10.17 12.93 13.01 14.79 12.38 15.16 13.82 12.46 13.77 14.05 9.901 11.84 12.14 12.88 B−R 1.192 0.917 0.382 0.854 1.223 0.92 0.953 0.415 1.269 1.628 0.919 1.669 1.068 0.94 0.761 0.947 0.918 0.94 0.759 1.185 0.901 1.434 1.331 0.94 1.125 1.185 1.622 0.77 1.237 0.919 Star # 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 MB 13.35 13.38 12.48 13.79 12.86 13.76 14.56 13.33 14.87 14.72 13.99 12.01 15.95 13.36 16.26 13.84 15.25 14.02 15.9 10.02 16.68 11.5 14.91 14.29 13.21 13.43 13.73 15.96 11.45 14.86 MR 12.46 12.6 10.89 12.83 11.92 12.81 13.6 12.4 13.71 13.63 13.06 11.3 14.75 11.88 14.95 12.87 14.19 13.11 14.56 10.04 15.45 9.641 13.99 13.37 12.38 12.5 12.75 15.01 11.22 13.8 B−R 0.895 0.772 1.588 0.966 0.943 0.948 0.962 0.929 1.164 1.095 0.924 0.71 1.202 1.476 1.305 0.974 1.063 0.914 1.344 -0.02 1.226 1.854 0.92 0.925 0.827 0.93 0.981 0.946 0.231 1.062 Note. B − R is the color index for the B and R filter, the margin of error is 0.05 Giant stars which have already left the main sequence appear above the main sequence in the upper right corner of Figure 3. They represent about 10% of the total number of stars in M67. White dwarfs should be located below the main sequence but they are not visible in our data and thus not present in the color-magnitude diagram. In previous studies, M67 is believed to be one of the oldest clusters so that the number of 69 Doan Duc Lam and Nguyen Anh Vinh white dwarfs and giant stars constitutes a considerable fraction of the total number of stars [1]. The lack of white dwarfs in our data could be due to equipment limitation and limited observation time. Blue stragglers, the brightest stars in the B filter, are located at the upper left corner of the color-magnitude diagram as shown in Figure 3. Their initial mass is a few times that of the solar masses. However, they don’t appear to be a part of the cluster’s evolution. The presence of blue stragglers in many clusters is still in question regarding stellar evolution. It has been suggested that blue stragglers came about as a result of interaction of binary stars. Almost all of the blue stragglers are found near the center of the cluster, a region crowded with many stars. This hypothesis agrees with many previous studies and does not support the idea that blue stragglers are in front of or behind clusters and not in a cluster. 3. Conclusion Using aperture photometry, we have measured magnitude and made color-magnitude diagrams for Messier 67. Suppose all of the stars in M67 were formed at the same time and located at the same distance from Earth. We estimate the number of stars in deferent phases of evolution. The result shows that the current phase of each star is dependent on its initial mass. This is in agreement with theories about stellar evolution. It is thought that the existance of Blue stragglers in the cluster is not a random coincidence. Acknowledgements. We are grateful to Georges Comte who provided the images of M67 for this research. REFERENCES [1] W.L. Sanders, 1977. Astron. Astrophys. Suppl., 27, 89. [2] Montgomery, K. A., Marschall, L. A., & Janes, K. A., 1993. AJ, 106, 181. [3] Bhat, P. N. et al., 1992. JApA, 13, 293. [4] Frogel, J. A, 1978. ApJ, 222, 165. [5] Nissen, Poul E.; Twarog, Bruce A.; Crawford, David L., 1987. AJ, 93, 634. [6] Sandquist, Eric L., 2004. MNRAS, 347, 101. [7] Mathieu, Robert D. et al., 2003. AJ, 125, 246. [8] Gilliland, Ronald L.; Brown, Timothy M., 1992. AJ, 103, 1945. [9] Anthony-Twarog, Barbara J., 1987. AJ, 93, 647. [10] Sanders, W. L., 1989. RMxAA, 17, 31. 70