由恆星自行運動辨別疏散星團NGC 7142及M45之成員星

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(1)國立臺灣師範大學地球科學研究所天文組 The Astronomy Group, Institute of Earth Sciences, National Taiwan Normal University 碩士論文. 由恆星自行運動辨別疏散星團 NGC 7142 及 M45 之成員星 The Membership Determination of Open Clusters NGC 7142 and M45 from Proper Motion Data. 研究生：謝翔宇 Hsiang-Yu Hsieh 指導教授：傅學海教授 Hsieh-Hai Fu 中華民國一○二年八月 August 2013.

(2) 誌謝在本研究當中，非常多的師長朋友給我多方面的協助。在這邊首先要感謝中央大學鹿林天文台之觀測助理蕭翔耀學長及中央大學天文所博士生鄭宇棋學長，在我臨時向兩位提出觀測需求時義不容辭地幫忙取得 NGC 7142 之天體影像，是本研究能夠完成的必要條件。而本研究會選擇此天體作研究，更是感謝莊孝爾學長在 2003 年的觀測資料，完整而詳細的影像在星等校準中扮演關鍵角色。這兩方面的協助是本研究的骨幹。也由於這些資料十分齊全，使我可以做更多的深入探討和研究，非常感謝他們。許多同學和朋友在研究中給予我直接和間接的協助。感謝育倫、憲隆和承峰在 meeting 時一起腦力激盪，以及那些一起努力和打混過的夜晚我是不會忘的。感謝冠州提供程式上的建議，還有皓元有耐心地教我使用 IRAF 一直到凌晨一點，經閔和明儀也時常熱情提醒我要專注於工作上。研究遇到問題時，晟庭和翔耀學長的論文是我最常拿起來參考的範本，學長們詳細的內容幫助我突破很多難關。在論文完稿階段，感謝萌婉、陳寬和維昕幫我進行繁複的論文校稿工作，還有可愛的 SUBARU 一直都很合作使得電腦沒被打翻過飲料。而碩士生涯中最意想不到的成就：第 11677 號行星。天文攝影聯展… 感謝有鴻選和所有的工作伙伴，我找到了一群可以一起追逐夢想的朋友。還有非常多的朋友們無法一一感謝，只能用一句我常說的話：要感謝的人太多了，不如就謝憨吧！在論文寫作階段，感謝家人對我作息日夜顛倒、常常在外過夜不回家的體諒，以及女友萌婉督促以及支持。對於口試完成後半年仍遲遲無法交出論文完稿，甚至拖延到人生的進程和規劃，我感到十分羞愧… 但此時更深刻感受到家人無悔的付出和愛，相較於我的壞脾氣和沒信用，家人給我最大的包容和接納，非常感謝他們。.

(3) 這裡要鄭重地感謝指導教授：傅學海老師。十年來和傅老的接觸和學習中，老師總能帶給我新的啟發和觀點。獨立判斷的能力、抽象思考的觀念、對新知開放的態度，以及注重科學精神的培養，乃至於教育學生的方法和課程規劃，老師全方位的示範了學者的研究和教育方法，對於想成為一位好老師的我是個很棒的鼓舞。碩士生涯中有幾次和老師有言語衝突，卻都是衝擊著我需要更新的觀念。即使我並不是個用功的學生，傅老仍對我依然充滿期許，令我十分感動。很榮幸身於傅 group 的一員！最後要感謝口試委員陳林文教授、陳岸立館長和傅學海教授，在我倉促交出論文初稿後短短的幾天內審查我的論文，而且老師們都有相當詳細的回應和批改，十分感謝口試委員們給我的建議。而在口試通過後修訂論文完稿，雖然我把時間一拖再拖，但傅老師交付的任務才使這篇論文更具價值，例如完成了 U 星等校準因而有雙色圖以供判斷紅化值，有了這些內容才能算是完整地將學長們辛苦取得的影像物盡其用。若本研究寫作上有太過冗長或過度詳細之處，是因為我想盡可能讓後人理解這個研究過程的細節，減少他們花費在摸索的時間上面。而本研究中若還有疏漏和不完整的地方，還請發揮您的想像力補完了… 希望日後的研究者能多多給我建議、討論和提醒，未來對這個領域能作出更多貢獻。 Email：[email protected]. 謹以此文獻給所有支持我走向研究之路的人.

(4) 摘要使用 Johnson UBV 光度資訊推算疏散星團的距離和年齡，首要工作是能夠分辦出星團的成員星。本研究使用雙變數常態分布之數學模式分析恆星自行運動，計算出每顆恆星為星團成員的或然率。本研究使用 2003 年 8 月 7 日和 2012 年 7 月 8 日在中央大學鹿林前山天文台的一米望遠鏡觀測之疏散星團 NGC 7142 之影像，得出其中 1697 顆恆星的 UBV 光度及自行運動資訊。而本研究同時也選擇擇了 M45(昴宿星團)作為距離、星際紅化消光已知之對照星團，採用 Tycho-2 星表記載之亮度及自行資訊。在分析恆星自行運動後得到每顆恆星為成員星之或然率，將或然率較高之恆星挑選出來繪製星等-星色圖和雙色圖。由星團之雙色圖擬合，以及文獻之 UBV 紅化斜率 0.72，得到 NGC 7142 的金屬豐度[Fe/H] = -0.37 以及紅化係數 E(B-V) = 0.36。而星等-星色圖由主序帶擬合之結果可得出距離模數(m-M)0 = 11.02，等同於 1600pc。而年齡由星等-星色圖之等時線擬合而得出，約為 79 億年。在 M45 之星等-星色圖代入文獻值 E(B-V) = 0.04、[Fe/H] = 0.022 以及(m-M)0 = 5.66 後，由等時線擬合得到年齡 1.26 億年。本研究比較 M45 和 NGC 7142 之主序帶星等，代入 M45 之文獻值 136pc，得到 NGC 7142 之距離為 1920pc。. 關鍵字：疏散星團、成員星、恆星自行、星等星色圖、雙色圖、 NGC 7142、M45.

(5) The Astronomy Group, Institute of Earth Sciences, National Taiwan Normal University. Master’s Thesis. The Membership Determination of Open Clusters NGC 7142 and M45 from Proper Motion Data. Researcher: Hsiang-Yu Hsieh Advisor: Hsieh-Hai Fu. August 2013.

(6) Abstract For accurately estimating the distance and age of an open cluster with Johnson UBV photometry, the membership of the cluster should be determine first.. The probability of membership for each star in the two. open clusters: NGC 7142 and M45 are determined in terms of the normal bivariate frequency function with proper motion data.. The database of. proper motion and UBV photometry calculated from the images for NGC 7142 observed in this work, and proper motion database of M45 taken from the Tycho-2 catalog. The observations of NGC 7142 carried out with the Lulin-One meter Telescope of National Central University on August 7, 2003 and July 8, 2012, respectively.. A catalog contained 1697 stars with UBV. photometry, proper motion and membership probability data is set up. From the main-sequence fitting on the Two-Color Diagram with the UBV reddening slope 0.72, the estimate value of the metallicity Z is 0.008, represents [Fe/H] = -0.37, and the E(B – V) = 0.36.. The distance. modulus is (m-M)0 = 11.02, represents distance is about 1600pc.. From. the isochrones fitting, the age of NGC 7142 is 7.9 Gyr. The M45 been chosen for comparison with almost no extinction and distance well known.. With the E(B-V) = 0.04, (m-M)0 = 5.66 and [Fe/H]. = 0.022, the isochrones fitting of age is 126 Myr.. From the main-. sequence comparison with the distance of M45 (136 pc), the distance of NGC 7142 is 1920 pc. Key words: open cluster, membership probability, proper motion, ColorMagnitude Diagram, Two-Color Diagram, NGC 7142, M45.

(7) Contents 1 Introduction. ------ 01. 1-1 Membership of open cluster. ------ 01. 1-2 NGC 7142 properties. ------ 02. 1-3 M45 properties. ------ 06. 2 Observation and Data. ------ 09. 2-1 LOT observations for NGC 7142. ------ 09. 2-2 Tycho-2 catalog for M45. ------ 17. 3 Data Reduction for NGC 7142. ------ 20. 3-1 Magnitude calibration. ------ 20. 3-2 Coordinate calibration and proper motion determine. ------ 35. 4 Membership determination. ------ 40. 4-1 Data selection. ------ 40. 4-2 Data process. ------ 43. 4-3 Membership probability. ------ 45. 5 Results and Discussion. ------ 54. 5-1 Color-Magnitude Diagram. ------ 54. 5-2 Two-Color Diagram. ------ 57. 5-3 Reddening and extinction. ------ 59. 5-4 Age and distance. ------ 63. 6 Conclusions. ------ 69. Acknowledgement. ------ 70. References. ------ 71. Appendixes A. M45 membership with PPMXL catalog. I. B. IRAF magnitude measure information. XI. C. IDL source code Program 1: Proper motion determine. XIV.

(8) Program 2: Membership determine D. E. F. XVIII. NGC 7142 catalog with membership Images. XXVIII. Text. XXXVIII. M45 images and catalog with membership Images. LXVIII. Text. LXXIII. DVD disk (with images, programs and data). LXXX.

(9) List of Figures Fig. 1-2.1. NGC 7142, taken by LOT. 4. Fig. 1-3.1. M45, taken by Yen Hung-Hsuan. 7. Fig. 2-1.1. NGC 7142 U band image (2003). 10. Fig. 2-1.2. NGC 7142 B band image (2003). 11. Fig. 2-1.3. NGC 7142 V band image (2003). 12. Fig. 2-1.4. NGC 7142 B band image (2012). 13. Fig. 2-1.5. NGC 7142 V band image (2012). 14. Fig. 2-1.6. Standard star field: SA 110, taken by LOT. 16. Fig. 2-1.7. Standard star field: SA 111, taken by LOT. 16. Fig. 2-2.1. M45 catalog from Tycho-2. 18. Fig. 2-2.2. M45 proper motion data from Tycho-2. 18. Fig. 2-2.3. M45 Tycho-2 data histogram, B and V mag.. 19. Fig. 2-2.4. M45 color-magnitude diagram. 19. Fig. 3-1.1. Errors of standard star field: SA 110.. 21. Fig. 3-1.2. Errors of standard star field: SA 111.. 21. Fig. 3-1.3. Color correlation in B and V calibration. 22. Fig. 3-1.4. B magnitude calibration. 23. Fig. 3-1.5. V magnitude calibration. 23. Fig. 3-1.6. Errors of inst. bv mag. in 2012 of NGC 7142. 24. Fig. 3-1.7. Errors of calibrated BV mag. in 2012 of NGC 7142. 25. Fig. 3-1.8. B mag. compare with reference. 26. Fig. 3-1.9. V mag. compare with reference. 26. Fig. 3-1.10. Color correlation in 2012B bright stars. 27. Fig. 3-1.11. Color correlation in 2012V bright stars. 27. Fig. 3-1.12. Calibrated B magnitude with reference. 28. Fig. 3-1.13. Errors of inst. BV mag. in 2003 of NGC 7142. 29. Fig. 3-1.14. 2003 B mag. calibration with reference mag.. 30. Fig. 3-1.15. 2003 V mag. calibration with reference mag.. 30.

(10) Fig. 3-1.16. 2003 U mag. calibration with reference mag.. 31. Fig. 3-1.17. Short time exposure in BV with reference. 32. Fig. 3-1.18. Final result of errors of 2003 BV magnitude. 33. Fig. 3-1.19. CMD, NGC 7142 with 2003 BV magnitude. 33. Fig. 3-1.20. TCD, NGC 7142 with 2003 UBV magnitude. 34. Fig. 3-1.21. CMD, NGC 7142 with 2003 UBV magnitude. 34. Fig. 3-2.1. Magnitude error with calibrated mag.. 35. Fig. 3-2.2. Histogram of 2003 BV mag.. 36. Fig. 3-2.3. Histogram of 2012 BV mag.. 36. Fig. 3-2.4. NGC 7142 catalog and position reference star. 38. Fig. 3-2.5. VPD, NGC 7142 proper motion distribution. 38. Fig. 3-2.6. VPD, NGC 7142 close up view. 39. Fig. 3-2.7. 3-D surface histogram of NGC 7142 proper motion. 39. Fig. 4-1.1. CMD, M45 after data BV selection. 41. Fig. 4-1.2. VPD, M45 proper motion. 41. Fig. 4-1.3. VPD, M45 proper motion zoom-in. 42. Fig. 4-1.4. 3-D surface histogram of M45 proper motion. 42. Fig. 4-2.1. VPD, NGC 7142 rotated. 44. Fig. 4-2.3. VPD, M45 rotated. 44. Fig. 4-3.1. VPD, NGC 7142 proper motion membership. 48. Fig. 4-3.2. Histogram of NGC 7142’s VPD. 48. Fig. 4-3.3. VPD, M45 proper motion membership. 49. Fig. 4-3.4. Histogram of M45’s VPD. 49. Fig. 4-3.5. Histogram of NGC 7142 membership probability. 51. Fig. 4-3.6. Histogram of M45 membership probability. 51. Fig. 4-3.7. Sky map of NGC 7142 membership with all stars. 52. Fig. 4-3.8. Sky map of NGC 7142 membership. 52. Fig. 4-3.9. Sky map of M45 membership with all stars. 53. Fig. 4-3.10. Sky map of M45 membership. 53. Fig. 5-1.1. CMD, of NGC 7142 membership > 70%. 55.

(11) Fig. 5-1.2. CMD, of NGC 7142 membership < 70%. 55. Fig. 5-1.3. CMD, of M45 membership > 95%. 56. Fig. 5-1.4. CMD, of M45 membership < 70%. 56. Fig. 5-2.1. TCD, of NGC 7142 membership > 90%. 57. Fig. 5-2.2. CMD, of membership suspected red giant stars. 58. Fig. 5-2.3. TCD, of membership suspected red giant stars. 58. Fig. 5-3.1. TCD, main-sequence fitting with different Z. 59. Fig. 5-3.2. TCD, main-sequence fitting with 4 different Z. 60. Fig. 5-3.3. TCD, main-sequence fitting with different E(B-V). 60. Fig. 5-3.4. TCD, main-sequence fitting with E(B-V). 61. Fig. 5-3.5. CMD, main-sequence fitting with E(B-V), (m-M)0. 62. Fig. 5-4.1. CMD, NGC7142 isochrones with different author. 63. Fig. 5-4.2. TCD, NGC7142 isochrones with different author. 64. Fig. 5-4.3. CMD, NGC 7142 fitting with different age. 65. Fig. 5-4.4. CMD, NGC 7142 fitting with 7.94 Gyr. 65. Fig. 5-4.5. CMD, M45 fitting with different age. 66. Fig. 5-4.6. CMD, M45 fitting with 126 Myr. 66. Fig. 5-4.7. CMD, NGC 7142 and M45 comparison. 68.

(12) List of Table Table 1-2.1 NGC 7142 properties with different researchers. 5. Table 1-3.1 M45 properties with different researchers. 7. Table 2-1.1 Observation information of NGC 7142. 9. Table 2-1.2 Standard star exposure time. 15. Table 2-1.3 Standard star airmass information. 15. Table 2-1.4 Standard stars of SA110. 15. Table 2-1.5 Standard stars of SA111. 16. Table 3-1.1 Standard star field magnitude information. 20. Table 4-3.1 NGC 7142 membership parameters. 47. Table 4-3.2 M45 membership parameters. 47.

(13) 1 Introduction In this study, the most important purpose is to identify the member stars of an open cluster from the field stars. Base on the statistic method first presented by Vasilevskis et al. (1958), the membership probability of each star could be compute, and two open clusters with different distance and age, NGC 7142 and M45, are chose for test and verify this method. 1-1 Membership of open cluster An open cluster is a group of stars, formed from a same giant molecular cloud with familiar chemical component, same age and dynamical property. The distance and age information of the open cluster related to the star formation, stellar type, and the evolution track on the color-magnitude diagram (Shapley 1920; Johnson and Morgan 1953; Sandage 1970; Montgomery et al. 1993; Bruzual and Charlot 2003).. However, the accurate photometry and. color of stars will be a big problem due to the extinction and reddening of stellar light that though interstellar medium (Sharov 1965). The main sequence of an open cluster is the key for fitting the isochrones, but the main sequence maybe too faint to determine for some older open clusters (van den Bergh 1962), and it’s less reliability to fit isochrones only for the red giants (Sharov 1968).. Many methods in terms of the photometry, CMD and. TCD, are used to find the members of the open clusters (for example, Castellani et al., 1992), and the method of determining the membership in terms of the proper motions is better method (Vasilevskis et. al. 1958; Sanders 1971). The dynamical property of members and field stars are different.. Thus, they. will separate in the Vector Point Diagram (VPD), and the separation appears often in nearby open cluster (van Leeuwen 2009).. However, nearby open. clusters usually has larger angular size, that caused many different dynamical 1.

(14) property field stars mixed into the member stars’ distribution on the VPD, that increase the difficulty of membership identification. Vasilevskis et al. (1957) assumed all the stars have a flat distribution of field stars and a normal distribution for membership stars on the VPD.. The first. mathematically rigorous procedure developed by Sanders (1971) with a statistical analysis of proper motion, and a mathematical procedure used to get the probability of membership for each star.. The CMD of the members will. plotted with the probability higher than a suitable value. The Sanders’ method only had little improvement in the past years.. Stetson. (1980) and Zhao and He (1990) improved the membership determination based on proper motions with different observed accuracies, and Sagar (1987) introduced additional parameters for cluster members’ elliptical distribution. Furthermore, Zhao and He added the correlation coefficient of the field star distribution in the Sanders’ equation, and Tian et al. (1998) added radial velocity data for this maximum likelihood method.. Base on the Sanders’ method,. Balaguer-Nunez et al. (1998) added the observed errors in the proper motion components of each star.. Bellini et al. (2009) and Sariya et al. (2012) also. used this improved method.. However, due to the lack of information of. observed errors in the IRAF program procedure in Chapter 3.. Furthermore,. the accuracy of the position of each star could not reach 2 mas.. Therefore, this. work still used the original Sanders’ equation in Chapter 4. 1-2 NGC 7142 Properties NGC 7142 (RA: 21h 45m 09s, Dec: +65° 46’ 30”) is a distant open cluster with aged ~7 Gyr (Janes and Hoq 2011), and it is also a small cluster containing a few hundred stars brighter than the 20th magnitude (van den Bergh 1962). This area has variable absorption explored in detail by Risley (1943).. A young. open cluster NGC 7129 (RA: 21h 42m, Dec: +66°06’, J2000) surrounded by a 2.

(15) reflecting nebula as a background, which partially obscures some regions in NGC 7142 that caused the absorption in the field is irregular (Punanova et al. 2012).. According to Trumpler (1930), the type of NGC 7142 was classified as. II 1m with an angular diameter of 11’.. However, Sharov (1965) extensive. amount of the stars in the field indicated that the total cluster diameter is greater than 75’, and he corrected the value to 23’ in 1968.. NGC 7142 as the target in. this work was due to that it located in the field with variable absorption (Sharov 1965) and it is probably not a real cluster (Jennens and Helfer 1975). There are many studies notices about the metallicity with different [Fe/H] value.. Most of them determine from spectroscopic measurements of giant. stars, like [Fe/H] = +0.14 gave from Jacobson et al. (2008).. Others determine. from main-sequence fitting on the CMD, like [Fe/H] = +0.13 gave from Sandquist et al. (2011). The first UBV survey of NGC 7142 done by van den Bergh and Heeringa (1970), and they listed 188 stars with precise BV magnitude and with few hundred stars has U magnitude.. Crinkdlaw and Talbert (1991) listed the first. CCD photometry, the BV photometry extends to V~18, and they estimated differential extinction across the cluster of 0.1 mag errors.. Adopting [Fe/H] =. -0.1 (Canterna et al. 1986) and E(B-V) = 0.41 (van den Bergh and Heeringa 1970), they derived distance modulus, (m-M)0 = 11.4 ±0.9 by fitting mainsequence with another old open cluster M 67. To search variable stars in NGC 7142, Punanova et al. (2012) made the latest CCD photometry of NGC 7142.. They used their VRI photometry of 2194. stars compare with the JHK photometry taken from 2MASS to improve the reddening calibration.. Although the value E(J-H) = 0.13±0.05 has good. agreement with previous studies, they did not estimate distance modulus due to short portion of main-sequence of cluster. 3.

(16) The latest complete discussion of distance and reddening of NGC 7142 had done by Sandquist et al. (2011).. They have BVI observations in 53 days. during the interval from August 2005 to July 2011.. From the previous studies,. most of the variable stars have good probability of being cluster members. Based on these variable stars fitting the isochrones on the CMD, they gave [Fe/H] = +0.13, the value has good agreement with Jacobson +0.14±0.01 from giants spectroscopic measurements, and E(B–V) = 0.32±0.06, (m-M)0 = 11.9 ±0.15. The properties of NGC 7142 taken from literature are list in Table 1-2.1.. Fig. 1-2.1 NGC 7142 BVR image, taken by Chuang Shao-Er (莊孝爾), with NCU Lulin One-meter Telescope, FOV=11.2’, August 7, 2003. 4.

(17) Table 1-2.1 NGC 7142 properties with different researchers Age Distance (m - M)0 E(B – V) Researcher 9 (10 yr) (pc) (Mag) (Mag) Johnson et al. 1961 1000 10.0 0.18 van den Bergh 1962. ~5. 1880. 11.37. Metallicity [Fe/H]. 0.46 ±0.05. Diameter (arcmin). 8.6. Sharov 1965. 75. Sharov 1968 van den Bergh & Heeringa 1970 Becker &Fenkart 1971 Jennens & Helfer 1975. 23 3200. 12.5. 0.41. 610. 8.93. 0.19. 5. 1500. 10.9. 0.29. -0.45. 3. 3000. (0.41). -0.1. 11. Friel et al. 1989 Crinklaw & Talbert 1991 Friel & Janes 1993. 3.4~4.5 3.1. (3000). (0.41). -0.23 ±0.13. Dias et al. 2002. 6.92. 2344. 0.32. +0.08 ±0.06. Friel et al. 2002. 4.4. 2950. Salaris et al. 2004. 4.04. WEBDA 2006 Jacobson et al. 2007 Jacobson et al. 2008. 1.89. Janes & Hoq 2011. 6.9. 11.85 ±0.05. 0.32 ±0.05. 3. 11.9±0.15. 0.32±0.06. +0.13. 11.02. 0.36. -0.37. Sandquist et al. 2011 Punanova et al. 2012 This work. 11.4 ±0.9. 0.35. 12. -0.10±0.10 +0.09. 1686. 0.397 (11.9). (0.41). (4). +0.08 ±0.06 +0.14 ±0.01. 3.6. (1683). 7.9. 1600 1920. 5. 8. (FOV=11).

(18) 1-3 Messier 45 Properties Messier 45 (Pleiades, RA:03h 47m 00s,, Dec:+24° 07’ 00”) is a famous open cluster.. Due to the large angular size and brightness, this cluster is the best-. known deep sky object.. Johnson and Mitchell (1958) used the M45 as the. sample of CMD without the effect of reddening and extinction.. From the. Hipparcos mission, the mean parallax of the M45 is 8.61±0.23 mas (van Leeuwen and Hansen Ruiz 1997), and it corresponds to a distance modulus of 5.32±0.06 mag or 116±3 pc (Pinsonneault et al. 1998).. They also gave the. metallicity, [Fe/H] = -0.03, and the E(B-V) is 0.04. The M45 represents an appropriate laboratory for stellar dynamics: bright and rich in stars, and old enough to evolve dynamical. (Adams et al. 2001). For the. M45, observations of lithium absorption in brown dwarfs have led to a precise age measurement of 120 Myr (Stauffer et al. 1998).. According to Adams et al.. (2001) studies, the cluster contains a total mass about 800 M⊙ from the N body simulation results, and it corresponds tidal radius to this mass estimate is approximately 13.1 pc. They also find the M45 is highly flattened in the Y Z plane.. That means, the Galactic tidal field plays the dominant role in open. cluster dispersal. The first membership determination in terms of proper motion data was present by Jones (1981) with the members to V~16.. Dobbie et al. (2002) gave. the deep I band survey ~22.5 provided the first constraint on the mass function of the M45 in the M < 0.04M⊙.. Furthermore, Deacon and Hambly (2004). obtained the membership down to R~21, and they gave the mass function of the cluster extended to lower masses. From the latest high precise measurement of parallax and radial velocity, the distance modulus of M45 from different researcher has good agreement about. 6.

(19) (m-M)0 = 5.6.. The properties of M45 taken from literature are list in Table 1-. 3.1.. Fig. 1-3.1 M45 (Pleiades). Taken by Yen Hung-Hsuan (顏鴻選), with Canon EOS 5D mark II and RC-reflector (D=204mm, f=1800mm), FOV=82’, October 2012. Table 1-3.1 M45 properties with different researchers. Researcher Age Distance (m - M)0 (106 yr) (pc) (Mag) Mitchell & Johnson 150 5.52 1957 Johnson et al. 1961 126 5.5 Mermilliod 1981. 78. Mazzei & Piggato 1989 Boesgaard & Friel 1990 Meynet et al. 1993. 150. Soderblom et al. 1993 van Leeuwen & Ruiz 1997 Pinsonneault et al. 1998. E(B – V) (Mag). Metallicity ([Fe/H]). 0.04. 70. -0.034 ±0.024. 100. 5.60 130. 5.6. 116 ±3. 5.32 ±0.06 5.60 ±0.04 7. 0.04. 0.04. -0.03.

(20) Stauffer et al. 1998 Narayanan & Gould 1999 Gatewood et al. 2000 King et al. 2000. 125. 130 5.58±0.18 5.59±0.12. 70~100. +0.06 ±0.05. Makarov 2002. 5.57±0.06. Munari et al. 2004. 132±2. Pan et al. 2004. 120. 132±4. 5.60±0.07. 130. 5.63. 135.6±3.67. 5.66±0.06. 0.02. 133.8±3. Soderblom et al. 2005 Southworth et al. 2005 Terrell 2005 WEBDA 2006. (0.035). 133 ~ 137. Zwahlen et al. 2004 COCD Kharchenko et al. 2005 Johns-Krull & Anderson 2005 Percival et al. 2005. 5.60±0.03. -0.4. 133. 5.65±0.05. 139.1±3.5. 5.72±0.05. 132 135. 150. 0.03. Paunzen & Netopil 2006 Groenewegen et al. 2007 An et al. 2007. 79 ±52. 133. 0.05 ±0.01. 138.0 ±1.5. 0.025 ±0.003. +0.06 ±0.23. 0.04 ±0.05. +0.022 ±0.014. Gebran & Monier 2008 Soderblom et al. 2009 Roser & Schilbach 2012 This work. 100. 5.66 ±0.01. +0.06 ±0.02 +0.03 ±0.04 5.50±0.13. 126. (136). (5.66). 8. (0.022). (+0.04).

(21) 2 Observation and Data This research had own observations of NGC 7142 in 2003 and 2012, using the NCU Lulin One-meter Telescope. For the M45, this research used the PPMXL catalog to get the deeper and precise proper motion data. 2-1 LOT observations of NGC 7142 The Lulin One-meter Telescope (LOT) of the National Central University locates on Mt. Front Lu-Lin, a 2862m peak in the Yu-Shan National Park.. The. average seeing is about 1.39 with average 200 clear nights annually (WeanShun Tsay 2001).. The field of view of LOT with PI1300 CCD camera (1340 x. 1300 pixels) is about 11’ x 11’; roughly the image scale is 0.512” per pixel. On August 7, 2003, the NTNU graduate student of astronomy group, Shao-Er Chuang (莊孝爾), observed NGC 7142, by B, V, and U band images.. Because. of the observation time limitation, he only had 768 seconds as the total exposure time in each band. On July 8, 2012, the NCU LOT observational operator Hsiang-Yao Hsiao (蕭翔耀) and the Graduate Institute of Astronomy, NCU graduate student Yu-Chi Cheng (鄭宇棋) observed NGC 7142 in B and V bands, and they also carried out the observations of standard star fields, 2 fields and 4 bands (BVRI). Table 2-1.1 Observation information of NGC 7142 images at LOT Time. Filter and total exposure time. Observer. 2003/08/07. U: 786s. B: 786s. V:786s. Chuang.. 2012/07/08. -. B: 1320s. V: 1020s. Cheng. and Hsiao.. 9.

(22) Fig 2-1.1 NGC 7142 in U band (color reversed), exposure time: 786s. FWHM=3.078 arcsec, details in Appendix B.. 10. Average star.

(23) Fig 2-1.2 NGC 7142 in B band (color reversed), exposure time: 786s. FWHM=3.078 arcsec, details in Appendix B.. 11. Average star.

(24) Fig 2-1.3 NGC 7142 in V band (color reversed), exposure time: 786s. FWHM=2.669 arcsec, details in Appendix B.. 12. Average star.

(25) Fig 2-1.4 NGC 7142 in B band (color reversed), exposure time: 1320s. Average star FWHM=2.812 arcsec, details in Appendix B.. 13.

(26) Fig 2-1.5 NGC 7142 in V band (color reversed), exposure time: 1020s. FWHM=2.713 arcsec, details in Appendix B.. 14. Average star.

(27) This research has two standard star field observations, and there had seven photometry standard stars in the field (Table 2-1.4). filter images: B, V, R and I (Table 2-1.2).. Each observation had four. The magnitude calibration. calculated using color and airmass, and the procedure shows in Section 3-1. The exposure and airmass information listed below: Table 2-1.2 Standard star field expose information of July 8, 2012. Standard star field. Filter and exposure time. SA110. B: 10s. V: 5s. R: 5s. I: 5s. SA111. B: 5s. V: 5s. R: 5s. I: 5s. Table 2-1.3 Standard star field airmass information of July 8, 2012. Standard star field. Filter and the airmass. SA110. B: 1.937151. V: 1.930752. SA111. B: 1.332727. V: 1.330531. Table 2-1.4 List of 7 standard stars of SA 110 field. Star. RA (J2000). Dec (J2000). 110 496. 18:42:59. +00:31:08. 13.004 1.040 0.737 0.607 0.681. 110 499. 18:43:07. +00:28:00. 11.737 0.987 0.639 0.600 0.674. 110 502. 18:43:10. +00:27:40. 12.330 2.326 2.326 1.373 1.250. 110 504. 18:43:11. +00:30:05. 14.002 1.248 1.323 0.797 0.683. 110 503. 18:43:11. +00:29:43. 11.773 0.671 0.506 0.373 0.436. 110 506. 18:43:19. +00:30:27. 11.312 0.568 0.059 0.335 0.312. 110 507. 18:43:19. +00:29:26. 12.440 1.141 0.830 0.633 0.579. 15. V. B-V. U-B. V-R. R-I.

(28) Fig. 2-1.6 Standard star field: SA 110, B band (left) and V band (right) image, taken by LOT. Open diamond for standard stars, circle for stars detected.. Table 2-1.5 List of 3 standard stars of SA 111 field. Star RA (J2000) Dec (J2000) V. B-V. U-B. V-R. R-I. 111 1925. 19:37:29. +00:25:01. 12.388. 0.395 0.262 0.221 0.253. 111 1965. 19:37:42. +00:26:50. 11.419. 1.710 1.865 0.951 0.877. 111 1969. 19:37:44. +00:25:48. 10.382. 1.959 2.306 1.177 1.222. Fig. 2-1.7 Standard star field: SA 111, B band (left) and V band (right) image, taken by LOT. Open diamond for standard stars, circle for stars detected.. 16.

(29) 2-2 Tycho-2 catalog of M45 Tycho-2 is a catalog including positions, proper motions of 2.5 million brightest stars in the sky. The positions and magnitudes of star listed on Tycho-2 based on the same observations of the Tycho-1 catalog (ESA SP-1200 1997), which collected by the ESA Hipparcos satellite.. This new reduction of. previous catalog obtained by a new analysis of 144 ground-based astrometry catalogs.. The completeness is 99% to magnitude of V band downing to 11.0,. and the completeness is about 90% to magnitude of V band downing to 11.5 (Hog 2000). In the Tycho-2, the mean error of proper motion data is about 2.5 mas/yr, and the catalog listed magnitude of Tycho-V(VT) and Tycho-B(BT ) of each star. The VT and BT will transfer to Johnson BV magnitude in the following equations: V = VT – 0.090 (B T – VT) B – V = 0.850 (B T – VT) This research selected 3 degrees of M45 region in this catalog with total star number of 1340.. From the histogram of magnitude of all stars, magnitude. completeness is down to 12.0 of B band and to 11.2 of V band, with 633 stars. The latest high precision proper motion data is PPMXL catalog with wavelength is from optical (B, R, and I-band) to near infrared (J, H, and Ks bands). Because the PPMXL did not have V band data, this research could not use it to compare with NGC 7142 CMD.. However, this research still do all the. membership determine process with PPMXL, and the process is shows in Appendix A.. 17.

(30) Fig. 2-2.1 M45 catalog from Tycho-2, radius 3 degree, 1340 stars.. Fig. 2-2.2 M45 proper motion data from Tycho-2 18.

(31) Fig. 2-2.3 M45 Tycho-2 data histogram with B (left) and V (right) magnitude, max number at B=12.0, V=11.2.. Fig. 2-2.4 M45 color-magnitude diagram, B-V to V magnitude. 19.

(32) 3 Data Reduction for NGC 7142 For the NGC 7142 images taken from LOT, the dark current, flat field, and bias calibration has been finished in the MaximDL program.. After the images. cleaned, this research used the IRAF program’s DAOPHOT procedure to get the position and magnitude information.. After calibration, a catalog with B. magnitude, V magnitude and the proper motion of each star of the NGC 7142 field is set up. 3-1 Magnitude calibration The magnitude of each image are calibrated with standard stars in the standard star field, and then, the magnitude of B and V band for each star are derived using transfer equations of photometry. There are two standard star fields, SA110 and SA111, observed in the LOT observational log in 2012, with seven standard stars in the SA110 and three standard stars in the SA111, respectively.. For each standard field, three bands. images, B, V and R observed with exposure time of 5 seconds. And with 10 seconds only for B band of SA110.. After cleaning images gotten, the IRAF. program DAOPHOT procedure used to determine the position (in pixel) and instrument magnitude of each star.. The IRAF program setting information has. shown in Appendix B. Table 3-1.1 Information of standard star field images. Total Standard Star number. 10 stars.. Standard stars V mag. 10.382 to 14.002.. Standard stars B mag. 11.880 to 15.250.. SA110 B filter total star number. 115 stars.. SA110 V filter total star number. 419 stars.. SA111 B filter total star number. 585 stars.. SA111 V filter total star number. 1135 stars.. 20.

(33) Fig. 3-1.1 Standard star field SA 110 (B and V) IRAF magnitude error with the instrument magnitude distribution, open diamonds are standard stars.. Fig. 3-1.2 Standard star field SA 111 (B and V) IRAF magnitude error with the instrument magnitude distribution, red squares are standard stars.. Before the magnitude calibration for standard stars, the images with a different exposure time have to calibrate into the same exposure time.. For. most of the standard star fields, the exposure time is 5 seconds, so that the exposure time of all images are calibrated into 5 seconds by using following equation: 21.

(34)  5 seconds   m cal  m inst  2.5 log exp. time  . Fig. 3-1.3 Standard stars b-v to B-b (left) and b-v to V-v (right), the color of stars (b-v) has relation to the difference from instrument and apparent magnitude.. Ten standard stars in the images used to get the instrument magnitude of each star using IRAF. The atmosphere extinction (X) and the color correction (b-v) are used to magnitude calibration, and instrument magnitude of b and v have transferred to magnitude of B and V using the following equations, respectively: (Bi - bi) = α1 + α2 Xj + α3 (bi - vi) (Vi - vi) = β1 + β2 Xj+ β3 (bi - vi) α1 , β1 are zero point of magnitude calibration. α2 , β2 are atmosphere correlation constant α3 , β3 are color correlation constant Xj is the airmass of the observation. The relationship between (b-v) and (B-b), and between (b-v) and (V-v) are shown on Fig. 3-1.3.. After regressed the six parameters, the magnitude. calibration equations are: Bcal = bi + 0.2840 - 0.1055 Xj + 0.2476 (bi - vi) Vcal = vi + 0.2207 - 0.1109 Xj - 0.0800 (bi - vi). 22.

(35) The standard error in B magnitude and V magnitude are 0.0322 and 0.0187, respectively (shown on the Fig. 3-1.4 and 5).. Fig. 3-1.4 Magnitude calibration of B band, the black circle shows the b magnitude after color and atmosphere extinction calibration.. Fig. 3-1.5 Magnitude calibration of V band, the black circle shows the v magnitude after color and atmosphere extinction calibration. 23.

(36) With both instrument magnitude of b and v of each star in the image of NGC 7142, b and v will be transfer to magnitude of B and V, respectively. The positions of ten stars have chosen to match B image and V image observed in 2012.. The transformation equations for position calibration as. follows: X=ax+by+c Y= d x + e y + f x and y are original coordinate X and Y are transposed coordinate. The constants, a, b, c and d, e, f called plate constants. The distance difference for the star between B image and V image less than 1 pixel (0.516 arcsec) as the same star.. Using regression function of IDL, the star catalog is. set up including coordinates, B magnitude and V magnitude.. Fig. 3-1.6 The IRAF instrument bv magnitude of NGC 7142 2012 observation. Upper figure shows B band image and the lower figure shows V band image.. The new catalog contained 1947 stars with coordinate(x, y), B magnitude, V magnitude and magnitude error, and there is no star had double recognized. The diagrams magnitude-magnitude error are shown on Fig. 3-1.7. 24.

(37) Fig. 3-1.7 The calibrated BV apparent magnitude of NGC 7142 2012 observation. Upper figure shows B band image and the lower figure shows V band image.. This research compare our result of B magnitude and V magnitude with the study of CCD photometric of NGC 7142 made by Crinklaw and Talbert (1991), B magnitude shown on Fig. 3-1.8 and V magnitude on Fig. 3-1.9.. For both B. and V image, stars which brighter than 15 have more difference than literature. The exposure time of single frame in these 2012’s images is 300 seconds so that caused the brighter stars over exposure.. In the magnitude calibration of. 2012’s images, the stars brighter than V = 15 are deleted from catalog, and replaced the magnitude with shorter exposure time images taken in 2003. For B magnitude, there was about 0.2 magnitude difference between this work and Crinklaw and Talbert (1991),.. Although the B band image has. longer exposure time (1320 sec) than V band image (1020 sec), the image quality still worse (see Fig. 2-1.4, Fig. 2-1.5).. For V band image (Fig. 3-1.9),. a good agreement (σ=0.023) with reference magnitude brighter than 15.. The. brighter stars also influence the relationship of b-v to b-B (Fig. 3-1.10) and v-V (Fig. 3-1.11).. Only V magnitude fainter than 15 has used for magnitude. calibration. 25.

(38) Fig. 3-1.8 The compare with apparent B magnitude and reference paper.. Fig. 3-1.9 The compare with apparent V magnitude and reference paper. 26.

(39) Fig. 3-1.10 The distribution of b-v to b-B compare with 2012’s and reference paper. Star’s B magnitude brighter than 15 which shows in diamond.. Fig. 3-1.11 The distribution of b-v to v-V compare with 2012’s and reference paper. Star’s V magnitude brighter than 15 which shows in diamond.. The result of B magnitude calibration has σ ~ 0.028 (Fig. 3-1.12).. This. research made a catalog contain the XY position, B and V magnitude and the magnitude error of each stars of 2012’s observation.. 27.

(40) Fig. 3-1.12 The result of regression of 2012 apparent B magnitude compare to the literature.. The observation of standard fields had carried out in 2012, but it had not carried out in 2003.. However, U band images had observed only in 2003.. The data from Crinklaw and Talbert (1991) used for B and V magnitude calibration and the data from van den Bergh and Heeringa (1970) used for U magnitude calibration. In this research, the magnitude calibrated from data of literature, the equations of magnitude calibration of 2003 did not consider about the airmass term.. The following functions use for calibrate the magnitude B and V: (Bi - bi) = α1 + α2 (bi - vi) (Vi - vi) = β1 +β2 (bi - vi) α1 , β1 are the zero point of magnitude calibration. α2 , β2 are the color correlation constant. Without counting the exposed bright stars, only the stars with V magnitude fainter than 15 are used for magnitude calibration. 28.

(41) Fig. 3-1.13 The IRAF instrument ubv magnitude of NGC 7142 2003 observation.. After manually matched 50 reference stars, the calibration functions of 2003’s B and V magnitude are: Bcal = bi + 4.092 + 0.308 (bi - vi) Vcal = vi +4.753 - 0.064 (bi - vi) Bcal and Vcal are apparent magnitude from Crinklaw and Talbert (1991) v i and bi are 2003’s instrument magnitude. The standard error for B magnitude and V magnitude are 0.030 (Fig.3-1.14). and 0.019 (Fig.3-1.15), respectively.. 29.

(42) Fig. 3-1.14 The apparent B magnitude compare with reference literature.. Fig. 3-1.15 The apparent V magnitude compare with reference literature. 30.

(43) For U magnitude calibration, , the transformation equation as following (van den Bergh and Heeringa, 1970): (Uref – u2003) = α1 + α2 (u2003 – b2003) α1 as the zero point of magnitude calibration α2 ,as the color correlation constant. The U magnitude of 18 stars are used to find the transfer equation (Fig. 2-1.1), and the standard error is 0.066 (Fig. 3-1.16).. The. Ucal = u2003 + 1.0468 + 0.1153 ( u2003 – b2003 ). Fig. 3-1.16 2003 U magnitude compare the reference magnitude from literature.. For the magnitude calibration of stars brighter 15, the images with short exposure taken in 2003 have calibrated.. The comparison of image with. different exposure time of 5 seconds and 1 second is shown on Fig. 3-1.17, and the images with exposure time of 5 seconds have used for magnitude calibration 31.

(44) of bright stars because of the smaller standard error.. The final catalog is. combined two databases, the stars brighter than 15 and the fainter stars in the images with long exposure of 2003. The magnitude with errors of the combined catalog shows in Fig. 3-1.18. Although the average error of short exposure image is larger than long exposure image, but the linear relation of bright stars is better that give the more accurate of magnitude.. Fig. 3-1.17 Calibrated magnitude compare with reference of different exposure time.. After the magnitude calibration, the Color-Magnitude Diagram (Fig.3-1.19) of NGC 7142 had plotted immediately, and the TCD (Fig.3-1.20) and CMD (Fig.3-1.21) had plotted only for 694 stars with U magnitude.. 32.

(45) Fig. 3-1.18 The magnitude errors distribution of the combined catalog.. Fig. 3-1.19 The Color-Magnitude Diagram of NGC 7142.. 33.

(46) Fig. 3-1.20 The Two-Color Diagram of NGC 7142.. Fig. 3-1.21 The Color-Magnitude Diagram of NGC 7142 with UBV data stars. 34.

(47) 3-2 Coordinate calibration and proper motion determine The precise of position measurement also relates with magnitude pointspread-function (PSF) fitting.. The stars with magnitude error larger than 0.1 in. V band are delete for analysis, and the magnitude to error relation shows in Fig 3-2.1.. Because fainter stars in the B and image have detected, the data taken. from B band image have used for determine the proper motion. The stars with V magnitude larger than 20.8 are also delete, because the histogram of magnitude shows the number of stars drop a lot beyond 20.8 (Fig. 3-2.2, Fig. 3-2.3).. Fig 3-2.1 The calibrated magnitude compare with errors. Dash line shows error 0.1 mag.. 35.

(48) Fig 3-2.2 The histogram of magnitude, 2003 B (top) and V (bottom). Fig 3-2.3 The histogram of magnitude, 2012 B (top) and V (bottom). 36.

(49) The total 1697 stars of the catalog is set up for the program of the proper motion, and the database include the coordinates, B magnitude, and V magnitude on the both the date of August 7, 2003 and July 8, 2012.. The. proper motion of each star had calculated in term of the fallowing equations: Δx = μx Δt and Δy = μy Δt, Where the Δt is the interval time between August 7, 2003 and July 8, 2012, and it is about 8 years and 11 months.. The coordinates of the image of July 8,. 2012 had transferred to the frame of August 7, 2003, and above equations become as following equations with plate constants, a, b, c and d, e, f: Δx = a x2012 + b y2012 + c – x2003 = μx Δt Δy =d x2012 + e y2012 + f – y2003 = μy Δt The process will be iteration for stars with the smallest proper motion to find suitable the value of plate constants and proper motions, μx and μy . First, use all stars as stars of reference frame, and after process, remove the stars with large value of μx and μy , and run the program again, until the value μx and μy of stars of reference frame smaller than 0.02 pixel.. The 94 stars used as reference. frame (Fig.3-2.4), and the proper motions of total 1697 stars, μx and μy , are calculated.. 37.

(50) Fig 3-2.4 NGC 7142 catalog and position reference stars after iteration.. Fig 3-2.5 NGC 7142 proper motion distribution, the Vector Point Diagram (VPD). 38.

(51) Fig 3-2.6 A close up view of NGC 7142 VPD.. Fig 3-2.7 3-D histogram of NGC 7142 proper motion data, bin in 2 mas/yr. 39.

(52) 4 Membership Determination The proper motion data is a physical component which independent from color or magnitude of star, but does related with membership of open cluster. To determine the membership of an open cluster, there are three steps: making the data selection, rotate the VPD, compute iteratively.. Once the membership. determined, the CMD of higher membership can be draw immediately. The Chapter 1 reviewed the progress of statistical method of membership determination from Sanders (1971) to Balaguer-Nunez (1998).. Although the. Balaguer-Nunez method is the most complete membership determination equation so far, the basic idea of statistical term still is the same with Sanders ’ method.. Our observation has relative larger difference in errors of position. measurement from the center to the rim of the image due to the distortion of optical system.. However, the original idea of Sanders (1971) used for this. work. 4-1 Data selection In the diagram of magnitude error and magnitude distribution (Fig 3-1.1), the stars had selected with the condition of magnitude errors and the histogram of magnitude.. Furthermore, the stars with proper motion higher than 100 mas/yr. are detected, because these stars should not be member of the cluster. In the catalog of NGC 7142, the original number of star is 1697.. Total 1402. stars have selected after deleting stars, V magnitude error larger than 0.1 and magnitude fainter than 20.8, and the figures were show in Section 3-2. In the catalog of M45, the original number of star is 1340.. Total 633 stars have. selected after deleting stars, B magnitude larger than 12.0, V magnitude larger than 11.2 and proper motion larger than 100 mas/yr.. 40.

(53) Fig 4-1.1 M45’s CMD after data selection. Fig 4-1.2 M45’s VPD after data selection 41.

(54) Fig 4-1.3 Zoom-in of M45’s VPD. Cross lines shows proper motion equal 0.. Fig 4-1.4 3-D histogram of M45 proper motion data, bin in 2 mas/yr.. 42.

(55) 4-2 Data process Assume the cluster members have a circular normal distribution, and field stars has elliptical distribution, the principal axes have to rotate alone the major axis of distribution of field stars. Vasilevskis et al. (1965) developed the VPD rotation method, the distribution of field stars on the VPD must be rotated alone the x and y-axis before normal distribution fitting.. Assume the distribution of. field stars with rotated are periodic, the maximum value will be the angle defined to be the direction of the major axis of the distribution (Vasilevskis et al. 1965).. After the rotation of the VPD, the center of the members and field stars. derived by the distribution of proper motion.. If the most probably cluster. members are obviously different from the field stars shown on M45 in Fig. 22.2 and Fig. 4-1.2, and these stars will remove before the VPD rotation procedure. After rotation, the VPD of all stars has shown on Fig. 4-2.2. Vasilevskis assume the ω is the angle between the original and rotating axes, ζ and η are modified proper motion in x and y, then the transfer function is: u(ω) = ζ cos(ω) – η sin(ω) v(ω) = ζ sin(ω) – η cos(ω) Then define a function S(ω) related with u(ω): S(ω) = Σ[u(ω)]2 The function S(ω) is periodic with a period of 180∘ and has a maximum value, and assume the following simple relationship: S(ω) = a + k cos(ω - Θ), or. S(ω) = a + b cos(ω) + c sin(ω),. where. b = k cos(Θ), c=k sin(Θ). The constants a, b and c could found from a lest-squares solution: tan(Θ) = c/b 43.

(56) An angle ω = Θ defining the direction of the major axis of the distribution can be found at the maximum value of S(ω) for ω=0∘ to 180∘.. Fig 4-2.1 The NGC 7142 rotated VPD.. Fig 4-2.2 The M45 rotated VPD.. 44.

(57) 4-3 Membership probability Sanders (1971) defined the function Φ to describe the distribution of open cluster on the VPD.. The model is one of overlapping normal bivariate. frequency function, one elliptical for the field and one circular for the cluster.  2   2  N μyi  μΥ0     1  μxi  μΧ0    Φμxi, μyi   exp      2π  x  y 2   x    y        2 2  n  1  μxi  μΧ0   μyi  μΥ0     exp       2  σ  2π 2  σ       N = number of field stars n = number of cluster stars, n is set equal to total star - N. Σx , Σy = Std. deviations of field in x and y σ = Std.deviation of cluster μΧ0 , μX0 = field centers μx0 , μy0 = cluster centers μxi , μyi = proper motion in x and y for the ith star. Using the method of maximum likelihood, NST.  i 1.  ln μxi, μyi   0, j  1,2,3,...,8 , j. for the equations above, with j being one of the 8 parameters.. Assumed:. 2 2   1  xi   0   yi   0       exp       2   x    y      .  1  xi  x 0  2  yi  y 0  2     exp        2        . A set of nonlinear equations of condition result will be derived as following. For field star number N, the equation: NST. 1.     i 1.    20 xy  . For field star x proper motion center, the equation: NST.      1. xi. i 1. 45.   0   0.

(58) For field star y proper motion center, the equation: NST.      1. yi.   0   0. i 1. For cluster member x proper motion center, the equation: NST.       1. xi.  x 0   0. i 1. For cluster member y proper motion center, the equation: NST.       1. yi.  y 0   0. i 1. For field star x proper motion sigma, the equation:  xi   0  2  1      1  0    x   i 1    NST. For field star y proper motion sigma, the equation:  yi   0  2  1       1  0  i 1    y   NST. For proper motion sigma of cluster member, the equation:  xi  x 0 2  yi  y 0 2  1     2  0   2  i 1   . NST. The initial approximate values for all parameters are given, and the eight equations iterated until final value of all eight the equations approach to zero. Th e iteration is sensitive with sum of eight parameters.. After iterations (about. 100 times), the eight parameters the final value for eight parameters will be calculated.. The initial value, adjust step for each iteration and final value. listed in Table 4-3.2 (NGC 7142) and Table 4-3.4 (M45), respectively.. 46.

(59) Table 4-3.1 Parameters for NGC 7142, the number of iteration = 88 Parameters. Initial value. Adjust step Result value. Cluster member number n. 752. ±1. 840. Field star center μ Χ0. 0.10. ±0.01. 0.14. Field star center μ X0. -0.20. ±0.01. -0.18. Cluster member center μ x0. -0.10. ±0.01. -0.10. Cluster member center μ y0. 0.10. ±0.01. 0.10. Field star sigma σ x. 0.70. ±0.01. 0.83. Field star sigma σ y. 0.70. ±0.01. 0.70. Cluster member sigma σ. 0.20. ±0.01. 0.20. Number of probability > 70%. 814. Table 4-3.2 Parameters for M45, the number of iteration = 47 Parameters. Initial value. Adjust step Result value. Cluster member number n. 146. ±1. 146. Field star center μ Χ0. 1.00. ±0.01. 1.29. Field star center μ X0. 0.00. ±0.01. 0.01. Cluster member center μ x0. 4.80. ±0.01. 4.80. Cluster member center μ y0. -0.30. ±0.01. -0.30. Field star sigma σ x. 1.20. ±0.01. 1.66. Field star sigma σ y. 1.50. ±0.01. 1.55. Cluster member sigma σ. 0.10. ±0.01. 0.22. Number of probability > 80%. 142. When all eight the parameters are determined, the elliptical and circular with the diameter of 3 sigma will be draw on the VPD.. Use the following function. to draw Fig. 4-3.2 and Fig. 4-3.4:   1  μxi  μΧ0 2   1  μxi  μΧ0 2   n  N    Φμxi    exp   exp         binsize  x 2π  2   x    2π  2  σ           1  μyi  μY0 2   1  μyi  μY0 2   n  N  ΦμYi    exp   exp         binsize  Y 2π  2   Y    2π  2  σ         47.

(60) Fig. 4-3.1 The NGC 7142 membership VPD. Dash circle shows the 3 sigma of cluster members’ distribution, dot ellipse shows the 3 sigma of field stars’ distribution. Fig. 4-3.2 The NGC 7142 proper motion distribution alone μx and μy axis, long dash line is the fitting result of field stars’ distribution, short dash line is cluster, and solid line is sum of field and cluster stars’ distribution. 48.

(61) Fig. 4-3.3 The M45 membership VPD. Red circle shows the 3 sigma of cluster members’ distribution, blue ellipse shows the 3 sigma of field stars’ distribution. Fig. 4-3.4 The M45 proper motion distribution alone μx and μy axis, long dash line is the fitting result of field stars’ distribution, short dash line is cluster, and solid line is sum of field and cluster stars’ distribution. 49.

(62) The probability of membership within the circle gets higher, and these stars thought to be members of the cluster.. The probability of membership for the. ith star, Pi will be calculated using following equation, i .  ic i. ,. where Φ ic and  i are the frequency function of the cluster and field, respectively. The probability of membership larger than 70% for NGC 7142 and 80% for M45 are plotted to see the relation on star position.. The VPD of the NGC. 7142 did not have obvious separation of members and fields, but there are obviously two distributions in the VPD of M45, and the characteristics do not change much with higher probability of membership. The result of membership determination is successful in both two clusters. For the cluster near and young like M45, the membership distribution separate very clearly (Fig. 4-3.4).. The method in this research does not present very. different efficiency from other method, such as cutting the dense and separated part of stars on the VPD.. For a distant and old cluster like NGC 7142 (Fig. 4-. 3.3), this method reveals a good result of membership determined. Using the information of the probability of membership, the distribution on the sky of stars with higher probability are shown on Fig. 4-3.6 and Fig. 4-3.8, the CMD is shown on Fig. 5-1.1 and Fig. 5-1.2, and the TCD is shown on Fig. 5-2.2.. 50.

(63) Fig. 4-3.5 The distribution of NGC 7142 membership probability. Fig. 4-3.6 The distribution of M45 membership probability. 51.

(64) Fig. 4-3.7 The distribution of NGC 7142 with membership > 70% in diamond.. Fig. 4-3.8 The sky map of NGC 7142 membership > 70%. 52.

(65) Fig. 4-3.9 The sky map of M45 with membership > 80% in diamond. Fig. 4-3.10 The sky map of M45 membership > 80%. 53.

(66) 5 Result and Discussion The method, finding the members of an open cluster, using the probability of membership of each star calculated from the proper motion data is a better method than that of using photometry. Using the stars with higher probability of membership, the reddening and extinction could derived in the TCD, and the age and distance of this open cluster could be determined from the CMD. 5-1 The Color-Magnitude Diagram The magnitude calibration is in Section 3-1. The CMD of NGC 7142 with stars having both B magnitude and V magnitude, is shown on Fig. 3-1.19, and stars having UBV magnitude is shown on Fig. 3-1.21.. The CMD of M45 has. shown in Fig.5-1.1. Using stars with higher probability of membership, the main-sequence with clearly turn-off point is shown on the CMD of NGC 7142 (Fig. 5-1.2), and a good main-sequence is shown on the CMD of M45 (Fig. 5-1.3).. For. comparison, the CMD of stars with lower probability of membership is plotted, CMD of NGC 7142 field and M45 field are shown Fig. 5-1.2 and Fig. 5-1.4, respectively, and there is no obvious main-sequence. NGC 7142 used to be though probably not a real cluster (Jennens and Helfer 1975), but after the membership determination in this work, it is more likely an old open cluster.. 54.

(67) Fig. 5-1.1 The CMD of NGC 7142 membership > 70%. Fig. 5-1.2 The CMD of NGC 7142 membership < 70%. 55.

(68) Fig. 5-1.3 The CMD of M45 membership > 80%. Fig. 5-1.4 The CMD of M45 membership < 80%. 56.

(69) 5-2 The Two-Color Diagram Since the M45 data from Tycho-2 catalog has no U photometry data, the reddening and extinction do not derived from TCD. However, the value of color excess, E(B-V) and distance modulus, (m-M)0 of M45 taken from the study of An et al., 2007. For the stars with higher probability of membership of NGC 7142, the turnoff point of main-sequence locate at about (B-V) ~ 0.95 and V ~ 16.5 in the CMD (Fig. 5-1.1), and there is a dense linear part at (B-V) = 0.9~1.5 in the TCD of NGC 7142 (Fig. 5-2.1).. Fig. 5-2.1 The TCD of NGC 7142 membership > 70%. On the CMD (Fig. 5-2.2), the suspect red giant stars lays on the region which V magnitude brighter than 16 and the B-V larger than 1.0, and the TCD of these stars have shown on Fig. 5-2.3.. The giant stars have different distribution. from main-sequence stars on the CMD and TCD (Deutschman et al. 1976), and the giant stars are removed for analysis of main-sequence fitting on the TCD in Chapter 6. 57.

(70) Fig. 5-2.2 The suspect red giant stars on the CMD of NGC 7142 membership > 70%.. Fig. 5-2.3 The suspect red giant stars on the TCD of NGC 7142 membership > 70%. 58.

(71) 5-3 Reddening and Extinction The reddening of an open cluster depends on different spectrum type of each star (Johnson and Morgan 1953, Racine 1973, Crawford and Mandwewala 1976).. However, Johnson and Morgan (1953) point out the typical ratio of. E(U-B) and E(B-V) is 0.72.. Although with some little further improvement,. this value has been introduce for many studies (Straizys, et al. 1976, Deutschman, et al. 1976).. E (U  B)  0.72 E(B  V ) The isochrones of dwarf stars or main-sequence stars are influence by X. metallicity, Z.. In the Two-Color Diagram of NGC 7142 (Fig. 5-3.1), the. isochrones with different value of metallicity are plotted, and the Isochrones data taken from CMD 2.4 online database.. The metallicity values transfer. function is: [Fe/H]=log(Z/Z☉), Z☉ = 0.019. Fig. 5-3.1 Main-Sequence Stars fitting on the TCD, 2 dash lines present the 0.72 ratio of E(UB)/E(B-V). Main-Sequence dwarf stars’ isochrones in the same age = 100 Myr, and different metallicity Z with different color are shown in figures. Left figure shows Z = 0.0001 to 0.0046, represents to [Fe/H] = -2.28 to -0.62. Right figure shows Z = 0.005 to 0.05, represents to [Fe/H] = -0.58 to 0.42.. 59.

(72) Fig. 5-3.2 Main-Sequence Stars fitting on the TCD, 2 dash lines present the 0.72 ratio of E(UB)/E(B-V). Main-Sequence dwarf stars’ isochrones in the same age = 100 Myr, and different metallicity Z = 0.006 to 0.012, represents to [Fe/H] = -0.50 to -0.20.. Fig. 5-3.3 Main-Sequence Stars fitting on the TCD. Main-Sequence dwarf stars’ isochrones in the same age = 100 Myr, and different metallicity Z = 0.006 (left) and 0.008 (right), represents to [Fe/H] = -0.50 and -0.37, with different color are shown with different E(B-V) = 0.25 to 0.40. 60.

(73) In the TCD of NGC 7142, the fitting curves of dwarfs with different value of Z are shown on Fig. 5-3.1, and the value of Z = 0.006 to 0.012 has the most likely with the UBV data (Fig. 5-3.2).. The best fitting curve of Z = 0.008 (Fig.. 5-3.3) is used for analysis.. Fig. 5-3.4 Main-Sequence Stars fitting on the TCD. Main-Sequence dwarf stars’ isochrones in the same age = 100 Myr, and constsant metallicity Z = 0.008, represents to [Fe/H] = -0.37, with different E(B-V) = 0.25 to 0.47 (left). The best result for E(B-V) = 0.36 (right). Solid line shows the non-reddening main-sequence.. The metallicity of NGC 7142 in different studies are listed on Table 1-2.1, and these values usually derived from the spectrum fitting of giant stars of the cluster (Canterna et al. 1986, Jacobson et al. 2008).. From this work, the value. of Z = 0.008, represents [Fe/H] = -0.37 by TCD main-sequence fitting.. Latest. reddening value of NGC 7142 is E(B-V) = 0.32 (Janes and Hoq, 2011) (Table 12.), and the best value of E(B-V) = 0.36 for the UBV data in Fig. 5-3.4.The extinction of stellar absorption AV given by Straizys, et al. (1976):. AV  RV E ( B  V ) RV  0.1707( B  V ) 0  3.3502 The distance modulus (m-M)0 = 11.02 comes from main-sequence fitting on the CMD (Fig. 5-3.5). 61.

(74) Fig. 5-3.5 Main-sequence fitting on the CMD, E(B-V)=0.36, (m-M)0 = 11.02.. 62.

(75) 5-4 Age and distance From the results of different researcher listed in Table 1-2.1, the isochrones with different metallicity and ages on the NGC 7142 CMD draw in Fig.5-4.1, and the TCD in Fig. 5-4.2.. Fig. 5-4.1 The isochrones of NGC 7142 from different researchers on the CMD.. 63.

(76) Fig. 5-4.2 The isochrones of NGC 7142 from different researchers on the TCD, most of them does not have agreement with the TCD in this work.. The age of NGC 7142 estimated is not consistence, from 1.89 Gyr (WEBDA 2006) to 6.9 Gyr (Janes and Hoq 2011).. From the isochrones fitting of this. work, the age of NGC 7142 is about 7.94 Gyr (Fig.5-4.4)for color excess, for E(B – V) = 0.36 and distance modulus (m-M)0 = 11.02. 7142 is 1600 pc, represents 5300 light years.. 64. The distance of NGC.

(77) Fig. 5-4.3 The isochrones fitting of NGC 7142 in this study. Fig. 5-4.4 The isochrones fitting of NGC 7142 with age: 7.94 Gyr.. 65.

(78) For the CMD of M45 shown on Fig. 5-3.5, the E(B-V) = 0.04 and (m-M)0 = 5.63 given by An et al.(2007).. The best result of isochrones at age = 126 Myr. (Fig. 5-3.6), agreement with many literature listed in Table 1-3.1.. Fig. 5-4.5 The isochrones fitting of M45. The value given by An et al. (2007), (mM)0 = 5.63, Z = 0.0218, represents [Fe/H] = -0.022.. Fig. 5-4.6 The best value of M45 isochrones fitting. Age = 126 Myr. 66.

(79) Comparison the isochrones of M45 and NGC 7142 in Fig. 5-4.7, the magnitude difference between the main-sequence of two clusters is about 5.75, represents the distance ratio between M45and NGC 7142 is 14.13. The distance of M45 is variable with different studies using parallaxes (van Leeuwen and Ruiz 1997; Meynet et al. 1993), radial velocities and proper motions (Roser and Schulbach 2012; Pan et al. 2004), main-sequence fitting (Percival et al. 2005; An et al. 2007), variable stars (Terrell 2005), or binary (Southworth, Maxted and Smalley 2005).. The range of the distance of M45 on. the literature is from 118 pc to 136 pc (Table 1-3.1).. The value of distance. modulus of 5.66 (An et al. 2007) is adopted, because this value is in excellent agreement with several geometric distance measurements. Substituting the value of 136 pc into distance ratio between M45 and NGC 7142, 5.75, the distance of NGC 7142 is about 1920 pc. For comparison, the distance of NGC 7142 is from 1500 pc to 3200 pc. (Table 1-2.1). 67.

(80) Fig. 5-4.7 The CMD comparison between M45 (red) and NGC 7142 (blue). The researcher also used the same procedure with high precise proper motion data of PPMXL, but in J-K magnitude.. The result shows in Appendix A.. 68.

(81) 6 Conclusion To determine the membership of an open cluster, the method with the proper motion data is more reliable than the method with neither position data nor photometry data.. From the CMD of higher probability of membership, the. main-sequence is more obvious, and the distribution of these star on the sky is also consistent. In this work, the UBV photometry of NGC 7142 agrees with the literature. The catalog of 1697 stars in the field of NGC 7142 with proper motion data down to V ~ 21 is set up, and the CMD of stars with higher probability of membership shows this cluster is more likely a very old cluster.. The distance. and age determination from this work conform to the literature.. Furthermore,. both the membership determination and the determination of age for M45 is consistence with the literature. The method for determination of the membership of open cluster is effective in not only the nearby cluster but also the distance one.. 69.

(82) Acknowledgement First, I have to make a huge acknowledgement to NCU LOT observation group, Hsiang-Yao Hsiao and Yu-Chi Cheng.. Without their generous. assistance of observed the 2012’s NGC 7142 images, it would be impossible to determine the proper motion data, this is the major part of this research. have great thanks for my senior colleague, Shao-Er Chuang.. I also. His observation. of UBV is so detailed that contained different exposure time and bands, that makes the photometry of bright stars could be possible in this work. Second, I would like to thank my close friends and colleagues. stand by me, and helping me when I need.. They always. I could not forget the lots of nights. spend on homework, research and play with Jalun, Steven and Chen-Fong. The great help and concern from Frodo, Jimmy and DHY are useful in every difficulty.. The most unforgettable memories are making the Planet 11677. Exhibition and album with Redscorpius, Chen-Kuan and the workshop group. There have many, many thanks of all my friends who spend on time with me. Third, I would like to thank my teachers, family and lover.. They gave me. lots of suggestion and kindly understanding of my slowly working step. Without their support and encourage I could not gone so far.. I will remember. their advices seriously and keep in working hard. In the end, I would like to give my deep thanks to my advisor, Professor Fu. Although in the past three years I was not always active in my studies, Prof. Fu always believes in my ability, and shows me the way to improve myself when I could not find the way out.. The atmosphere of freedom and trust in the Fu. group makes me learned so much. graduated in the Fu group.. I am very proud to be one of the students. Thank you my teacher again.. To all the people who help me in this thesis, THANK YOU! 70.

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(86) Website database CMD 2.5 online database http://stev.oapd.inaf.it/cmd COCD online database http://heasarc.gsfc.nasa.gov/W3Browse/all/cocd.html DSS Plate Finder http://stdatu.stsci.edu/cgi-bin/dss_plate_finder STScI Digitized Sky Survey http://archive.stsci.edu/cgi-bin/dss_form Tycho-2 homepage: http://www.astro.ku.dk/~erik/Tycho-2 WEBDA online open cluster database http://www.univie.ac.at/webda/. 74.

(87) Appendix A: M45 membership from PPMXL catalog PPMXL is a catalog including positions, proper motions of 900 million stars and galaxies, aiming to complete down to about V=20 full-sky.. It. is the result of a re-reduction of USNO-B1.0 together with 2MASS (Two Micron All Sky Survey) to the International Celestial Reference Frame (ICRS). It’s wavelength from optical (B,R, and I-band) to near-infrared (J, H, and Ks bands). The typical individual mean errors of the proper motions range from 4 mas per year to more than 10 mas per year.. The. mean errors of positions at J2000 are 80~120mas. (Roeser, Demleitner, and Schilbach, 2010) We selected M45 region in this catalog by radius 1 degree.. The total. star number in this region were 32159, and the histogram of magnitude showed the magnitude completeness down to J=16.4, K=15.2.. Because. not all the stars have the magnitude in all bands (B, R, I, J, H, and K), we picked up the stars which have J and K magnitude.. There’re 12270 stars. both have J and K magnitude.. M45 catalog from PPMXL, radius 1 degree. 12270 stars with J and K magnitude, the star size determine by J magnitude. I.

(88) M45 PPMXL data histogram with J mag, max number at J=16.4.. M45 PPMXL data histogram with K mag, max number at K=15.2.. M45 proper motion data from PPMXL.. II.

(89) M45color-magnitude diagram, J-K to J magnitude. In M45’s catalog, the original number of star is 32159.. After the data. selection, there’s 7642 stars. But the M45 has obvious main-sequence on the CMD at 5~12 in J magnitude, and it’s brighter than most stars in the catalog.. Therefore we make another condition for M45 is J. magnitude must smaller than 14.0.. The final number of star after. selection is 2331.. M45’s CMD after data selection III.

(90) M45’s VPD after data selection. M45’s VPD 3D surface after data selection. IV.

(91) The M45 rotated VPD.. M45 membership parameters Iteration time = 429 Parameters. Initial value Result value. Cluster member number n. 324. 324. Field star center μΧ0. -1.00. -1.00. Field star center μX0. -1.00. -0.93. Cluster member center μx0. -3.50. -3.56. Cluster member center μy0. -4.00. -3.97. Field star sigma Σx. 1.00. 1.21. Field star sigma Σy. 1.00. 1.25. Cluster member sigma Σ. 0.10. 0.46. Probability > 80% number. 398. V.

(92) The M45 membership probability. The M45 membership VPD. VI.

(93) The M45 membership histogram. The sky map of M45 with membership > 80% in diamond VII.