年老疏散星團NGC 7142之運動學與動力學

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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 Master’s Thesis. 年老疏散星團 NGC 7142 之運動學與動力學 The Kinematics and Dynamics of an Old Open Cluster NGC 7142. 研究生：徐逸翔 Researcher: Yi-Hsinag Hsu 指導教授：傅學海 Advisor: Hsieh-Hai Fu. 中華民國一○五年八月 August, 2016.

(2) 中文摘要星團由於其生成的機制，其成員星皆有著相似的特性，例如其化學組成、年齡、金屬豐度、自行量等。星團可作為研究銀河系之結構與演化的工具。而疏散星團則是由於其分布於接近銀盤，可代表銀盤面的特性與演化過程。以 UBV 測光系統估計疏散星團之年齡、金屬豐度、紅化消光等需要成員星的資訊。我們測量各恆星自行量，並進行視差影響之修正。再從各恆星的自行量在向量點圖中的分布，擬和二元常態分布的方式得出各星之成員星概率。我們以這些可能的成員星以參考文獻之方法進行紅化消光的修正，最後再以金氏模型之曲線擬合的方式獲得疏散星團的質量與潮汐半徑。本文以鹿林山天文台取得 NGC 7142 疏散星團之觀測，選擇 774 顆恆星為可能成員星並進行紅化消光修正。結果顯示當金屬豐度 Z = 0.010 時平均紅化值為 E(B−V) = 0.42、E(U−B) = 0.45，平均消光值 Av = 1.30。所得星團距離模式 (m−M) = 11.60 ± 0.1，對數年齡約落在 9.50 ~ 9.80 之間。金氏模型求得潮汐半徑 22 秒差距、質量約 4100 太陽質量。. 關鍵字：疏散星團，NGC 7142，恆星自行，成員星，星等星色圖，雙色圖，紅化，消光，金氏模型。. I.

(3) Abstract The forming mechanisms of star clusters make the member stars of open cluster have similar properties such as chemical composition, age, metallicity and motion in the sky. Star clusters are useful tools in studying structure and evolution of Milky Way Galaxy because of these properties. Open cluster is a type of star cluster that mainly located on the Galactic disk which can represent the Galactic disk evolution progress and characteristic. Using UBV image photometry to estimate the age, metallicity and reddening/extinction correction require membership information. Our method bases on astrometry measurement of proper motion with parallax correction. From fitting the bivariate distribution of proper motion in vector-point diagram, we acquire the membership probability of each star. We estimate the reddening and extinction with the probabilities information and UBV photometry data by literature method. Finally, we also estimated the mass and tidal radius the open cluster by the King model. To our target we observed with Lulin one-meter telescope, NGC 7142, we selected 774 star as probable member stars and result shows the average reddening E(B−V) = 0.42, E(U−B) = 0.45, mean extinction Av = 1.30 while Z = 0.010, distance modulus (m−M) = 11.60 ± 0.1 and logarithm age = 9.50 ~ 9.80. Tidal radius of the cluster is ~ 22 pc. The mass of NGC 7142 from the King model is ~ 4100 Mʘ.. Keyword: open clusters, NGC7142,. proper. motion,. membership. probability, color-magnitude diagram, two-color digram, reddening, extinction, King model. II.

(4) 誌謝「因為要感謝的人太多了，那就謝天吧」，陳之籓《謝天》文中. 的這句話，正如我在寫本文途中想要表達的心情。除了要感謝給予我協助以及陪伴的人們，當然身為一個靠天吃飯的天文學家，也得感謝老天給觀測時段好天氣，否則又要拖上好一陣子才能完成研究了。首先，感謝中央大學鹿林天文台的蕭翔耀學長的協助讓我得以申請鹿林的觀測時段。也感謝鹿林天文台助理林啟生先生幫忙進行觀測事宜。還要感謝前輩們所拍攝的 12 年以及 03 的觀測影像，讓本研究得以獲得所需的資料。再來要謝謝天文館的陳岸立館長以及成大的許瑞榮老師做我的口試委員，謝謝老師們對我的論文詳盡的審查，以及對於論文內容提出深入的問題和建議，使得這粗糙的論文得以更加的完善。在師大的六年期間，走過各種風風雨雨。感謝幾位摯友：佳宇、梓興、威凱、煒翔，我們這幾個過去常一起騎著機車衝山、拍星星、瞎哈啦的好夥伴不時的支持與鼓勵我，成為我強大的後盾。要感謝憲隆、育倫、翔宇這幾個過去 Fu’s Group 的學長，從你們身上我學到了很多；同是現任 Fu’s Group 的鴻選、延駿、孟嫻、柏諺，我們一起在傅老師底下共患難的日子是段有趣的時光。還要謝謝研究所好同學雅雯、明錡、康華們的鼓勵。自從高中一年級以來同班九年的炳權，天文讀書小組的彧珉、家萱、琨富、若雅、芷郁、亭安，還有其他地科系的一些要好的學長姐、同學、學弟妹們如梓舜、禹函、孟澔、南州、玠堯、緯鈞、裕國、采玟、偉任、唯嫚、美雅等， III.

(5) 以及打工時的老闆兼好友的源智，有這些人的一路陪伴，是我完成論文過程的精神支柱。最後要謝我的父母及家人，謝謝你們在背後給我加油打氣，在我處於各種壓力時，讓我有個避風港。再來要鄭重感謝我的指導老師：傅學海老師，老師的教學總是能以不同的角度切入問題，讓我能思考，雖然得讓老師忍受我的愚鈍。我知道老師對我期待很高，但是我總是做出讓老師頭大的問題出來，尤其是做老師通識課助教這段期間。謝謝老師五年多來的教誨，我非常榮幸、也非常幸運能成為 Fu’s Group 的一員，我想在未來能成為老師所期待的眾所矚目新星。最後，還要在這向老師說：老師退休快樂！. 謹以此文獻給所有支持鼓勵我走向研究之路的人們. IV.

(6) Contents 中文摘要 ................................................................................................. I Abstract ................................................................................................. II 誌謝 ...................................................................................................... III Contents ................................................................................................. V List of Tables ...................................................................................... VII List of Figures .................................................................................. VIII 1.. Introduction ................................................................................ 1 1.1 Membership Probability of Stars in an Open Cluster ................................... 1 1.2 Reddening and Extinction Correction ......................................................... 2 1.3 King’s Model ............................................................................................. 4 1.4 NGC 7142 .................................................................................................. 5. 2.. Observations and Data ............................................................... 7 2.1 LOT images ................................................................................................ 7. 3.. Data Process and Reduction .................................................... 12 3.1 Magnitude Calibration .............................................................................. 12 3.2 Coordinate Calibration, Proper Motion and Parallax ................................. 23. 4.. Membership Determination..................................................... 28 4.1 VPD Axis Rotation ................................................................................... 28 4.2 Membership Determining by Maximum-likelihood Method ..................... 30. 5.. Reddening and Extinction Correction .................................... 36. 6.. King Model Fitting ................................................................... 41. V.

(7) 7.. Result, Conclusions and Discussion......................................... 51. References ............................................................................................ 53 Appendixes ........................................................................................... 55 Appendix A: IRAF information ...................................................................... 55 Appendix B: Python source code .................................................................... 61 Appendix C: Stars List of NGC 7142 ............................................................. 97. VI.

(8) List of Tables Table 1-1. Properties of NGC 7142 comparison to different literatures ...................... 6 Table 2-1. Observation Parameter of LOT ................................................................. 8 Table 3-1 Standard stars from Crinklaw & Talbert (1991) ....................................... 13 Table 3-2 Standard stars from van den Bergh (1970) ............................................... 14 Table 3-3 Constants of magnitude calibration .......................................................... 15 Table 4-1 Iteration parameters ................................................................................. 32 Table 6-1 The ring count and stars number density in NGC 7142 ............................ 44 Table 6-2 The ring count and stars number density in NGC 7142 ............................ 49. VII.

(9) List of Figures Figure 2-1. NGC 7142 in V band of the observation 2003 ......................................... 8 Figure 2-2. NGC 7142 in V band of the observation 2012 ......................................... 9 Figure 2-3. NGC 7142 in U band of the observation 2015O....................................... 9 Figure 2-4. NGC 7142 in B band of the observation 2015O ..................................... 10 Figure 2-5. NGC 7142 in V band of the observation 2015O..................................... 10 Figure 2-6. NGC 7142 in V band of the observation 2015N..................................... 11 Figure 3-1 Magnitude Calibration of V and B band.. ............................................... 15 Figure 3-2 B-V to B-b diagram and b-v to B-b diagram ........................................... 16 Figure 3-3 B-V to V-v diagram and b-v to V-v diagram .......................................... 17 Figure 3-4 Magnitude Calibration of U band.. ......................................................... 18 Figure 3-5 U-B to U-u diagram and u-b to U-u ........................................................ 19 Figure 3-6 The u magnitude to U magnitude calibrated result of all stars ................. 20 Figure 3-7 The b magnitude to B magnitude calibrated result of all stars ................. 21 Figure 3-8 The v magnitude to V magnitude calibrated result of all stars ................. 21 Figure 3-9. CMD of NGC 7142 region in 2015O image .......................................... 22 Figure 3-10. TCD of NGC 7142 region in 2015O image .......................................... 22 Figure 3-11 Mannal selected reference stars in 2012 image. .................................... 23 Figure 3-12. Final reference stars in 2012 image. ..................................................... 24 Figure 3-13. VPD of NGC 7142 region. .................................................................. 26 Figure 3-14. Histogram of parallax .......................................................................... 26 Figure 3-15 VPD of NGC 7142 region.(without the parallax correction.)................. 27 Figure 4-1 VPD of selected stars having both photometry and proper motion info ... 28 Figure 4-2. Proper motion VPD after rotated. ω = -20.51° ....................................... 30 VIII.

(10) Figure 4-3. Maximum-likelihood iteration result...................................................... 33 Figure 4-4. Histogram of membership probability of stars ....................................... 33 Figure 4-5. VPD of NGC 7142 region. probability > 85% ....................................... 34 Figure 4-6. The VPD of membership stars and field stars ........................................ 34 Figure 4-7. CMD of NGC 7142 region. probability > 85% ...................................... 35 Figure 4-8.The CMD of membership stars and field stars ........................................ 35 Figure 5-1. Linear fitting of red giant branch selected stars and unreddening standard curve of Z = 0.010 ................................................................................................... 39 Figure 5-2. CMD of NGC 7142. Unreddening corrected and reddening corrected. ... 39 Figure 5-3. TCD of NGC 7142. Unreddening corrected and reddening corrected ..... 40 Figure 5-4 CMD of NGC 7142 after reddening, extinction and distance modulus corrected with isochrones of different age................................................................ 40 Figure 6-1. Histogram of V magnitude count. .......................................................... 43 Figure 6-2. All Membership stars field of NGC 7142 .............................................. 43 Figure 6-3. Rings of radius from 0.35 to 4.90 arcmin with steps of 0.35 arcmin. ...... 44 Figure 6-4. Standard King model curve of and NGC 7142 curve fitting. .................. 46 Figure 6-5 Histogram of star number to x pixel in 2015O image .............................. 47 Figure 6-6 Histogram of star number to y pixel in 2015O image .............................. 47 Figure 6-7 Histogram of V magnitude count. Binsize = 0.2 mag and magnitude limit V=20.4 .................................................................................................................... 48 Figure 6-8 Rings of radius from 0.35 to 8.40 arcmin with steps of 0.35 arcmin. ....... 48 Figure 6-9 Radius to star density diagram. ............................................................... 50 Figure 6-10 Standard King model curve of and NGC 7142 curve fitting. ................. 50. IX.

(11) 1.. Introduction. The open clusters, or known as galactic clusters, which located on the disk of our Milky Way Galaxy, play an important role on the research of stellar evolution and galactic dynamics. The members of an open cluster formed in the same nebula in a relativity short time. Therefore, they have similar properties such as age, chemical composition, metallicity, interstellar extinction and spatial movement. For past decade, our group has been studying in open cluster including the. photometry,. dynamics,. interstellar. reddening. and. membership determination. This research combines and refines the methods based on the studies completed by our group (Hsieh, 2013; Huang, 2006; Wu, 2004; Wang, 2003) and applies on our target NGC 7142. 1.1 Membership Probability of Stars in an Open Cluster Determining the member star of a cluster is not an easy task, especially an open cluster. The open clusters are located on the galactic disk, where the stars concentrated. In the field of image frame, there are cluster members and field stars. We can tell the member stars from some particular properties, such as photometry, spectroscopy and astrometry. These properties, however, cannot absolutely differentiate these member stars from the field stars because the random field star may have similar properties. Therefore, we can only use the statistic method to tell their likelihood to be a member, i.e. the membership.. 1.

(12) The better way to determine the membership of star belonging to a cluster is measuring the stellar proper motion. The cluster members are forming in the same giant molecule cloud. Their spatial total movement should similar to each other member (Shapley, 1917). Basing on previous fact, Vasileskis et al. (1957) derived the probabilities of each star in the image from proper motion vector-point diagram (VPD). Lately, Vasileskis et al. (1965) assumed the proper motion VPD of cluster members and the field stars distribute in circular and elliptical bivariate frequency function, respectively. As the computing ability progress of computer, Sanders (1971) used mathematical procedure to improve the method proposed by Vasileskis with computer iterating the variable which derived by maximum-likelihood method. This method has been widely used on open clusters research. The maximum-likelihood method has continuously improved by follow-up research. Slovak (1977) first added the correlation coefficient of field star distribution. The uncertainties of proper motion of the membership stars and field stars distribution on the VPD were introduced (Zhao and He, 1990), and the dispersion in image has also been considered (Balaguer-Núňez, Tian & Zhao, 1998) 1.2 Reddening and Extinction Correction The light coming from a distant star goes through the interstellar median (ISM). ISM particles scatter, absorb and reflect these lights and cause the light reaching the observer less than expectation. Therefore, the brightness and color of distance stars should be corrected to find out their intrinsic magnitude and color. The reddening research in Johnson system had started from 1950s Johnson and Morgan (1953) introduce the broad-band filter system 2.

(13) known as UBV photometry system or Johnson system. The color index B−V represents the blackbody continuum line which implies the surface temperature and spectra of stars, and color index U−B represent the Balmer limit as the brightness and temperature. Johnson and Morgan observed the spectra of OB stars and measured the reddening of them to find the reddening relation in two color indexes, B−V and U−B in twocolor diagram (TCD).They introduced the color excesses which are difference of the apparent color and intrinsic color: E(U − B) ≡ (U − B) − (U − B)0 (1.1) E(B − V) ≡ (B − V) − (B − V)0 (1.2) The (U−B) and (B−V) is observed color and (U−B)0 and (B−V)0 is intrinsic color. Johnson and Morgan also discovered the OB stars distributing in TCD are near linear and the slope, which is ratio of color excesses E(U−B)/E(B−V), is near a constant 0.72. The further research by Hiltner and Johnson (1956) gave the correlation of OB stars between E(U−B)/E(B−V) and E(B−V) should be: E(U−B) E(B−V). = 0.72 + 0.05 E(B − V). (1.3). Furthermore, in the research Lindholm (1957) presented that the E(U−B)/E(B−V) have variation with (B−V). Striazys (1976) gave the curve E(U−B)/E(B−V) to (B−V) and the table in his research is an important literature to this work. The most widely used way to measure extinction of interstellar medium is the totally-to-selective extinction ratio Rv:. Rv =. AV E(B−V). 3. (1.4).

(14) The Rv value is variated in many research, but most of them are around 3.0 to 3.1 (e.g. Morgan, Harris and Johnson, 1953; Blanco, 1955; Savage and Mathis, 1979; Cardelli, Clayton and Mathis, 1989) The previous research of our group by Huang (2003) used the mathematic approach to solve the reddening. The main concept of Huang’s research is fitting the linear part in TCD of main sequence stars, which are OB and GK type stars, and compare them to the theoretical isochrones. The detail of how to correct reddening is present at Section 5. 1.3 King’s Model King model is a serial research by King (1962, 1965, 1966a, 1966b, 1968) that describe the dynamics inside the clusters. Jeans (1916) derived density law of clusters and supported the by Bailey’s (1915) observation. King, however, disapproved Jeans’ prove and he point out that Jeans based on the wrong assumptions which could cause non-zero potential at large radius of clusters, and the observation poorly support Jeans’ point. Therefore, King proposed his empirical law of density function, known as King’s model or King’s law, based on globular cluster observational result. The density function depends on 3 kind of radius: 1) radius from center of the clusters 2) the central part radius where the density is near linear called core radius and 3) tidal radius that stars can be hold by gravitational force of clusters. King has also tested his model to on the other spherical system such as open cluster and elliptical galaxy and proofed the model can fit for those systems. The theoretical proof was also published at the third of the sequence paper (King, 1966). In this research, we derived the mass and radius of the target cluster by King’s empirical function and presented it in Section 6.. 4.

(15) 1.4 NGC 7142 NGC 7142 (RA: 21h45m09s, Dec: +65º46ʼ30”, l: 105.35°, b: 9.49°) is an open cluster located at constellation Cepheus whose age is about 3Gyr (Straižys et al., 2014), being one of old open clusters. It contains few hundred stars brighter than 20 magnitude (van den Bergh, 1962) and the diameter is about 12 arcmin (Dias, 2002), which is about the field of view of an one-meter telescope. There are several nebulas nearby the location where NGC 7142 is, causing the uneven absorption across the cluster. The photometry studies of NGC 7142 started in 1960s. The Hoag et al. (1961) finished the first color-magnitude diagram (CMD) and twocolor diagram (TCD) of NGC 7142. The CMD shows NGC 7142 have giant branch extended from around V = 16 despite the limitation of magnitude being around V = 17. Van den Bergh (1962) analyzed the data from Hoag’s observation and concluded the distance modulus m−M = 12.75, total reddening E(B−V) = 0.46 magnitude, extinction AV = 1.38 magnitude and approximate age of NGC 7142 is ~5 Gyr. Van den Bergh & Heeringa (1970) also publish their observation data of photograph photometry using Hale and Kitt peak 2.15m telescope. Their result shows the E(B−V) = 0.41 while E(U−B) = 0.30, and distance modulus m−M = 12.5. The CCD photometry of NGC 7142 was first done by Criklaw and Talbert (1991), the magnitude limit extended to V ~ 18. The most resent research is in 2014, Straižys et al. used Vilnius photometry system to obtain the interstellar extinction, distance and age. About the size of NGC 7142, Sharov (1965) used distribution of star to estmatate the radius of NCG 7142 and concluded it in 1968 that the radius of NGC 7142 is about 12 arcmin.. 5.

(16) Table 1-1. Properties of NGC 7142 comparison to different literatures. Research Van den Bergh (1962). E(B−V) (mag) 0.46. AV. m−M. Age. (mag) (mag). (Gyr). 1.38. 11.37. [Fe/H]. Radius (arcmin). ~5. Sharov (1965). 75. Sharov (1968). 23. Van den Bergh & Heeringa (1970). 0.41. 12.5. Jennes & Helfer (1975). 0.29. 10.9. 5. 11.4. 3.4~4.5. 12.36. 1.89. Cinklaw & Talbert (1991) WEBDA. 0.397. -0.45. +0.04. Jacobson et al. (2007). +0.08±0.06. Jacobson et al. (2008). +0.14±0.01. Straižys et al. (2014). 0.35. 1.1. 11.8. 3.0. This work. 0.50. 1.35. 11.9. 3.2~6.3. 6. +0.20. 12.0.

(17) 2.. Observations and Data. This research use 4 set of image taken from Lulin One-meter Telescope (LOT). The properties of these images will describe in the following sections. 2.1 LOT images The LOT is the largest telescope for now in Taiwan operating by National Central University (NCU). LOT is located on Mt. Lu-lin, a peak in the Yushan National Park. LOT is a Ritchey-Chretien type telescope with focal length 8000 mm. The four LOT image sets are taken in 2003, 2012 and 2015. The details of each image are shown in Table 2-1. In order to calculate the proper motion, we only take the V images of 2003 and 2012 image sets. The 2003 image set was taken by the NTNU graduated student, Shao-Er Chaung (莊孝爾) on August 7,2003. The observation include U, B and V band. Each of them has 786 second exposure because the limitation of observing time. The 2012 image set was taken by LOT operater Hsiang-Yao Hsiao (蕭翔耀) and NCU graduated student Yu-Chi Cheng (鄭宇棋) on July 8, 2012 with B and V band. Both of 2003 and 2012 image set were taken with Princeton Instrument VersArray:1300B CCD which have 1340 × 1300 pixels in 20 × 20 μm per pixel. The field of view (FOV) is about 11 × 11 arcmin and the pixel scale is 0.51 arcsec/pixel. The two 2015 image sets were taken on October 14 and November 11, 2015 by the LOT operator Chi-Shang Lin (林啟生). With the Apogee 7.

(18) U42 CCD and 0.5x focus reducer, the image frame size come to 2048 × 2048 pixel in 13.5 × 13.5 μm with pixel scale 0.63 arcsec/pixel. The October image set contains U, V and B images. Because the poor weather condition at the appointed observation time, the November set exposed only 600 second in V band and observed the NGC 7142 at 21° altitude. Table 2-1. Observation Parameter of LOT FOV. Pixel Scale. (arcmin). (“/pixel). V: 186s. 11 × 11. 0.51. PI 1300B. V: 1020s. 11 × 11. 0.51. 2015/10/14. U42. U:3600s B:900s V:900s. 21.5 × 21.5. 0.63. 2015/11/11. U42. V: 600s. 21.5 × 21.5. 0.63. Obs. ID. Obs. Date. CCD. Exp. Time. 2003. 2003/08/07. PI 1300B. 2012. 2012/07/08. 2015O 2015N. Figure 2-1. NGC 7142 in V band of the observation 2003 8.

(19) Figure 2-2. NGC 7142 in V band of the observation 2012. Figure 2-3. NGC 7142 in U band of the observation 2015O 9.

(20) Figure 2-4. NGC 7142 in B band of the observation 2015O. Figure 2-5. NGC 7142 in V band of the observation 2015O. 10.

(21) Figure 2-6. NGC 7142 in V band of the observation 2015N. 11.

(22) 3.. Data Process and Reduction. The NGC 7142 images from LOT are dark current, flat field and bias calibrated by the Maxim DL program. After images been cleaned, the DAOPHOT package from IRAF program was used to get the position and magnitude information. The following data process and reduction are proceeded by Python 2.7 code we made with Python modules Numpy, Matplotlib, Scipy and Astropy. 3.1 Magnitude Calibration Because the 2015O observation has entire UBV data, we selected them to calculate the photometry. Without observing standard star field, we calibrated the BV magnitude using the CCD photometry data by Crinklaw & Talbert (1991) and UB magnitude with van den Bergh (1970) as our standard stars (Table 3-1, 3-2). The instrumental magnitude of reference stars are calibrated by equation below where the k1, k2 of each equation are found by regression (Table 3-3). U − u = k1U + k2U (u − b). (3.1). B − b = k1B + k2B (b − v). (3.2). V − v = k1V + k2V (b − v). (3.3). 12.

(23) Table 3-1 Standard stars from Crinklaw & Talbert (1991). Star 9 12 37 54 58 76 87 91 98 99 109 118 150 154 162 169 178 190 217 220 224 230 233 393 235 237 243 252 280 303 308 318 331 341 342 343 364 372 375 420. V 16.37 17.10 17.07 14.60 16.28 18.45 18.33 16.13 16.25 16.29 16.00 16.59 17.76 16.42 17.69 16.81 16.76 19.44 17.58 15.75 15.03 14.82 18.43 14.56 17.21 16.69 17.64 15.81 16.87 14.87 15.78 18.18 14.19 15.94 14.37 18.06 18.78 15.89 17.55 16.37 13. B-V 1.16 1.13 1.56 0.45 0.86 0.97 0.88 0.90 0.78 0.86 0.89 1.52 1.26 0.99 1.39 0.99 0.93 0.88 1.10 0.96 0.93 1.65 1.46 0.96 0.99 0.89 0.87 1.51 0.81 1.73 0.97 1.18 1.47 1.29 1.52 1.19 1.55 0.96 1.06 0.89.

(24) Table 3-2 Standard stars from van den Bergh (1970). Star G H I J K L M N O P Q R S T U a 36 39 43 45 50 52 54 58. V 13.12 13.44 13.45 13.42 13.59 13.58 13.52 13.70 13.71 14.04 14.60 15.07 14.90 15.38 15.05 15.10 15.88 15.90 15.39 15.08 15.76 12.12 15.14 15.53. B-V 1.62 0.70 1.39 0.62 0.79 0.70 0.63 1.29 1.40 1.47 1.02 0.86 0.67 0.91 1.15 0.96 0.77 0.83 0.89 1.03 0.78 1.87 0.90 0.85. 14. U-B 1.68 0.09 1.10 0.08 0.15 0.17 0.12 1.06 1.02 1.36 0.41 0.25 0.41 0.24 0.29 0.22 0.31 0.23 0.20 0.24 0.38 2.06 0.14 0.25.

(25) Table 3-3 Constants of magnitude calibration Filter. U. B. V. k1. 1.610. 4.749. 5.322. k2. 0.206. 0.357. -0.077. Figure 3-1 Magnitude Calibration of V and B band. The blue dots are instrumental magnitude and the black dots are calibrated magnitude.. 15.

(26) Figure 3-2 Standard star color to magnitude difference diagram (top panel) and instrumental color to magnitude difference (Button panel) of B band. 16.

(27) Figure 3-3 Standard star color to magnitude difference diagram (top panel) and instrumental color to magnitude difference (Button panel) of V band. 17.

(28) Figure 3-4 Magnitude Calibration of U band. The blue dots are instrumental magnitude and the black dots are calibrated magnitude.. 18.

(29) Figure 3-5 Standard star color to magnitude difference diagram (top panel) and instrumental color to magnitude difference (Button panel) of U band. 19.

(30) If we subtract equation (3.3) from (3.2) and replace (B-V) with (3.3), we get calibrated V magnitude: Vcal = v + k1V +. (k1B −k1V )+(b−v) [1−(k2B −k2V )]. (3.4). We can also get B and U magnitude by similar method: Bcal = b + k1B +. (k1B −k1V )+(b−v) [1−(k2B −k2V )]. Ucal = u + k1U +. (k1U −k1B′ )+(u−b) [1−(k2U−k2B′ )]. (3.5) (3.6). The results after calibration are shown in color-magnitude diagram (CMD) and two-color diagram (TCD) next page.. Figure 3-6 The u magnitude to U magnitude calibrated result of all stars. 20.

(31) Figure 3-7 The b magnitude to B magnitude calibrated result of all stars. Figure 3-8 The v magnitude to V magnitude calibrated result of all stars. 21.

(32) Figure 3-9. CMD of NGC 7142 region in 2015O image. Figure 3-10. TCD of NGC 7142 region in 2015O image. 22.

(33) 3.2 Coordinate Calibration, Proper Motion and Parallax The position and coordinates measurement are taken from the V image frame of all LOT observation. To match the stars from the selected frame, we converted the star coordinates to the reference frame coordinates, which we selected the 2012 observation. We manually selected 23 random stars (Figure 3-11) as reference stars so that we can derive the convert coefficients by regression from the following equation: x2012 = axi + byi + c. (3.7). y2012 = dyi + exi + f. (3.8). Where (a, d) are scale, (b, e) are rotation term and (c, f) are shift constant. After getting these coefficients, we converted the star coordinates of each observation and matched the stars.. Figure 3-11 Mannal selected reference stars in 2012 image. The reference stars are marked in red circle 23.

(34) Figure 3-12. Final reference stars in 2012 image. The reference stars are marked in red circle.. The stars are not fixed on the celestial sphere. They will move to a certain direction because stars revolve around the Galactic center at different orbit and cause relative motion to the observer which called proper motion. The proper motion is a function of time that is: Δα = μα Δt cos δ. (3.9). Δδ = μδ Δt. (3.10). Where the Δα, Δδ is displacement of right ascension and declination, Δt is the interval of time. The Earth also revolves around the sun on the ecliptic plane at the same time, causing another relative movement named parallax. To cancel out the inference by parallax, we have to add parallax term in the equation. The total displacement of stars should be like following equation:. 24.

(35) ∆α∗ = (xi − xref ) × scale = μ∗α ∆t + Pα πα (3.11) ∆δ = (yi − yref ) × scale = μδ ∆t + Pδ πδ. (3.12). The Δx, Δy are displacement of a star in x and y direction, respectively. The xi, yi are coordinates of stars of each image and xref, yref are coordinates from reference image. The μx, μy are proper motion in x and y direction. Δt is the interval of observation time. Px, Py are parallax factors and πx, πy are parallax terms. The parallax factors are equation of solar longitude, which are shown in equation 3.13 and 3.14. Pα = cos α cos ϵ sin ⨀ − sin α cos ⨀ (3.13) Pδ = (sin ϵ cos δ − cos ϵ sin α sin δ) sin ⨀ − cos α sin δ cos ⨀ (3.14) The α and δ are the stars position of observing time. ϵ is the inclination of Earth’s rotation axis and ⨀ is solar true ecliptic longitude, both of ϵ and ⨀ are calculated from NOAA’s solar calculator for years. After the coordinates of stars in each image been derived from previous section, we match the stars with their coordinates. We calculated proper motion and parallax of the matched stars with equation 3.11 and 3.12. The randomly selected reference stars is possibly having large displacement therefore we reassigned reference stars which have proper motion < 10 mas/yr and parallax < 10 mas after we had derived the result, and repeated the previous procedure from coordinates calibration to derive proper motion for 10 times. In the end, there are totally 1148 stars have been matched and their proper motion and parallax are derived and their proper motion VPD and histogram of πα, πδ are shown in Figure 313 and 3-14. 25.

(36) Figure 3-13. VPD of NGC 7142 region.. Figure 3-14. Histogram of parallax. 26.

(37) The Figure 3-14 indicates that the parallax distribution has serious problem and it probabily cause by only 3 observation intervals. The literature of NGC 7142 shows the distance is over 1000 pc, meaning the parallax of most stars should less the 0.001 arcsec. Therefore, we redo the coordinates calibration. But this time we did not using proper motion combining parallax correction program, we did it only with proper motion program instad. The result of VPD we demonstrate in Figure 3-15.. Figure 3-15 VPD of NGC 7142 region. The proper motion are without the parallax correction.. 27.

(38) 4. Membership Determination. The proper motion of stars are derived from V image frame of all observation set. In order to get the CMD of the membership stars of NGC 7142, we matched the photometry data and proper motion data and selected the stars both having BV photometry and proper motion information (Figure 4-1).. Figure 4-1 VPD of selected stars having both photometry and proper motion info. 4.1 VPD Axis Rotation The distribution of field stars is an elliptic bivariate distribution. The principal axis of elliptic dispersion, however, is not always along the direction of right ascension and declination. Therefore, we need to rotate 28.

(39) the orientation of axis to align where the principal axis is, and Vasilevskis (1965) provided an approach to solve this rotation angle. This approach is to find the maximum or minimum dispersion along the axes. Here we assume the difference between each star and mean point in x and y directions are ξ and η: ξi = μαi − μ ̅̅̅, μδ α ηi = μδi − ̅̅̅. (4.1). These two components then were transformed into rotated axes coordinate system: u(ω) = ξ cos(ω) + η sin(ω). (4.2). v(ω) = η cos(ω) − ξ sin(ω). (4.3). where the ω is rotation angle. Then, we define the function S(ω): S(ω) = Σ[u(ω)]2. (4.4). The function S(ω) is total dispersion along the rotated axes and is a periodic function from ω=0° to 180°. We computed the S(ω) with interval 0.01° and relationship is: S(ω) = a + k cos(ω − Θ). (4.5). where the Θ is the angle between original axes and principal axes of the elliptic dispersion. The equation 4.1.5 can be expended as: S(ω) = a + b cos(ω) + c sin(ω) , b = k cos(Θ), c = k sin(Θ) (4.6) The constants a, b and c can be found by least-square, and from the equation 4.6: tan(Θ) = b⁄c. 29. (4.7).

(40) The Θ can be derived from tan-1. In case of our data, the rotation angle of principal axes ω = -20.51° The VPD after rotated is presented in Figure 4-2.. Figure 4-2. Proper motion VPD after rotated. ω = -20.51°. 4.2 Membership Determining by Maximum-likelihood Method The approach we used base on Sanders (1971) with modification of the measurement error by Zhao and He (1991). The bivariate distribution function of each star is: Φ(μxi , μyi ) = Nc 2π(σ2 +ϵ2i ). 1 (μxi −μxf )2. Nf 2π√Σ2x +ϵ2i √Σ2y +ϵ2i. exp {− [. 1 (μxi −μxf )2. exp {− [ 2. σ2 +ϵ2i. 2. +. Σ2x +ϵ2i. (μyi −μyf) σ2 +ϵ2i. +. (μyi −μyf ) Σ2y +ϵ2i. 2. ]} +. 2. ]}. (4.8). Where the Nc and Nf are number of cluster and field stars, and the summation of them equal to total number of stars. Σx and Σy are half length of major and minor axis of elliptical field stars dispersion while σ 30.

(41) is circular cluster stars dispersion. The μxf, μyf, μxc and μyc are the center of the bivariate distribution of field stars and cluster stars, respectively. ϵ is the error of proper motion of individual star that from the proper motion fitting in Section 3.2. To solve these parameters, we use the maximum-likelihood method which presents in the equation 4.9: ∑NST i=1. ∂Φ(μxi ,μyi ) ∂uj. = 0, j = 0,1,2, … ,8. (4.9). The uj means the each of the 8 parameter. Here we let: 1 (μxi −μxf )2. α = exp {− [ 2. Σ2x +ϵ2i. 1 (μxi −μxf )2. β = exp {− [ 2. σ2 +ϵ2i. +. (μyi −μyf ). 2. Σ2y +ϵ2i. ]}. (4.10). ]}. (4.11). 2. +. (μyi −μyf ) σ2 +ϵ2i. And the results are 8 non-linear equations which are: NST. Nc : ∑ i=1. 1 Φ. α (. √Σx2. NST. μxf : ∑ i=1. NST. μyf : ∑ i=1. 1 ∙ Φ. 1 ∙ Φ. +. −. ϵ2i √Σy2. +. ϵ2i. (σ2. β + ϵ2i ). α(μxi − μxf ) √(Σx2. +. ϵ2i )3 √Σy2. NST. μxc : ∑ i=1 NST. μyc : ∑ i=1. +. ϵ2i √(Σy2. +. =0 +. =0 ϵ2i )3. 1 β(μxi − μxc ) ∙ =0 Φ (σ2 + ϵ2i )2 1 β(μyi − μyc ) ∙ =0 Φ (σ2 + ϵ2i )2. 31. ). ϵ2i. α(μyi − μyf ) √Σx2. =0.

(42) NST. Σx : ∑ i=1. 1 ∙ Φ. α[. (μxi − μxf )2 − 1] Σx2 + ϵ2i. √(Σx2. +. ϵ2i )3 √Σy2. +. =0. ϵ2i. 2. NST. Σy : ∑ i=1. 1 ∙ Φ. (μyi − μyf ) α[ − 1] Σy2 + ϵ2i √Σx2. +. ϵ2i √(Σy2. +. =0. ϵ2i )3. NST. 2. (μxi − μxc )2 − (μyi − μyc ) 1 β [ σ=∑ ∙ 2 − 2] = 0 Φ (σ + ϵ2i )2 σ2 + ϵ2i i=1. These 8 parameters are solved by the iteration. We iterated for 1000 times and the initial value and final result of iteration are show in the Table 4-1 Table 4-1 Iteration parameters Parameter. Initial value. Adjust step. Result value. Number of cluster stars. 562. ±1. 910. Center for cluster stars. (0.10, 0.10). ±0.01. (0.12, -0.04). Sigma for cluster stars. 5.00. ±0.01. 0.23. Center for field stars. (0.10, 0.10). ±0.01. (6.22, -0.36). Sigma for field stars. (10.00, 10.00). ±0.01. (9.14, 8.61). The membership probability of each star can be described as: Pc =. Φc. (4.12). Φf +Φc. The follow figure show the histogram of probability, and the bin size is 5%. We take the stars whose probability > 85% as the membership stars, and the membership stars are present on VPD, CMD and TCD. 32.

(43) Figure 4-3. Maximum-likelihood iteration result. Red circle is cluster stars and blue ellipse is field stars distribution function. Green line is orientation of Galactic plan, yellow arrow is direction of solar apex and the brown arrow is the principal axis of field star ellipse.. Figure 4-4. Histogram of membership probability of stars 33.

(44) Figure 4-5. VPD of NGC 7142 region. The red dots are the stars whose probability > 85%. Figure 4-6. The VPD of membership stars (left panel) and field stars (right panel). 34.

(45) Figure 4-7. CMD of NGC 7142 region. The red dots are the stars whose probability > 85%. Figure 4-8.The CMD of membership stars (left panel) and field stars (right panel) 35.

(46) 5. Reddening and Extinction Correction. The reddening and extinction correction to each star we used is done by Huang (2003). In the Straizys’ (1976) research, he list a table of the relation between the reddening curve slope E(U−B)/E(B−V) and spectral type, and the B−V range as well. Most of the curve whose B−V within 0.5 to 1.5 in E(U−B)/E(B−V) to B−V diagram are near linear. Huang used the part of table for main sequence stars to find the relation of E(U−B)/E(B−V) to B−V. However, in the case of NGC7142, which is an old open cluster, is no longer contain early-type stars and have more stars in giant branch. In the case of that we take the red giant instead. We have derived that the slope of reddening curves at giant branch with spectral type G0III to K3III (0.57 < B−V < 1.31) from Straizys’ Table 3 should be: E(U−B) E(B−V). = k1 (B − V)0 + k2. (5.1). k1 = 0.278, k2 = 0.675 We selected the red giant branch part in the CMD of probable membership stars which (B-V) > 1.2 and V < 16.5 and did the linear fitting to these stars in TCD. The isochrones came from the CMD 2.8 data archive. Hereafter we replace the (U−B) and (B−V) to y and x respectively. Assuming the reddening curve is linear, the line equation of reddening curve line should be: y=. E(U−B) E(B−V). x+C. and the isochrone and data fitting lines are: 36. (5.2).

(47) y0 = a 0 x 0 + b 0. (5.3). y1 = a1 x1 + b1. (5.4). Where the (x0, y0) and (x1, y1) are [(B−V)0, (U−B)0] and (B−V, U−B). The (x0, y0) and (x1, y1) are two points on the reddening curve and we get the simultaneous equations:. {. y0 = y1 =. E(U−B). x E(B−V) 0 E(U−B) x E(B−V) 1. +C. (5.5). +C. By Subtracting the two equation and replacing the E(U−B)/E(B−V) and y, we get: y0 − y1 = (a0 x0 + b0) − (a1 x1 + b1) = (k1 x0 + k2 )(x0 − x1 ) (5.6) And we sort the equation: k1 x02 + (k2 − k1 x1 − a0 )x0 + (a1 x1 − k2 x1 + b1 − b0) = 0 (5.7) Solve the x0 then we get the intrinsic color (B−V)0. The (U−B)0 can be found by bring the x0, x1 and y1 into the equation. And the extinction AV can be derived from RV, here we take average interstellar extinction RV = 3.1. However, the different metallicities can make the different isochrones, which will effect on the result of reddening correction (Figure 5-1). First, we set the age at 5 Gyr which is approximate age of NGC 7142. We have tested from Z = 0.01 to 0.06 with step 0.001 and compared the standard curves to reddening-corrected data both in CMD and TCD, and the fitting result at Z = 0.010 is best-matched line in the. 37.

(48) TCD and CMD. The result suggests that the distance modulus is 11.60 ± 0.1 magnitude and the logarithm age is 9.50~9.80.. 38.

(49) Figure 5-1. Linear fitting of red giant branch selected stars (blue dot) and unreddening standard curve of Z = 0.010 (black line). Figure 5-2. CMD of NGC 7142. Black dots are unreddening corrected and red dot are reddening corrected. 39.

(50) Figure 5-3. TCD of NGC 7142. Black dots are unreddening corrected and red dot are reddening corrected. Figure 5-4 CMD of NGC 7142 after reddening, extinction and distance modulus corrected with isochrones of different age.. 40.

(51) 6. King Model Fitting. King (1962) derived the empirical dynamics model to express the relation of stars number density, and tidal radius rt as a function of radius, which is: 1. 1 2. r. rt. f(r) = f1 [ − ]. (6.1). where the f(r) is star number density distribution depending on r from the cluster center. f1 is a constant. At the central part of the cluster, the function of density at center area can be express as:. f(r) =. f0. r 2 1+( ) rc. (6.2). King combined the Eq. 6.1 and 6.2 and the equation would be: 2 1. f(r) = k. − 2. r √1+( ) rc. [. 1 √1+(rc) rt. (6.3). 2. ]. From the Eq 6.2, we invers each side of the equation and we get:. 1 f(r). =. r2 f0 rc. +. 1 f0. (6.4). the f0 is the stars density of concentrated part. The f0 and rc can be derived by least-square fitting. And the rt can be obtain from the Eq 6.1 with slightly change: 1. √f(r) = √𝑓1 ( ) − 𝑟. 41. √𝑓1 (6.5) 𝑟𝑡.

(52) The mass of the the cluster can be derive by Eq. 6.6. rt = [. GMc. 4A(A−B). 1 3. ]. (6.6). Where the A and B are Oort constants. Here we take the Oort constants from Feast & Whitelock (1997) where A = 14.82 ± 0.84 km s-1 kpc-1, B = −12.37 ± 0.64 km s-1 kpc-1. There are 774 membership stars we have selected in previous section. The following figure shows the stars field of the membership stars positions. Here we plotted the number count of V magnitude to estimate the completeness, and completeness is at V = 19.4 mag. There are total 498 stars have been selected. Considering the approximate diameter of NGC7142 is ~10 arcmin, Table 6-1 shows the star count of different radius rings from center to 4.9 arcmin with steps of 0.35 arcmin, and as well the density of each ring area. The center of cluster is fit by Gaussian distribution of each axis.. 42.

(53) Figure 6-1. Histogram of V magnitude count. Binsize = 0.2 mag and magnitude limit V = 19.4. Figure 6-2. All Membership stars field of NGC 7142 43.

(54) Figure 6-3. Rings of radius from 0.35 to 4.90 arcmin with steps of 0.35 arcmin. The Stars are selected from completeness V < 19.4. Table 6-1. The ring count and stars number density in NGC 7142 Ring. Inner radius Outter radius Star count. Sky area. Density. 1. 0.00. 0.35. 2. 0.3848. 5.197. 2. 0.35. 0.70. 13. 1.1545. 11.260. 3. 0.70. 1.05. 22. 1.9242. 11.433. 4. 1.05. 1.40. 30. 2.6939. 11.136. 5. 1.40. 1.75. 32. 3.4636. 9.239. 6. 1.75. 2.10. 33. 4.2333. 7.795. 7. 2.10. 2.45. 45. 5.0030. 8.995. 44.

(55) Ring. Inner radius Outter radius Star count. Sky area. Density. 8. 2.45. 2.80. 51. 5.7727. 8.835. 9. 2.80. 3.15. 57. 6.5424. 8.712. 10. 3.15. 3.50. 53. 7.3121. 7.248. 11. 3.50. 3.85. 55. 8.0817. 6.805. 12. 3.85. 4.20. 63. 8.8514. 7.117. 13. 4.20. 4.55. 62. 9.6211. 6.444. 14. 4.55. 4.90. 61. 10.3908. 5.817. The fitting result of rc, f0 are 5.31 arcmin and 10.91 stars/arcmin2 respectively. However, the log(f/f0) to log(r/rc) diagram shows in our case which lack of outer ring star count, the tidal radius rt can not be obtained from Eq. 6.5 (Figure 6-4). The standard King’s model curve fitting shows there are quite large range of possible rt/rc ratio. Unless having further observation, here we are not able to conclude the mass of NGC 7142 with the membership stars.. 45.

(56) Figure 6-4. Standard King model curve of log(r/rc) = 1.0 ~ 2.0 and NGC 7142 curve fitting.. Therefore, we try another approach to fit the King module. We take the 2015O V image to count the stars. The center of the cluster was found from histogram of star number of x and y axes by using Gaussian curve fitting. We count all of the stars instad the membership star. The star count and density are shown in Table 6-2. The cluster star density are the total star density subtracting the background star density which fitting by a Gaussian function in radius to density diagram. The fitting process are the same as the previous part of this chapter. The result of rc, f0, and rt are 2.12 arcmin, 11.55 stars/arcmin2 and 36.37 arcmin, respectively. The log(rt/rc) = 1.23. The mass of NGC 7142 that obtain by Eq. 6.6 is about 4100 Mʘ.. 46.

(57) Figure 6-5 Histogram of star number to x pixel in 2015O image. Figure 6-6 Histogram of star number to y pixel in 2015O image. 47.

(58) Figure 6-7 Histogram of V magnitude count. Binsize = 0.2 mag and magnitude limit V=20.4. Figure 6-8 Rings of radius from 0.35 to 8.40 arcmin with steps of 0.35 arcmin. The Stars are selected from completeness V < 20.4 48.

(59) Table 6-2 The ring count and stars number density in NGC 7142 Ring. Inner radius Outter radius. Star count. Sky area. Density. 1. 0. 0.35. 8. 0.385. 20.788. 2. 0.35. 0.7. 30. 1.155. 25.984. 3. 0.7. 1.05. 42. 1.924. 21.827. 4. 1.05. 1.4. 61. 2.694. 22.644. 5. 1.4. 1.75. 75. 3.464. 21.654. 6. 1.75. 2.1. 87. 4.233. 20.551. 7. 2.1. 2.45. 83. 5.003. 16.59. 8. 2.45. 2.8. 104. 5.773. 18.016. 9. 2.8. 3.15. 109. 6.542. 16.661. 10. 3.15. 3.5. 124. 7.312. 16.958. 11. 3.5. 3.85. 126. 8.082. 15.591. 12. 3.85. 4.2. 151. 8.851. 17.059. 13. 4.2. 4.55. 153. 9.621. 15.903. 14. 4.55. 4.9. 157. 10.391. 15.109. 15. 4.9. 5.25. 151. 11.161. 13.53. 16. 5.25. 5.6. 162. 11.93. 13.579. 17. 5.6. 5.95. 181. 12.7. 14.252. 18. 5.95. 6.3. 188. 13.47. 13.957. 19. 6.3. 6.65. 203. 14.239. 14.256. 20. 6.65. 7. 220. 15.009. 14.658. 21. 7. 7.35. 188. 15.779. 11.915. 22. 7.35. 7.7. 223. 16.548. 13.476. 23. 7.7. 8.05. 211. 17.318. 12.184. 24. 8.05. 8.4. 214. 18.088. 11.831. 49.

(60) Figure 6-9 Radius to star density diagram. The red dash line is the background density.. Figure 6-10 Standard King model curve of log(r/rc) =1.03 ~ 1.43 and NGC 7142 curve fitting. 50.

(61) 7. Result, Conclusions and Discussion. We have presented the procedure which can obtain dynamical properties such as radius and mass of cluster from the kinematics and CCD photometry. The membership stars of NGC 7142 were selected from the proper motions of the open cluster that we measured in 4 epochs of observations. The proper motion of stars was corrected from the effect of parallax. The membership probabilities belonging to NGC 7142 were derived from maximum-likelihood method with proper motion error and there are 774 stars have been selected as membership stars. The interstellar reddening and extinction correction were found by fitting the TCD red giant branch line and the result showed the average reddening E(B−V) = 0.42, E(U−B) = 0.45 and mean extinction Av = 1.30. while Z = 0.010 or [M/H] = -0.28. By fitting the standard evolution curve we got the distance modulus (m−M) = 11.60 ± 0.1 and logarithm age = 9.5 ~ 9.8, or distance ~ 2100 pc while age = 3.2 ~ 6.3 Gyr. The tidal radius from King model fitting is 36.37 arcmin, or about 22 pc. The mass of NGC 7142 is about 4100 Mʘ. There still have several problems going to be solved in this research. Although the observations passed a decade is enough to derived proper motion, the 4 epochs are still a little insufficient to fit parallax properly. Secondly, the NGC 7142 area have uneven distributing interstellar medium that cause the reddening correction more complex and difficult, but here we still using general correction method and that might have more uncertainty on reddening and extinction correction. From the V images of 2015O and 2015N, there are circular ring-shape nebula seems locate across the NGC 7142. The reddening correction should consider 51.

(62) these nebula but we are not able to correct it for now. The concept of correcting these reddening and extinction causing by the nebula is that to fit these ring and correct the reddening and extinction ring by ring separately. The King model profile fitting shows the data point are concentrate at core part and density at outer part of the cluster have large error. It is probably due to small field of view in 2003 and 2012 images. The further observations with better FOV are required in the future work. The procedure we used is currently aimed to one open cluster, NGC7142. In the future, we hope there will be more open cluster observation in different age, location, distance, metallicity or even spiral arm to know how the open cluster evolve, how they relate to the evolution of Milky Way Galaxy and mapping 3 dimensional distribution of open cluster.. 52.

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(65) Appendixes. Appendix A: IRAF information We use the DAOPHOT package of IRAF program to progress the position and photometry data. There are the parameters in DAOPHOT of each image frame. File 1. 20030807V.fits IRAF file name: 20030807V IMEXAM: STDDEV: 6.65 6.78 7.69 6.98 7.56 7.21 6.88 7.04 7.58 6.42 avg 7.04 ≈ 7.0 IMEXAM: FWHM: 2.54 2.56 2.31 2.75 2.67 2.58 2.9 2.47 2.43 2.79 avg 2.60 DAOFIND: FWHMPSF = 2.6 SIGMA = 7.0 THRESHOLD = 4 PHOT: CALGORITHM = none SALGORITHM = mode ANNULUS = 10 DANNULUS = 10 APERTURE = 2.6 PSF: FUNCTION = gauss VARORDER = 1 PSFRAD = 11 FITRAD = 6 20 psf stars. par1: 1.250444. par2: 1.311973 ALLSTAR: RECENTER = yes GRPSKY = yes FITSKY = yes ANNULUS = 10 DANNULUS = 10 MAXGROUP = 60 txdump > 20030807V.als id,xcen,ycen,mag,merr. 55.

(66) file 2. 20120708V.fits IRAF file name: 20120708V IMEXAM: STDDEV: 35.4 49.6 42.5 35.8 31.2 47.5 39.2 34.5 42.1 46.7 avg 40.45 ≈ 40.5 IMEXAM: FWHM: 2.84 2.56 3.11 2.89 2.97 2.58 2.94 2.47 3.01 2.68 avg 2.8 DAOFIND: FWHMPSF = 2.8 SIGMA = 40.5 THRESHOLD = 4 PHOT: CALGORITHM = none SALGORITHM = mode ANNULUS = 12 DANNULUS = 12 APERTURE = 2.8 PSF: FUNCTION = gauss VARORDER = 1 PSFRAD = 12 FITRAD = 6 18 psf stars. par1: 1.441058 par2: 1.540893 ALLSTAR: RECENTER = yes GRPSKY = yes FITSKY = yes ANNULUS = 12 DANNULUS = 6 MAXGROUP = 60 txdump > 20120708V.als id,xcen,ycen,mag,merr. 56.

(67) file 3. 20151014V.fits IRAF file name: 20151014V IMEXAM: STDDEV: 43.5 39.6 42.5 45.8 41.2 47.5 39.2 54.2 42.1 46.7 avg 44.5 ≈ 45 IMEXAM: FWHM: 2.54 2.33 2.22 2.47 2.41 2.36 2.42 2.51 2.36 2.34 avg 2.39 DAOFIND: FWHMPSF = 2.39 SIGMA = 45 THRESHOLD = 4 PHOT: CALGORITHM = none SALGORITHM = mode ANNULUS = 10 DANNULUS = 10 APERTURE = 2.4 PSF: FUNCTION = gauss VARORDER = 1 PSFRAD = 11 FITRAD = 7 14 psf stars. par1: 1.278506 par2: 1.17063 ALLSTAR: RECENTER = yes GRPSKY = yes FITSKY = yes ANNULUS = 10 DANNULUS = 10 MAXGROUP = 60 txdump > 20151014V.als id,xcen,ycen,mag,merr. 57.

(68) file 4. 20151014B.fits IRAF file name: 20151014B IMEXAM: STDDEV: 30.2 36.5 25.0 26.7 31.2 27.6 32.6 29.8 24.1 35.1 avg 29.9 ≈ 30 IMEXAM: FWHM: 2.54 2.77 2.36 2.76 2.84 2.68 2.73 2.61 2.46 2.34 avg 2.61 DAOFIND: FWHMPSF = 2.61 SIGMA = 30 THRESHOLD = 4 PHOT: CALGORITHM = none SALGORITHM = mode ANNULUS = 12 DANNULUS = 7 APERTURE = 2.6 PSF: FUNCTION = gauss VARORDER = 1 PSFRAD = 12 FITRAD = 7 17 psf stars. par1: 1.393874 par2: 1.326062 ALLSTAR: RECENTER = yes GRPSKY = yes FITSKY = yes ANNULUS = 12 DANNULUS = 7 MAXGROUP = 60 txdump > 20151014B.als id,xcen,ycen,mag,merr. 58.

(69) file 5. 20151014U.fits IRAF file name: 20151014U IMEXAM: STDDEV: 20.3 24.7 25.0 19.3 21.3 24.2 18.9 19.6 24.1 21.5 avg 21.9 ≈ 22 IMEXAM: FWHM: 2.54 2.77 2.36 2.76 2.84 2.68 2.73 2.61 2.46 2.34 avg 3.2 DAOFIND: FWHMPSF = 3.2 SIGMA = 22 THRESHOLD = 4 PHOT: CALGORITHM = none SALGORITHM = mode ANNULUS = 18 DANNULUS = 6 APERTURE = 3.2 PSF: FUNCTION = gauss VARORDER = 1 PSFRAD = 15 FITRAD = 9 15 psf stars. par1: 1.940688 par2: 1.454519 ALLSTAR: RECENTER = yes GRPSKY = yes FITSKY = yes ANNULUS = 18 DANNULUS = 6 MAXGROUP = 60 txdump > 20151014U.als id,xcen,ycen,mag,merr. 59.

(70) file 6. 20151014U.fits IRAF file name: 20151014U IMEXAM: STDDEV: 20.3 24.7 25.0 19.3 21.3 24.2 18.9 19.6 24.1 21.5 avg 21.9 ≈ 22 IMEXAM: FWHM: 2.98 3.31 3.57 3.34 2.84 3.44 2.73 2.89 3.46 3.61 avg 3.2 DAOFIND: FWHMPSF = 3.2 SIGMA = 22 THRESHOLD = 4 PHOT: CALGORITHM = none SALGORITHM = mode ANNULUS = 18 DANNULUS = 6 APERTURE = 3.2 PSF: FUNCTION = gauss VARORDER = 1 PSFRAD = 15 FITRAD = 9 15 psf stars. par1: 1.940688 par2: 1.454519 ALLSTAR: RECENTER = yes GRPSKY = yes FITSKY = yes ANNULUS = 18 DANNULUS = 6 MAXGROUP = 60 txdump > 20151014U.als id,xcen,ycen,mag,merr. 60.

(71) Appendix B: Python source code Here we present the Python code made by our own. There are 6 programs and they are sorted by the order of the sequence. 1. photometry.py # -*- coding: utf-8 -*""" UBV phtometry @author: Yi-Hsiang Hsu """ import numpy as np import matplotlib.pyplot as plt path="C:\\Users\\Yi-Hsiang\\Documents\\Astro\\Research\\NGC7142\\OBS\\20151014\\" fltr=["V","B"] fin=[] star=[] for i in range(len(fltr)): fin.append(open(path+"20151014"+fltr[i]+".als", "r").readlines()) star.append([]) for j in range(len(fin[i])): star[i].append(np.array(fin[i][j].split(),dtype=np.float)) fin=[] ref=[] for i in range(len(fltr)): fin.append(open(path+"20151014BV_"+fltr[i]+".photref", "r").readlines()[2:]) ref.append([]) for j in range(len(fin[i])): ref[i].append(np.array([fin[i][j].split()[0],fin[i][j].split()[1]],dtype=np.float)) #match ref in star and get info rstar=[] coorx=[] coory=[] for i in range(len(fltr)): rstar.append([]) for j in range(len(ref[i])): for k in range(len(star[i])): if (np.abs(star[i][k][1]-ref[i][j][0]) < 2) and (np.abs(star[i][k][2]ref[i][j][1]) < 2): rstar[i].append(star[i][k][1:]) break rstar[i]=np.array(rstar[i]).T coorx.append(np.polyfit(rstar[i][0],rstar[0][0],deg=1)) coory.append(np.polyfit(rstar[i][1],rstar[0][1],deg=1)) #phot ref phot=[] for i in range(len(ref[0])): phot.append(np.array((open(path+"NGC7142BV.photref", "r").readlines())[i].split()[1:],dtype=np.float)). 61.

(72) phot[i]=np.array([phot[i][0],phot[i][0]+phot[i][1]]) phot=np.array(phot).T vfit=np.polyfit(rstar[1][2]-rstar[0][2],phot[0]-rstar[0][2],deg=1) Vfit=np.polyfit(phot[1]-phot[0],phot[0]-rstar[0][2],deg=1) bfit=np.polyfit(rstar[1][2]-rstar[0][2],phot[1]-rstar[1][2],deg=1) Bfit=np.polyfit(phot[1]-phot[0],phot[1]-rstar[1][2],deg=1) magcal=[Vfit[0]/(1-Bfit[0]+Vfit[0]),Bfit[0]/(1-Bfit[0]+Vfit[0])] bvcalv=[vfit[0]/Vfit[0],(vfit[1]-Vfit[1])/Vfit[0]] bvcalb=[bfit[0]/Bfit[0],(bfit[1]-Bfit[1])/Bfit[0]] plt.figure(figsize=[8,10]) plt.subplot(211) plt.plot(range(14,21),range(14,21), "r-") plt.plot(rstar[0][2],phot[0],"bo",label=r"$v$") plt.plot((rstar[0][2]+magcal[0]*(Bfit[1]-Vfit[1]+(rstar[1][2]-rstar[0][2])\ *bvcalv[0]+bvcalv[1])+Vfit[1]), phot[0],"k.",label=r"$V_{cal}$") plt.xlabel(r"$v\ &\ V_{cal}$",fontsize=14) plt.ylabel(r"$V$",fontsize=14) plt.legend(loc=2) plt.subplot(212) plt.plot(range(14,22),range(14,22), "r-") plt.plot(rstar[1][2],phot[1],"bo",label=r"$b$") plt.plot((rstar[1][2]+magcal[1]*(Bfit[1]-Vfit[1]+(rstar[1][2]-rstar[0][2])*\ bvcalb[0]+bvcalb[1])+Bfit[1]),phot[1],"k.",label=r"$B_{cal}$") plt.xlabel(r"$b\ &\ B_{cal}$",fontsize=14) plt.ylabel(r"$B$",fontsize=14) plt.legend(loc=2) plt.tight_layout() path2="C:\\Users\\Yi-Hsiang\\Documents\\Astro\\Research\\NGC7142\\OBS\\" plt.savefig(path2+"magcalBV.png") plt.clf() plt.figure(figsize=[8,10]) plt.subplot(211) plt.plot(phot[1]-phot[0],phot[0]-rstar[0][2], "ko") plt.plot(range(3),Vfit[0]*np.array(range(3))+Vfit[1],"r-") plt.xlabel(r"$(B-V)$",fontsize=14) plt.ylabel(r"$V - v$",fontsize=14) plt.text(1, 5.3, "V-v = %.3f (B-V) + %.3f"%(Vfit[0], Vfit[1]), color="r") plt.subplot(212) plt.plot(rstar[1][2]-rstar[0][2],phot[0]-rstar[0][2], "ko") plt.plot(range(3),vfit[0]*np.array(range(3))+vfit[1],"r-") plt.xlabel(r"$(b-v)$",fontsize=14) plt.ylabel(r"$V - v$",fontsize=14) plt.text(1, 5.3, "V-v = %.3f (b-v) + %.3f"%(vfit[0], vfit[1]), color="r") plt.tight_layout() path2="C:\\Users\\Yi-Hsiang\\Documents\\Astro\\Research\\NGC7142\\OBS\\" plt.savefig(path2+"magcalV2.png") plt.clf() plt.subplot(211) plt.plot(phot[1]-phot[0],phot[1]-rstar[1][2], "ko") plt.plot(range(3),Bfit[0]*np.array(range(3))+Bfit[1],"r-") plt.xlabel(r"$(B-V)$",fontsize=14) plt.ylabel(r"$B - b$",fontsize=14) plt.text(0.25, 5.3, "B-b = %.3f (B-V) + %.3f"%(Bfit[0], Bfit[1]), color="r") plt.subplot(212) plt.plot(rstar[1][2]-rstar[0][2],phot[1]-rstar[1][2], "ko") plt.plot(range(3),bfit[0]*np.array(range(3))+bfit[1],"r-") plt.xlabel(r"$(b-v)$",fontsize=14) plt.ylabel(r"$B - b$",fontsize=14) plt.text(0.25, 5.3, "B-b = %.3f (b-v) + %.3f"%(bfit[0], bfit[1]), color="r"). 62.

(73) plt.tight_layout() path2="C:\\Users\\Yi-Hsiang\\Documents\\Astro\\Research\\NGC7142\\OBS\\" plt.savefig(path2+"magcalB2.png") #%% #allstar match and phot for i in range(len(ref)): ref[i]=np.array(ref[i]).T coeffx=[] coeffy=[] for i in [0,1]: coeffx.append(np.linalg.lstsq(np.array([ref[i][0],np.ones(40)]).T,ref[0][0])[0]) coeffy.append(np.linalg.lstsq(np.array([ref[i][1],np.ones(40)]).T,ref[0][1])[0]) vstar=np.array(star[0]).T bstar=np.array(star[1]).T bx,by=bstar[1]*coeffx[0][0]+coeffx[0][1],bstar[2]*coeffy[0][0]+coeffy[0][1] bstar[1],bstar[2]=bx,by vstar=np.array(vstar).T bstar=np.array(bstar).T #%% bvmatch=[[],[]] for i in range(len(vstar)): for j in range(len(bstar)): if np.abs(vstar[i][1]-bstar[j][1]) < 4 and np.abs(vstar[i][2]-bstar[j][2]) < 4: bvmatch[0].append(np.append(vstar[i],i+1)) bvmatch[1].append(np.append(bstar[j],j+1)) break for i in range(2): bvmatch[i]=np.array(bvmatch[i]).T vcal=bvmatch[0][3]+magcal[0]*(Bfit[1]-Vfit[1]+(bvmatch[1][3]bvmatch[0][3])*bvcalv[0]+bvcalv[1])+Vfit[1] bcal=bvmatch[1][3]+magcal[1]*(Bfit[1]-Vfit[1]+(bvmatch[1][3]bvmatch[0][3])*bvcalb[0]+bvcalb[1])+Bfit[1] fout=open(path2+"Thesis_bv.txt", "w") fout.writelines("%4s %5s %5s\n"%("ID", "B", "V")) for i in range(len(vcal)): fout.writelines("%4d %5.2f %5.2f\n"%(bvmatch[0][5][i], bcal[i], vcal[i])) fout.close() fout=open(path2+"Thesis_bvxy.txt", "w") fout.writelines("%4s %7s %7s %7s\n"%("ID", "V", "X", "Y")) for i in range(len(vcal)): fout.writelines("%4d %7.3f %7.3f %7.3f\n"%(bvmatch[0][5][i], bvmatch[0][1][i], bvmatch[0][2][i])) fout.close(). vcal[i],. #%% plt.clf() plt.figure(figsize=[20,16]) plt.gca().set_aspect('equal', adjustable='box') plt.title("Stars field of NGC7142 in 2015O image",fontsize=20) for i in range(len(vcal)): plt.plot(bvmatch[0][1][i], bvmatch[0][2][i], "ko", ms=10**((-0.1)*vcal[i]+2.5)). 63.

(74) plt.xlim(400,1600) plt.ylim(500,1700) plt.savefig(path2+"starfieldV.png") plt.clf() plt.figure(figsize=[8,8]) plt.plot(bvmatch[0][3],vcal,"k.") Vvfit=np.polyfit(bvmatch[0][3],vcal, deg=1) plt.plot(np.linspace(4,20),np.linspace(4,20)*Vvfit[0]+Vvfit[1]) plt.xlabel("v") plt.ylabel("V") plt.savefig(path2+"magV.png") plt.clf() plt.plot(bvmatch[1][3],bcal,"k.") Bbfit=np.polyfit(bvmatch[1][3],bcal, deg=1) plt.plot(np.linspace(6,20),np.linspace(6,20)*Bbfit[0]+Bbfit[1]) plt.xlabel("b") plt.ylabel("B") plt.savefig(path2+"magB.png") plt.clf() plt.cla() plt.figure(figsize=[8,8]) plt.plot(bcal-vcal, vcal, "k.", ms=2) plt.title("CMD in NGC 7142 region",size=14) plt.xlabel("B-V",fontsize=12) plt.ylabel("V",fontsize=12) plt.tick_params(labelsize=12) plt.ylim(22,10) plt.xlim(-.5,2.5) plt.savefig(path2+"CMD.png") plt.clf() plt.figure(figsize=[8,8]) plt.subplot(211) plt.hist(vcal,bins=np.array(range(80,250,2))*0.1,color='green',label="V band\nV$_{lim}$=20.4") plt.plot([20.4,20.4],[0,350],"r-") plt.title("Histogram of Magnitude", fontsize=12) plt.xlabel("V magnitude") plt.ylabel("N star") plt.legend() plt.subplot(212) plt.hist(bcal,bins=np.array(range(80,250,2))*0.1,color='blue',label="B band\nB$_{lim}$=21.8") plt.plot([21.8,21.8],[0,300],"r-") plt.xlabel("B magnitude") plt.ylabel("N star") plt.legend() plt.tight_layout() plt.savefig(path2+"completeness_bv.png") #%% """ UBV calibration and photometry """ fltr=["B","U"] fin=[] star=[] for i in range(len(fltr)): fin.append(open(path+"20151014"+fltr[i]+".als", "r").readlines()) star.append([]) for j in range(len(fin[i])):. 64.

(75) star[i].append(np.array(fin[i][j].split(),dtype=np.float)) fin=[] ref=[] for i in range(len(fltr)): fin.append(open(path+"20151014"+fltr[i]+".photref", "r").readlines()[2:]) ref.append([]) for j in range(len(fin[i])): ref[i].append(np.array([fin[i][j].split()[0],fin[i][j].split()[1]],dtype=np.float)) phot=[] for i in range(len(ref[0])): phot.append(np.array((open(path+"NGC7142.photref", "r").readlines()[1:])[i].split()[1:],dtype=np.float)) phot[i]=np.array([phot[i][0],phot[i][0]+phot[i][1],phot[i][0]+phot[i][1]+phot[i][2] ]) phot=np.array(phot).T #match ref in star and get info rstar=[] coorx=[] coory=[] for i in range(len(fltr)): rstar.append([]) for j in range(len(ref[i])): for k in range(len(star[i])): if (np.abs(star[i][k][1]-ref[i][j][0]) < 2) and (np.abs(star[i][k][2]ref[i][j][1]) < 2): rstar[i].append(star[i][k][1:]) break rstar[i]=np.array(rstar[i]).T coorx.append(np.polyfit(rstar[i][0],rstar[0][0],deg=1)) coory.append(np.polyfit(rstar[i][1],rstar[0][1],deg=1)) #phot ref phot=[] for i in range(len(ref[0])): phot.append(np.array((open(path+"NGC7142.photref", "r").readlines()[1:])[i].split()[1:],dtype=np.float)) phot[i]=np.array([phot[i][0],phot[i][0]+phot[i][1],phot[i][0]+phot[i][1]+phot[i][2] ]) phot=np.array(phot).T bufit=np.polyfit(phot[2]-phot[1],phot[1]-rstar[0][2],deg=1) ufit=np.polyfit(rstar[1][2]-rstar[0][2],phot[2]-rstar[1][2],deg=1) Ufit=np.polyfit(phot[2]-phot[1],phot[2]-rstar[1][2],deg=1) umagcal=Ufit[0]/(1-Ufit[0]+bufit[0]) ubcal=[ufit[0]/Ufit[0],(ufit[1]-Ufit[1])/Ufit[0]] #allstar match and phot for i in range(len(ref)): ref[i]=np.array(ref[i]).T coeffx=np.linalg.lstsq(np.array([ref[1][0],np.ones(24)]).T,ref[0][0])[0] coeffy=np.linalg.lstsq(np.array([ref[1][1],np.ones(24)]).T,ref[0][1])[0] ustar=np.array(star[1]).T ux,uy=ustar[1]*coeffx[0]+coeffx[1],ustar[2]*coeffy[0]+coeffy[1]. 65.

(76) ustar[1],ustar[2]=ux,uy ustar=np.array(ustar).T plt.figure(figsize=[8,5]) plt.plot(range(14,19),range(14,19), "r-") plt.plot(rstar[1][2],phot[2],"bo",label=r"$u$") plt.plot(rstar[1][2]+umagcal*(Ufit[1]-bufit[1]+(rstar[1][2]-rstar[0][2])*\ ubcal[0]+ubcal[1])+Ufit[1]+0.6,phot[2],"k.",label=r"$U_{cal}$") plt.xlabel(r"$u\ &\ U_{cal}$",fontsize=14) plt.ylabel(r"$U$",fontsize=14) plt.legend(loc=2) plt.tight_layout() path2="C:\\Users\\Yi-Hsiang\\Documents\\Astro\\Research\\NGC7142\\OBS\\" plt.savefig(path2+"magcalU.png") plt.clf() plt.figure(figsize=[8,10]) plt.subplot(211) plt.plot(phot[2]-phot[1],phot[2]-rstar[1][2], "ko") plt.plot(range(4),Ufit[0]*np.array(range(4))+Ufit[1],"r-") plt.xlabel(r"$(U-B)$",fontsize=14) plt.ylabel(r"$U - u$",fontsize=14) plt.text(1.5, 1.7, "U-u = %.3f (U-B) + %.3f"%(Ufit[0], Ufit[1]), color="r") plt.subplot(212) plt.plot(rstar[1][2]-rstar[0][2],phot[2]-rstar[1][2], "ko") plt.plot(range(3,7),ufit[0]*np.array(range(3,7))+ufit[1],"r-") plt.xlabel(r"$(u-b)$",fontsize=14) plt.ylabel(r"$U - u$",fontsize=14) plt.text(4.5, 1.7, "U-u = %.3f (u-b) + %.3f"%(ufit[0], ufit[1]), color="r") plt.tight_layout() path2="C:\\Users\\Yi-Hsiang\\Documents\\Astro\\Research\\NGC7142\\OBS\\" plt.savefig(path2+"magcalU2.png") #%% #match ubvmatch=[[],[],[]] for i in range(len(vstar)): for j in range(len(bstar)): if np.abs(vstar[i][1]-bstar[j][1]) < 4 and np.abs(vstar[i][2]-bstar[j][2]) < 4: break for k in range(len(ustar)): if np.abs(bstar[j][1]-ustar[k][1]) < 4 and np.abs(bstar[j][2]-ustar[k][2]) < 4 and j != len(bstar)-1: ubvmatch[0].append(np.append(vstar[i],i+1)) ubvmatch[1].append(np.append(bstar[j],j+1)) ubvmatch[2].append(np.append(ustar[k],k+1)) break for i in range(3): ubvmatch[i]=np.array(ubvmatch[i]).T vcal=ubvmatch[0][3]+magcal[0]*(Bfit[1]-Vfit[1]+(ubvmatch[1][3]ubvmatch[0][3])*bvcalv[0]+bvcalv[1])+Vfit[1] bcal=ubvmatch[1][3]+magcal[1]*(Bfit[1]-Vfit[1]+(ubvmatch[1][3]ubvmatch[0][3])*bvcalb[0]+bvcalb[1])+Bfit[1] ucal=ubvmatch[2][3]+umagcal*(Ufit[1]-bufit[1]+(ubvmatch[2][3]ubvmatch[1][3])*ubcal[0]+ubcal[1])+Ufit[1]+0.6 fout=open(path2+"Thesis_ubv.txt", "w") fout.writelines("%4s %5s %5s %5s\n"%("ID", "U", "B", "V")) for i in range(len(ucal)):. 66.

(77) fout.writelines("%4d %5.2f %5.2f %5.2f\n"%(ubvmatch[0][5][i], ucal[i], bcal[i], vcal[i])) fout.close() fout=open(path2+"Thesis_ubvxy.txt", "w") fout.writelines("%4s %7s %7s %7s\n"%("ID", "V", "X", "Y")) for i in range(len(vcal)): fout.writelines("%4d %7.3f %7.3f %7.3f\n"%(ubvmatch[0][5][i], ubvmatch[0][1][i], ubvmatch[0][2][i])) fout.close() #%% plt.clf() plt.figure(figsize=[8,8]) plt.plot(ubvmatch[2][3],ucal,"k.") Uufit=np.polyfit(ubvmatch[2][3],ucal, deg=1) plt.plot(np.linspace(8,20),np.linspace(8,20)*Uufit[0]+Uufit[1]) plt.xlabel("u") plt.ylabel("U") plt.savefig(path2+"magU.png") plt.clf() plt.figure(figsize=[8,8]) plt.subplot(311) plt.hist(vcal,bins=np.array(range(80,250,2))*0.1,color='green',label="V band\nV$_{lim}$=18.0") plt.plot([18,18],[0,120],"r-") plt.title("Histogram of Magnitude", fontsize=12) plt.xlabel("V magnitude") plt.ylabel("N star") plt.legend() plt.subplot(312) plt.hist(bcal,bins=np.array(range(80,250,2))*0.1,color='blue',label="B band\nB$_{lim}$=18.8") plt.plot([18.8,18.8],[0,120],"r-") plt.xlabel("B magnitude") plt.ylabel("N star") plt.legend() plt.subplot(313) plt.hist(ucal,bins=np.array(range(80,250,2))*0.1,color='violet',label="U band\nU$_{lim}$=19.0") plt.plot([19.2,19.2],[0,140],"r-") plt.xlabel("U magnitude") plt.ylabel("N star") plt.legend() plt.tight_layout() plt.savefig(path2+"completeness_ubv.png") plt.clf() plt.figure(figsize=[8,8]) plt.plot(bcal-vcal, ucal-bcal, "k.", ms=2) plt.title("TCD in NGC 7142 region",fontsize=16) plt.xlabel("B-V",fontsize=12) plt.ylabel("U-B",fontsize=12) plt.tick_params(labelsize=12) plt.ylim(3,-2) plt.xlim(-.5,2.5) plt.savefig(path2+"TCD.png") plt.clf() plt.figure(figsize=[30,30], dpi=300) img=plt.imread(path+"\\20151014Urev.jpg") plt.imshow(img,cmap='gray'). 67. vcal[i],.

(78) for i in range(len(vcal)): if ucal[i]-bcal[i] < -0.2: plt.plot((ubvmatch[2][1][i]-coeffx[1])/coeffx[0], coeffy[1])/coeffy[0],"ro",fillstyle="none",mew=1,ms=6) plt.xlim(0,2048) plt.ylim(0,2048) plt.savefig(path2+"position_ub_lt-0.2.png"). 68. (ubvmatch[2][2][i]-.

(79) 2. match.py # -*- coding: utf-8 -*""" This program read the IRAF all star file and reference stars list, match stars and calculate the coordinates coversion coefficient @author: Yi-Hsiang Hsu """ #import modules import numpy as np import matplotlib.pyplot as plt from astropy.coordinates import SkyCoord """ coefficients part """ #set file parameter path="C:\\Users\\Yi-Hsiang\\Documents\\Astro\\Research\\NGC7142\\OBS\\" #directory name dirlist=["20030807","20120708","20151014","20151111"] #allstar files, reference star file name iraflist=["20030807V","20120708V","20151014V","20151111V"] #obs data in JD obstime=np.array([2452858.95903,2456117.95139,2457310.0609028,2457338.17847222]) imgsize=np.array([1340,1300]) #XYsize pixscl=0.51 #pixel scale in arcsec, in the ref frame imgctr=np.array([SkyCoord("21h45m13s", "+65d46m25s").ra.degree,\ SkyCoord("21h45m13s", "+65d46m25s").dec.degree]) pixctr=imgsize/2. #open the ref star files fin=[] for i in range(len(dirlist)): fin.append(open(path+"\\"+dirlist[i]+"\\"+iraflist[i]+".ref", "r")) #read the files datalist=[] for i in range(len(fin)): datalist.append(fin[i].readlines()[2:]) fin[i].close() #close file #solar opsition calculator Jc=(obstime-2451545.)/36525. #Julian century GMLS=(280.46646+Jc*(36000.76983 + Jc*0.0003032))%360. #Geom Mean Long Sun (deg) GMAS=357.52911+Jc*(35999.05029 - 0.0001537*Jc) #Geom Mean Anom Sun (deg) SEqC=np.sin(np.deg2rad(GMAS))*(1.914602-Jc*(0.004817+0.000014*Jc))+\ np.sin(np.deg2rad(2*GMAS))*(0.019993-0.000101*Jc)+\ np.sin(np.deg2rad(3*GMAS))*0.000289 #Sun equation of center STL=GMLS+SEqC #sun true long (deg) STA=GMAS+SEqC #sun true anom Ecc=0.016708634-Jc*(0.000042037 + 0.0000001267*Jc) #Eccent Earth Orbit AU=(1.000001018*(1-Ecc*Ecc))/(1+Ecc*np.cos(np.deg2rad(STA))) Incl=23.+(26.+((21.448-Jc*(46.815+Jc*(0.00059-Jc*0.001813))))/60.)/60. dt=[] #time differenciation for i in range(len(obstime)): if i != 1: dt.append(obstime[i]-obstime[1]) dt=np.array(dt)/365.25. 69.