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研究生： 蔣承峰
Chiang, Cheng-Fong
論文名稱： 使用雙星觀測數據推算雙星軌道參數
Determination of the orbital parameters of binaries from the observed data
指導教授： 傅學海
Fu, Hsieh-Hai
學位類別： 碩士
Master
系所名稱： 地球科學系
Department of Earth Sciences
論文出版年： 2016
畢業學年度： 104
語文別： 英文
論文頁數： 72
中文關鍵詞： 雙星 、軌道參數 、Koval’skij-Olevi´c方程式 、克卜勒方程式 、第四干涉星表 、軌道星表第六版
英文關鍵詞： binary, orbital parameter, Koval’skij-Olevi´c Formulae, Kepler's laws, The Fourth Catalogue Of Interferometric Measurements Of Binary Stars, The Sixth Catalog of Orbits of Visual Binary Stars
DOI URL： https://doi.org/10.6345/NTNU202204410
論文種類： 學術論文
相關次數： 點閱：136  下載：15 
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    • 摘要

    如果能得知恆星的確切質量，便能推算出成分，並且推算出恆星的演化過程，進而得知此恆星何時與如何誕生，壽命多長，還有未來何去何從。而目前能最精確測得恆星質量方式便是由雙星軌道參數推算獲得。雙星軌道通常以另顆星為焦點之一繪出的橢圓形運行，如果他們正面面對我們，只需精確觀測即可得知軌道參數。但由於軌道平面通常不會剛好正面朝著我們，因此我們看到的通常是個歪掉的軌道，被繞行的恆星也會因此不在其焦點上。現有數以萬計的已觀測雙星，能定其參數的卻還有限，仍有大量的雙星待計算。大多數的方法需要先猜測部分的參數再予以計算，本篇試圖避免使用猜測步驟。在電腦能力遠勝於以往之後，本篇希望能藉由電腦的運算能力增快計算的速度，並且使此類問題不必再經由外國人之程式而得。

    本校對雙星軌道觀測已有數次用中小型望遠鏡以散斑技術觀測經驗，並藉由影像重建對數十雙星分別算出一到四個觀測記錄，該讓這些觀測數據試著幫助此次軌道運算了。這次論文使用一個三步驟的程式首先用最小平方法，然後用Koval’skij-Olevi´c 方法求出半長軸、離心率與三個尤拉角，最後再用克卜勒方程式以三種方法算雙星掃過面積的近似值求出週期並以此求得近星點時間並將一次運算到底。首先以三個參考文獻使用之雙星檢驗程式，然後試著讓本校學長們嘗試觀測過之四十三個雙星加入計算並得知本程式能得以計算出軌道的條件與極限。資料點會從第四干涉星表找出我們觀測過雙星的觀測資料，算出結果將與第六干涉星表資料比對結果並分別畫出。

    　　對四十三個雙星計算結果，本程式得到了十二個相當好的結果，其中部分也比較出本校學長觀測成果相當優異。七個還不錯的軌道，其問題大多在週期的計算。六個假軌道，是因為錯誤的觀測資料使原本無法算出也不該出現的軌道出現。而其他大約十八個未成功之案例，原因大多是觀測點不足或分布未能跨越超過180度，這也是假軌道之所以從無法算出一個橢圓變成出現一個彷彿像答案的結果關鍵所在，其他成敗的原因則有待繼續研究。

    Abstract

    Star is a huge mass object in the sky. People want to know how heavy it is. If the mass of star can be gotten exactly, the star’s ingredient will be found. After that, how and when the star was born, how long will it stay in the main sequence and how the star will go to die will be known. Binary star is a key of stars mass. The best of star mass detect is gets the mass of binary stars with getting their orbit’s parameters. Binary star’s orbit is one of the two stars which go around another by a conic section shape. If they do not just pass one time, the shape of orbit should be an ellipse or cycle. If all of the binary stars are faced on earth, the parameters of orbits just need some skill of observation to get. But that is too good to be true. Because they almost do not face on us, the one of two stars do not stay in the focus of ellipse. Thousands of binary stars have been observed, but not as much as of binary stars’ parameters have been determined. Several methods of binary parameters’ determination have to guess one of those parameters. Computer’s power is much more then themselves decades ago. This work writes a program which we hope can use to determine the parameters of binary stars by just one time. And it can bring the program to Taiwan, Republic of China so that do not have to request from foreigners.
    Our group of professor Hsieh-Hai Fu has observed a number of binary stars with speckle several times. They have detected dozens of stars and gotten one to four points of position for each binary star. They reconstructed the image of binary stars which gotten with speckle-masking method and used the parameters they found from the Sixth Catalog Of Orbits Of Visual Binary Stars to check it. It is time to try using those data to help the program of the methods to get the better results. This work uses a program of three steps, which including the Least Square Method, Koval’skij-Olevi´c method and use the Kepler's laws of planetary motion to get the periods with three kinds of area calculation and use them to get the periastrons, just in one time. The paper used three binary stars of reference paper to check the program before use at all of 43 binary stars which our group has observed. It will find how those observation data can help and what the limit the program is. It is going to be used binary stars which in Fourth Catalogue Of Interferometric Measurements Of Binary Stars which our group has observed and be check by Sixth Catalog of Orbits of Visual Binary Stars.
    Finally, the program makes 12 binary star orbits well and some of them our group has good observed data, 7 binary star orbits not bad but 6 fake orbits with the wrong data they should have failed. Some of fail due to not enough data, others because they did not cross more than two close quadrants, still others reason we still have to research.

    Contents 0. Abstract ------------------------------------------------------------------------------------------------------------- 04 1. Introduction --------------------------------------------------------------------------------------------------------- 07 2. Method Used in This Paper -------------------------------------------------------------------------------------- 09 2-1. Formulation of the Least Square Method ----------------------------------------------------------------- 09 2-2. Koval’skij-Olevi´c ------------------------------------------------------------------------------------------- 10 2-3. The period and the apse time ------------------------------------------------------------------------------- 13 3. Observations -------------------------------------------------------------------------------------------------------- 16 4. Results --------------------------------------------------------------------------------------------------------------- 18 4-1. Calculated results from the previous paper --------------------------------------------------------------- 18 4-2. Calculated results from fourth catalogs and our laboratory students observed results -------------- 20 4-3. Discussion ----------------------------------------------------------------------------------------------------- 24 4-4. Result without big problem --------------------------------------------------------------------------------- 24 4-5. Results have something wrong ----------------------------------------------------------------------------- 34 4-6 Fail results ----------------------------------------------------------------------------------------------------- 36 5. Conclusion ---------------------------------------------------------------------------------------------------------- 38 References ------------------------------------------------------------------------------------------------------------- 39 Appendixes A1. The Dynamics of Two Bodies --------------------------------------------------------------------------------- 40 A2. The coordinate transformation between the real orbit and observed orbit ------------------------------- 41 A3. The transformation between the true orbit and apparent orbit --------------------------------------------- 43 A4. The IDL code source for the determination of the orbit of binary ---------------------------------------- 45 A5. The list of abandoned data ------------------------------------------------------------------------------------- 60 List Of Tables Table 1 List of observed binaries ----------------------------------------------------------------------------------- 16 Table 2 The results compare between the reference paper and this program -------------------------------- 18 Table 3 The summary table of calculated and catalog result --------------------------------------------------- 20 Table 4 Result without big problem ------------------------------------------------------------------------------- 24 Table 5: Problem with the periods --------------------------------------------------------------------------------- 31 Table 6 Different results from the sixth catalog ------------------------------------------------------------------ 33 Table 7 Results have something wrong ---------------------------------------------------------------------------- 34 Table 8 Not very close from the Fourth Catalogue Of Interferometric Measurements Of Binary Stars -- 34 Table 9 The fake success orbits result ----------------------------------------------------------------------------- 36 List Of Pictures Picture 1 The Euler Angles ------------------------------------------------------------------------------------------ 08 Picture 2 The Ellipse Parameters ----------------------------------------------------------------------------------- 08 Picture 3 The Euler Angles 2 ---------------------------------------------------------------------------------------- 10 Picture 4 The Euler Angles 3 ---------------------------------------------------------------------------------------- 10 Picture 5 Method With Heron's Formula -------------------------------------------------------------------------- 14 Picture 6 Method With Sectors Shape ----------------------------------------------------------------------------- 14 Picture 7 ADS 11520 ------------------------------------------------------------------------------------------------ 18 Picture 8 ADS 9982 -------------------------------------------------------------------------------------------------- 19 Picture 9 ADS 10786 ------------------------------------------------------------------------------------------------- 19 Picture 10 STT 185 --------------------------------------------------------------------------------------------------- 25 Picture 11 STT 185 --------------------------------------------------------------------------------------------------- 25 Picture 12 STT 208 --------------------------------------------------------------------------------------------------- 26 Picture 13 STT 208 --------------------------------------------------------------------------------------------------- 26 Picture 14 STF 1670 AB -------------------------------------------------------------------------------------------- 27 Picture 15 STF 1670 AB -------------------------------------------------------------------------------------------- 27 Picture 16 BU 151 AB ----------------------------------------------------------------------------------------------- 28 Picture 17 AGC 13 AB ---------------------------------------------------------------------------------------------- 28 Picture 18 STF 1998AB --------------------------------------------------------------------------------------------- 29 Picture 19 STF 1998AB --------------------------------------------------------------------------------------------- 29 Picture 20 STF 2084 -------------------------------------------------------------------------------------------------- 30 Picture 21 STF 2084 -------------------------------------------------------------------------------------------------- 30 Picture 22 STF 1523AB ----------------------------------------------------------------------------------------------31 Picture 23 BU 612AB ------------------------------------------------------------------------------------------------ 32 Picture 24 STF 2272AB --------------------------------------------------------------------------------------------- 32 Picture 25 STF 3121 AB --------------------------------------------------------------------------------------------- 33 Picture 26 STF 3121 AB --------------------------------------------------------------------------------------------- 33 Picture 27 STF 1937 AB --------------------------------------------------------------------------------------------- 34 Picture 28 Hu 879 ----------------------------------------------------------------------------------------------------- 35 Picture 28 Hu 879 ----------------------------------------------------------------------------------------------------- 35 Picture 28 Castor ----------------------------------------------------------------------------------------------------- 37 Picture 28 STF 1728 AB --------------------------------------------------------------------------------------------- 37 Picture 28 The binary's interaction --------------------------------------------------------------------------------- 44 List of Abbreviations a Semi – major axis e Eccentricity P Period T Periastron Ω The Euler Angle Ω i The Euler Angle i ω The Euler Angle ω ρ The angle separation of binary form the observatory θ The companion azimuth of binary form the observatory's north r The angle separation of binary form the observatory when it face to observatory ψ The companion azimuth from the periastron to the connection of binary L.O.T. The one-meter-diameter telescope at the Lulin Observatory R.C.16 Ritchey–Chrétien telescope with a diameter of 16 inches C.14 Cassegrain catadioptric telescope with a diameter of 14 inches N.C.U. National C University N.T.N.U. National Taiwan Normal University F.R.O. Flat Roof Observatory ( in National Taiwan Normal University ) Fourth catalog Fourth Catalogue Of Interferometric Measurements Of Binary Stars Sixth catalog Sixth Catalog Of Orbits Of Visual Binary Stars

    Reference:

    1. Bob Argyle, ed., 2004, Observing and Measuring Double Stars, pp. 10–11
    2. William Herschel, 1803, Philosophical Transactions of the Royal Society of London,93,pp. 339–382
    3. Kam-Ching Leung, 1997, ASPC, 130,291
    4. D. Belorizky, 1949, On Kowalsky’s formulae for binary orbits.
    5. A. Rambant, 1984, M.A On the determination of double star orbits.
    6. Heinrich K. Eichhorn, 1989, An improved algorithm for the determination of the system parameters of a visual binary.
    7. Cheng Zong-Yi and Yan Lin-Shan, 1993,The area factors of the conic section and its application on orbit determinaton of visual binary.
    8. D.Pourbaix,1992, A trial—and—error approach to the determination of the orbital parameters of visual binaries.
    9. D.Olevic and Cvetkovic, 2003, Orbits of 10 interferometric binary systems calculated by using the improved Koval’skij method.
    10. Budi Dermawan, Moch. Irfan, Suryadi Siregar, et alia, 2008, ICMNS,1276,1280
    11. Diomar Cesar Lobгo, Determination of The Orbital Elements of a Visual Binary Star
    12. Hsien-Lung Cheng, 2012, Image Reconstruction of Selected Binaries by Speckle-Masking Method
    13. En-Wei Lin, 2009, Speckle Observation of Binary Stars at NTNU
    14. Andreas Alzner The Orbital Elements of a Visual Binary Star Chapter
    15. Hartkopf, William I.; McAlister, Harold A.; Mason, Brian D.; Barry, Donald J.; Turner, Nils H.; Fu, Hsieh-Hai, 1994, ICCD speckle observations of binary stars
    16. Fu, Hsieh-Hai; Hartkopf, William I.; Mason, Brian D.; McAlister, Harold A.; Dombrowski, Edmund G.; Westin, Tor; Franz, Otto G. ICCD Speckle Observations of Binary Stars.
    17. Michael Perryman, The Exoplanet Handbook

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