modified orbital elements of the close visual binary

‫جامعة آل البيت‬ Al al-Bayt University

MODIFIED ORBITAL ELEMENTS OF THE CLOSE VISUAL BINARY SYSTEMS, HIP11352, HIP70973, HIP72479 ‫عناصر مدارية معدلة لألنظمة النجمية الثنائية البصرية‬ HIP11352, HIP70973, HIP72479 by

Yamam Mahmoud Al-Tawalbeh ‫يمام محمود الطوالبة‬ Supervisor:

Prof. Mashhoor A. Al-Wardat

This Thesis was Submitted in Partial Fulfillment of the Requirements for the Master’s Degree in Physics Deanship of Graduate Studies Al al-Bayt University © Copyright | AABU, Mafraq, 2017

‫نموذج رقم (‪)1‬‬ ‫نموذج تفويض‬

‫جامعة آل البيت‬ ‫عمادة الدراسات العليا‬ ‫أنا‬

‫يمام محمود فواز الطوالبة‬

‫أفوض جامعة آل البيت بتزويد نسخ من رسالتي‪ ،‬للمكتبات أو المؤسسات أو الهيئات أو األشخاص عند طلبهم حسب التعليمات‬ ‫النافذة في الجامعة‪.‬‬

‫التاريخ‪:‬‬

‫التوقيع‪............................ :‬‬

‫‪ii‬‬

‫نموذج رقم (‪)2‬‬

‫جامعة آل البيت‬ ‫عمادة الدراسات العليا‬ ‫نموذج إقرار والتزام بقوانين جامعة آل البيت وأنظمتها وتعليماتها لطلبة الماجستير والدكتوراه‪.‬‬ ‫أنا يمام محمود فواز الطوالبة‬

‫الرقم الجامعي‪1520402012 :‬‬

‫تخصص‪ :‬فيزياء‬

‫كلية‪ :‬العلوم‬

‫أُعلنُ بأني قد التزمت بقوانين جامعة آل البيت وأنظمتها وتعليماتها وقراراتها السارية المفعول المتعلقة بإعداد رسائل الماجستير‬ ‫والدكتوراه عندما قمت شخصيا ً بإعداد رسالتي بعنوان‪:‬‬ ‫‪Modified Orbital Elements Of The Close Visual Binary Systems HIP11352, HIP70973,‬‬ ‫‪HIP72479‬‬ ‫عناصر مدارية معدلة لألنظمة النجمية الثنائية البصرية‬ ‫‪HIP11352, HIP70973, HIP72479‬‬ ‫وذلك بما ينسجم مع األمانة العلمية المتعارف عليها في كتابة الرسائل واألطاريح العلمية‪ .‬كما أنني أُعلن بأن رسالتي هذه غير‬ ‫منقولة أو مستلة من رسائل أو كتب أو أبحاث أو أي منشورات علمية تم نشرها أو تخزينها في أي وسيلة إعالمية‪ ،‬وتأسيسا ً على ما‬ ‫تقدم فإنني أتحمل المسؤولية بأنواعها كافة فيما لو تبين غير ذلك بما فيه حق مجلس العمداء في جامعة آل البيت بإلغاء قرار منحي‬ ‫التخرج مني بعد صدورها دون أن يكون لي الحق في التظلم أو االعتراض أو‬ ‫الدرجة العلمية التي حصلت عليها وسحب شهادة‬ ‫ّ‬ ‫الطعن بأي صورة كانت في القرار الصادر عن مجلس العمداء بهذا الصدد‪.‬‬

‫التاريخ‪:‬‬

‫التوقيع ‪............................‬‬

‫‪iii‬‬

iv

Dedication

To my loving parents To my dear sisters and To my loving family

v

Acknowledgments I would like to thank my supervisor Prof. Mashhoor Ahmad Al-Wardat, for his sincere advice and guidance provided throughout my research and thesis preparation. Special thanks for my family; my parents, my sisters and my wife for their encouragement. Special thanks to my friend and dear colleague Abdulrahman Ghuzlan for his constant encouragement and Communication and to advise me when I need that.

vi

Table of Contents Table of Contents )1( ‫نموذج رقم‬...................................................................................................................................... ii )2( ‫نموذج رقم‬..................................................................................................................................... iii Dedication ....................................................................................................................................... v Acknowledgments.......................................................................................................................... vi Table of Contents .......................................................................................................................... vii List of Tables ................................................................................................................................. xi List of Figures ............................................................................................................................... xv Abstract (English) ...................................................................................................................... xviii Abstract (Arabic) ......................................................................................................................... xix Chapter 1 Introduction................................................................................................................. 1 1 Introduction ................................................................................................................................ 1 1.1 Historical Background: .................................................................................................... 1 1.2 What is a Star? ................................................................................................................. 7 1.3 Hertzsprung-Russell Diagram: ........................................................................................ 8 1.4 Properties of Stars: ......................................................................................................... 10 1.5 Binary Stars: .................................................................................................................. 15 1.6 Discovery and Evolution: ............................................................................................... 16 1.7 Classification of Binary Stars: ....................................................................................... 17 1.7.1

Visual Binaries: ................................................................................................... 17

1.7.2

Spectroscopic Binaries: ...................................................................................... 18

1.7.3

Eclipsing Binaries: .............................................................................................. 20

1.7.4

Astrometric binaries: .......................................................................................... 20 vii

1.7.5

"Exotic" Types: ................................................................................................... 22

1.8 Previous Studies: ............................................................................................................. 22 Chapter 2 ..................................................................................................................................... 25 The Visual Binaries Hip11352, Hip70973, and Hip72479 ....................................................... 25 2 Introduction: ........................................................................................................................... 25 2.1 Hip11352 System:............................................................................................................ 26 2.1.1

Triangulum Constellation(The Home of Hip11352):....................................... 26

2.1.2

Location of Hip11352: ........................................................................................ 26

2.1.3

Proper Motion of Hip11352: .............................................................................. 27

2.1.4

The Age and The star's Galacto-Centric Distance of Hip11352: .................. 27

2.1.5

The Apparent Magnitude and The Orbit of Hip11352: .................................. 27

2.1.6

Another Identifiers and The Parameters of Its Components: ........................ 28

2.1.7

Orbital elements of the system Hip11352: ........................................................ 31

2.1.8

Data From Fourth Catalogue of Hip11352: ..................................................... 32

2.2 Hip70973 System:............................................................................................................ 33 2.2.1

Virgo Constellation(The Home of Hip70973): ................................................. 33

2.2.2

Location of Hip70973: ........................................................................................ 33

2.2.3


2.2.4

Distance to Hip70973: ......................................................................................... 34

2.2.5


2.2.6


2.2.7


2.3 Hip72479 System:............................................................................................................ 39 2.3.1

Boötes Constellation(The Home of Hip72479): ................................................ 39

2.3.2

Location of Hip72479: ........................................................................................ 39 viii

2.3.3


2.3.4

Distance To Hip72479: ....................................................................................... 40

2.3.5


2.3.6


2.3.7


2.4 Goals of the study: .......................................................................................................... 46 Chapter 3 Theoretical Background and Methodology ............................................................ 47 3 Introduction ............................................................................................................................ 47 3.1 The Two-Body Problem: ................................................................................................ 48 3.2 The orbital shape: ........................................................................................................... 51 3.3 Time-Dependent Orbits: ................................................................................................ 54 3.4 The Orbital Elements: .................................................................................................... 59 Chapter 4 Procedures and Computational Work .................................................................... 64 4 Introduction ............................................................................................................................ 64 4.1 What is IDL program? ................................................................................................... 64 4.2 Benefits of IDL: ............................................................................................................... 65 4.3 What is Origin Software? ............................................................................................... 66 4.4 Fourth Catalogue of Interferometric Measurements of Binary Stars: ...................... 66 4.5 Sixth Catalogue of Orbits of Visual Binary Stars: ...................................................... 67 4.6 Algorithm: ....................................................................................................................... 67 4.6.1

Start:..................................................................................................................... 67

4.6.2

Input:.................................................................................................................... 67

4.6.3

Process: ................................................................................................................ 70

4.6.4

Output: ................................................................................................................. 71

4.6.5

End: ...................................................................................................................... 71 ix

4.7 Procedure: ....................................................................................................................... 72 Chapter 5 Results and Conclusion ............................................................................................ 82 5 82 5.1 Results: ............................................................................................................................. 82 5.1.1

Relative Positional Measurements In Fourth Catalogue: ............................... 82

5.1.2

Modified Orbits and Comparison with Orbits Published In The Sixth Catalogue: ............................................................................................................ 87

5.1.3

Modified Orbital Elements: ............................................................................... 91

5.1.4

Ephemerides and Mass Sum: ............................................................................. 96

5.2 Conclusions:..................................................................................................................... 99 5.3 Recommendations: ........................................................................................................ 102 Appendix A ............................................................................................................................... 104 Astronomical units: ........................................................................................................ 104 Constants: ........................................................................................................................ 105 Appendix B ................................................................................................................................ 106 Method of Least Square .................................................................................................. 106 References .................................................................................................................................. 108

x

List of Tables Table 1-1: B-V Color index as a function of temperature for the Main-Sequence stars (Al-Wardat, 2017) ............................................................................................................................................. 14 Table 2-1: The parameters of the components of Hip11352 (Al-Wardat, 2009).......................... 28 Table 2-2: Strömgren and Tycho magnitudes for Hip11352 (Al-Wardat, 2008) ........................ 29 Table 2-3: The main body of astrometric results from the data set(Horch et al., 2002) ............... 29 Table 2-4: The fastest-moving objects, namely, those having moved more than 100 mas in separation between the Hipparcos measure and the latest measure in Table 2 in Horch's paper (Horch et al., 2002) ....................................................................................................................... 30 Table 2-5: As it was mentioned in Balega's paper for Hip11352 (Balega et al., 2001)................ 30 Table 2-6: The orbital elements of Hip11352 (Al-Wardat, 2009, p. 11352) ................................ 31 Table 2-7: The data of the observations of the system Hip11352 in Fourth Catalogue (“Fourth Interferometric Catalogue - RA 2 hr,” 2017) ................................................................................ 32 Table 2-8: The parameters of the components of Hip70973(Al-Wardat, 2012)........................... 35 Table 2-9: Strömgren and Tycho magnitudes(Al-Wardat, 2008) ................................................ 36 Table 2-10: The values and the mean values of the parallaxes of Hip70973 (BartKevicius and Gudas, 2001a, 2001b, 2001c) ....................................................................................................... 36 Table 2-11: The individual values of the proper motions of Hip70973 (BartKevicius and Gudas, 2001d) ........................................................................................................................................... 36 xi

Table 2-12: The mean values of the proper motions of Hip70973 (BartKevicius and Gudas, 2001e) ........................................................................................................................................... 36 Table 2-13: The orbital elements of Hip70973 (Al-Wardat, 2012, p. 70973) .............................. 37 Table 2-14: The data of the observations of the system Hip70973 in Fourth Catalogue (“Fourth Interferometric Catalogue - RA 14 hr,” 2017a). ........................................................................... 38 Table 2-15: The parameters of the components of Hip72479(Al-Wardat, 2012)......................... 40 Table 2-16:Strömgren and Tycho magnitudes and color indices(Al-Wardat, 2008)................... 41 Table 2-17: The values and the mean values of the parallaxes of Hip72479 (BartKevicius and Gudas, 2001a, 2001b, 2001c) ....................................................................................................... 41 Table 2-18: The individual values of the proper motions of Hip72479 (BartKevicius and Gudas, 2001d) ........................................................................................................................................... 42 Table 2-19: The mean values of the proper motions of Hip72479 (BartKevicius and Gudas, 2001e) ........................................................................................................................................... 42 Table 2-20: The individual values of the radial velocities of Hip72479 (BartKevicius and Gudas, 2001g) ........................................................................................................................................... 42 Table 2-21: The individual values of the radial velocities of Hip72479 (BartKevicius and Gudas, 2001h) ........................................................................................................................................... 42 Table 2-22: The Kinematics parameters of Hip72479 in (𝑘𝑚/𝑠) (BartKevicius and Gudas, 2001i) ............................................................................................................................................ 42 xii

Table 2-23: The orbital elements of Hip72479 (Al-Wardat, 2012, p. 70973) .............................. 43 Table 2-24: Orbital elements of Hip72479 by Söderhjelm1999 (“Sixth Orbit Catalog,” 2018) .. 44 Table 2-25: The data of the observations of the system Hip72479 in Fourth Catalogue (“Fourth Interferometric Catalogue - RA 14 hr,” 2017b) ............................................................................ 44 Table 5-1: Positional measurements of the system Hip11352 from the Fourth Interferometric Catalogue after modifying ............................................................................................................ 84 Table 5-2: Positional measurements of the system Hip70973 from the Fourth Interferometric Catalogue after modifying ............................................................................................................ 85 Table 5-3: Positional measurements of the system Hip72479 from the Fourth Interferometric Catalogue after modifying ............................................................................................................ 86 Table 5-4: Orbital elements of the system Hip11352 built by Balega (“Sixth Orbit Catalog (frames version),” 2018) and this work ....................................................................................... 92 Table 5-5: Orbital elements of the system Hip70973 built by Docobo (“Sixth Orbit Catalog (frames version),” 2018) and this work ........................................................................................ 93 Table 5-6: Orbital elements of the system Hip72479 built by Söderhjelm(1999) (“Sixth Orbit Catalog (frames version),” 2018), Docobo(2000), and this work .............................................. 94 Table 5-7: Ephemerides of Hip11352 .......................................................................................... 96 Table 5-8: Ephemerides of Hip70973 .......................................................................................... 97 Table 5-9: Ephemerides of Hip72479 .......................................................................................... 98 xiii

Table 5-10: Orbital elements of the binary system Hip11352 of this work................................ 100 Table 5-11: Orbital elements of the binary system Hip70973 of this work................................ 100 Table 5-12: Orbital elements of the binary system Hip72479 of this work................................ 100

xiv

List of Figures Figure 1-1: Hertzsprung-Russell Diagram (Wikipedia, 2018). ...................................................... 9 Figure 1-2: The Absolute Visual Magnitude (Cosmos, 2017). ................................................. 12 Figure 1-3: UBVRcIc Johnson-Cousins Filters (“Johnson-Cousins UBVRI filter curves,” 2018). ....................................................................................................................................................... 13 Figure 1-4: Orbits of stars in a binary system (Introduction to Binary Stars, 2018) .................... 15 Figure 1-5: Visual Binaries (Australia Telescope National Facility, 2016). ................................ 18 Figure 1-6: Spectroscopic Binaries (Australia Telescope National Facility, 2016). .................... 19 Figure 3-1: Barycenter coordinate description of a binary system (Binacquista, 2013)............... 48 Figure 3-2: Elliptical orbit with the origin centered on one star (Binacquista, 2013). ................. 53 Figure 3-3: Properties of an ellipse (Binacquista, 2013). ............................................................. 55 Figure 3-4: the different angles for the orbital elements of a binary system (Binacquista, 2013).60 Figure 3-5: The leaning orbit is the true relative orbit of the system in space. The horizontal orbit representing the relative apparent orbit as seen in sky (“Taitus Software | Orbital elements,” 2018) ............................................................................................................................................. 60 Figure 4-1: The initial form of the program of Hip11352 system on Notepad with abstract observations from the Fourth Catalogue before modification ...................................................... 68

xv

Figure 4-2: The Input file of Hip11352 system after modifying according to the previous steps mentioned to be ready to draw on IDL ......................................................................................... 69 Figure 4-3: Observations of Hip11352 as in Fourth Catalogue .................................................... 73 Figure 4-4: Observations of Hip70973 as in Fourth Catalogue .................................................... 73 Figure 4-5: Observations of Hip72479 as in Fourth Catalogue .................................................... 74 Figure 4-6: IDL program for Hip70973 ........................................................................................ 75 Figure 4-7: IDL program for Hip72479 ........................................................................................ 75 Figure 4-8: IDL page with compilation the file (orbitwithaxis.pro) ............................................. 76 Figure 4-9: The command of plotting the orbit ............................................................................ 76 Figure 4-10: The screen orbit for Hip70973 system ..................................................................... 77 Figure 4-11: The command of the orbit file as appears in the main page of IDL......................... 78 Figure 4-12: The command of orbit file in(Command Line) ........................................................ 78 Figure 4-13: Executing the plotting command by IDL as an image ............................................. 78 Figure 4-14: The Image of orbit file of Hip70973 ........................................................................ 79 Figure 4-15: Ephemeris for Hip70973 from(2018) to(2028) by 20 points ................................... 80 Figure 4-16: Mass sum command for Hip70973 under IDL ........................................................ 81 Figure 5-1: The modified orbit of Hip11352 by using the new observations(this work) ............. 87 xvi

Figure 5-2: The modified orbit of Hip70973 by using the new observations(this work) ............. 87 Figure 5-3: The modified orbit of Hip72479 by using the new observations(this work) ............. 88 Figure 5-4: Image of Hip11352 orbit by Balega published in Sixth Catalogue at 2005(Balega, 2005) ............................................................................................................................................. 88 Figure 5-5: Image of Hip70973 orbit by Docobo published in Sixth Catalogue in 2000(Docobo, 2000) ............................................................................................................................................. 89 Figure 5-6: Image of Hip72479 orbit by Soderhjelm published in Sixth Catalogue at 1999(Sod, 1999) ............................................................................................................................................. 89 Figure 5-7: The built orbit of Hip11352 using the results of this work presented in a solid line, with the orbit built using the results of Balega 2005 presented by a dashed line. ........................ 90 Figure 5-8: The built orbit of Hip70973 using the results of this work presented in a solid line, with the built orbit using the results of Docobo 2000 presented by a dashed line. ...................... 90 Figure 5-9: The built orbit of Hip72479 using the results of this work presented in a solid line, with the built orbit using the results of Soderhjelm 1999 presented by a dashed line. ................. 91

xvii

Abstract (English) Modified Orbital Elements Of The Close Visual Binary Systems HIP11352, HIP70973, HIP72479 A Master Thesis By Yamam Mahmoud Al-Tawalbeh Supervisor: Prof. Mashhoor A. Al-Wardat

Department of Physics, Al al-Bayt University, 2017

Abstract Al-Tawalbeh, Yamam Mahmoud. Modified Orbital Elements Of The Close Visual Binary Systems HIP11352, HIP70973, HIP72479. Master of Science Thesis, Department of Physics, Al al-Bayt University, 2018 (Supervisor: Prof. Mashhoor Ahmad Al-Wardat). This study aims at modified orbital elements of the close visual binary systems HIP11352, HIP70973, HIP72479. In this study, the solution of the two body problem and Kepler’s equations are used to build an orbit for the binary systems HIP11352, HIP70973, HIP72479. These orbits are fitted to the observational data available in the Fourth Catalogue of interferometric measurements, using the Least Square Method. Keywords: stars, orbital motion, parameters, binaries, visually close binary systems, orbital elements, HIP11352, HIP70973, HIP72479, two body problem.

xviii

‫)‪Abstract (Arabic‬‬

‫عناصر مدارية معدلة لألنظمة النجمية الثنائية البصرية‬ ‫‪HIP11352, HIP70973, HIP72479‬‬ ‫رسالة ماجستير قُدمت من قبل‬ ‫يمام محمود الطوالبة‬ ‫المشرف‪:‬‬ ‫أ‪.‬د‪ .‬مشهور أحمد الوردات‬

‫قسم الفيزياء‪ ،‬جامعة آل البيت‪2017 ،‬م‬

‫ملخص‬ ‫يمام محمود الطوالبة‪ .‬عناصر مدارية معدلة لألنظمة النجمية الثنائية البصرية‬ ‫تهدف هذه الدراسة إلى إجراء تعديل على العناصر المدارية لألنظمة النجمية الثنائية البصرية اآلتية ‪:‬‬ ‫‪ .HIP11352, HIP70973, HIP72479‬يتم استخدام حل مسألة الجسمين المرتبطين بمركز كتلة واحد‪ ،‬وحل‬ ‫معادالت كبلر لبناء مدار معدل لألنظمة الثنائية ‪ . HIP11352, HIP70973, HIP72479‬هذه المدارات تعدل‬ ‫بناء على بيانات الرصد المتوافرة في الكتالوج الرابع لألرصاد الفلكية باستخدام طريقة التربيع األقل‪.‬‬

‫‪xix‬‬

Modified Orbital Elements of CVBSs Hip11352,Hip70973, Hip72479; Chapter 1

1

Chapter 1

Introduction 1

Introduction

1.1

Historical Background: Astronomy began with the formation of human consciousness around this world, and it was important in ancient times to know the changes of weather and weather related to the movement of the earth around the sun and the impact of the sun and moon on the movement of wind and clouds on the ground. The Egyptians, for example, set tables to determine the times of the Nile flood and what the peasants should do to avoid corruption of the agricultural season (Al-Taei, 2003). This science was also associated with predicting future events in what was known as Astrology, and although it was not adopted on clear scientific grounds, but it contributed to the development of star monitoring machines. Old humans used the relatively stable map of the sky for the guidance in the land and the sea. As for the modernity, the importance of this science is reflected in the realization of a more accurate understanding of the world in all its forms of energy and matter and its laws, and to benefit from this understanding in the development of material life on Earth. The astronomy was also used to investigate historical incidents recorded and associated with astronomical events such as the death of Ibrahim, the son of the Prophet Muhammad (peace be upon him) and it was narrated in Hadith books that the sun was eclipse on the day of his death (Al-Taei, 2003). Astronomy was used to search for life or effects of life on other planets and many efforts were made in this aspect.

M.Sc. Thesis • Yamam Al-Tawalbeh •Department of Physics • AABU University • 2017


2

Ancient astronomers built accurate calendars based on observing the movement of stars in the sky. They collected them in star clusters called constellations, and used them to track the movement of the sun and planets. The earliest Babylonian put The oldest star Catalogues dating from about 1200 BC. Many of the names of the stars mentioned in the Sumerian language reach the early Bronze Age and this indicates the old of their contributions to astronomy with their Babylonian contemporaries (“History of astronomy,” 2017). The Greek astronomer Aristyllus was the first one who created the first astronomical Catalogue in the Greeks around 300 BC, with the help of TimOcharis (Wikipedia, 2017). Apollonius of Perga put The first such model, and further developments in it in the 2nd century BC were attributed to Hipparchus of Nicea (Astronomy, 2017). The first measurement of precession and the compilation of the first star Catalogue were other contributions of Hipparchus in Astronomy, which paved the way for our modern system of the apparent magnitudes of stars (Al-Wardat, 2017). Many of the constellations and star names in use today are derived from Greek astronomy. Chinese astronomers were the first to know supernova in history, and they called it "guest stars" by observing their appearance suddenly among the fixed stars. Their first recording of it was in the Astrological Annals of the Houhanshu in 185 AD (“History of astronomy,” 2017). Crab Nebula is an example of a supernova "guest star" which was observed by Chinese astronomers in 1054, which had not been recorded by their European contemporaries. Medieval Islamic astronomers also had a role in progressing astronomy. The development of astronomy continued with their contributions, especially as they needed to know the five prayer times accurately and the beginning of the lunar M.Sc. Thesis • Yamam Al-Tawalbeh •Department of Physics • AABU University • 2017


3

months to see the entry of the seasons of worship, such as Ramadan, Hajj and other worship related to various astronomical phenomena such as prayer of eclipse. Some of their contributions were the designation of certain Arabic names to many stars that are still in use today. The first observatory was built in Baghdad During the reign of Caliph al-Mamun al-Rashid, and subsequent observatories were built around Iraq and Iran In the 8th century (Stirone, 2017). These observatories also prepared star Catalogues, solar and lunar tables, making it easier to locate planetary positions, lunar phases and eclipses (“Muslim Astronomers in the Islamic Golden Age,” 2012). Since this was before the telescope had been developed, the astronomers of the time invented observational sextants. These tools, some as large as 40 meters, were critical to the study of the angle of the sun, movement of the stars, mainly for the purpose of producing Zij1 star Catalogues. In his Book of Fixed Stars The astronomer Abd al-Rahman al-Sufi (Azophi) carried out observations on the stars and described their positions, magnitudes, brightness, and colour and drawings for each constellation (“History of astronomy,” 2017). He also gave the first descriptions and pictures of the Andromeda Galaxy "A Little Cloud" as he called it, and the first mention of the Large Magellanic Cloud was also attributed to al-Sufi. In 1006, Ali ibn Ridwan observed the brightest supernova in history "SN 1006", and documented an accurate description of the temporary stars (“History of astronomy,” 2017). In Khorasan, Abu Rayhan

1

Zij is the generic name of Islamic astronomical books that tabulate parameters used for astronomical calculations concerning the positions of the Sun, Moon, stars, and planets. The name is derived from a Persian term meaning cord (Amelia, 2013). M.Sc. Thesis • Yamam Al-Tawalbeh •Department of Physics • AABU University • 2017


4

Biruni observed and described the solar eclipse on 8 April 1019, and the lunar eclipse on 17 September 1019, and gave the exact latitudes of the stars during the lunar eclipse. He assumed the Milky Way galaxy to be a collection of abundant ambiguous stars (“Al-Biruni,” 2018). Moreover, Ibn Bajjah(Avempace) considered "the Milky Way to be a phenomenon both of the spheres above the moon and of the sublunar region." The Stanford Encyclopedia of Philosophy describes his theory and observation on the Milky Way as follows: "The Milky Way is the light of many stars which almost touch one another. Their light forms a “continuous image” (khayâl muttasil) on the surface of the body which is like a “tent” (takhawwum) under the fierily element and over the air which it covers". In 500 AH /1106-1107 AC Ibn Bajjah watched the conjunction of two planets, Jupiter and Mars, and he benefited from it to define the continuous image as the result of refraction, and he saw them having an elongate figure although their figure is circular. (Josép, 2018). In 1584 Giordano Bruno differentiates between "suns" which produce light and self-heat and there are objects revolving around them, and the celestial bodies, which derive light and heat from suns and revolve around them and he called these celestial bodies "earths". Bruno considered The majority of fixed stars as we see them are in fact suns. According to astrophysicist Steven Soter, he was the first person to grasp that "stars are other suns with their own planets" (“Giordano Bruno,” 2017). By the following century, the idea of the stars being the same as the Sun was reaching a consensus among astronomers. To explain why these stars exert no net gravitational pull on the Solar System, Isaac Newton suggested that the stars are M.Sc. Thesis • Yamam Al-Tawalbeh •Department of Physics • AABU University • 2017


5

equal distributed in every direction, Newton based his cosmology on the cosmological principle, that asserts that all points and directions in the universe are more or less equivalent on sufficiently large scales (“The Study of the Universe,” n.d.). That was a big change from the Ptolemaic/Aristotelian cosmology. Not only Newton suggested this idea, but William Herschel try to search and

determine how stars were distributed in the sky, so he was the first

astronomer who try to search in this aspect practically. In 1774 Herschel constructed his first large telescope, after which he spent nine years carrying out sky surveys to investigate double stars (“William Herschel,” 2018). The resolving power of the Herschel telescopes revealed that the nebulae in the Messier Catalogue were clusters of stars. Herschel published Catalogues of nebulae in 1802 (2,500 objects) and in 1820 (5,000 objects). By using prisms and temperature

measuring equipment

Herschel

was

the

pioneer of

using

the astronomical spectrophotometry as a diagnostic tool, to measure the wavelength distribution of stellar spectra (“William Herschel,” 2018). Herschel's discoveries formed a scientific astronomical race, that after he published astronomical Catalogues of more than 800 double or multi-star systems - most of them physical rather than optical - no new Catalogues were published until 1820 by Friedrich Wilhelm Struve, James South and John Herschel. His theoretical and observational

work

provided

the

foundation

for

modern

binary

star

astronomy(admin,2013). Before the HR (Hertzsprung-Russell) diagram2 was created and before we understood the true range of luminosities of stars, Herschel could estimate the distance to each star from the Earth, depending on the fact that

2

There is explanation about HR diagram at page 8 from this thesis



6

all stars had the same intrinsic luminosity as he assumed. Using his data, he created a map of the whole sky that incorporated the number of stars he counted and their distances from the Earth. William Herschel observed that some stars do not merely lie along the same line of sight, but are physical companions that form binary star systems, which was one of his important achievements (“Stars, Dwarfs,” 2017). Observation of double stars gained increasing importance during the 19th century. In 1819, Bessel determined the position of over 50,000 stars using a meridian circle from Reichenbach, assisted by some of his qualified students. The most prominent of them was Friedrich Wilhelm Argelander. In 1834, Friedrich Bessel observed changes in the proper motion of the star Sirius, and inferred a hidden companion (“Friedrich Bessel,” 2017). In 1882, Edward Pickering developed a method to photograph the spectra of multiple stars simultaneously by putting a large prism in front of the photographic plate. In 1890, Mizar A was discovered to itself be a binary, being the first binary to be discovered using spectroscopy. In 1908, spectroscopy revealed that Mizar B was also a pair of stars, making the group the first-known quintuple star system (“Edward Charles Pickering,” 2017). The last part of the twentieth century saw rapid technological advances in astronomical instrumentation. Optical telescopes were growing ever larger, and employing adaptive optics to partly negate atmospheric blurring. New telescopes were

launched

into

space,

and

began

observing

the

universe

in

the infrared, ultraviolet, x-ray, and gamma ray parts of the electromagnetic spectrum, as well as observing cosmic rays (“Observational astronomy,” 2017). Interferometer arrays produced the first extremely high-resolution images M.Sc. Thesis • Yamam Al-Tawalbeh •Department of Physics • AABU University • 2017


7

using aperture synthesis at radio, infrared and optical wavelengths. Karl Schwarzschild discovered that the color of a star, and hence its temperature, could be determined

by comparing the visual magnitude against the photographic

magnitude (Appenzeller, 2016). The development of the photoelectric photometer allowed for very precise measurements of magnitude at multiple wavelength intervals. In 1921 Albert A. Michelson invented a measurement which, for the first time made possible the direct determination of a stellar diameter (“Stars, Dwarfs,” 2017).

1.2

What is a Star? The star is a huge ball of inflamed incandescent gases, consisting mainly of hydrogen and helium. Nuclear fusion reactions occur in its core, where huge energy production, and very high temperatures, for example our Sun which is the nearest star of us, its surface temperature is 6000 Kelvin, while its core temperature of 15 million Kelvin because of those interactions (Sharp et al., 2017). The external pressure of the flammable gases by fusion is balanced by pulling the inner gravity of the star components themselves, leaving the star in hydrostatic equilibrium. Energy reaches the outside of the star by radiation and convection when the energy gets high enough in the atmosphere. This radiation is shown in the form of light for all wave-lengths, as well as stellar winds, and allows the passage of light into outer space that the area above star's atmosphere is transparent. The stars rotate and vary in their luminosity, although they may seem fixed to us as viewers on the Earth. There are hundreds of billions of stars in our galaxy Milky Way, our Sun is one of them(Khan Academy Website, 2017). The temperature at which the nuclear reactions ignite is about 5 million Kelvin, and in order to reach that degree the mass of a proto-star must be at least



8

80 times the mass of Jupiter. The star fuses hydrogen into helium in most of its life, called during this period the dwarf star, and is classified in the HertzsprungRussell diagram by its temperature and its luminosity(or spectral type) on a continuum called the main sequence (“Stars at Dictionary,” 2018). After the star consumes all the hydrogen in its core, it evolves depending on its mass to one of several forms other than the main sequence. The stars, ranging from 8 percent to 1.4 of the mass of the Sun, turn into red giants, ending their lives as white dwarfs after their outer layers explode. Stars that are more than 1.4 of the mass of Sun become super-giants and end their lives either to neutron stars if their masses are less than a 3. 4 of the mass of Sun, or as black holes if their masses are greater than 3. 4 of the mass of Sun, after it explodes into supernova (“Lecture 16 & 17 Star Birth & Death.pdf,” n.d.).

1.3

Hertzsprung-Russell Diagram: To understand the classification of stars generally, we must know what HR diagram,

how we classify stars into this diagram, and what star's properties we depend on them to classify stars in it. Firstly HR diagram was put by two astronomers Ejnar Hertzsprung (1893–1967) of Denmark and Henry Norris Russel (1877–1957) of the U.S. in the early 1900s (Patterson, 2013)to facilitate the study of the stars, know the stages of their evolution, and nature of the sequence of those stages, which helps us to understand the stages of life of the star, and we can predict the end of life of the stars based on their masses, temperatures and luminosities. The diagram explains the relation between effective temperature(spectral type) on the horizontal axis and luminosity (absolute magnitude) on the vertical axis, where if you chose M.Sc. Thesis • Yamam Al-Tawalbeh •Department of Physics • AABU University • 2017


9

a star randomly, and fall it on HR diagram, you will find it in definite regions and you can predict the future stages of it, and what is its fate after those stages that it will pass.

Figure 1-1: Hertzsprung-Russell Diagram (Wikipedia, 2018). Figure 1-1 An HR diagram showing many well known stars in the Milky Way galaxy. The main sequence band contains 90% of stars, starts from the upper left (hot, bright blue giants) to the lower right (cool, faint dwarfs). The other 10% of stars lie into the upper right region (cool, bright giants and supergiants) or in the lower left one (hot, lowluminosity white dwarfs) (Patterson, 2013).



10

We can determine the radius of any star by its luminosity that luminosity(L) is proportional to the square of its radius(R) times the fourth power of its surface temperature (T). Only our Sun we can measure its diameter directly because of its proximity to us, which enables astronomers to measure its size directly, so HR diagram provides sufficient information to calculate star sizes in this way and easily.

1.4

Properties of Stars: In this section some properties of stars such as the magnitude, luminosity and temperature are going to be discussed: • Apparent magnitude of a star: The apparent magnitude of a star is the brightness observed from its real distance from the observer. For example, the sun has the greatest apparent magnitude of all stars observed from the Earth, although there are many stars in the space more bright than it because it is the nearest star to us and naturally the apparent magnitude of it will be the largest. Two stars may be equal to the apparent magnitude, although their luminosities are different, because they occur on two different distances but the farthest ones are brighter than the nearest ones. They appear to observer on the Earth with the same brightness, and generally the closer star has the greater apparent magnitude, the farther has the less. To describe the brightness of the stars, classification by brightness, study their characteristics accurately, compare between them, and linking that to the stages of star life and its development, Hipparchus (around 150 BC) put a scale of apparent magnitude made the greatest fate of rank 1 and lowest rank 6. Later in 1830 William Herschel determined that the ratio between the brightness of a star of the first group and the brightness of a star of the sixth group is 100:1.



11

Accordingly, in 1856 the Oxford astronomer Norman R. Pogson proposed that a difference of five magnitudes be exactly defined as a brightness ratio of 100 to 1. This convenient rule was quickly adopted. One magnitude thus corresponds to a brightness difference of exactly the fifth root of 100, or very close to 2.512 — a value known as the Pogson ratio. (if B is the brightness of stars with the magnitudes m and n then

𝐵𝑚 𝐵𝑛

= 100.4(𝑚−𝑛) )(Al-Wardat, 2017).

• Absolute magnitude (absolute visual magnitude): M is the magnitude the star would have if it was placed at a distance of 10 parsecs(32.6 light years) from Earth. If we want to study the intrinsic characteristics of the stars, it is necessary to convert the apparent intensity of the luminance to the virtual absolute, since the apparent magnitude does not reflect the real brightness of the star, that the giant star may have a weak apparent magnitude because of its distance from the earth, but another small one appears by strong apparent magnitude because it is close to the Earth (Al-Wardat, 2017). To convert the observed brightness of a star (the apparent magnitude, m) to an absolute magnitude, we need to know the distance, d, to the star. Alternatively, if we know the distance and the apparent magnitude of a star, we can calculate its absolute magnitude. Both calculations are made using:

𝑚 − 𝑀 = 5 log (

𝑑 ) 10



12

with m – M known as the distance modulus and d measured in parsecs (Cosmos, 2017).

Figure 1-2: The Absolute Visual Magnitude (Cosmos, 2017). Figure 1-2 shows the comparison between the apparent visual magnitude and the absolute visual magnitude for the same star. • Apparent and absolute magnitudes within a specific range: Practically, the apparent magnitude is measured within a specific range known to astronomers as the amount of light coming from the star passing through a particular filter, using filters that allow a specific wavelength of light to reach the detector (Al-Wardat, 2017). Many filters have been manufactured, the most famous and most commonly used filters Johnson-Cousins(UBVRcIc Johnson-Cousins), as well as Strömgren filters(uvby Strömgren), where the symbols refer to: U: refers to ultraviolet, B: refers to blue, V: refers to visual, R: refers to red, I: refers to infrared (Al-Wardat, 2017). Thus, it is expressed as apparent magnitude,



13

depending on the filter used in the photometer as follows: mV, mB, mR, mu, mv, and for absolute magnitude: MV, MB, MR, Mu, Mv.

Figure 1-3: UBVRcIc Johnson-Cousins Filters (“Johnson-Cousins UBVRI filter curves,” 2018). • Bolometric

magnitude:

The bolometric magnitude Mbol,

takes

into

account electromagnetic radiation at all wavelengths. It includes those unobserved due to instrumental pass-band, the Earth's atmospheric absorption, and extinction by interstellar dust. It is defined based on the luminosity of the stars. In the case of stars with few observations, it must be computed assuming an effective temperature. Classically, the difference in bolometric magnitude is related to the luminosity ratio according to: 𝑀𝑏𝑜𝑙 − 𝑀𝑏𝑜𝑙☉ = 2.5[𝐿☉ ⁄𝐿] (“Properties of stars,” 2017). Where : L⊙ is the Sun's luminosity (bolometric luminosity) L is the star's luminosity (bolometric luminosity) Mbol⊙ is the bolometric magnitude of the Sun Mbol, is the bolometric magnitude of the star. M.Sc. Thesis • Yamam Al-Tawalbeh •Department of Physics • AABU University • 2017


14

To calculate the bolometric magnitude from the visual magnitude (𝑚𝑣 ) by the magnitude of the bolometric correction (𝐵𝐶) we use the following relationship (Al-Wardat, 2017): 𝑀𝑏𝑜𝑙 = 𝑚𝑣 − 𝐵𝐶 • Color Indices: The color index is the difference between two color magnitudes. Color Indices are used as an indicator of the temperature of the star's surface, where the results of astronomical observations are given in terms of photo-color magnitude (V), and color Indices (U-B , B-V), for example look the next table: Table 1-1: B-V Color index as a function of temperature for the Main-Sequence stars (Al-Wardat, 2017) T(K)

B–V

35000 21000 13500 9700 8100 7200 6500 6000 5400 4700 4000 3300

- 0.45 - 0.31 - 0.17 0.00 + 0.16 + 0.30 + 0.45 + 0.57 + 0.70 + 0.84 + 1.11 + 1.39

• Effective Temperature: This property is one of the fundamental parameters that characterizes a star. The effective temperature of a star is the temperature of a Black Body of the same size as the star and that would radiate the same total amount of electromagnetic luminosity as emitted by the star. If L is the luminosity radiated by a star of radius R, then Teff is derived by applying the 4 Stefan–Boltzmann law : 𝐿 = 4𝜋𝑅 2 𝜎𝑇eff (Rouan, 2011).



15

1.5 Binary Stars: Multiple star systems are star systems consisting of two or more stars orbiting around particular mass center, whereas binary systems consist of two stars orbiting around their common barycenter. These systems are generally difficult to detect except by precise monitoring. They are visible to the naked eye as a single bright spot, especially if they are more distant, they are then discovered as binary or multiple systems by other means. The importance of studying such systems is the fact that about half of the visible stars or more are multi-star systems, as astronomical researchs have indicated over the last two centuries (Binary stars, 2016).

Figure 1-4: Orbits of stars in a binary system (Introduction to Binary Stars, 2018) Figure 1-4 shows how the two stars in a binary system revolve around each other in elliptical orbits (can be almost circular in some cases) (Introduction to Binary Stars, 2018). Another important factor in the study of binary systems in astrophysics is that masses of stellar systems can be determined by the orbital calculations of these systems, and the determination of their orbital elements, which in turn allows other stellar parameters, such M.Sc. Thesis • Yamam Al-Tawalbeh •Department of Physics • AABU University • 2017


16

as radius and density, to be indirectly estimated. This also determines an empirical massluminosity relationship (MLR) from which the masses of single stars can be estimated. In addition, the frequent study of binary systems, their relationship with each other, and the determination of their orbital elements gives us very important informations about the conditions of stellar birth, the stages of its evolution, and the stages of dynamical evolution of the Galaxy (Introduction to Binary Stars, 2018).

1.6

Discovery and Evolution: Double stars are the broader term that includes binary systems and stars close to each

other. Not all two stars close to each other must have a binary system; where they can be a double star, meaning that they are independent of each other and have different distances for us, but appear close to each other as seen by the observer. the first discoverer of the binary systems was the Italian scientist Galileo Galilei in 1617, when he spotted by his telescope the second star from the end of the handle of the Big Dipper, to show it then as a two stars revolve around a common center of mass and not a single star, and this system was visual binary, so the visual binaries was the first discovery (“Binary Star Systems,” 2016). Some astronomers then began studying the characteristics of binary systems, how they were formed, and drawing orbits suggested for them based on their own astronomical observations. The first was astronomer John Michel. In 1767, he proposed that these systems could have physically attached with each other, explained that double stars come by chance alignment are unlikely. He was followed by William Herschel, who began in 1779 by studying binary systems, and he changed the relative positions of a number of M.Sc. Thesis • Yamam Al-Tawalbeh •Department of Physics • AABU University • 2017


17

double stars over 25 years, and concluded that they must be binary systems. But in spite of all this, no orbit of these star systems was calculated until 1827 by Felix Saffari when the orbit of the star Xi Ursae Majoris was calculated, and then followed the calculations of the double systems to be organized in special Catalogues(Binary stars, 2016). One of the most important Catalogues is Washington double star Catalogue, which includes database for 100,000 double stars, collected by the US Naval Observatory, includes optical doubles and binary systems. But the orbits are drawn for only a few thousands of them (Binary stars, 2016).

1.7

Classification of Binary Stars: Binary systems are classified according to their monitoring method into four main

types: visually, if direct; spectroscopically, depending on the difference in the type of spectrum; eclipsing, where its brightness varies due to eclipse, or astrometric, where the deviation in the star's position is measured because of an invisible companion. Many binary systems may belong to more than one types, such as spectroscopic binaries, many of which are also eclipsing.

1.7.1

Visual Binaries: Until the binary star is classified as visual, the angular separation between its

components must be sufficient to observe it as a double star with the telescope, or even with some high- powered binoculars. There are important factors to detect these visual binaries, the most important of which is the angular accuracy of the telescope, that the larger angular accuracy leads to more detection of new visual binaries, and their other



18

important factor is the relative brightness of the stars of the binary system, which glow from the brighter star may hinder the detection of the fainter element or make it difficult. We consider the brighter star to be the primary, and the fainter one is secondary in the visual binaries. In some references (especially older ones), the secondary is called companion. If the two stars are of the same brightness, the selection of the discoverer for the primary is acceptable (Binary stars, 2016).

Figure 1-5: Visual Binaries (Australia Telescope National Facility, 2016). Figure 1-5 shows an example of visual binary star system and the separation between the components of it in milliarcseconds.

1.7.2

Spectroscopic Binaries: In some cases, the binary star cannot be distinguished by direct observation, because

the angular separation between the two stars is very small and its orbital velocity is very high. The only evidence of its existence is the Doppler effect, that the binary consists of a pair of stars, one of which moves away from us, so its spectral lines shift Towards the red, and the other approaching us shift towards the blue during its motion with the period of their common orbit.



19

The radial velocity of the system varies periodically, and since the radial velocity of the spectra can be measured by the shifting of the spectral lines according to Doppler effect, these binaries are known as spectroscopic binaries. Most of these binaries can not be resolved as visual binaries even with higher power detection telescopes (Binary stars, 2016).

Figure 1-6: Spectroscopic Binaries (Australia Telescope National Facility, 2016). Figure 1-6 The observed combined spectrum shows periodic splitting and shifting of spectral lines. The amount of shift is a function of the alignment of the system relative to us and the orbital speed of the stars



1.7.3

20

Eclipsing Binaries:

In this type of binaries, the orbit plane of the two stars lies in the observer's line of vision, subjecting its stars to mutual eclipses. One of the most famous examples of this type of binaries Algol star. This type is also spectroscopic binaries because the spectrum from each star constantly changes due to eclipses. The extragalactic eclipsing binaries' are of great importance in measuring distances to galaxies, using the 8 m class telescopes, this is more accurate than using the standard candles method. These binaries provide a direct way to measure the distance to galaxies to an improved level of accuracy of 5%, and have been used recently (last decade) to calculate estimated distances to the Large Magellan Cloud (LMC), the Smale Magellan Cloud (SMC), Andromeda Galaxy and Triangulum Galaxy (Binary stars, 2016). These binaries are variable stars because of eclipses, not because the light of their individual components is different. The optical curve of an eclipsing binary is practically constant with periodic decreases in intensity. If the stars are different in size, the smaller will be obscured by a total eclipse, but larger by an annular eclipse.

1.7.4

Astrometric binaries:

Some astronomers have observed that some stars revolve arround an empty area in space, without a visible companion, and without observing a different optical spectrum. But the application of mathematics used in ordinary binaries enables us to calculate the mass of the missing companion, which may be very dim, or emit a little electromagnetic radiation such as the neutron star for example, or its optical spectrum is obscured by the glare of the primary star. M.Sc. Thesis • Yamam Al-Tawalbeh •Department of Physics • AABU University • 2017


21

Based on the above, since we cannot detect the position of the companion by direct observation, or difference of the optical spectrum, we can infer the existence of the companion because of the impact of its attractiveness on the position of the primary star, where it can be seen that periodic shifts occur on its motion resulting from The influence of companion star. But this kind of measurement can only be done on nearby stars that are no more than 10 parsecs, which makes the observer able to observe that oscillation in the path of this type of binaries. To determine accurate information about the comet's mass and its orbital period, it must be large enough to make significant changes in the position of the star. Although it is not visible, we can determine its properties using Kepler's laws. An important use of this method in detecting binaries is to locate planets outside the solar system orbiting a star. But the conditions for these measurements are very accurate, due to the large difference in the mass ratio, and the long period of planets around their stars. The detection of changes in the position of the star is very accurate, and difficult to achieve the necessary accuracy, so it is preferable to use space telescopes to avoid the blurring effect of Earth's atmosphere, which gives us greater accuracy in measurements (Binary stars, 2016).



1.7.5

22

"Exotic" Types:

Recently, some strange binaries have been discovered, which have surprised scientists and opened up new horizons in astronomy and its association with the general relativity of Einstein. One of the most exciting celestial bodies is the first known pulsed binary star PSR J0737-3039, which was discovered in late 2003 on the Parkes radio telescope. It has a 23 millisecond pulsar PSR J0737-3039A and another pulsar, PSR J0737-3039B, which spin once every 2.8 seconds with 2.4 hours period. The high orbital velocity of this binary star makes it an excellent opportunity to study the principles of general relativity and the search for gravitational waves. However, this system is not the first of its kind where it can be attached to eclipsing binaries. The relativistic effects of its orbit means that its orbit will tend to an eclipsing orbit over the next ten years or so (Australia Telescope National Facility, 2016).

1.8

Previous Studies: In the past century, and over thirty years, the method of speckle interferometry has

been the main method that has continued and expanded the work of the visual binary star observers (Horch, 2003). The scientists applied astronomical methods on the components of the speckle interferometric binary systems as Atmospheric modeling and dynamical analysis,

to

determine the physical and geometric parameters of the individual components of these systems.



23

Model atmospheres were used to compute the individual synthetic spectral energy distribution (SED) for each component separately. These models were constructed using a grid of Kurucz’s solar metallicity blanketed models. To compute the entire SED for the systems from the net luminosities of the components A and B located at a distance d from the Earth, SEDs were combined together (Al-Wardat, 2014). Al-Wardat’s complex method for analyzing CVBS was used to be a reverse method of building the individual and entire synthetic SEDs of the binary systems, beginning with the entire observational spectral energy distribution (SED), and the magnitude difference between the subcomponents. To calculate the new orbits This was combined with Docobo’s analytic method. Although possible short (approximately 9 years) and long period (of about 18 years) orbits could be considered taking into account the similar results of the stellar masses obtained for each of them (Al-Wardat et al., 2017). The technique of estimating the orbital elements of a stellar binary system – either a visual close one or a spectroscopic one – from a set of observational relative position measurements and radial velocities (in the case of spectroscopic binaries) goes back to several decades. Of course, different groups of research around the world use different methods of calculations. The revolution of high speed computers has greatly improved the situation, and reduced the time required for the calculations. In the website of the Sixth Catalogue of Orbits of Visual Binary Stars you can find the names of the top 25 orbit calculators. Here we mention the main orbit calculators: 1- Mason, Brian D., Hartkopf, William I. and Miles, Korie N.(Mason et al., 2017) 2- Balega, Yuri. (Balega et al., 2017) M.Sc. Thesis • Yamam Al-Tawalbeh •Department of Physics • AABU University • 2017


3- Tokovinin, Andrei (Tokovinin, 2017) 4- Al-Wardat, Mashhoor (Al-Wardat et al., 2016) 5- Docobo, José A.(Docobo et al., 2017) 6- Couteau, Paul (Couteau, 1997) 7- Zirm, H. (Zirm, 2008)


24


25

Chapter 2

The Visual Binaries Hip11352, Hip70973, and Hip72479 2

Introduction: We will talk in this chapter about the three visual binary systems Hip11352, Hip70973, and Hip72479, which are the subject of our research in this thesis. Firstly there is a general introduction about every system, then its location as in Simbad, the parameters of components of every system as I found in the different papers which talk about these systems, the orbital elements of every system which have an important factor in building the orbit of system by our procedure, and lastly table of astronomical observations published in the Fourth Catalogue of each of these systems, which are the basis of building the program to draw the system's orbit on IDL software. It is worth noting that I have made every effort to include all the informations published on these three systems, their properties, their parameters, and their orbital elements, but there may be some informations that I could not find. There is no perfection in any human action, but the informations we have are rich and sufficient to give a clear picture of each system, and enable us to build modified orbits for previously published orbits.



2.1

26

Hip11352 System: Hip11352 is a white to yellow star that can be located in the constellation of

Triangulum. The star cannot be seen by the naked eye, you need a telescope to see it. The star has an estimated age of 13.10 Billion of Years but could be as young as 8.10 to 16.60 according to Hipparcos (Universe Guide, 2018).

2.1.1

Triangulum Constellation(The Home of Hip11352): The Triangulum constellation is one of the ancient Greek constellations that were

first classified by the Greek astronomer Ptolemy in the second century AD. This constellation is located in the northern sky. Its name comes from the Latin word (triangle) in relation to the brightest three stars in it (alpha, beta and gamma) which form a long and narrow triangle. This constellation is home to several galaxies, the nearest and best known of us is the Triangulum Galaxy(Messier 33) (“Triangulum Constellation,” 2018).

2.1.2

Location of Hip11352:

We can determine the coordinates of any point on Earth using latitude and longitude, looks like that for the planetarium the Right Ascension (R.A.) and Declination (Dec.) to determine the location of the star in the galaxy. The Right Ascension is how far in time (hh:mm:ss) the star is along the celestial equator, if it is positive then its eastwards, negative westwards. The Declination is how far in degrees the star locates north or south the celestial equator (Universe Guide, 2018). For Hip11352 its Right Ascension is: 02h 26m 09s.5895, the Declination is: +34˚ 28ˊ 10˝.031(D. SIMBAD, 2017).



2.1.3

27

Proper Motion of Hip11352:

Proper Motion details the movements of the stars in the constellation to each other and are measured in milliarcseconds. If we saw Hip11352 in the horizon, we will find that it is moving -50.85 ± 0.51 miliarcseconds/year towards the north and 117.87 ± 0.99 miliarcseconds/year east. The Radial Velocity of it is -2.20000 km/s with an error of about 0.30 km/s, this velocity means the speed at which the star is moving away/towards us (Universe Guide, 2018).

2.1.4

The Age and The star's Galacto-Centric Distance of Hip11352:

The star is 7,430.00 Parsecs from the Galactic Centre, and is equivalent to 24,233.94 Light Years. The Abundance of Iron 3 in the star is -0.14 with an error rate of 0.08 Fe/H compared to 1 in the Sun (Universe Guide, 2018). The stars age according to Hipparcos data files is about 13.10 Billion years, but could be between 8.10 and 16.60 Billion years old. In comparison, the Sun's age is about 4.6 Billion Years Old. If we compare the recorded age of the star with the age of the Universe, we find that the age of the star is greater than the age of our knowm Universe, which justifies scepticism in this age (Universe Guide, 2018).

2.1.5

The Apparent Magnitude and The Orbit of Hip11352: Hip11352 has an apparent magnitude of 8.00(Universe Guide, 2018). Hipparcos was

a European Space Agency satellite (E.S.A.) operation launched in 1989 for four years (Universe Guide, 2018).

3

Astronomers define an ‘abundance ratio’ as the logarithm of the ratio of two metallic elements in a star relative to their ratio in the Sun, abundance ratios contain useful information regarding the source of the gases making up the star (“Abundance Ratio | COSMOS,” 2018). M.Sc. Thesis • Yamam Al-Tawalbeh •Department of Physics • AABU University • 2017


28

Hoffmann said in his thesis at 2000 (Hoffmann, 2000)"A search for binary stars using speckle interferometry" that the secondary star in this system (Hip11352) has completed nearly a full orbit since the Hipparcos discovery observations. The orbit for this system appear to be less than 15 years. The data for this star system are collected from the Hipparcos satellite in 1991 and data from the WIYN telescope over the last four years. These data were used to plot the position of the secondary star versus the primary star and an arc of the orbit was found. Hip11352 is 43 parsecs from Earth and has an estimated orbital period of 12 years (Hoffmann, 2000), however, this information contradicts what is published and documented in the Sixth Catalog of Orbits of Binary System by Balega, as the period mentioned there is 6.85 years.

2.1.6

Another Identifiers and The Parameters of Its Components:

Some another identifiers to this binary system are HD 15013(Henry Draper Designation) , and WDS J02262+3428AB, and 𝜋 = 22.13 𝑚𝑎𝑠, with G5V spectral type (D. SIMBAD, 2017), but the most recent value of Parallax for this system in the latest release of the recently launched Gaya Archive is 𝜋 = 22.515589732059386 (“Gaia Archive,” 2018). The parameters of the components of this system were derived by Al-Wardat's paper (Al-Wardat, 2009)"Parameters of the components of visually close binary systems: Hip11352" as: Table 2-1: The parameters of the components of Hip11352 (Al-Wardat, 2009) 𝑎 (𝑘) 𝑇𝑒𝑓𝑓

5650 ± 50

𝑏 (𝑘) 𝑇𝑒𝑓𝑓

5550 ± 50

log 𝑔𝑎

4.47 ± 0.13

log 𝑔𝑏

4.49 ± 0.13

𝑅𝑎 (𝑅ʘ )

0.92 ± 0.03

𝑅𝑏 (𝑅ʘ )

0.88 ± 0.02

𝑀𝑎 (𝑀ʘ )

0.93 ± 0.05

𝑀𝑏 (𝑀ʘ )

0.89 ± 0.05

Spa

G7

Spb

G9



29

Al-Wardat in this paper also calculated The synthetic magnitudes of both components under Johnson-Cousins, Strömgren, and Tycho photometrical systems. And as already about Strömgren and Tycho magnitudes from Al-Wardat's paper (AlWardat, 2008) we find for this system Strömgren magnitudes, and Tycho magnitudes as in this table: Table 2-2: Strömgren and Tycho magnitudes for Hip11352 (Al-Wardat, 2008) VJ 8.01

v 9.15

b 8.47

y 7.96

v-b 0.68

b-y 0.51

BT 8.94

VT 8.1

(B - V )T 0.84

Horch produces in his paper (Horch et al., 2002) "Speckle Observations Of Binary Stars With The Wiyn Telescope. II. Relative Astrometry Measures During 1998-2000" presented five hundred and twelve relative astrometry measures for 253 double stars, including 53 double stars discovered by Hipparcos. In 15 cases, relative astrometry was reported for the first time for newly confirmed pairs. In this paper we find these magnitudes for our system Hip11352 as : Table 2 in the paper contains the main body of astrometric results from the data set as follows : The observation date in fraction of the Besselian year , the observed position angle (θ), with north through east defining the positive sense of h; the observed separation (ρ); the center wavelength of the filter (λ) used to make the observation, and the full width at halfmaximum of the filter passband(Δλ). Table 2-3: The main body of astrometric results from the data set(Horch et al., 2002) Date

θ(deg)

ρ(arcsec)

λ(nm)

Δλ(nm)

1998.9246

205.8

0.124

648

41



30

Table 5 in the same paper presents the fastest-moving objects, namely, those having moved more than 100 in position angle or 100 mas in separation between the Hipparcos measure and the latest measure in Table 2 (mentioned in Table 2-3 in this thesis), from this table we find for our system Hip11352: the system apparent magnitude (V) , the system parallax , the composite spectral type of the pair , the change in position angle from 1991.25 to the latest measure in Table 2, the change in separation from 1991.25 to the latest measure in Table 2: Table 2-4: The fastest-moving objects, namely, those having moved more than 100 mas in separation between the Hipparcos measure and the latest measure in Table 2 in Horch's paper (Horch et al., 2002) π(mas) 23.19

Total V 8.00

Spectrum G5

Δθ(deg) 287

Δρ(mas) -71

Absolute magnitudes and spectral types of binary star system Hip11352 as it was mentioned in Balega's paper (Balega et al., 2001)"Speckle interferometry of nearby multiple stars": Table 2-5: As it was mentioned in Balega's paper for Hip11352 (Balega et al., 2001) MA

MB

Filter

SpA

SpB

5.5

5.7

V

G8

G9



2.1.7

31

Orbital elements of the system Hip11352:

The orbital elements of any binary system is necessary to build its orbit, and we find in Al-Wardat's paper (Al-Wardat, 2009, p. 11352)" Parameters of the components of visually close binary systems: Hip11352" the orbital elements of Hip11352 as in the next table: Table 2-6: The orbital elements of Hip11352 (Al-Wardat, 2009, p. 11352) 𝑃(𝑦𝑟)

6.85 ± 0.05

T

1992.12 ± 0.06

e

0.284 ± 0.006

a (mas)

100 ± 1

i (deg)

50.0 ± 0.6

Ω (deg)

15.1 ± 0.9

ω(𝑑𝑒𝑔)

4.4 ± 1.6

𝑀𝑡𝑜𝑡𝑎𝑙 (𝑀ʘ )

1.71 ± 0.27

Where: i : Angle of inclination

Ω: Longitude of the ascending node

ω: Longitude of the periastron

a: Semimajor axis

e: Eccentricity

T: Time of periastron

P: Period of the system These elements were published by Balega in 2005 in Sixth Catalogue of Orbits of Binary Stars, in this study we will go to modify these elements and the orbital resulting of it based on the new observations after 2005 in Fourth Catalogue of Interferometric Measurements of Binary Stars, as they are included in the next section.



2.1.8

32

Data From Fourth Catalogue of Hip11352:

From Fourth Catalogue of Interferometric Measurements of Binary Stars we find these data about our system Hip11352: Table 2-7: The data of the observations of the system Hip11352 in Fourth Catalogue (“Fourth Interferometric Catalogue - RA 2 hr,” 2017) Epoch of observation

Theta(θ)(mas)

Error in theta(Δθ) (mas)

Rho(ρ)(pc)

Error in Ro(Δρ)(pc)

Diameter of telescope

Source

1991.25

177

.

0.133

.

0.3

HIP1997a

1997.8242

186.8

.

0.122

.

3.5

Hor1999

1998.7747

.

.

.

.

6

Plz2005

1998.7747

202.9

0.5

0.127

0.002

6

Bag2002

1998.7773

.

.

.

.

6

Plz2005

1998.7773

202.7

0.5

0.126

0.002

6

Bag2002

1998.9246

.

.

.

.

3.5

Hor2004

1998.9246

205.8

.

0.124

.

3.5

Hor2002a

1999.813

228.3

0.3

0.0965

0.0004

6

Bag2004

1999.8856

.

.

.

.

3.5

Hor2004

1999.8856

50.1

.

0.092

.

3.5

Hor2002a

2000.7622

.

.

.

.

3.5

Hor2004

2000.7622

103.7

.

0.062

.

3.5

Hor2002a

2000.8757

293

1.1

0.057

0.002

6

Bag2006b

2001.7528

4.2

0.5

0.07

0.001

6

Bag2006b

2001.7622

4.2

.

0.07

.

6

Bag2005

2001.7643

188

.

0.07

.

3.5

Hor2008

2002.7993

85.5

.

0.059

.

6

Bag2005

2002.7993

87.6

4.5

0.059

0.005

6

Bag2013

2003.629

153.9

.

0.077

.

3.5

Hor2008

2003.7885

158

.

0.091

.

6

Bag2005

2003.9249

163.5

.

0.095

.

6

Bag2005

2003.9249

163.2

.

0.094

.

6

Bag2005

2003.9249

162.8

0.7

0.096

0.002

6

Bag2013

2003.9249

163.1

0.6

0.095

0.001

6

Bag2013

2004.8157

185.4

0.4

0.124

0.002

6

Bag2007b

2007.818

.

.

< 0.054

.

2.5

Hrt2009

2007.8256

296.7

.

0.059

.

3.5

Hor2010

2010.0074

111.1

.

0.0579

.

3.5

Hor2011

2010.0074

.

.

.

.

3.5

Hor2011

2010.72

156.4

.

0.0876

.

3.5

Hor2017

2010.72

156.4

.

0.0885

.

3.5

Hor2017

2012.6777

202.7

0

0.124

0

1.5

RAO2015



33

The observations that did not fit into the published orbit of this star in 2005 by Balega were shaded by gray, as these new observations are the ones that will modify the orbit of this star. By using this data and the orbital elements of this system, we can build a new orbit of its motion by IDL software as will be discussed later in chapter4.

2.2

Hip70973 System: HIP 70973 is a white to yellow star that can be located in the constellation of Virgo.

The star cannot be seen by the naked eye, you need a telescope to see it (“HIP 70973, Universe Guide,” 2018).

2.2.1

Virgo Constellation(The Home of Hip70973):

Virgo is one of the constellations of the zodiac. Its name is Latin for virgin. After Hydra constellation Virgo considered to be the second-largest constellation in the sky, where is occupying an area of 1294 square degrees (“Virgo (constellation), Wikipedia,” 2018). It lies in the southern sky. First catalogued by the Greek astronomer Ptolemy in the 2nd century. It is located in the third quadrant of the southern hemisphere (SQ3) and can be seen at latitudes between +80° and -80°, contains one of the brightest stars in the night sky (Spica) star (“Virgo Constellation,” 2018).

2.2.2


Its Right Ascension is: 14h 31m 00s.62017 , and its Declination is: -05˚ 48ˊ 08˝.4641 (SIMBAD, 2017a).



2.2.3

34


Proper Motion details the movements of the stars in the constellation to each other and are measured in milliarcseconds. If we saw Hip70973 in the horizon, we will find that it is moving -21.23 ± 0.51 miliarcseconds/year towards the north and -207.03 ± 0.89 miliarcseconds/year east. The Radial Velocity of it is -21.50000 km/s with an error of about 0.20 km/s, this velocity means the speed at which the star is moving away/towards us (“HIP 70973, Universe Guide,” 2018).

2.2.4

Distance to Hip70973:

The distance from the Galactic Centre to the star is 7,374.00 Parsecs, or terms of Light Years is 24,051.28. From Earth the calculated distance to HIP 70973 is 134.17 light years away or 41.14 parsecs by using parallax of 24.31. This distance does not remain constant as it is recalculated from time to time, because the star moves away or close to us according to its proper motion (“HIP 70973, Universe Guide,” 2018).

2.2.5

Another Identifiers and The Parameters of Its Components:

Some another identifiers to this binary system are HD 127352, and WDS J143100548AB(SIMBAD, 2017a). The binary system Hip70973 is well-known VCBSs. Atmospheric modelling of the components of the visually close binary system Hip70973 was used to estimate the individual physical parameters of its components, and the components of this system as were derived by Prof. Al-Wardat (Al-Wardat, 2012) are:



35

Table 2-8: The parameters of the components of Hip70973(Al-Wardat, 2012) 𝑎 (𝑘) 𝑇𝑒𝑓𝑓

5700 ± 75


5400 ± 75

log 𝑔𝑎

4.50 ± 0.05

log 𝑔𝑏

4.50 ± 0.05

𝑅𝑎 (𝑅ʘ )

0.98 ± 0.07

𝑅𝑏 (𝑅ʘ )

0.89 ± 0.07

𝐿𝑎 (𝐿ʘ )

0.91 ± 0.08

𝐿𝑏 (𝐿ʘ )

0.61 ± 0.05

Spa

G4

Spb

G9

𝜋( 𝑚𝑎𝑠)

23.59

But I find in Simbad a new values for spectral types as K0V, and for parallax 𝜋 = 24.31 𝑚𝑎𝑠 (SIMBAD, 2017a), but the most recent value of Parallax for this system in the latest release of the recently launched Gaya Archive is 𝜋 = 24.35404566949619 (“Gaia Archive,” 2018). The apparent magnitude of Hip70973 is 7.68, and its absolute magnitude is 4.61. The star's Iron Abundance is -0.06 with an error value of 0.08 Fe/H (“HIP 70973, Universe Guide,” 2018) see Footnote 3 page 27.

There are different methods to estimate the parameters of the individual components. The two most important methods are: Al-Wardat’s method , which depends on the atmospheres modeling of the individual components to estimate the complete parameters of the system, and Sowell–Wilson’s method which uses the integrated magnitude and the ratio of the luminosities at selected wavelengths (Strömgren passbands) to obtain the intrinsic brightness and color of each component (Al-Wardat, 2008). The Strömgren vby and Tycho BV passbands have been used to calculate magnitudes of the binary systems. These data when combined with the magnitude differences of the sub-components from speckle interferometry observations will allow the finding of the M.Sc. Thesis • Yamam Al-Tawalbeh •Department of Physics • AABU University • 2017


36

parameters of the individual components. These parameters will improve our knowledge about binary system formations and evolution. For our system Hip70973 we find its data from Al-Wardat's paper as follows : Strömgren magnitudes and Tycho magnitudes: Table 2-9: Strömgren and Tycho magnitudes(Al-Wardat, 2008) VJ

v

b

y

v-b

b-y

BT

VT

(B - V )T

7.63

8.91

8.14

7.59

0.77

0.56

8.68

7.73

0.95

In the paper of "Kinematics of Hipparcos Visual Binaries" by A. BartKevicius and A. Gudas, we find nearby magnitudes as follows: Table 2-10: The values and the mean values of the parallaxes of Hip70973 (BartKevicius and Gudas, 2001a, 2001b, 2001c) V

B-V

𝜋(𝑚𝑎𝑠)

𝜎𝜋 (𝑚𝑎𝑠)

𝜋(𝑚𝑎𝑠)(𝑚𝑒𝑎𝑛 𝑣𝑎𝑙𝑢𝑒)

𝜎𝜋 (𝑚𝑎𝑠)(𝑚𝑒𝑎𝑛 𝑣𝑎𝑙𝑢𝑒)

7.68

0.775

26.04

1.04

25.97

0.07

and we find in the same paper the individual values of the proper motions: me

Table 2-11: The individual values of the proper motions of Hip70973 (BartKevicius and Gudas, 2001d) 𝜇𝛼 (𝑚𝑎𝑠/𝑦𝑟)

𝜎𝜇𝛼 (𝑚𝑎𝑠/𝑦𝑟)

𝜇𝛿 (𝑚𝑎𝑠/𝑦𝑟)

𝜎𝜇𝛿 (𝑚𝑎𝑠/𝑦𝑟)

−203.98

1.18

−21.64

0.74

and the mean values of the proper motions: Table 2-12: The mean values of the proper motions of Hip70973 (BartKevicius and Gudas, 2001e) 𝜇𝛼 (𝑚𝑎𝑠/𝑦𝑟)




−205.02

1.84

−20.80

2.49



37

And the transverse velocity of star without radial velocity: 𝑉𝑡 = 37.62 𝑘𝑚/𝑠 , 𝜎𝑉𝑡 = 0.35 𝑘𝑚/𝑠 (BartKevicius and Gudas, 2001f).

2.2.6

Orbital elements of the system Hip70973: Hip70973 System Explorer was the astronomer Rossiter in 1938 (exactly 1938.51) at

the Lamont–Hussey Observatory using the 27-inch telescope (0.69 m). orbits of the system had been built by several astronomers: Couteau in 1960, Morel in 1970, Heintz in 1981, the last one built two orbits: the first with 45.4 years and dynamical parallax .020 arcsec, the second with 22.4 years, and dynamical parallax 0.029 arcsec, the second orbit is more correct (Al-Wardat, 2012). We find in Al-Wardat's paper (Al-Wardat, 2012, p. 70973)" Physical Parameters of the Visually Close Binary Systems Hip70973 and Hip72479" the orbital elements of Hip70973 as in the next table: Table 2-13: The orbital elements of Hip70973 (Al-Wardat, 2012, p. 70973) 𝑃(𝑦𝑟)

22.98 ± 0.30

T

1993.62 ± 0.02

e

0.499 ± 0.010

a (arcsec)

0.243 ± 0.002

i (deg)

49.1 ± 2.0

Ω (deg)

13.8 ± 2.0

ω(𝑑𝑒𝑔)

121.0 ± 2.5

These orbital elements were set by Docobo in 2000, and are therefore considered the most recent, and result from the orbit published in the Sixth Catalog in the same year, where we go to modify it in this study.



2.2.7

38

Data From Fourth Catalogue of Hip70973:

From Fourth Catalogue of Interferometric Measurements of Binary Stars we find these data about our system Hip70973: Table 2-14: The data of the observations of the system Hip70973 in Fourth Catalogue (“Fourth Interferometric Catalogue - RA 14 hr,” 2017a). Epoch of observation

Theta(θ) (mas)


Rho(ρ) (pc)

Error in Ro(Δρ) (pc)


1989.309

10.9

.

0.255

.

4

1990.3437

19.6

.

0.219

.

4

1991.25

29

.

0.187

.

0.3

1992.4572

68.3

.

0.102

.

4

1993.0905

106.7

.

0.082

.

4

1997.3939

241.2

2

0.177

0.003

6

2001.2737

295.1

0.5

0.218

0.003

6

2001.2737

295.1

0.5

0.216

0.003

6

2001.498

116.6

.

0.217

.

3.5

2001.498

116.9

.

0.219

.

3.5

2002.2545

305.8

0.3

0.229

0.001

6

2002.3255

306.3

0.3

0.23

0.001

6

2002.3255

306.4

0.3

0.23

0.001

6

2004.196

320.5

0.7

0.267

0.004

1.55

2006.4376

335

0.9

0.292

0.005

6

2007.317

340.4

.

0.291

.

2.5

2008.5449

347.3

0

0.3024

0.0001

4

2008.5449

347.3

0

0.3023

0.0001

4

2014.3032

34.3

0

0.1644

0.0004

4.2

Source McA19 90 Hrt199 3 HIP19 97a Hrt199 6b Hrt200 0a Bag19 99a Bag20 06b Bag20 06b Hor20 08 Hor20 08 Bag20 13 Bag20 13 Bag20 13 Hrt200 8 Bag20 13 Hrt200 9 Tok20 10 Tok20 10 Tok20 15c

Sc Sc Hh Sc Sc S S S S S S S S Su S Su S S St

Gray-shaded observations are the new observations that are incorporated into the modified orbit construction for the first time, and on which the orbital elements of the system orbit are modified M.Sc. Thesis • Yamam Al-Tawalbeh •Department of Physics • AABU University • 2017


2.3

39

Hip72479 System: HIP 72479 is an orange to red main sequence dwarf star that can be located in the

constellation of Boötes. The star cannot be seen by the naked eye, you need a telescope to see it (“HIP 72479, Universe Guide,” 2018).

2.3.1

Boötes Constellation(The Home of Hip72479):

One of the largest star constellations in the sky, is located in the northern hemisphere between 0° and +60° declination, and 13 and 16 hours of right ascension on the celestial sphere. Its name comes from the Greek word (Boōtēs) which means plowman, or herdsman (“Boötes, Wikipedia,” 2018). Boötes is the 13th largest constellation in the night sky, occupying an area of 907 square degrees. The brightest star in the constellation is Arcturus, Alpha Boötis, which is also the third brightest star in the night sky (“Boötes Constellation,” 2018).

2.3.2


Its Right Ascension is: 14h 49m 13s.62103 , and its Declination is: +10˚ 12ˊ 52˝.0630 (SIMBAD, 2017b).

2.3.3


If we saw Hip72479 in the horizon, we will find that it is moving -185.23 ± 0.74 miliarcseconds/year towards the north, and 138.84 ± 1.23 miliarcseconds/year east. The Radial Velocity of it is -89.40000 km/s with an error of about 0.30 km/s , this velocity means the speed at which the star is moving away/towards us (“HIP 72479, Universe Guide,” 2018).



2.3.4

40

Distance To Hip72479:

The distance from the Galactic Centre to the star is 7,377.00 Parsecs, or terms of Light Years is 24,061.07 . From Earth the calculated distance to HIP 72479 is 144.38 light years away or 44.27 parsecs by using parallax of 22.59 (mas). This distance does not remain constant as it is recalculated from time to time, because the star moves away or close to us according to its proper motion (“HIP 72479, Universe Guide,” 2018).

2.3.5

Another Identifiers and The Parameters of Its Components: Some another identifiers to this binary system are HD 130669, and WDS

J14492+1013AB (SIMBAD, 2017b). Atmospheric modelling of the components of the visually close binary system Hip72479 was used to estimate the individual physical parameters of its components, and the components of this system as were derived by Prof. Al-Wardat (Al-Wardat, 2012) are: Table 2-15: The parameters of the components of Hip72479(Al-Wardat, 2012) 𝑎 (𝑘) 𝑇𝑒𝑓𝑓

5400 ± 50


5180 ± 50

log 𝑔𝑎

4.50 ± 0.05

log 𝑔𝑏

4.60 ± 0.05

𝑅𝑎 (𝑅ʘ )

0.89 ± 0.07

𝑅𝑏 (𝑅ʘ )

0.80 ± 0.07

Spa

G9

Spb

K1

𝜋( 𝑚𝑎𝑠)

23.59



41

It is clear from the parameters of the components of the systems and their positions on the evolutionary tracks that they are solar-type main-sequence stars in the early stages of their life (Al-Wardat, 2012). The absolute magnitudes calculated from the Hipparcos photometric system, +6.28 and +6.19, agree better with the WDS spectral type, K2 V, than do those calculated from WDS apparent magnitudes, +5.34 and +5.43. The calculated total mass is 1.45 𝑀ʘ (Docobo et al., 2000). The star's Iron Abundance is -0.10 with an error value of 0.05 Fe/H with the Sun has a value of 1 (“HIP 72479, Universe Guide,” 2018) see Footnote 3 page 27.

And as already about Strömgren and Tycho magnitudes from the previous AlWardat's paper we find for this system : Strömgren and Tycho magnitudes: Table 2-16:Strömgren and Tycho magnitudes and color indices(Al-Wardat, 2008) VJ

v

b

y

v-b

b-y

BT

VT

(B - V )T

8.38

9.84

8.94

8.32

0.91

0.62

9.56

8.49

1.07

In the paper of "Kinematics of Hipparcos Visual Binaries" by A. BartKevicius and A. Gudas, we find nearby magnitudes as follows: Table 2-17: The values and the mean values of the parallaxes of Hip72479 (BartKevicius and Gudas, 2001a, 2001b, 2001c) V

B–V

𝜋(𝑚𝑎𝑠)

𝜎𝜋 (𝑚𝑎𝑠)

𝜋(𝑚𝑎𝑠)(𝑚𝑒𝑎𝑛 𝑣𝑎𝑙𝑢𝑒)

𝜎𝜋 (𝑚𝑎𝑠)(𝑚𝑒𝑎𝑛 𝑣𝑎𝑙𝑢𝑒)

8.42

0.866

24.21

1.29

23.38 me

0.90



42

And we find in the same paper the individual values of the proper motions: Table 2-18: The individual values of the proper motions of Hip72479 (BartKevicius and Gudas, 2001d) 𝜇𝛼 (𝑚𝑎𝑠/𝑦𝑟)




−139.17

1.01

−187.90

1.20

And the mean values of the proper motions: Table 2-19: The mean values of the proper motions of Hip72479 (BartKevicius and Gudas, 2001e) 𝜇𝛼 (𝑚𝑎𝑠/𝑦𝑟)




−138.69

0.62

−186.14

1.47

And the individual values of the radial velocities of this system : Table 2-20: The individual values of the radial velocities of Hip72479 (BartKevicius and Gudas, 2001g) 𝑉𝑟 (𝑘𝑚/𝑠)

𝜎𝑉𝑟 (𝑘𝑚/𝑠)

−138.69

0.62

And the mean values of the radial velocities: Table 2-21: The individual values of the radial velocities of Hip72479 (BartKevicius and Gudas, 2001h) 𝑉𝑟 (𝑘𝑚/𝑠)

𝜎𝑉𝑟 (𝑘𝑚/𝑠)

−90.80

0.80

And the Kinematics parameters of this system with known radial velocities: Table 2-22: The Kinematics parameters of Hip72479 in (𝑘𝑚/𝑠) (BartKevicius and Gudas, 2001i) 𝑈

𝜎𝑈

𝑉

𝜎𝑉

𝑊

𝜎𝑊

𝑉𝑡𝑜𝑡

𝜎𝑉𝑡𝑜𝑡

𝑉𝑡

𝜎𝑉𝑡

𝑉𝑟𝑜𝑡

−33.32

1.34

−47.72

1.89

−68.84

0.62

102.27

1.10

47.07

1.83

172.28



43

2.3.6 Orbital elements of the system Hip72479: The

system

Hip72479

(ADS9397)

was

discovered

by

Aitken

in1916

(exactly1916.40) at the Lick Observatory. Its orbits had been calculated by van den Bos (1954, 1945, 1964), Eggen (1965, 1967; different orbits using photometrical parallax 0.026 arcsec) (Al-Wardat, 2012). WDS 14492+ 1013(Hip72479) orbits had been calculated by van den Bos (1945, 1954, 1964), Ekenberg (1945), and Eggen (1967 ; two orbits, I and II). The solution was published by Söderhjelm (1999) . Radial velocity measurements have been published by Heintz (1981b) and in the Wilson-Evans-Batten Catalogue (Docobo et al., 2000). We find in Al-Wardat's paper (Al-Wardat, 2012, p. 70973)" Physical Parameters of the Visually Close Binary Systems Hip70973 and Hip72479" the orbital elements of Hip72479 as in the next table: Table 2-23: The orbital elements of Hip72479 (Al-Wardat, 2012, p. 70973) 𝑃(𝑦𝑟)

9.98 ± 0.04

T

1988.059 ± 0.03

e

0.491 ± 0.001

a (arcsec)

0.127 ± 0.001

i (deg)

45.8 ± 2.0

Ω (deg)

142.3 ± 2.0

ω(𝑑𝑒𝑔)

156.8 ± 3.0

These elements were set by Docobo in 2000, but I don't find a published orbit for these elements, that the published orbit in the Sixth Catalogue for Orbits of Binary Systems was by Söderhjelm in 1999 but without errors, these orbital elements are in the next table(Table 2-24):


44


Table 2-24: Orbital elements of Hip72479 by Söderhjelm1999 (“Sixth Orbit Catalog,” 2018) Orbital Element P (yr) T (yr) e a (arcsec) Ω (degree) ω (degree) i (degree)

Söderhjelm 1999 9.97 1988.2 0.51 0.122 143.0 338.0 40.0

In this study we will modify these elements based on the new observations after 1999 which published in Fourth Catalogue of Interferometric Measurements of Binary Stars.

2.3.7

Data From Fourth Catalogue of Hip72479: From Fourth Catalogue of Interferometric Measurements of Binary Stars we find

these data about our system Hip72479: Table 2-25: The data of the observations of the system Hip72479 in Fourth Catalogue (“Fourth Interferometric Catalogue - RA 14 hr,” 2017b) Epoch of observation

Theta(θ) (mas)


Rho(ρ) (pc)

Error in Ro(Δρ) (pc)


source

1984.1866

146

2.2

0.166

0.013

2

Bnu1986

S

1984.1974

147.5

2.2

0.167

0.013

2

Bnu1986

S

1985.1862

150.2

.

0.157

.

6

Bag1987

S

1985.4895

153.5

.

0.154

.

3.8

McA1987c

Sc

1985.4978

153

.

0.15

.

3.8

McA1987c

Sc

1985.5226

153.2

.

0.151

.

3.6

McA1987a

Sc

1986.407

173.9

.

0.109

.

3.8

Hrt2000a

Sc

1986.4076

174.5

.

0.117

.

6

Bag1989a

S

1987.3801

220.2

.

0.065

.

6

Bag1989a

S


45


1987.3801

220.2

.

0.065

.

6

Bag1991b

S

1989.2274

212

.

0.078

.

3.8

McA1990

Sc

1989.2301

212.8

.

0.076

.

3.8

McA1990

Sc

1989.3063

212.8

.

0.076

.

4

McA1990

Sc

1990.2732

260.5

.

0.11

.

3.8

Hrt1992b

Sc

1990.276

258

.

0.114

.

3.8

Hrt1992b

Sc

1991.25

288

.

0.155

.

0.3

HIP1997a

Hh

1991.319

286.8

.

0.143

.

3.8

Hrt1994

Sc

1991.3297

283.4

.

0.148

.

3.8

Hrt1994

Sc

1992.3074

296.2

.

0.173

.

3.8

Hrt1994

Sc

1993.1974

307.1

.

0.182

.

3.8

Hrt1994

Sc

1995.4365

:332.1

.

: 0.151

.

2.5

Hrt1997

Sc

1997.3939

234

2

0.059

0.003

6

Bag1999a

S

2001.2709

103.9

0.5

0.144

0.003

6

Bag2006b

S

2001.2709

103.8

0.5

0.144

0.003

6

Bag2006b

S

2001.2709

103.7

0.5

0.145

0.003

6

Bag2006b

S

316.4

0.7

0.207

0.003

1.55

Hrt2008

Su

2007.4844

.

.

.

.

6

Doc2010h

S

2008.443

.

.

.

.

0.7

Gii2012

S

2008.4716

159.4

2.2

0.065

0.003

3.5

Hor2012a

S

2008.5476

162.9

0.6

0.069

0.0008

4

Tok2010

S

2008.5476

159.7

0.7

0.055

0.0004

4

Tok2010

S

2009.2629

214.2

0.3

0.073

0.0005

4.1

Tok2010

S

2009.2629

214.1

0.2

0.069

0.0004

4.1

Tok2010

S

2014.1859

316.9

0.2

0.182

0.0001

4.2

Tok2015c

St

2014.2244

137.6

.

0.177

.

4.3

Hor2015b

S

2014.2244

137.7

.

0.178

.

4.3

Hor2015b

S

2015.3381

330.4

0.4

0.158

0.0002

4.1

Tok2016a

St

2004.196

Gray-shaded observations are the new observations that are incorporated into the modified orbit construction for the first time, and on which the orbital elements of the system orbit are modified. M.Sc. Thesis • Yamam Al-Tawalbeh •Department of Physics • AABU University • 2017


2.4

46

Goals of the study:

The main goal of this study is to get the complete set of the orbital of the selected CVBSs Hip11352, Hip70973, and Hip72479. Which will enable us to determine their masses in a precis way and to come closer to the understand of the formation and evolution of these systems and CVBS in general. The systems have been selected according to the following requirements: 1- The system has a separation less than 0.5 arcsec (where systems of higher separation usually studied by other simple techniques). 2- The system has new relative position measurements that can be used to modify its orbit and orbital elements.



47

Chapter 3 Theoretical Background and Methodology

3

Introduction In this chapter, we are going to describe the Two-body proplem which related to the

central force between two bodies rotating around their center of mass . We define the binary system as a system consisting of two stars rotating around a common center of mass on the line connecting them, and under the influence of the gravitational force between them on the same line (central force). This type of problems (two-body problem) is very important in physics, especially in studying two major areas of physics: • Certain two-body nuclear interactions: the scattering of particles by nuclei . . . • Celestial bodies: planets, moons, comets, binary stars ”double stars” . . . Johannes Kepler (1571 – 1630) put three laws resulting from his studying of the motion of the planets of our solar system which was known as Kepler's laws (“Johannes Kepler,” 2018): 1. The orbit of the planet around the Sun is elliptical and the sun is located in one of its foci. 2. The line between the planet and the Sun sweeps equal areas during equal intervals of time. This means that the orbital velocity of the planet increases as the planet approaches from the Sun (at the periastron) and decreases as it moves away (at the apastron). M.Sc. Thesis • Yamam Al-Tawalbeh •Department of Physics • AABU University • 2017

48


3. The square of the periodic time of a planet is proportional to the cube of the semimajor axis of its orbit around the Sun.

3.1

The Two-Body Problem: The two-body problem is a problem of two bodies rotating around their center of mass

under influence of central force like gravity between two masses. To find the motion of two bodies under central force, we can use the Lagrangian, which can be expressed as:

ℒ=

1 2

𝑚1 𝑣21 +

1 2

𝑚2 𝑣22 +

𝐺𝑚1 𝑚2

| 𝑟2 − 𝑟1 |

(3-1)

Figure 3-1: Barycenter coordinate description of a binary system (Binacquista, 2013). We choose a barycentric coordinate system, so that: 𝑚1 𝑟1 + 𝑚2 𝑟2 = 0

(3-2)

and therefore : 𝑚1 𝑟1 = 𝑚2 𝑟2 M.Sc. Thesis • Yamam Al-Tawalbeh •Department of Physics • AABU University • 2017

(3-3)

49


We define : 𝑟 = 𝑟1 + 𝑟2

(3-4)

and we can get from these two equations:

𝑟1 =

𝑚2 𝑟, M

(3-5)

𝑟2 =

𝑚1 𝑟, M

(3-6)

Where 𝑀 = 𝑚1 + 𝑚2 Since the orbits are at one level, we have: m2 2 v12 = r12̇ + r12 θ12̇ = ( ) (r 2̇ + r 2 θ2̇ ) M

(3-7)

m1 2 v22 = r22̇ + r22 θ22̇ = ( ) (r 2̇ + r 2 θ2̇ ) M

(3-8)

When we put the two previous quantities in Lagrange equation:

ℒ=

1 m1 m22 2̇ 1 m2 m12 2̇ Gm1 m2 2 2̇ (r + r θ ) + (r + r 2 θ2̇ ) + 2 2 2 M 2 M r

=

1 m1 m2 2̇ 1 m1 m2 2 2̇ Gm1 m2 M r + r θ + 2 M 2 M Mr

=

1 2̇ 1 GμM μr + μr 2 θ2̇ + 2 2 r


(3-9)


50

Since ℒ is independent of θ, we have:

d ∂ℒ ∂ℒ − =0 dt ∂θ̇ ∂θ ∂ℒ = μr 2 θ̇ = Pθ = angular momentum ̇ ∂θ

(3-10)

We know that the total energy is conserved, and is given by the following relationship: 1 1 Gm1 m2 1 GμM m1 v12 + m2 v22 − = μ(ṙ 2 + r 2 θ2̇ ) − = 𝐸 2 2 r 2 r

(3-11)

but

θ2̇ =

Pθ 2 μ2 r 4

(3-12)

So 1 2̇ 1 Pθ 2 GμM E = μr + − 2 2 μr 2 r

(3-13)

(Binacquista, 2013)



3.2

51

The orbital shape: To determine the shape of the orbit, we use Eq (3-13) to obtain the time dependence of

the radius of the orbit, and then we can obtain the time dependence of the orbital angle. We first make the variable substitution (q = 1/r), so that: dq dθ

= q́ = −

ṙ =

1 dr

dr

r2 dθ

dθ

= −r 2 q́

dr Pθ Pθ θ̇ = −r 2 q́ = − q́ dθ μr 2 μ

(3-14)

(3-15)

Substitution of Eq. (3-15) into Eq. (3-13) we gain: Pθ 2 ́2 Pθ 2 2 E= q + q − GμMq 2μ 2μ

(3-16)

Let 𝓀 = 𝑃𝜃 2 ⁄𝐺𝜇 2 𝑀 , so that 𝑃𝜃 2 ⁄𝜇 = 𝓀𝐺𝜇𝑀 , and

𝐸=

1 1 𝐺𝑀𝜇𝓀𝑞́2 + 𝐺𝑀𝜇𝓀𝑞 2 − 𝐺𝜇𝑀𝑞 2 2

(3-18)

Then we can obtain 2𝐸𝓀 𝓀2 𝑞́2 + 𝓀2 𝑞 2 − 2𝓀𝑞 + 1 = +1 𝐺𝜇𝑀


(3-19)


We define

𝑒2 =

2𝐸𝓀 +1 𝐺𝜇𝑀

(3-20)

and make 𝑥 = 𝓀𝑞 − 1 , then we have 𝑥́2 + 𝑥 2 = 𝑒 2

(3-21)

𝑥́ = √𝑒 2 − 𝑥 2

(3-22)

by integration 𝑥

∫

𝑑𝑥

𝑥0 √𝑒

2

− 𝑥2

𝜃

= ∫ 𝑑𝜃

(3-23)

𝜃0

𝑥 𝑥0 sin−1 ( ) − sin−1 ( ) = 𝜃 − 𝜃0 𝑒 𝑒

(3-24)

|x| ≤ |e| in order for the arcsin to make any sense. For 𝜃0 = 0 and x(0) = e we obtain 𝑥(0) 𝑥0 sin−1 ( ) − sin−1 ( ) = 0 𝑒 𝑒

(3-25)

𝑥0 𝜋 sin−1 ( ) = sin−1(1) = 𝑒 2

(3-26)

𝑥 = 𝑠𝑖𝑛(𝜃 + 𝜋⁄2) = cos 𝜃 𝑒

(3-27)

𝑥 = 𝑒 cos 𝜃


(3-28)

52


53

Then

𝑟=

𝓀 (1 + 𝑒𝑐𝑜𝑠𝜃)

(3-29)

which is the equation of ellipse. We conclude from this that the shape of the orbit is an ellipse, has a point of periastron at θ =0, and a point of apastron at θ =𝜋. And know we can define the semimajor axis (a) of an ellipse as half of the long axis, which is also half the sum of the minimum distance and the maximum distance (periastron and apastron). Thus,

Figure 3-2: Elliptical orbit with the origin centered on one star (Binacquista, 2013).

𝑟𝑚𝑖𝑛 = 𝑟(0) =

𝓀 (1 + 𝑒)

𝑟𝑚𝑎𝑥 = 𝑟(𝜋) =

𝓀 (1 − 𝑒)


(3-30)

(3-31)


54

and

𝑎=

1 𝓀 (𝑟𝑚𝑖𝑛 + 𝑟max ) = (1 − 𝑒 2 ) 2

(3-32)

𝓀 = 𝑎(1 − 𝑒 2 )

(3-33)

The periastron and apastron can now be expressed in terms of the semimajor axis as 𝑟𝑚𝑖𝑛 = 𝑎(1 − 𝑒),

(3-34)

𝑟𝑚𝑎𝑥 = 𝑎(1 + 𝑒),

(3-35)

The value of e is found to be the eccentricity of the elliptical orbit (Figure 3-2), that we will show the total energy of this system is less than zero( E ˂ 0 ), and when we refer to Eq(320), we find the eccentricity (0 ˂ e ˂1), and this value is of the elliptical orbit. (Binacquista, 2013)

3.3

Time-Dependent Orbits: We need to determine the position of the binary system components as a function of

time knowing the so-called Kepler equation, as our access to this equation enables us to measure the orbit and determine the orbital speed of the system. To derive this equation, it is necessary to study the geometry of an ellipse. Firstly if we take an ellipse with semimajor axis (a) that is surrounded by a circle of radius (a), as shown in Figure 3-3. From the figure, we can define the following straight pieces and angles: OΠ= 𝑎 = semimajor axis ,

SΠ = 𝑎(1−e) = periastron ,

OS = ae



Figure 3-3: Properties of an ellipse (Binacquista, 2013). and • The angle θ is called the true anomaly. • The angle E is called the eccentric anomaly. We want to find the time dependence of the eccentric anomaly, E. The auxiliary circle has the property that 𝑃𝑅 ⁄𝑄𝑅 = 𝑏⁄𝑎 = √1 − 𝑒 2 , so 𝑟 cos 𝜃 = −𝑅𝑆 = 𝑂𝑅 − 𝑂𝑆 = 𝑎 cos 𝐸 − 𝑎𝑒,

𝑟 sin 𝜃 = 𝑃𝑅 = (√1 − 𝑒 2 ) 𝑄𝑅 = 𝑎 sin 𝐸 √1 − 𝑒 2 ,

(3-36) (3-37)

And 𝑟 = √𝑟 2 (cos 𝜃)2 + 𝑟 2 (sin 𝜃)2

= √𝑎2 𝑒 2 − 2𝑎2 𝑒cos𝐸 + 𝑎2 (cos 𝐸)2 + 𝑎2 (sin 𝐸)2 − 𝑎2 𝑒 2 (sin 𝐸)2


55


= √𝑎2 𝑒 2 (1 − (𝑠𝑖𝑛 𝐸)2 ) − 2𝑎2 𝑒𝑐𝑜𝑠𝐸 + 𝑎2

= 𝑎√𝑒 2 (𝑐𝑜𝑠 𝐸)2 − 2𝑒𝑐𝑜𝑠𝐸 + 1 = 𝑎(1 − 𝑒𝑐𝑜𝑠𝐸).

(3-38)

We use the equation for angular momentum per mass: 𝑟 2 𝑑𝜃 = 𝐿 𝑑𝑡

(3-39)

where L = 𝑃𝜃 ⁄𝜇 , and by noting that : 𝑑 𝑑𝐸

sinθ = cos 𝜃

𝑑𝜃 𝑑𝐸

(3-40)

We obtain an equation for θ by using:

sin 𝜃 =

=

𝑎 sin 𝐸 √1 − 𝑒 2 𝑟

𝑎 sin 𝐸 𝑏 𝑎(1 − 𝑒𝑐𝑜𝑠𝐸) 𝑎

=

𝑏 sin 𝐸 𝑎(1 − 𝑒𝑐𝑜𝑠𝐸)

(3-41)

and differentiating with respect to E gives d 𝑏 cos 𝐸 − 𝑒 (sin θ) = d𝐸 𝑎 (1 − 𝑒𝑐𝑜𝑠𝐸)2


(3-42)

56


so that cos θ 𝑑𝜃 =

𝑏 (cos 𝐸 − 𝑒)𝑑𝐸 . 𝑎(1 − 𝑒𝑐𝑜𝑠𝐸)2

(3-43)

Now using −𝑎(𝑒 − cos 𝐸) −(𝑒 − cos 𝐸) = 𝑟 (1 − 𝑒 cos 𝐸)

(3-44)

(1 − 𝑒𝑐𝑜𝑠𝐸) (cos 𝐸 − 𝑒)𝑏 𝑏 𝑑𝐸 𝑑𝐸 = 2 (𝑒 − cos 𝐸) (1 − 𝑒𝑐𝑜𝑠𝐸) 𝑎 𝑎(1 − 𝑒𝑐𝑜𝑠𝐸)

(3-45)

cos θ =

We find that

dθ = −

Finally, we have

𝑎2 (1 − 𝑒𝑐𝑜𝑠 𝐸)2

𝑏 𝑑𝐸 = 𝐿𝑑𝑡 𝑎(1 − 𝑒𝑐𝑜𝑠𝐸)

(3-46)

𝐿 𝑑𝑡 𝑎𝑏

(3-47)

𝐿 ∫ 𝑑𝑡 𝑎𝑏

(3-48)

or

(1 − 𝑒𝑐𝑜𝑠𝐸)𝑑𝐸 =

Integrating this equation gives

∫(1 − 𝑒𝑐𝑜𝑠𝐸)𝑑𝐸 =

𝐸 − 𝑒 𝑠𝑖𝑛𝐸 =

𝐿 𝑡+𝑘 𝑎𝑏


(3-49)

57


58

To determine the integration constant, we define T to be the time at periastron passage and we note that E = 0 at periastron, so

𝑘= −

𝐿 𝑇 𝑎𝑏

(3-50)

𝐿 (𝑡 − 𝑇) 𝑎𝑏

(3-51)

and


From Kepler’s second law, we have

2𝜋

∫ 0

1 2

1

𝑟 2 𝑑𝜃 = 𝑑𝐴 = 2 𝐿𝑑𝑡, so

1 2 1 𝑟 𝑑𝜃 = 𝜋𝑎𝑏 = 𝐿𝑃 2 2

(3-52)

𝐿 2𝜋 = =𝜔 𝑎𝑏 𝑃

(3-53)

where P is the period of the elliptical motion, and then


2𝜋 (𝑡 − 𝑇) 𝑃

(3-54)

This equation is generally solved using numerical techniques. (Binacquista, 2013)



3.4

59

The Orbital Elements: The system's orbital elements can determine the orientation of the system, since the

binary system does not fall into one level (the two levels of the stars are aligned at an angle). These elements are determined by angular momentum vector and total energy of the orbit. To determine the axes z and x in the orbital plane of the binary system, we can describe the binary rotation using the direction of the total angular momentum vector and the perigee(priastron) direction, respectively. A coordinate system is determined by the tangent plane of the celestial sphere at the location of the binary, to measure the previous directions for it. A Cartesian coordinates are determined by the line of sight from the observer to the binary, and the tangent to a great circle that connects the binary to the northern celestial pole. The angle of inclination is then defined as the angle between the orbital plane and the tangent plane of the celestial sphere. The ascending node (N) is the line defined by the intersection of the plane of the orbit and the tangent plane and points in the direction where the binary passes from inside the celestial sphere to outside the celestial sphere. Figure 3-4 shows the orientation of the orbit relative to the tangent plane and the three angles that define this orientation. These three angles are:

Angle of inclination

Longitude of the ascending node Ω

i

Longitude of the periastron ω

The shape of the orbit is then given by three quantities: Semimajor axis a

Eccentricity e

Time of periastron T



60

Figure 3-4: the different angles for the orbital elements of a binary system (Binacquista, 2013).

Figure 3-5: The leaning orbit is the true relative orbit of the system in space. The horizontal orbit representing the relative apparent orbit as seen in sky (“Taitus Software | Orbital elements,” 2018) These six quantities are called the orbital elements. If the orbital elements can be measured, then the masses of the binary can be determined. The orbit will always appear to M.Sc. Thesis • Yamam Al-Tawalbeh •Department of Physics • AABU University • 2017


61

be an ellipse when viewed on the sky, but unless i = 0, the center of mass of the system will not lie at the focus of this apparent ellipse . The angular momentum and total energy are also related to the orbital period and orbital shape. To obtain these relations we begin by noting that the kinetic energy is

𝐾=

1 1 1 𝑚1 𝑣12 + 𝑚2 𝑣22 = 𝜇𝑣 2 2 2 2

(3-55)

where 𝑣 2 = 𝑟 2̇ + 𝑟 2 𝜃 2̇ and 𝑟 and 𝜃 are relative separation variables. Now, using 𝑟 = 𝓀⁄(1 + 𝑒 cos 𝜃), we find that 𝑟2 𝐿 𝑒 sin 𝜃 = 𝑒 𝑠𝑖𝑛 𝜃 𝓀 𝓀

(3-56)

𝑟 2 𝜃̇ 𝐿 𝐿 = = (1 + 𝑒 cos 𝜃) 𝑟 𝑟 𝓀

(3-57)

𝑟̇ = 𝜃̇ and 𝑟 𝜃̇ = From here we get

𝐿 2 2 𝑣 = ( ) [𝑒 (sin 𝜃)2 + 1 + 2𝑒 cos 𝜃 + 𝑒 2 (cos 𝜃)2 ] 𝓀 2

𝐿 2 2 𝐿 2 2 = ( ) [𝑒 + 1 + 2𝑒 cos 𝜃)] = ( ) [𝑒 − 1 + 2(1 + 𝑒 cos 𝜃)] 𝓀 𝓀 𝐿 2 2𝓀 𝐿 2 2 (1 − 𝑒 2 ) 2 = ( ) [ − (1 − 𝑒 )] = [ − ] 𝓀 𝑟 𝓀 𝑟 𝓀 𝐿2 2 1 = [ − ] 𝓀 𝑟 𝑎


(3-58)


where we have used 𝑎 = 𝓀/(1 − 𝑒 2 ) in the last step. Now from Kepler’s second law, we have 𝐿 = 2𝜋𝑎𝑏/𝑃 , where P is the orbital period. Noting that b2 = a2(1−e2) we find 4𝜋 2 𝑎2 𝑏 2 4𝜋 2 𝑎3 𝐿 = = 𝑎(1 − 𝑒 2 ) 𝑃2 𝑃2 2

= 𝐺𝑀𝑎(1 − 𝑒 2 ) = 𝐺𝑀𝓀

(3-59)

where we have used Kepler’s third law. Finally, we have 2 1 𝑣 2 = 𝐺𝑀 [ − ] , 𝑟 𝑎

(3-60)

and so the kinetic energy is 𝐾=

1 2 1 𝑚1 𝑚2 2 1 𝜇𝑣 = 𝐺𝑀 [ − ] 2 2 𝑀 𝑟 𝑎

=

𝐺𝑚1 𝑚2 𝐺𝑚1 𝑚2 − 𝑟 2𝑎

(3-61)

Now, the potential energy is PE = − Gm1m2 / r, so the total energy is 𝐸 = 𝐾 + 𝑃𝐸 = −

𝐺𝑚1 𝑚2 2𝑎

𝐿2 = 𝐺𝑀𝑎(1 − 𝑒 2 ) This gives: 𝑃𝜃 =

1 (𝑚 𝑚2 + 𝑚2 𝑚12 )√𝐺𝑀𝑎(1 − 𝑒 2 ) 𝑀2 1 2


(3-62)

(3-63)

62


63

𝐺𝑎(1 − 𝑒 2 ) √ = 𝑚1 𝑚2 𝑀

=

2𝜋 𝑚1 𝑚2 𝑎2 √(1 − 𝑒 2 ) 𝑃 𝑀

(3-64)

Thus, the total energy is fixed by the masses and the semimajor axis, while the total angular momentum also depends upon the period and the eccentricity (Binacquista, 2013).



64

Chapter 4 Procedures and Computational Work

4

Introduction As we said at the beginning of this thesis that we are going to modify the orbital

elements of the systems Hip11352, Hip70793, and Hip72479 by using the Least square method depending on the observations of Fourth Catalogue and comparing that with the orbital drawing in Sixth Catalogue for every system. It was pointed out that the software algorithm used to construct the modified orbit was based on the theoretical basis of Kepler's laws and the two-body problem described in the preceding chapter. When we go to draw the new orbit of any system we will do that by using IDL program, so firstly we will give a brief introduction of this program , its benefits, and how it works.

4.1

What is IDL program? IDL is an abbreviation for (Interactive Data Language) is an integrated computing

environment for the interactive analysis and visualization of data. The program combines powerful and advanced language with many graphical display techniques and analysis techniques. This program summarizes the long time and effort when compared to other computer languages such as Fortran and C++, where in hours can accomplish what needs days or months in traditional languages. Users can interactively explore data in the form of commands using IDL, and then create complete applications through programs under IDL (“idl_basics.pdf,” n.d.).



4.2

65

Benefits of IDL:

1. IDL is a complete, structured language that can be used both interactively and to create sophisticated functions, procedures, and applications. 2. Operators and functions work on entire arrays (without using loops), simplifying interactive analysis and reducing programming time. 3. Immediate compilation and execution of IDL commands provides instant feedback and “hands-on” interaction. 4. Rapid 2D plotting, multi-dimensional plotting, volume visualization, image display, and animation allow you to observe the results of your computations immediately. 5. Many numerical and statistical analysis routines—including Numerical Recipes routines—are provided for analysis and simulation of data. 6. IDL’s flexible input/output facilities allow you to read any type of custom data format. Support is also provided for common image standards (including BMP, GIF, JPEG, and XWD) and scientific data formats (CDF, HDF, and NetCDF). 7. IDL programs run the same across all supported platforms (Unix, VMS, Microsoft Windows, and Macintosh systems) with little or no modification. This application portability allows you to easily support a variety of computers. 8. Existing FORTRAN and C routines can be dynamically linked into IDL to add specialized functionality. Alternatively, C and FORTRAN programs can call IDL routines as a subroutine library or display “engine” (“idl_basics.pdf,” n.d.).



4.3

66

What is Origin Software? Origin is an interactive computer program for interactive scientific graphs and data

analysis. This program was produced by Origen Lab Corporation, and works on Microsoft Windows. Origin supports 2D and 3D graphics support, and can import data files in different formats such as ASCII text, Excel, NI TDM, DIADem, NetCDF, SPC, etc, and export graphics to different image file formats such as JPEG, GIF, EPS, TIFF, etc (“Origin (software),” 2018).

4.4 Fourth Catalogue of Interferometric Measurements of Binary Stars: This Catalogue began at the Georgia State University Center for High Angular Resolution Astronomy (CHARA) in 1982. The aim of this Catalogue was to detect binary star observations using the speckle interferometry technique in the patches by the group's speckle camera. The Catalogue did not stop at that point, but it soon developed its objective and its observations to include other efforts which included all the astronomical and optical data of binary stars obtained by other angular precision techniques. Later results of surveys based on infrared techniques or imaging techniques were added. In the 1980s, two printed editions of this Catalogue were published(McAlister & Hartkopf 1984, 1988), and web versions have been available since the early 1990s (“Fourth Interferometric Catalogue,” 2018).



4.5

Sixth Catalogue Binary Stars:

of

Orbits

of

The Sixth Catalog of Orbits

of

Visual Binary Stars

67

Visual is

a

catalogue for

publication the binary orbits published in the Fourth Catalog. The Sixth Catalog includes the orbital elements of each binary system and the name of the astronomer that deployed the orbit and the year of deployment of the orbit. It also includes an ephemerids of each star for its projected future movement based on its orbital motion and orbital elements. The catalog contains important notes and information about each system (“Sixth Orbit Catalogue,” 2018).

4.6

Algorithm:

4.6.1 Start: IDL works on the Windows system without any problems or complications, and without the need to start with certain commands such as programs running on DOS. After running the program

icon and selecting

(Workbench) we run the program

(orbitwithaxis.pro), then we make compilation for the program to start running programs written to draw the orbits of the systems under study.

4.6.2 Input: Input files entered for drawing in the edit program are written in advance on Notebad, where the orbital elements of the star system are inserted into specific fields, and then the observations are entered from the Fourth Catalogue of the binary system we want, Otherwise, the Input file will not be true and acceptable for drawing under IDL, and there are some notes about preparing the Input file to be acceptable for drawing: a) The point that has no observed angle must be deleted from observations and its row. M.Sc. Thesis • Yamam Al-Tawalbeh •Department of Physics • AABU University • 2017


68

b) The columns which we need in the Input file only: Epoch Theta Rho

Error in Rho

I1(type of data observational )

Source of observation

, but remain columns which contain errors in Theta and others must be deleted. c) The observations that do not have an error in Rho in the catalog are placed with the highest error rate of Rho in all observations, and (0.001) was adopted to be the lowest error exists for Rho to any point has an error less than (0.001). To illustrate the previous steps, and the difference between abstract observations of the Fourth Catalogue and the program is ready to draw according to IDL, I enclosed here illustrations figures comparing the abstract observations, and the Input file for one of the three systems under study.

Figure 4-1: The initial form of the program of Hip11352 system on Notepad with abstract observations from the Fourth Catalogue before modification M.Sc. Thesis • Yamam Al-Tawalbeh •Department of Physics • AABU University • 2017


69

Figure 4-2: The Input file of Hip11352 system after modifying according to the previous steps mentioned to be ready to draw on IDL By a simple look at the two previous figures and comparing them, we can distinguish the differences and modifications that have been made as we have explained previously. After preparing the Input file, then we must save it by any name which we choose with (.inp) extension. The first 13 fields of the Input file are filled with the orbital elements of the system as follows: Object: Name of the binary system. RA: Right ascension in term of HH MM SS. Dec: Declination in term of DD MM SS. P: Period In years, the difference between two dates have the same Angle and Ro. M.Sc. Thesis • Yamam Al-Tawalbeh •Department of Physics • AABU University • 2017


70

T: Epoch of passage through periastron in years. e: Eccentricity. a: Semi-major axis. W : Position angle of the line of nodes(degrees). w : Longitude of the periastron(degrees). i : Inclination of the orbit (degrees). K1 : Semi-amplitude of the velocity of the primary in km/s. K2 : Semi-amplitude of the velocity of the secondary in km/s. V◦ : Velocity of the center of mass in km/s.

But the last three elements are not used in our systems in this thesis, because these elements for studying the red and blue shifts of the light curved of the components which doesn't included in our study, so we put before them a star or (c) letter to cancel their effect on drawing, and we can make the same thing(putting (c) letter before the point) for any observed point we need canceling its weight because of its big error or any other reason.

4.6.3 Process: The main purpose of using IDL software is to build assumed orbit for our binary systems depending on data in Fourth Interferometric Catalogue, and determination the orbital elements for the new orbit by using the Method of Least Squares. With this procedure we have reached the main goal of our study by modifying the orbital elements of our visual binary systems and modifying their orbits based on new observations. The errors are calculated as the root mean square (RMS) of the error in observing ρi and θi. M.Sc. Thesis • Yamam Al-Tawalbeh •Department of Physics • AABU University • 2017


71

4.6.4 Output: The IDL software gives three different files as follow: 1.

OUT file containing the new orbital elements calculated with its errors, contains the observations that rank like the order of the observations in the original document, and can be used as a new INPUT file to give an orbit and then make adjustments.

2.

EPH file contains ephemeris calculated from the estimated elements. Specifies the beginning and end of this ephemeris by the user, and determines the number of points he wants to calculate during a certain period.

3.

POS file contains an image of the modified orbit that contains all the old and new points added to the orbit and modified the orbit based on its existence.

4.6.5 End: We can plot the estimated orbit with Origin Pro2016 software which contains advanced options enables us to show the old and new orbits (modified) in one image to compare them, and so the observations are plotted as: x =ρ sinθ y= - ρ cosθ



4.7

72

Procedure: I summarize here all the steps I have taken from the beginning of searching for the

basic information of the star system, until the orbit is drawn and modified, and product results as images of a new orbit and ephemeris: ST1: We look for the star system by its name through Simbad site in the field called search (by identifier), then the site link us to the basic information of the star such as Right Ascension, Declination, parallax of star, the spectral type and others, and there is an image of the system taken by one of the astronomical telescopes. ST2: By clicking on the (display) icon on the same page we reach all the published papers about the system under study, and from these papers we can get all the orbital elements of the system, properties, parameters of its components, and other informations. ST3: Using the Right Ascension (especially the first two digits), we enter the Fourth Catalogue site to the hour the system belongs to. For example, the system Hip11352 has a Right Ascension(02h 26m 09s), and its hour is (02h), after entering it we look for its name with a key (F3) on the keyboard to get its published observations in Fourth Catalogue as explained in the following figure



73

Figure 4-3: Observations of Hip11352 as in Fourth Catalogue Figure 4-3 gives us the observations of Hip11352 as in Fourth Catalogue where the columns are arranged as follow (“Fourth Interferometric Catalogue - RA 2 hr,” 2017): Epoch

θ

σθ

ρ σρ

Data used in other studies

Source of observations

For the other two systems the figures belong to them are:

Figure 4-4: Observations of Hip70973 as in Fourth Catalogue



74

Figure 4-5: Observations of Hip72479 as in Fourth Catalogue ST4: Input file for every system had been developed on Notepad page containing firstly the orbital elements of the system, and then the observations were arranged in columns in the following order. Data that is not needed in our study was excluded and may be used in other studies: Epoch

θ ρ

Error in ρ

I1(type of data observational )

Source of observation

some observed values have been modified by adding or subtracting 180 degrees from the angle to align with the nearby points. The Input file must be with (inp) extension to be acceptable for implementation under IDL . Lastly the ready Input files to be drawn under IDL for systems Hip70973, and Hip72479 are as in the following figures (Figure 4-6, Figure 4-7):



75

Figure 4-6: IDL program for Hip70973

Figure 4-7: IDL program for Hip72479 ST5: When we start IDL program, we choose the Workbench option, then open a file (orbitwithaxis) and compile it. Before we give the graphic command, we have to copy the Input file to (C:\Users\dell\) directory, and then give the command (xorb, 'filename.inp') in the (Command Line) in the program page, then click enter, for example (xorb, M.Sc. Thesis • Yamam Al-Tawalbeh •Department of Physics • AABU University • 2017


76

'HIP70973X.inp'), then the new orbit will be built In a separate screen containing its modified orbital elements and the possibility of fitting on the new orbit based on new observations. Look at these figures which I explained in it the previous steps related to working on IDL:

Figure 4-8: IDL page with compilation the file (orbitwithaxis.pro) Figure 4-8 appears the main page of IDL program with Workbench option, and after open the file (orbitwithaxis.pro) and making compilation to it

Figure 4-9: The command of plotting the orbit Figure 4-9 appears the drawing command of Hip70973 in (Command Line) at the bottom of the program page.



77

Figure 4-10: The screen orbit for Hip70973 system Figure 4-10 appears the screen orbit for Hip70973 after we give the command in (Command Line) at the bottom of the main page. The orbital elements appears beside the image of orbit, if we click (Fit) the program will make simple modifying on the orbit and the orbital elements too, and beside every element we can see errors in these elements. (Save) icon give us a new file containing the new observations and the new orbital elements after fitting(modified observations and orbital elements), and carry the same name of the origin one by (out) extension saved in the same directory(C:\Users\dell\), this new file can be used for building orbit under IDL too. ST6: To save an image of the modified orbit to be included in the Master's thesis or in a scientific paper or similar we click(POS) which appears under The orbit chart in the previous figure (Figure 4-10), then the main page of the program will ask us to enter an



78

orbit file command (orbplot, /ps, speckle=0.02) to insert it into (Command Line) as I explain in the next figure(Figure 4-11):

Figure 4-11: The command of the orbit file as appears in the main page of IDL Then we put the command into (Command Line) and click enter to gain a file containing the image of the orbit as explaining in the next figures(Figure 4-12, Figure 4-13):

Figure 4-12: The command of orbit file in(Command Line)

Figure 4-13: Executing the plotting command by IDL as an image



79

As we see in Figure 4-13, IDL gives us the plotting as an image by (ps) extension with the same name of the file addition to it ( _POS) as it appears in the previous figure (HIP70973X_POS.ps). This file will be found in the same directory(C:\Users\dell\) to put it in any place we want, and the image in this file appears as in the following Figure 4-14:

Figure 4-14: The Image of orbit file of Hip70973 ST7: We can produce ephemeris to every system by click on (analysis) which appears beside the orbit plot in Figure 4-10 and choose (Ephemerid) then the program will ask us to put (Number of points) in (Command Line), after we put the number we need and click enter it will ask us to put(Start time), and after that (End time) to give us expected data for the movement of the system during the period of time we have entered, and the number of points which were provided to the program .



80

For example if we applied these commands on (Hip70973) by 20 points from (2018) to (2028) we will gain data consisting of [Epoch , θ , ρ ] for this period as in the following Figure 4-15:

Figure 4-15: Ephemeris for Hip70973 from(2018) to(2028) by 20 points As we see in Figure 4-15 the data give us three columns the first for (Epoch), the second for (Theta), and the third for (Rho). This data can be used to build a new system file of Hip70973 to apply it under IDL too.



81

ST8: We can also from the list of analysis obtain the total mass of the system in a solar mass unit, by choosing (Mass sum), the program asks us to introduce the parallax to the system in (arcsec) unit, and after clicking enter gives us the sum of the mass system based on the new orbit and its modified orbital elements, look Figure 4-16:

Figure 4-16: Mass sum command for Hip70973 under IDL After these steps we have obtained the data and information we need for the star systems, orbits, and their modified orbital elements.



82

Chapter 5 Results and Conclusion

5 5.1

Results: Kepler's laws and solutions of two-body problem were used to build the orbit of the

binary systems under study, which consisting of two stars orbiting around their center of mass. The published observations in the Fourth Catalogue has been adopted as a reference and basis for orbital construction, which includes the site coordinates of the system (ρ , θ) to obtain the best solution for the modified orbit. Method of The least squares was used to reach the modified orbital elements with the least errors. This study studied the three star systems(Hip11352, Hip70973, Hip72479), all of which are binary systems, in order to modify their orbits and geometrical elements according to modern astronomical observations.

5.1.1 Relative Positional Measurements In Fourth Catalogue: There are 33 relative positional measurements in the Fourth Catalogue of Interferometric Measurements of the system(Hip11352), covering more than one period(376˚), but we used only 26 relative positional measurements of them to built the orbit because the other points do not have an observed angle. For the system(Hip70973) there are 19 relative positional measurements, covering more than one period(384˚), and for the system(Hip72479) there are 37 relative positional



83

measurements, covering more than one period(544˚), but I excluded 2 relative positional measurements of them for the same previous reason. These observations were included as in the Fourth Catalogue in the tables (Table 2-7, Table 2-14, Table 2-25) respectively in Chapter 2. The new observations that were first introduced to modify the orbit in this study were grayed out to distinguish them from the old observations, as follows: The last seven points for the system(Hip11352) from 2007.818 until 2012.6777(which cover about 265˚), but I excluded two points of them have not an observed angle, the last thirteen points of the system(70973) from 2001.2737 until 2014.3032 (which cover about 99˚), and the last fifteen points of the system(72479) from 2001.2709 until 2015.3381 (which cover about 227˚).



Table 5-1: Positional measurements of the system Hip11352 from the Fourth Interferometric Catalogue after modifying Epoch

θ

ρ

1991.25

177

0.133

1997.8242

186.8

0.122

1998.7747

202.9

0.127

1998.7773

202.7

0.126

1998.9246

205.8

0.124

1999.813

228.3

0.0965

1999.8856

230.1*

0.092

2000.7622

283.7*

0.062

2000.8757

293

0.057

2001.7528

4.2

0.07

2001.7622

4.2

0.07

2001.7643

8*

0.07

2002.7993

85.5

0.059

2002.7993

87.6

0.059

2003.629

153.9

0.077

2003.7885

158

0.091

2003.9249

163.5

0.095

2003.9249

163.2

0.094

2003.9249

162.8

0.096

2003.9249

163.1

0.095

2004.8157

185.4

0.124

2007.8256

296.7

0.059

2010.0074

111.1

0.0579

2010.72

156.4

0.0876

2010.72

156.4

0.0885

2012.6777

202.7

0.124

* These points were modified by 180◦ to become consistent with the nearby points.


84


Table 5-2: Positional measurements of the system Hip70973 from the Fourth Interferometric Catalogue after modifying Epoch

θ

ρ

1989.309

10.9

0.255

1990.344

19.6

0.219

1991.25

29

0.187

1992.457

68.3

0.102

1993.091

106.7

0.082

1997.394

241.2

0.177

2001.274

295.1

0.218

2001.274

295.1

0.216

2001.498

296.6*

0.217

2001.498

296.9*

0.219

2002.255

305.8

0.229

2002.326

306.3

0.23

2002.326

306.4

0.23

2004.196

320.5

0.267

2006.438

335

0.292

2007.317

340.4

0.291

2008.545

347.3

0.302

2008.545

347.3

0.302

2014.303

34.3

0.164

* These points were modified by 180◦ to become consistent with the nearby points.


85


Table 5-3: Positional measurements of the system Hip72479 from the Fourth Interferometric Catalogue after modifying Epoch 1984.187 1984.197 1985.186 1985.49 1985.498 1985.523 1986.407 1986.408 1987.380 1987.380 1989.227 1989.23 1989.306 1990.273 1990.276 1991.25 1991.319 1991.33 1992.307 1993.197 1995.437 1997.394 2001.271 2001.271 2001.271 2004.196 2008.472 2008.548 2008.548 2009.263 2009.263 2014.186 2014.224 2014.224 2015.338

θ 146 147.5 150.2 153.5 153 153.2 173.9 174.5 220.2 220.2 32.0* 32.8* 32.8* 80.5* 78* 108* 106.8* 103.4* 116.2* 127.1* 152.1* 234 103.9 103.8 103.7 136.4* 349.4* 342.9* 349.7* 34.2* 34.1* 136.9* 137.6 137.7 150.4*

ρ 0.166 0.167 0.157 0.154 0.15 0.151 0.109 0.117 0.065 0.065 0.078 0.076 0.076 0.11 0.114 0.155 0.143 0.148 0.173 0.182 0.151 0.059 0.144 0.144 0.145 0.207 0.065 0.069 0.056 0.074 0.069 0.183 0.178 0.179 0.158

* These points were modified by 180◦ to become consistent with the nearby points. M.Sc. Thesis • Yamam Al-Tawalbeh •Department of Physics • AABU University • 2017

86


87

5.1.2 Modified Orbits and Comparison with Orbits Published In The Sixth Catalogue: Now I will list the modified orbits that were built based on the new observations in the previous tables(Table 5-1, Table 5-2, Table 5-3) using Origin2016 software:

Figure 5-1: The modified orbit of Hip11352 by using the new observations(this work)

Figure 5-2: The modified orbit of Hip70973 by using the new observations(this work) M.Sc. Thesis • Yamam Al-Tawalbeh •Department of Physics • AABU University • 2017


88

Figure 5-3: The modified orbit of Hip72479 by using the new observations(this work)

For comparison, I include the images of the orbits built and published in the Sixth Catalogue, who built every orbit of these, and the date of its publishing:

Figure 5-4: Image of Hip11352 orbit by Balega published in Sixth Catalogue at 2005(Balega, 2005) M.Sc. Thesis • Yamam Al-Tawalbeh •Department of Physics • AABU University • 2017


Figure 5-5: Image of Hip70973 orbit by Docobo published in Sixth Catalogue in 2000(Docobo, 2000)

Figure 5-6: Image of Hip72479 orbit by Soderhjelm published in Sixth Catalogue at 1999(Sod, 1999)


89


90

For Comparison I included in the next figures the new orbit of this work with the published orbit to see the difference between them, and the done modifying with new observations in this work(the orbit by this work presented in a solid line, the other by a dashed line).

Figure 5-7: The built orbit of Hip11352 using the results of this work presented in a solid line, with the orbit built using the results of Balega 2005 presented by a dashed line.

Figure 5-8: The built orbit of Hip70973 using the results of this work presented in a solid line, with the built orbit using the results of Docobo 2000 presented by a dashed line. M.Sc. Thesis • Yamam Al-Tawalbeh •Department of Physics • AABU University • 2017


91

Figure 5-9: The built orbit of Hip72479 using the results of this work presented in a solid line, with the built orbit using the results of Soderhjelm 1999 presented by a dashed line.

5.1.3 Modified Orbital Elements: In this section I will include the modified orbital elements for each system which I built the new orbits in this work based on it, with making comparison with the published orbital elements in Sixth Catalogue before modifying:

5.1.3.1

Modified Orbital Elements of Hip11352:

The orbital elements of this system were published by Balega at 2005 in the Sixth Orbit Catalogue of Binary Stars, and I included the built orbit by Balega in Figure 5-4 in this chapter, before it I included the built orbit by this work in Figure 5-1 to make comparison between them, and the two orbits in the same figure to appear the difference



92

between them in Figure 5-7. In next Table 5-4 I included the orbital elements which were published by Balega(2005), and the modified orbital elements by this work: Table 5-4: Orbital elements of the system Hip11352 built by Balega (“Sixth Orbit Catalog (frames version),” 2018) and this work Orbital Element

Balega 2005

This Work

P (yr)

6.85 ± 0.05

6. 94010 ± 0.01114

T (yr)

1995.12 ± 0.06

1995.0606 ± 0.0111

e

0.284 ± 0.006

0.2895 ± 0.0032

a (arcsec)

0.100 ± 0.001

0.0999 ± 0.0006

Ω (degree)

15.1 ± 0.9

13.65 ± 0.68

ω (degree)

4.4 ± 1.6

7.53 ± 0.79

i (degree)

50.0 ± 0.6

51.07 ± 0.56

As mentioned above, I have used 26 points of the Fourth Catalog (covering about 376˚) with five new points (covering about 265˚). This means that the built orbit contains more than one period of the orbit of the star and will be more accurate using the new observations, that we notice from the previous table the error ratios of the modified orbital elements are less than the error ratios of the previously published elements. The RMS(Root Mean Square) values for theta, rho of all observations used to built the modified orbit are: ( 1.7735 , 0.0015 ) respectively. By using parallax value of Hip11352 from Simbad (𝜋= 22.13 mas), we find mass sum of it: (1.9104459 M⨀ ), but if we excluded the point of Hipparcos(1991.25) the RMS values will decrease to: (1.0943 , 0.0015), and the mass sum of the system will be (1.9032006 M⨀ ). If we use the most recent value of Parallax for this system

in

the latest

release of the recently launched Gaya

Archive

(𝜋 =

22.515589732059386), we will gain the mass sum: (1.8140571 M⨀ ), and we notice from this value that the mass sum decreased by this recent value of parallax. M.Sc. Thesis • Yamam Al-Tawalbeh •Department of Physics • AABU University • 2017


5.1.3.2

93


The orbital elements of this system were published by Docobo at 2000 in the Sixth Orbit Catalogue of Binary Stars, and I included the built orbit by Docobo in Figure 5-5 in this chapter, the built orbit by this work in Figure 5-2 to make comparison between them, and the two orbits in the same figure to appear the difference between them in Figure 5-8. In next Table 5-5 I included the orbital elements which were published by Docobo(2000), and the modified orbital elements by this work: Table 5-5: Orbital elements of the system Hip70973 built by Docobo (“Sixth Orbit Catalog (frames version),” 2018) and this work Orbital Element P (yr) T (yr) e a (arcsec) Ω (degree) ω (degree) i (degree)

Docobo 2000 22.98 ± 0.30 1993.62 ± 0.02 0.499 ± 0.010 0.243 ± 0.002 13.8 ± 2.0 121.0 ± 2.5 49.1 ± 2.0

This Work 22.88803 ± 0.03064 2016.5644 ± 0.0446 0.5010 ± 0.0046 0.2395 ± 0.0013 11.03 ± 0.61 123.98 ± 0.92 50.55 ± 0.37

As mentioned above, I have used 19 points of the Fourth Catalog (covering about 384˚) with five new points (covering about 99˚). This means that the built orbit contains more than one period of the orbit of the star and will be more accurate using the new observations, that we notice from the previous table the error ratios of the modified orbital elements are more less than the error ratios of the previously published elements especially in period(P), inclination(i), longitude of the ascending node(Ω), longitude of the periastron(ω), and eccentricity(e), only error ratio in time of periastron(T) was more than the value published by Docobo, and it does not have much impact on the outcome of the study. By using parallax value of Hip70973 from Simbad (𝜋= 24.31 mas), we find mass M.Sc. Thesis • Yamam Al-Tawalbeh •Department of Physics • AABU University • 2017


94

sum of it: (1.8244165 M⨀ ), but if we use the most recent value of Parallax for this system in the latest release of the recently launched Gaya Archive (𝜋= 24.35404566949619 mas), we will gain the mass sum: (1.8145356 M⨀ ), we notice that the mass sum is more nearly to previous value, and there is no big difference between them. The RMS(Root Mean Square) values for theta, rho of all observations used to built the modified orbit are: ( 0.3948 , 0.0018 ) respectively, these values are small and indicate high accuracy in the study results of the modified orbit.

5.1.3.3


The orbital elements of this system were published by Söderhjelm at 1999 in the Sixth Orbit Catalogue of Binary Stars, and I included the built orbit by Söderhjelm in Figure 5-6 in this chapter, the built orbit by this work in Figure 5-3 to make comparison between them, and the two orbits in the same figure to appear the difference between them in Figure 5-9. In next Table 5-6 I included the orbital elements which were published by Söderhjelm(1999), Docobo(2000) (Al-Wardat, 2012, p. 72479), and the modified orbital elements by this work: Table 5-6: Orbital elements of the system Hip72479 built by Söderhjelm(1999) (“Sixth Orbit Catalog (frames version),” 2018), Docobo(2000), and this work Orbital Element P (yr) T (yr) e a (arcsec) Ω (degree)

Söderhjelm 1999 9.97 1988.2 0.51 0.122 143.0

Docobo 2000 9.98 ± 0.04 1988.059 ± 0.03 0.491 ± 0.001 0.127 ± 0.001 142.3 ± 2.0

This Work 10.05651 ± 0.01520 1987.9919 ± 0.0292 0.5127 ± 0.0037 0.1234 ± 0.0004 143.47 ± 0.95



95

ω (degree) 338.0 156.8 ± 3.0 156.61 ± 1.17 i (degree) 40.0 45.8 ± 2.0 45.48 ± 0.67 As mentioned above, I have used 35 points of the Fourth Catalog (covering about 544˚), with fifteen new points (covering about 227˚). This means that the built orbit contains more than one period of the orbit of the star and will be more accurate using the new observations. Unfortunately, in the sixth catalog I did not find the errors of the orbital elements published by Söderhjelm as we see in the previous table, but there are orbital elements of the same system published with error ratios by Docobo(Al-Wardat, 2012), and I have posted them in the table to compare them with the modified orbital elements of this work. We notice from the previous table that the error ratios of the modified orbital elements are less than the error ratios of the previously published elements by Docobo except eccentricity. The decrease appears to be more pronounced in the error ratios in the elements of period(P), inclination(i), longitude of the ascending node(Ω), and longitude of the periastron(ω). By using parallax value of Hip72479 from Simbad (𝜋 = 22.59 mas), we find mass sum of it: (1.6416592 M⨀ ). The RMS values for theta, rho of all observations used to built the modified orbit are: ( 1.2978 , 0.005 ) respectively.



96

5.1.4 Ephemerides and Mass Sum: The Ephemerides for the three systems based on the new observations, and modified orbital elements are included in the next tables(Table 5-7, Table 5-8, Table 5-9) from 2018 to 2028 with 20 points for every system, and after every table the Mass sum of two stars consisting the binary system in solar mass unit: Table 5-7: Ephemerides of Hip11352 Epoch

θ

ρ

2018.000

169.36

0.1033

2018.526

182.36

0.1205

2019.053

192.83

0.1282

2019.579

202.84

0.1261

2020.105

214.03

0.1147

2020.632

228.85

0.0955

2021.158

252.4

0.0729

2021.684

292.58

0.0585

2022.211

338.51

0.064

2022.737

13.36

0.0711

2023.263

50.09

0.0607

2023.789

102.59

0.0553

2024.316

144.34

0.0744

2024.842

166.43

0.0993

2025.368

180.24

0.118

2025.895

190.98

0.1276

2026.421

200.96

0.1272

2026.947

211.8

0.1174

2027.474

225.68

0.0995

2028.000

247.03

0.077

By using parallax value of Hip11352 from Simbad (𝜋= 22.13 mas), we find mass sum of it: (1.9105425 M⨀ ). The period of this system is 6.93887 years (i.e. 6 years 11 M.Sc. Thesis • Yamam Al-Tawalbeh •Department of Physics • AABU University • 2017


97

months and about 8 days), with periastron passage ( i. e. the two stars reach the closest point) on 1995.0628 (i.e. January 23d, 1995).

Table 5-8: Ephemerides of Hip70973 Epoch

θ

ρ

2018.000

197.95

0.1482

2018.526

209.01

0.1584

2019.053

219.01

0.1643

2019.579

228.46

0.1679

2020.105

237.57

0.1706

2020.632

246.41

0.1732

2021.158

254.95

0.1765

2021.684

263.14

0.1807

2022.211

270.92

0.1859

2022.737

278.24

0.1922

2023.263

285.06

0.1994

2023.789

291.37

0.2073

2024.316

297.21

0.2158

2024.842

302.59

0.2247

2025.368

307.56

0.2338

2025.895

312.16

0.2429

2026.421

316.43

0.2518

2026.947

320.41

0.2603

2027.474

324.14

0.2684

2028.000

327.67

0.276

By using parallax value of Hip70973 from Simbad (𝜋= 24.31 mas), we find mass sum of it: (1.8244165 M⨀ ). The period of this system is 22.88803 years (i.e. 22 years 10 months M.Sc. Thesis • Yamam Al-Tawalbeh •Department of Physics • AABU University • 2017


98

and about 20 days), with periastron passage ( i. e. the two stars reach the closest point) on 2016.5644 (i.e. July 25th, 2016).

Table 5-9: Ephemerides of Hip72479 Epoch

θ

Ρ

2018.000

291.41

0.0543

2018.526

339.23

0.0648

2019.053

18.67

0.0682

2019.579

51.67

0.0789

2020.105

74.58

0.0969

2020.632

89.91

0.1169

2021.158

100.81

0.1358

2021.684

109.18

0.1523

2022.211

116.04

0.1655

2022.737

122.01

0.1751

2023.263

127.48

0.1808

2023.789

132.72

0.1824

2024.316

138

0.1796

2024.842

143.59

0.172

2025.368

149.9

0.1592

2025.895

157.59

0.1408

2026.421

168.12

0.1162

2026.947

185.35

0.0861

2027.474

221.7

0.0568

2028.000

286.36

0.0532

By using parallax value of Hip72479 from Simbad (𝜋 = 22.59 mas), we find mass sum of it: (1.6416592 M⨀ ). The period of this system is 10.05651 years (i.e. 10 years, and about M.Sc. Thesis • Yamam Al-Tawalbeh •Department of Physics • AABU University • 2017


99

20 days), with periastron passage ( i. e. the two stars reach the closest point) on 1987.9919 (i.e. December 28th, 1987).

5.2

Conclusions: The study of binary systems is important to study the formation and evolution of

galaxies and star clusters, especially as they form about half of the star systems in the universe, and thus give us a perception of the development of the universe and the stages of its formation in general. Calculating the orbital elements of binary systems gives us an indirect way of calculating their masses, comparing them with other stars known to us as the Sun. The importance of studying the spatial investigation of the orbital planes of binary systems in star clusters or in some regions of the galaxy stems from the fact that they play an important role in the dynamics of interacting galaxies. The solution of two-body problem and Kepler's laws are used in this study to obtain the modified orbital elements, based on a new observations in Fourth Catalogue of Interferometric Measurements of Binary Stars. In addition to the theoretical basis, IDL program was used to calculate the modified orbital elements of the three binary systems(Hip11352, Hip70973, Hip72479). Origin2016 software was used to draw the modified orbits where included in this study into figures(Figure 5-1, Figure 5-2,Figure 5-3 ) and compare them with the orbits published in the Sixth Catalog of Orbits of Visual Binary Stars where included into figures(Figure 5-4,



100

Figure 5-5, Figure 5-6). The final results of the modified orbital elements of this study were as follows:

Table 5-10: Orbital elements of the binary system Hip11352 of this work Orbital Element

Results of This Work

Period (P (yr))

6. 94010 ± 0.01114

Periastron epoch(T(yr))

1995.0606 ± 0.0111

Eccentricity(e)

0.2895 ± 0.0032

Simi-major axis(a(arcsec))

0.0999 ± 0.0006

Position angle of the line of nodes(Ω(deg))

13.65 ± 0.68

Longitude of periastron (ω(deg))

7.53 ± 0.79

Inclination (i(deg))

51.07 ± 0.56

Table 5-11: Orbital elements of the binary system Hip70973 of this work Orbital Element Period (P (yr)) Periastron epoch(T(yr)) Eccentricity(e) Simi-major axis(a(arcsec)) Position angle of the line of nodes(Ω(deg)) Longitude of periastron (ω(deg)) Inclination (i(deg))

Results of This Work 22.88803 ± 0.03064 2016.5644 ± 0.0446 0.5010 ± 0.0046 0.2395 ± 0.0013 11.03 ± 0.61 123.98 ± 0.92 50.55 ± 0.37

Table 5-12: Orbital elements of the binary system Hip72479 of this work Orbital Element Period (P (yr)) Periastron epoch(T(yr))

Results of This Work 10.05651 ± 0.01520 1987.9919 ± 0.0292



Eccentricity(e) Simi-major axis(a(arcsec)) Position angle of the line of nodes(Ω(deg)) Longitude of periastron (ω(deg)) Inclination (i(deg))

101

0.5127 ± 0.0037 0.1234 ± 0.0004 143.47 ± 0.95 156.61 ± 1.17 45.48 ± 0.67

For system Hip11352 by using parallax (𝜋= 22.13 (mas)) we find mass sum of the system: (1.9105425 M⨀ ), and from the properties of individual components of the system as effective temperature, radius, spectral type, gravity, and the mass of each individual component as stated in Al-Wardat's paper(Al-Wardat, 2009)(see Table 2-1), we conclude that both components are similar to Sun and they are most likely belong to the series of the main sequence of stars in Hertzsprung-Russell diagram(solar-type main sequence star).

For system Hip70973 by using parallax (𝜋= 24.31 (mas)) we find mass sum of the system: (1.8244165 M⨀ ), and from the properties of individual components of the system, and the mass of each individual component as stated in Al-Wardat's paper(Al-Wardat,2012) (see Table 2-8), we conclude that both components are similar to Sun and they are most likely belong to the series of the main sequence of stars in Hertzsprung-Russell diagram (solar-type main sequence star).

For system Hip72479 by using parallax (𝜋= 22.59 (mas)) we find mass sum of the system: (1.6416592 M⨀ ), and from the properties of individual components of the system, and the mass of each individual component as stated in Al-Wardat's paper(Al-Wardat,2012) (see Table 2-15), we conclude that both components are similar to Sun and they are most M.Sc. Thesis • Yamam Al-Tawalbeh •Department of Physics • AABU University • 2017


102

likely belong to the series of the main sequence of stars in Hertzbrung-Russell diagram (solar-type main sequence star).

5.3

Recommendations: To gain better results, and to develop the study of binary system movement, and building more accurate orbits of them, the following recommendations are made: 1- Continuous monitoring of new points of

binary stars movement using

Interferometric measurements, or optical techniques is recommended to build new, more accurate orbits and estimate new dynamical parallax of the system. 2- The development of more accurate telescopic satellite systems for more precise measurements, and modified orbital elements with a very small error rate. 3- Treatment of problems of observations in Fourth Catalogue, which does not contain the error rate in theta or rho, or contains error in the value of theta because of the personal errors in determining directions on the telescope. 4- The continuous search for new ways to study the orbital movement of binary systems, because they are important in understanding the stages of formation and evolution the stars and galaxies and their interaction with each other, and thus form a comprehensive perception of the evolution of the entire universe.



103

5- Using more sophisticated software, easier to use to solve the two-body problem and Kepler's laws, to give more accurate results. 6- Continue to modify the published orbits of binary systems in Fourth Catalogue based on the new observations, and add modified orbits and modified orbital elements to Sixth catalog, to further contribute to this important and influential work in the development of astronomical catalogues in general. 7- Building orbits for binary systems that do not have published orbits, and have observations in the Fourth Catalogue. 8- Suggest improvements and additions to the Fourth and Sixth catalogues to become clearer and easier to use. 9- To provide financial and moral support for scientific research efforts in our universities in the field of astronomical observations in general, and study the movement of binary stars in particular, and to seek to build a scientific observatory in Jordan certified, developed, and able to provide a database and observations of binary systems added to the global catalogues.




104


Appendix A Constants and Astronomical units Astronomical units:

360 degrees (◦) = 24 hour = 2𝜋 radians = circle 1 degree (◦) = 60 arcminutes (ˊ) = 3600 arcseconds ( ˝ ) 1 radian (rad) = 57.296◦ = 206265( ˝ ). 1 arcsecond ( ˝ )= 1000 mas 1 Astronomical Unit (AU) = 1.4960 × 108 Km = 9.2975 ×106 miles. 1 light-year (Ly.) = 9.4605 × 1012 Km = 5.8797 × 1012 miles. 1 parsec (pc) = 3.0856 × 1013 Km = 3.2616 Ly. = 206265 AU.


105


Constants: Sun’s apparent magnitude (m⨀)= -28.8 Sun’s absolute magnitude (Mabs⨀)= 4.8 Sun’s bolometric magnitude (Mbol⨀)=4.8 Sun’s luminosity (L⨀)=3.84 × 1030 Watt Sun’s effective temperature (Teff⨀)=5777 K Sun’s radius (R⨀)=6.961 × 108 m Sun’s mass (M⨀)=1.99 × 1030 Kg Sun’s mass density (ρ⨀)=1.41 × 103 Kgm−3 Sun’s surface gravity (log g⨀) = 4.43 Stefen Boltzmann (σ) =5.670373 × 10−8 W m−2K−4


106


107

Appendix B Method of Least Square The Method of Least Squares is a procedure to determine the best fit line to data; the proof uses simple calculus and linear algebra. The basic problem is to find the best fit straight line y = ax + b given that, for 𝑛 ∈ {1, … … , 𝑁}, the pairs (𝑥𝑛 , 𝑦𝑛 ) are observed. The method easily generalizes to finding the best fit of the form 𝑦1 = 𝑎1 𝑓(𝑥) + … . + 𝑎𝑘 𝑓𝑘 (𝑥) The functions are not necessarily to be linearly in the variable( x), only y to be a linear combination of these functions (Miller, 2010). For data {(𝑥1 , 𝑦1 ) , …….. , (𝑥𝑁 , 𝑦𝑁 )}, we define the error to saying 𝑦 = 𝑎𝑥 + 𝑏 , by 𝑁

𝐸(𝑎, 𝑏) = ∑(𝑦𝑛 − (𝑎𝑥𝑛 + 𝑏))

2

𝑛=1

Note that the error is a function of two variables, no difference here whether we studied the variance or N times the variance, and note that the error is a function of two variables. The purpose is to find values of a and b that minimize the error. This requires us to find values a and b so that: 𝜕𝐸 =0, 𝜕𝑎

𝜕𝐸 =0 𝜕𝑏

Differentiating E(a; b) yields: 𝑁

𝜕𝐸 = ∑ 2(𝑦𝑛 − (𝑎𝑥𝑛 + 𝑏)) ∗ (−𝑥𝑛 ) = 0 𝜕𝑎 𝑛=1



108

𝑁

𝜕𝐸 = ∑ 2(𝑦𝑛 − (𝑎𝑥𝑛 + 𝑏)) ∗ 1 = 0 𝜕𝑏 𝑛=1

We can rewrite these equations as: 𝑁

𝑁

𝑁

∑ 𝑥𝑛 𝑦𝑛 =

∑(𝑥𝑛2 ) 𝑎

𝑛=1

𝑛=1

𝑛=1

𝑁

𝑁

𝑁

+ ∑(𝑥𝑛 ) 𝑏

∑ 𝑦𝑛 = ∑(𝑥𝑛 ) 𝑎 + ∑(1) 𝑏 𝑛=1

𝑛=1

𝑛=1

Then we can satisfy these matrix equations to obtain the values of (a and b): 𝑁

𝑁

∑ 𝑥𝑛 𝑦𝑛 𝑛=1 𝑁

(

=

∑ 𝑦𝑛

)

𝑛=1

𝑁

∑(𝑥𝑛2 )

∑(𝑥𝑛 )

𝑛=1 𝑁

𝑛=1 𝑁

∑(𝑥𝑛 ) (𝑛=1

𝑎 ( ) 𝑏

∑(1) 𝑛=1

)

So, to obtain (a , b): 𝑁

𝑎 ( )= 𝑏

∑𝑁 (𝑥𝑛2 ) 𝑀 = ( 𝑛=1 ∑𝑁 𝑛=1(𝑥𝑛 )

𝑁

∑(𝑥𝑛2 )

∑(𝑥𝑛 )

∑ 𝑥𝑛 𝑦𝑛

𝑛=1 𝑁

𝑛=1 𝑁

𝑛=1 𝑁

∑(𝑥𝑛 ) (𝑛=1

Let

−1

𝑁

∑(1) 𝑛=1

)

(

∑ 𝑦𝑛 𝑛=1

)

∑𝑁 𝑛=1(𝑥𝑛 ) ) , we can easily prove by finding det 𝑀 if all the 𝑥𝑛 are ∑𝑁 𝑛=1(1)

not equal, det 𝑀 will be non-zero and 𝑀 will be invertible, then the values of (a , b) will be obtained by solving a linear system of equations in the last equation. (Miller, 2010)



109

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modified orbital elements of the close visual binary

modified orbital elements of the close visual binary

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