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University of Baghdad College of Science Department of Astronomy and Space Science COMPUTING THE PERTURBATION EFFECTS ON ORBITAL ELEMENTS OF THE MOON A Thesis Submitted to the Department of Astronomy and Space Science College of Science University of Baghdad In partial fulfillment for the requirements of the Degree of Master in Astronomy and Space Science By Hayder Ridha Ali Al-Ali B.Sc. 2004 Supervised by: Assist professor Dr. Abdul-Rahman H.S. Al-Mohammedi 2011 A.D. 1432 H.D. Supervisor Certification I certify that this thesis was prepared by Hayder Ridha Ali under my supervision at the Department of Astronomy and Space, College of Science, University of Baghdad as a partial fulfillment of the requirements needed to award the degree of Master of Science in Astronomy and Space. Signature Name Title Address : : : : Date : Dr. Abdul – Rahman H. S. Assist. Professor Department of Astronomy and Space, College of Science, University of Baghdad / / 2011 Certification of the Head of the Department In view of the available recommendation, I forward this thesis for debate by the examination committee. Signature Name Title Address : : : : Date : Dr. Kamal M. A. Assist Professor Head of Astronomy and Space Department, College of Science, University of Baghdad / / 2011 Examination Committee We, members of the Examining Committee, certify that after reading this thesis and examining the student (Hayder Ridha Ali) in its contents and, in our opinion that it is adequate for the award of the degree of Master of Science in Astronomy and Space. Signature: Signature: Name: Dr. Layth Mahmood Karim Name: Dr. Mohamed Jaafar Al-Bermani Title: Professor Title: Assist Professor Date: Date: / /2011 / /2011 (Member) (Chairman) Signature: Signature: Name: Dr. Salman Zaidan Khalaf Name: Dr. Abdul-Rahman Al-Mohammedi Title: Assist Professor Title: Assist Professor Date: Date: / /2011 (Member) / /2011 (Supervisor) Approved by the University Committee of Graduate studies Signature: Name: Professor Dr. Saleh Mahdi Ali Dean of College of Science Address: University of Baghdad, College of Science Date: / /2011 ﺃﻟَﻢ ﺗَﺮﹶﻭﺍ ﻛَﻴﻒﹶ ﺧﹶﻠَﻖﹶ ﺍﻟﻠﱠﻪ ﺳﹶﺒﻊﹶ ﺳﹶﻤﹶﻮﹶﺍﺕ ﻃﹺﺒﹶﺎﻗًﺎ ١٥ﻭﹶﺟﹶﻌﹶﻞَ ﺍﻟْﻘَﻤﹶﺮﹶ ﻓﹺﻴﻬِﻦُ� ﻮﺭﺍ ﻭﹶﺟﹶﻌﹶﻞَ ﺍﻟﺸﻤﺲﹶ ﺳﹺﺮﹶﺍﺟﺎ ١٦ ﺳﻮرة ﻧﻮح ﺖ ﹺﺑﻬﺎ ﺲ ﺿﻴﺎﺀً ،ﻭ ﺧﹶﻠ ﹾﻘ ﺸ ﻤ ﺖ ﺍﻟ ﺲ ﻭ ﺟ ﻌ ﹾﻠ ﺸ ﻤ ﺖ ﹺﺑﻬﺎ ﺍﻟ » ﻭ ﺧﹶﻠ ﹾﻘ ﺐ ﻭ ﺟ ﻌ ﹾﻠﺘﻬﺎ ﺖ ﹺﺑﻬﺎ ﺍﹾﻟﻜﹶﻮﺍ ﻛ ﺖ ﺍﹾﻟ ﹶﻘ ﻤ ﺮ ﻧﻮﺭﺍﹰ ،ﻭ ﺧﹶﻠ ﹾﻘ ﺍﹾﻟ ﹶﻘ ﻤ ﺮ ﻭ ﺟ ﻌ ﹾﻠ ﻕ ﺖ ﻟﹶﻬﺎ ﻣﺸﺎ ﹺﺭ ﻧﺠﻮﻣﹰﺎ ﻭﺑﺮﻭﺟﹰﺎ ﻭﻣﺼﺎﺑﻴ ﺢ ﻭﺯﻳﻨ ﹰﺔ ﻭ ﺭﺟﻮﻣﺎﹰ ،ﻭ ﺟ ﻌ ﹾﻠ ﺖ ﻟﹶﻬﺎ ﹶﻓﻠﹶﻜﹰﺎ ﻱ ،ﻭ ﺟ ﻌ ﹾﻠ ﺖ ﻟﹶﻬﺎ ﻣﻄﺎﻟ ﻊ ﻭﻣﺠﺎ ﹺﺭ ﺏ ﻭ ﺟ ﻌ ﹾﻠ ﻭﻣﻐﺎ ﹺﺭ ﺖ ﺗﻘﹾﺪﻳﺮﻫﺎ، ﺴﻨ ﻭﻣﺴﺎﺑﹺـ ﺢ ﻭﹶﻗ ﺪ ﺭﺗﻬﺎ ﰲ ﺍﻟﺴﻤﺎ ِﺀ ﻣﻨﺎ ﹺﺯ ﹶﻝ ﹶﻓﺄ ﺣ ﻚ ﹺﺇﺣﺼﺎ ًﺀ ﺼﻴﺘﻬﺎ ﹺﺑﹶﺄﺳﻤﺎﺋ ﺖ ﺗﺼﻮﻳﺮﻫﺎ ﻭﹶﺃ ﺣ ﺴﻨ ﺻ ﻮ ﺭﺗﻬﺎ ﹶﻓﹶﺄ ﺣ ﻭ ﺨ ﺮﺗﻬﺎ ﹺﺑﺴﻠﹾﻄﺎ ﻥ ﺖ ﺗﺪﺑﲑﻫﺎ ﻭ ﺳ ﺴﻨ ﻚ ﺗﺪﺑﲑﹰﺍ ﻓﺄ ﺣ ﺤ ﹾﻜ ﻤﺘ ﻭ ﺩﺑ ﺮﺗﻬﺎ ﹺﺑ ﲔ ﻭﺍﹾﻟﺤﺴﺎﺏﹺ، ﺕ ﻭ ﻋ ﺪ ﺩ ﺍﻟﺴﻨ ﺍﻟﻠﱠﻴ ﹺﻞ ﻭ ﺳ ﹾﻠﻄﺎ ﻥ ﺍﻟﻨﻬﺎ ﹺﺭ ﻭﺍﻟﺴﺎﻋﺎ ﺉ ﻭﺍﺣﺪﹰﺍ «. ﺱ ﻣ ﺮ ﺖ ﺭ ﺅﻳﺘﻬﺎ ﻟﺠﻤﻴ ﹺﻊ ﺍﻟﻨﺎ ﹺ ﻭ ﺟ ﻌ ﹾﻠ l^ÛŠÖ]ð^Â Dedication To My Mother Acknowledgment I am indebted to God for helping me to finish what I started, and for helping me to present this work. I would like to express my thanks to my supervisor Assist. Professor Dr. Abdul-Rahman Hussein for his guidance, encouragement, and valuable advice throughout the period of preparation of this thesis. My deep thanks to the head of the Department of Astronomy and Space, academic staff, colleagues at the College of Science, University of Baghdad for their encouragement, and friends who encouraged me with a piece of advice, invocation of God, or a smile. I would like to express my deep gratitude to my family, specially my wife for their support and patience throughout the tough times. I would like to thank the dean of the College of Science. Finally thanks are due to everyone who contributed in my successful journey throughout my life. Hayder Abstract In this research a study of the equation of Keplerian motion to understand the solution of the two-body problem with and without perturbation for the Moon's orbit as a sample of elliptical orbits using two techniques: The first technique depends on actual formula for the Moon coordinates distance and its variation with time in a month then calculate the components of the position and velocity to calculate momentum component to find the orbital elements, which consider the motion with perturbation. The second technique depends on the solution of Kepler's equation of motion by the eccentric anomaly to calculate the Moon geocentric coordinates and its variation with time then calculate the components of the position and velocity to calculate momentum component after convert them to the Earth plane by inverse Gauss matrix to find the orbital elements, which consider the motion without perturbation. The perturbation was found in each element by comparing the results which calculated from two method above. And its variation with the time for four different months in the year 2010. It can be noted that in all elements varies with time by perturbation. In addition that, this research gave an explanation and discussion for the common forces that perturb an objects orbit such as, the atmosphere drag, non-spherical earth, solar radiation pressure and third body attraction, where the last one is the main effect of the Moon orbital elements. Also, the time of conjunct between the Sun and the Moon was calculated for all months of the year 2010 to determine the start conjunct astronomical month, which can be used to determine the start of Hegree months. CONTENTS Acknowledgment Abstract Contents List Of Figures List Of Tables List Of Symbols Page No. I III IV Chapter One 1.1. 1.2. 1.3. 1.4. 1.5. "INTRODUCTION" Introduction The Moon’s Orbit Moon's Month Literature Survey The Aim Of The Present Work 2 3 7 9 16 Chapter Two "COORDINATE SYSTEMS AND MOON COORDINATE" 2.1. Coordinate Systems 2.1.1. The Horizontal (alt– azimuth) System 2.1.2. The Equatorial System 2.1.3. The Ecliptic System 2.2. Transformation Of One Coordinate System Into Another 2.3. Date And Julian Date 2.3.1. Computation Of Julian Date (JD) From Calendar Date 2.3.2. Compute Of The Calendar Date From JD 2.4. Calculating The Julian Day For The Crescent Moon 2.5. Moon Elliptical Coordinate 2.6. 2.7. 2.8. Calculating The Moon Distance Moon Coordinate Conversion Calculating The Moon Velocity Component 18 18 21 23 25 26 26 27 29 30 31 31 32 Chapter Three "TWO BODY PROBLEM WITHOUT PERTURBATION" 3.1. Introduction 3.2. Equations Of Motion 3.3. The Solution Of The Two-Body Problem 3.4. The Energy Integral 3.5. The Velocity Of a Planet In Its Orbit 3.6. The Orbital Element 3.7. The Period Of Revolution Of a Planet In Its Orbit 3.8. The Orientation Of The Orbital Plane 3.9. Calculating The Orbital Elements 3.10. Solution Of Kepler's Equation 34 37 40 40 42 43 46 50 50 52 Chapter Four 4.1. 4.2. 4.2.1. 4.2.2. 4.2.3. 4.2.4. 4.3. "PERTURBATIONS" Introduction Orbital Perturbations Atmospheric Drag Non-Spherical Gravitational Field Of The Earth Solar Radiation Pressure (SRP) Third Body Attractions Other Perturbation Forces 56 57 59 61 62 64 66 Chapter Five "RESULTS, DISCUSSION, CONCLUSIONS AND FUTURE WORK" 5.1. Introduction 5.2. Results and Discussion 5.3. Conclusions 5.4. Future Work References Appendix A Appendix B "flowchart" 68 69 85 86 88 List Of Figures Figures No. Descriptions Page No. 1.1. 1.2. The Moon's orbit around the Earth orbit. The Moon’s phases result from the relative orientation of the Moon and the Sun, as seen from the Earth. 4 6 2.1. 2.2. 2.3. 2.4. Horizon coordinate system. The observer’s celestial sphere. The equatorial system. Ecliptic coordinates. 19 20 21 24 3.1. 3.2. An orbital ellipse. The two-body problem presented in a rectangular coordinate frame. The velocity of a planet in an elliptical orbit. The orbital elements. The shape of the orbit for different values of the eccentricity. 35 38 3.6. The geometry of an elliptical orbit. 47 4.1. Secular and periodic variations of an orbital element. Vector definitions for third body attraction. 58 3.3. 3.4. 3.5. 4.2. 5.1. The Moon distance (Rm) variation with date for four different months of the year 2010. The Moon's velocity (V) variation with date for four different months of the year 2010. The semi major axis (a) of the Moon in km with date for four different months of the year 2010. The eccentricity (e) of the Moon as a function of time for four different months of the year 2010. The effect of alignment of the major axis of the Moon's orbit with the Earth - Sun line. The effective eccentricity of the lunar trajectory as a function of time of the year 1980. The instantaneous eccentricity of the lunar orbit, 1996 to 1998. 5.2. 5.3. 5.4. 5.5. 5.6. 5.7. 42 44 45 64 71 72 73 75 75 76 76 I 5.8. 5.9. 5.10 5.11 5.12 The inclination angle (i) in degree for the Moon with the date using two techniques for four different months in the year 2010. The longitude of ascending node (Ω) of the Moon as a function of the date for four months using two techniques. The eccentric anomaly (E) in degree of the Moon against the time using two techniques for four different months in the year 2010. 78 The mean anomaly (M) in degrees of the Moon against the time using two techniques for four different months in the year 2010. The argument of perigee (ω) of the Moon against time for four different months in the year 2010 using two techniques. 81 79 80 83 II List Of Tables Table No. Description (1.1) (3.1) (5.1) (5.2) (5.3) (5.4) (5.5) (5.6) (5.7) (5.8) Page No. Developed critical condition for crescent visibility. Classical orbital elements The maximum, minimum and average the Moon distance by two techniques for four different months in the year 2010 and the difference between two methods for each month. The maximum, minimum and average of the semi major axis (a) of the Moon by two techniques for four different months and the difference between them. The maximum, minimum and average values of actual eccentricity (e) of the Moon orbit for four different months using first technique (perturbed) and the difference each month. The maximum, minimum and average values of inclination (i) of the Moon's orbit by first method and inclination (i) by second method for four different months of the year 2010 and the difference between them for each month. The maximum, minimum and average values of longitude of ascending node for the Moon orbit for four difference months using two techniques. The maximum, minimum and average values of argument of perigee of the Moon for four different months in the year 2010 using two techniques and the difference between them for each month. 10 46 70 The period of the conjunct months of the Moon in the year 2010 A.D. The date and time of the new Moon. 84 73 74 77 79 82 84 III List Of Symbols Symbol A a Aa ad aJ2 al As asolar b Batm C c CD Dd e E F f FD Fm Fθ G H h , , i J2 JD JDm l l' M Definition unite The azimuth angle The semi-major axis The effective surface area of body The perturb acceleration due to the gravitational attraction of a third body The perturb acceleration due to J2 The altitude angle The cross-sectional area of body The perturb acceleration due to solar radiation pressure The minor major axis The barometric coefficient The energy constant The speed of light The drag coefficient The difference between the mean longitudes of the Sun and the Moon The eccentricity The eccentric anomaly The force of attraction The true anomaly The magnitude of the drag force The Moon's argument of latitude The solar energy flux The gravitation constant The hour angle The angular momentum of the orbit The function of the altitude The component of angular momentum The inclination angle of orbit The zonal coefficient of the Earth gravitational field The Julian Date The Julian Day for the crescent moon The Moon's mean anomaly of the year 2000 A.D The Sun's mean anomaly The mean anomaly degree Km m2 m / S2 m / S2 degree m2 m / S2 Km without unite without unite Km / S without unite degree without unite degree Newton degree Newton degree W / m2 m3 / Kg.S2 hr mn ss m2 / S2 without unite m2 / S2 degree without unite day day degree degree degree IV Mm The Moon's mean anomaly degree The Sun's mean anomaly at time JD of degree year 1900 A.D The mean motion rad / S The momentum flux from the Sun W S / m3 Ms N PS R-1 Re Rm rp, ra Rx, Ry, Rz T tp u V, VP, VA Vx, Vy, Vz xw ,yw α β βm δ ε θ λm µ π Ω ω ωv , o The elements of Gaussian matrix without unite The inverse of Gauss matrix The Earth radius The Moon's distance from the center of the Earth Perihelion, aphelion geocentric radial distance The components distance of the Moon The number of Julian centuries Time of the perigee passage The argument of the latitude angle The velocity, velocity at perihelion and velocity at aphelion The velocity component of the Moon The cartesian coordinate of the Moon in its orbit The right ascension angle The ecliptic latitude The Moon's elliptical latitude The declination angle The obliquity angle The angular distance The Moon's elliptical longitude The Gravitational parameter The constant ratio The longitude of the ascending node angle The argument of the perigee angle The angular velocity The ecliptic longitude The atmospheric density, the initial density at perigee point The observer latitude without unite Km Km Km Km without unite hour degree Km / S Km / S Km degree degree degree degree degree rad degree Km3 / S2 without unite degree degree rad / S2 degree Kg / m3 degree V Chapter One INTRODUCTION Chapter one Introduction 1.1. Introduction The Moon's Earth is a natural satellite. Sometimes it is called “Luna” but that name conjures up visions of madness and worship and is not used by astronomers [1]. It is the first astronomical object to see in anyone's eye at night sky, therefore the observations are begin as long as 3000 years ago [2] because it is the brightest object in the night sky and second important sky body, where many events on the Earth depend on the Moon position phase, such as the night shining, the solar and lunar eclipse, and affect activities such as deep-sea fishing and navigation [3]. Also for some people the Moon birth used to determine the dates as (Hegree date for Islamic countries). It is moves eastward by about one diameter per hour or 13o per day [4] or at an average speed of slightly more than 30' per hour [2] and the apparent Sun moves to the east about 1o per day, this means that the Moon rise and Moon set events are both grow later between (36 min - 52 min) every day [5]. It is apparent diameter is about half a degree. And an actual diameter of 3476 km [6], or 3480 km, it is about 27.2 percent that of Earth's diameter [1]. Its density is 3.34 g/cm3, and mass is 7.3483×1022 Kg [6] which means it has 1/81.30 of the Earth’s mass and 0.0203 of its volume [7,8]. The Earth and the Moon revolve about their common centre of gravity, a point called the barycentre. If the Earth and the Moon were equal in all respects, then the barycentre would be positioned in space exactly between the Earth and the Moon. But the Moon’s mass is about 2% that of the Earth, which offsets the barycentre considerably in the Earth’s direction, so much so that the common centre of gravity is actually located within the Earth’s mantle [9], around 4671 km [8] or 4700 km from the centre of the Earth [9]. It is so bright, (the full Moon has apparent visual magnitude of -12.7), although its surface rocks are dark, and the Moon’s albedo is only 0.07 [5,7]. 2 Chapter one Introduction The rotation time of the Moon is equal to the sidereal month (the same as its axial rotation period), so the observer on the Earth always sees the same side of the Moon always faces the Earth. Such synchronous rotation is common among the satellites of the solar system [10,11]. The visible side is called the nearside, and the side invisible from the Earth is the farside. In fact, the face of the Moon presents to us does vary slightly because of a number of effects known collectively as libration [7]. Owing to the libration, a total of 59% of the surface area can be seen from the Earth. The libration or "rocking motion" [8] is quite easy to see if one follows some detail at the edge of the lunar limb [10]. And it occurs because of the variation of orbital speed of Moon according to Kepler's second law, and because the Moon’s mass is not uniformly distributed within the globe, and Earth’s gravity has managed, over millions of centuries, to tug the Moon’s rotation rate into near-perfect lock step with its revolution [1]. 1.2. The Moon’s Orbit Planets move around the Sun in elliptical orbits, and their satellites follow elliptical orbits around them. An ellipse is a closed curve with two focal points lying on its main axis; the Earth lies at one focal point of the Moon’s orbital ellipse. The Moon’s orbit around the Earth appears almost circular, with the Earth positioned very close to the centre of the circle. Careful measurement will reveal that the figure is really an ellipse with a mean eccentricity of 0.055 [9] or for more accuracy is 0.054900489 [8,12] which is defined as the ratio of the difference between the major and minor axes to the major axis, with the Earth lying slightly to one side of the centre, positioned over a focus of that ellipse [9]. 3 Chapter one Introduction A celestial observer viewing the Solar System from a great distance would not see the Moon making loops in space about the Earth. It is seen in orbit around the Sun, as is the Earth, and that the effect of the Earth's influence is to make the Moon's orbit wiggle a little as the relative positions of Earth and Moon change (Figure 1.1). This is because the Sun's gravitational force on the Moon is much greater than that of the Earth, even though the latter is nearer. It is hardly surprising that the orbit of the Moon is so difficult to calculate since it is regulated by two bodies, not one, and the two bodies are themselves tied in orbit about each other [13]. Sun Moon's orbit about the Sun Earth's orbit about the Sun Figure (1.1): The Moon's orbit around the Earth orbit [13]. The Moon's sidereal path crosses the ecliptic twice at each month with mean angle 5o9' and the crossing point ascending and descending nodes move westward covering about 20o over a year [14]. Orbiting at an average distance from the Earth of 381000 km [1], 384000 km [10], 384400 km [8,12], 384401 km [9], the Moon lies about 30 times the Earth’s diameter away [1,9]. Light (electromagnetic radiation) takes an average of 1.3 seconds to cross the space between the Moon and the Earth. Radar signals bounced off the Moon enable its distance to be determined to an accuracy of less than half a kilometer. 4 Chapter one Introduction The Moon’s surface does not reflect radio waves as strongly as it would were it covered by large expanses of solid rock, and it was known to be covered by a thick layer of soil long before the first soft-landing probes touched down on its surface. The most accurate means of measuring the distance of the Moon is to aim short pulses of laser light at a reflecting point at a known location on the lunar surface and accurately time the returning light signals. Measurements made by aiming lasers at the passive laser reflectors left at the Apollo landing sites give a Moon distance accurate to within a few meters, and observations over the years have demonstrated that the Moon’s mean distance from the Earth is slowly increasing [9]. The new moon is that instant when the Moon is in conjunction with the Sun. Almanacs define the phases of the Moon in terms of ecliptic longitudes; the longitudes of the new moon and the Sun are equal. Usually the new moon is slightly north or south of the Sun because the lunar orbit is tilted 5o with respect to the ecliptic. About 2 days after the new moon, the waxing crescent moon can be seen in the western evening sky. About 1 week after the new moon, the first quarter follows, when the longitudes of the Moon and the Sun differ by about 90◦. The right half of the Moon is seen lit (left half when seen from the Southern hemisphere). The full moon appears a fortnight after the new moon, and 1 week after this the last quarter. Finally the waning crescent moon disappears in the glory of the morning sky [10] as shown in figure (1.2). 5 Chapter on ne Inttroduction F Figure (1.2): The Moonn’s phases reesult from the t relative orientation of the Moon and a the Sun,, as seen froom Earth [1]. The Mooon's orbitt is variab ble in shappe and sizze becausee of otherr body attrraction suuch as the Sun and some s plannets these problems are moree distance and thee complicaate becauuse the variation v of the Earth-Sun E planetaryy positionn. The orbbital elem ments havee some vaariations with w time.. The minimum disttance of thhe Moon from the centre of the Earth (perigee)) the closeest approaach to the Earth’s su urface is 356400 3 km m, or is 34 48294 km m [4]. Andd the maxximum disstance (ap pogee) thee furthest distance from thee Earth’s surface iss 406700 km, or is i 3985811 km [4]. This rep presents a differencce of 13.55 percent of o the Mo oon’s meaan distancee [1]. Thiss range iss larger thaan the onee calculateed from th he semi-m major axis aand the ecccentricityy [10]. Thherefore thhe angulaar diameteer of the Moon as seen from m Earth'ss surface varies v betw ween 33′299″ and 29′′23″ [9], 33'.5 3 and 229'.4 [10] at a averagee it is 31′005″. Havinng an apparent diam meter abouut 12% laarger at peerigee, thee apparent area of thhe perigee Moon is 29% 2 largerr than at appogee [9]. 6 Chapter one Introduction The line that joins the points of apogee and perigee – effectively the major axis of the Moon’s elliptical orbit – is called the “line of apsides”. The line of apsides rotates (with respect to the stars) in a prograde fashion every 8.85 years. The plane of the Moon’s orbit around the Earth is inclined to the plane of the Earth’s orbit around the Sun (the plane of the ecliptic) by 5°8′43″. The two points at which these planes intersect are called the ascending and descending nodes. The ascending node is the point on the ecliptic where the Moon moves to the north of the ecliptic; the descending node marks the point where the Moon moves to the south of the ecliptic. The line of nodes rotates (with respect to the stars) in a retrograde fashion every 18.61 years [9,10]. In an astronomical sense, the recession of the line of nodes is important, since solar and lunar eclipses are dependent on the nodes’ position in relation to the Sun and the Earth [9] i.e. the solar eclipse happens at the nearest angle between the Sun and the Moon and lunar eclipse happens at about 180o between them [13]. 1.3. Moon's Month There are different types of months for the Moon depending on the start point to measure the period that Moon takes it to return to the same point which start from it. These are: 1) Draconic month (nodical month): Time taken for the Moon to complete a single revolution around the Earth, measured relative to its ascending node; it is equivalent to 27.21222 days of mean solar time [2,7,8,10] or 27d5h5m34.1s [15]. 2) Tropical month: Time taken for the Moon to complete a single revolution around the Earth, measured relative to the first point of Aries; it is equivalent to 27.32158 days of mean solar time [2,7] or 27d7h34m4.74s [15]. 7 Chapter one Introduction 3) Sidereal month: Time taken for the Moon to complete a single revolution around the Earth, measured relative to a fixed star; it is equivalent to 27.32166 days of mean solar time [6,7,8,10,13]. Which equals to 27d7h43m11.5s [9,15]. 4) Anomalistic month: Time taken for the Moon to complete a single orbit around the Earth, measured from perigee to perigee. An anomalistic month is shorter than the more commonly used synodic month, being equivalent to 27.55455 days of mean solar time [2,7,10,16] or 27d13h18m37.4s [15]. 5) Synodic month (lunar month): Period between successive new or full moons. This is the same duration as one lunation and is equivalent to 29.53059 days of mean solar time [2,7,10,13,16] or 29d12h44m2.9s [15]. The difference between the sidereal and synodic months in period because during this time the Earth moves on along its own orbit so that the Sun's position changes with respect to the stars. Hence the Moon has some extra distance to make up to regain its position relative to the Sun. The interval defined by the time taken for the Moon to return to the same position relative to the Sun. Although in any revolution of the Moon in its orbit these months may differ by a few hours from the mean values given above, the mean values remain steady over many centuries to within one second. 8 Chapter one Introduction 1.4. Literature Survey For the ancient Babylonian, Greek and Egyptian astronomers the major interest was to predict the positions of the Sun, the Moon and the planets on the celestial sphere, because the main tasks for them were to provide an exact calendar and the precise determination of the date of the eclipses in advance. This was accomplished with long and difficult observations to detect the different periods of the motion of the celestial bodies in the sky [17]. Many Arab and Muslim old astronomers studied the Moon orbit and developed the lunar visibility such as: Yaqupe bin Tariq, Habash, Al-Khwarzmi, Al-Tabari, Al-Farghani, Thabet bin Qurrah, Al-Battani, Ibn Maimon, Al-Biaruni, Abdul Rahmman Alsufi, Ibn Sina, Al-Tusi and Al-Kashani [18,19]. The Moon visibility criteria are important and developed from the biging of Islamic date as in table (1). From the nineteenth century, the lunar visibility crescent developed by many astronomer such as: Fotheringham 1910, Maunder 1911, Carl Schoch, Bruin 1977, B.E. Schaefer 1988, 1991, 1994, 1996 Shaukat, M. Ilyas 1979, 1981, 1982, 1983, 1984, 1988 and 1993, Bernard Yallop 1997 [18,20], Muhammad Shahid Qureshi [21]. Also the Arab astronomer such as: Z. Sardar 1982, Al-Abadee 1991 [22], Odeh M. [23], N. Guessoum and K. Meziane [24], A. H. Sultan [25]. Furthermore some observatories were involved such as: Royal Greenwich Observatory (RGO), South African Astronomical Observatory (SAAO) [20]. In Iraq, Astronomers work on the lunar visibility, these are: H. M. Alneamy et al 1987, 1989, 1995, 1994, M. M. Jarad, Abdul AlRahmman H. S. 1997, F. M. Abdulla 2001, and D. M. Al-Fead 2002 [11,18]. 9 Chapter one Introduction The Moon visibility criteria developed in the past time are illustrated in table (1.1). Table (1.1): Developed critical condition for crescent visibility. Date B.C Astronomer Babylonians Chinese and Japanese B.C 500 A.D Hindus 767‐778 Yaqub bin Tariq A.D 740‐840 Habash A.D 836 A.D Al‐Khwarizmi 731‐861 Mousa bin A.D Maimon 850‐929 Al‐Battani A.D 826‐901 Thabet bin A.D Qurra Abdul‐Rahman 986 A.D Al‐Sufi 973‐ Al‐Biaruni 1048 A.D 1258‐ Nasir Al‐Deen 1274 A.D Al‐Tusi 15th century Ghiyath Al‐Din Al‐Kashani Fotheringham and Maunder 1977 A.D Bruin 1910 A.D Islamic conference in Istanbul 1978 A.D 1981‐84 A.D Ilyas Critical condition Comment o a>= 12 Based on observation Depend on Babylonians observation Depend accurate o a>= 12 observation and evolved liner relation Put crescents calculate table Calculated suit modified a>9.5o 9o<=a<=24o a < 12o In general it was suitable in spring and autumn Calculate suit modified 11o<=a<=25o a >= 12o Depend on Babylon rules Depend on Hapash and Al‐Battani rules o a>= 12 a(∆Z) >= f(Z,A) a(∆Z) >= f(Z,W) a>8o l>5o a(∆Z) >= f(a,Z) Depend on Babylon system 24 minute a er Sun set Or a>=11o‐12o where ∆A=0 from observa on Theoretical (a) is angular distance for Moon from Sun. (l) is height of Moon from horizontal Compare between two critical independent case 10 Chapter one Introduction 1983 A.D Ilyas Age>=f(lat, season, year) Simpler approximate for critical case 1984‐88 A.D Ilyas a>=f(lat, season) More accurate Hameed + Sameer 1985 A.D a>7o l>4o Hameed + Abdul Age>=10h & a>5o 1993 A.D Rahman & l>3o Age>7h+a>=5o + 1997 A.D Abdul Rahman I>3o+Makth>10min 2001 A.D Fuad Mahmud 2002 A.D Dhaha Al‐Faidh P= 2.075Age + 2.471Makth + 2.484Alt +0.659l Depended on modern astronomical calculate with observation All condition should be verify The effect of geographic sit on the coefficient critical Foundation the equation of probability for lunar crescent visibility The motion of Moon can be considered as a two-body problem but in fact it is a system with many body problems. Johannes Kepler (1571–1630) who found that the motion of Mars is an ellipse, with the Sun in its focus, did computations using an empirical mathematical equation depending on Tycho Brahe (1546–1601) observations [17]. He published his famous first and second law in 1609 and third law in 1619. He proved to that the orbits of planets including the Earth are ellipses instead of circles with the Sun at one focus. Although requiring a single extra parameter for each orbit (eccentricity), Kepler's law agreed with observations perfectly without the need for epicycles [26] therefore he was the first to solve two-body gravitation problem and he put the mathematical roles to solve ellipse equation [27]. These three laws are considered as the basis of all future work in celestial mechanics. To understand the motion of the planets, Sun and Moon the universal law of gravitation should be used, which was discovered by Isaac Newton (1642–1727) in 1687 [17], Newton's contribution was a huge triumph for 11 Chapter one Introduction astronomy, physics and mathematics. His universal law of gravity is still used today to guide spacecraft flying to the outer Solar System and to model the motion of the planets and the Moon to exquisite accuracy [26]. The knowledge of the two-body motion is of such importance because [17]: 1. The motion of a single planet around the Sun (the two-body problem) is the only astrodynamical problem where we have a complete and general solution (although it is not as simple as it looks like) besides very special cases of the three-body problem not exactly realized in nature. 2. For many problems in the dynamics of celestial bodies it is a very good first approximation. 3. It is the starting point of analytical theories from which the astronomers work to develop the solution to higher orders with respect to small parameters involved like the eccentricity, the inclination of the orbit and the small masses of the other planets, which perturb the elliptic motion of a planet around the Sun. The Restricted Three Body-Problem (R3BP) dates back to Leonhard Euler (1707–1783), who worked on a lunar theory. His main contribution to the R3BP was the introduction of a synodic coordinate system, where the two massive bodies have fixed positions. He also solved a special case of the R3BP that is called two fixed centre problem, where two fixed centre of force act on a third one [17]. In 1750 he recognized that these equations of motion are valid for any mass element and thus define a “new” mechanical principle [28]. The most complete study on the R3BP was published by Szebehely [17,29]. George William Hill (1838–1914) developed his lunar theory by studying the actual motion relative to a periodic solution of the 3BP Earth-Moon-Sun [29]. He wanted to prove the stability of the restricted problem by representing an actual orbit as 12 Chapter one Introduction infinitesimally close to a suitable periodic solution, found that for some of the resonant motions the opposite was true. It often happens that a problem, which is simplified in the attempt to find simple approximations, is no longer strongly related to the original problem [28]. Between the 1890s and the 1930s, George Darwin (1897–1911), George Hill (1898), Henry Plummer (1903), Forest Moulton (1920), Elis Strömgren (1935), and their colleagues contributed to the discovery of the first known periodic orbits in the circular restricted three-body problem [29]. Joseph Louis Lagrange (1736–1813) besides his foundational work in classical mechanics and his creation of the variation calculus, found the Lagrange points while attempting to solve the three-body problem, worked out a method to determine a comet's orbit with only three observations, and did additional important work on orbital precession and stability. Pierre-Simon Laplace (1749–1827), through a series of memoirs to the Academy of Science in Paris, addressed the stability of the Solar System by showing that the changes of the orbital mean motions of Jupiter and Saturn were periodic and due to their near-resonance orbits. He also spent a significant amount of time in the study of lunar motion perturbed by a non-spherical Earth, and of the oceanic tides induced by the Sun and the Moon. His most significant contribution was the compilation of the five volume Celestial Dynamics (1799–1825), which "offer a complete solution of the great mechanical problem presented by the Solar System, and bring theory to coincide so closely with observation that empirical equations should no longer find a place in astronomical tables." Laplace included most of his work on planetary orbits and perturbations, as well as problems solved by earlier astronomers [26]. Research on celestial dynamics achieved a real predictive triumph when the British astronomer John Cough Adams (1819–1892) and the French astronomer Le Verrier (1846) independently discovered Neptune by analytical calculation of its 13 Chapter one Introduction perturbation on the orbit of Uranus. Galle (1846) later found the planet only 1o off Le Verrier's prediction [26,28]. In the 1890s, Poincaré contributed substantial advancement to the understanding of the three-body problem, most notably by describing the existence of chaos in the dynamical system. The existence of chaos helped to explain why a solution to the general three-body problem was so evasive [29]. Many new studies on different kinds of perturbations to the motion of planets and satellites were conducted. Most of the analytical works focused on three-body problems (e.g., expansion of the disturbing function by Boquet, 1889), or low-order approximations for systems with a few more objects and additional perturbations (e.g., secular frequencies in the Solar System by Brouwer et al., 1950). Darwin (1879, 1880) also began to pioneer the analysis of the lower-order effects of tides and tidal friction. The use of computers for numerical integration opened a new window on the subject in the 1960s, and made it possible to handle more complicated systems for a long period of time and to study the formation and evolution of the whole Solar System. One key numerical integration of the outer Solar System for 120,000 years was undertaken by Cohen and Hubbard (1965) [26]. The studies of the influence of a perturbing third body are: Kaula (1962) expressed the disturbing function in osculating Keplerian elements in a fashion similar to the work he did for the terrestrial gravitational field [30,31]. From the mid-1960s to the early 1970s, good progress was made in the application of computer techniques to the manipulation of analytical expressions. The basic principles of computerized manipulation of series expansion were developed by Danby, Deprit, and Rom (1965), Broucke and Garthwaite (1969), Keesey (1971), Ananda (1973), and Chao (1976) applied the computerized series-expansion system designed by Broucke for obtaining solutions of mutual perturbations among planets, perturbations 14 Chapter one Introduction resulting from Oblateness, and the motion of the Galilean satellites of Jupiter for obtaining solutions of, respectively, mutual perturbations among planets [31]. Giacaglia (1977) developed the geopotential, the third body and the solar pressure disturbing functions in nonsingular variables. Some recent works include Prado (2003) and Broucke (2003). Both used a double averaging of the third-body potential. But Prado used this potential to analyze the evolution of orbits around major natural moons of the Solar System, while Broucke’s motivation was to study the long-term third-body effects on the stability of an Earth satellite. Solórzano and Prado (2004) published a paper in which they studied the long term evolution of the orbital elements using a single average model [30]. Ke Zhang (2007) studied satellite orbits, to understand how orbits evolve over the time of the Solar System depending on their timescale, he classified orbital interactions as either short-term (orbital resonances) or long-term (secular evolution) [26]. Nadege Pie (2008) used a simplified model of the third-body problem. The main body with mass (M) is at the origin of the system. The orbit of the perturbing body of mass (m) is assumed to be unperturbed Keplerian and elliptical [30] . 15 Chapter one Introduction 1.5. The Aim Of The Present Work 1- To study the two-body problem with and without perturbation. 2- To calculate the Moon geocentric coordinates (λ, β, Rm) and its variation with time by using two methods: The first technique using empirical formula with perturbation which agrees with the observations. The second technique solving Kepler's equation for the Moon elliptic orbit (without perturbation). 3- To calculate the value of orbital elements (M, a, e, Ω, ω, i) as well as the position, velocity, and eccentric anomaly (r, v, E) and its variation with time through some months of the year from the above two methods. 4- To calculate the perturbation in the orbital elements by comparing these elements from the two above method. 5- To discuss the perturbation with orbital elements and how can be used to determine the exactly Moon orbit which can be used with practical formula to determine the lunar crescent visibility criteria. 16 Chapter Two COORDINATE SYSTEMS AND MOON COORDINATE Chapter two Coordinate Systems and The Moon Coordinate 2.1. Coordinate Systems Observing or calculating the position and velocity of any celestial object requires a coordinate system and the first requirement for describing an orbit is a suitable inertial reference frame [8]. In the 4000 years ago during which astronomy was developed, various coordinate systems have been introduced because of the wide variety of problems to be solved. It has particular reference to great circles by which the direction of any celestial body can be defined uniquely at a given time. The choice of origin of the system also depends on the particular problem. It may be the position of observer on Earth's surface (a topocentric system) or from the centre of the Earth (a geocentric system) or from the centre of the Sun (a heliocentric system) or centre of the Moon (a selenocentric system) or in the case of satellite problems, the centre of a planet (a planetocentric system) [14] or even the galactic centre in stellar dynamics [12]. Some celestial coordinate systems can be classified as the following: 2.1.1. The Horizontal (alt– azimuth) System The most primitive system, most immediately related to the observer’s impression of being on a flat plane and at the centre of a vast hemisphere across which the heavenly bodies move [14]. The horizon is defined as the dividing line between the Earth and the sky, as seen by an observer on the ground [32]. 18 Chapter two Coordinate Systems and The Moon Coordinate As shown in the drawing below figure (2.1), the origin of this coordinate system is the observer (O), The fundamental plane of the system contains the observer and the horizon, The horizon is that line on the celestial sphere which is everywhere 90° from the zenith [33], (Z) which is the point straight overhead, perpendicular to the horizon plane, and nadir is the point directly under the observer. A vertical circle through an object in the sky and the zenith is the object circle. The coordinates of the object are given by the azimuth (A), which is the horizontal angle from north (N) clockwise to the object circle from (0o – 360 o), and the altitude or elevation angle (al), which is measured upward from the horizon to the object, which is change between (0o – 90 o). The great circle through the north and south points out the horizon and the zenith is called the meridian [32]. Z al S N O A E Figure (2.1): Horizon coordinate system [32]. 19 Chapter tw wo Coordinate Systems and The Moon C Coordinate The observer’s ceelestial sph here is shoown in figgure (2.2) where w (Z)) is the zeenith, (O) the obserrver, (P) is i the nortth celestiaal pole an nd OX thee instantanneous direection of a star. Thee great cirrcle througgh (Z) and d (P) cutss the horizzon NESA AW at thee north (N N) and souuth (S) pooints. Another greatt circle WZE W at righht angles to t the great circle NPZS N cutss the horizzon in thee west (W W) and eaast (E) pooints. Thee arcs ZN N, ZW, Z ZA, etc, are a calledd verticals. The poinnts N, E, S and W are a the caardinal poiints [12,14 4]. It is too be notedd that west is alwayys on the left hand of an obsserver faciing north.. The vertticals throough east and west are called prime vverticals; ZE is thee prime veertical eastt, ZW is thhe prime vertical v weest [14]. The twoo angle thaat specify the positiion of (X)) in this system aree the azim muth, (A) ,aand the alltitude, (al). An alterrnative cooordinate to t altitudee is the zennith distannce, (Z), of o (X), indiicated by ZX Z in figuure (2.2.) [12,14]. al = 90o – Z al Figgure (2.2): The T observerr’s celestiall sphere [12,14]. 20 2 Chapter two Coordinate Systems and The Moon Coordinate The main disadvantage of the horizontal system of coordinates is that it is purely local. Two observers at different points on the Earth’s surface will measure different altitudes and azimuths for the same object at the same time. Also an observer will find the object’s coordinates changing with time as the celestial sphere appears to rotate [12,14]. 2.1.2. The Equatorial System If the plane of the Earth’s equator is extended it will cut the celestial sphere in a great circle called the celestial equator. It intersects the horizon circle in two points (W) and (E) (figure 2.3). The (W) and (E) are the west and east points [12,14]. Since the angle between equator and zenith is the observer’s latitude it is seen that the altitude of the north celestial pole (P) is the latitude ( of the observer [12]. Points (P) and (Z) are the poles of the celestial equator and the horizon respectively. But (W) lies on both these great circles so that (W) is 90° from the points (P) and (Z). Hence, (W) is a pole on the great circle ZPN and, therefore, be 90° from all points on it in particular from (N) and (S). Hence, it is the west point. By a similar argument (E) is the east point. Any great semicircle through (P) and (Q) is called a meridian. The meridian through the celestial object (X) is the great semicircle PXBQ cutting the celestial equator in (B) (see figure 2.3). The meridian PZTSQ is the observer’s meridian [14]. Figure (2.3): The equatorial system [12,14]. 21 Chapter two Coordinate Systems and The Moon Coordinate An observer viewing the sky will note that all natural objects rise in the east, climbing in altitude until they transit across the observer’s meridian then decrease in altitude until they set in the west. A star will follow a small circle parallel to the celestial equator in the arrow’s direction. Such a circle (UXV in the diagram) is called a parallel of declination and provides us with one of the two coordinates in the equatorial system [12,14]. The declination, (δ), of the star is the angular distance in degrees of the star from the equator along the meridian through the star. It is measured north and south of the equator from 0° to 90°, being taken to be positive when north [12,14]. The declination of the celestial object is thus analogous to the latitude of a place on the Earth’s surface, and the latitude of any point on the surface of the Earth when a star is in its zenith is equal to the star’s declination. A quantity called the north polar angle of the object, is the arc PX. Obviously, [14] north polar angle = 90° − declination angle The second coordinate is the angle ZPX is called the hour angle, H, of the star and is measured from the observer’s meridian westwards (for both north and south hemisphere observers) to the meridian through the star from 0h to 24h or from 0° to 360°. Consequently, the hour angle increases by 24h each sidereal day for a star [12,14]. If a point ( ), fixed with respect to the stellar background, is chosen on the equator, its angular distance from the intersection of the meridian through (X) and the equator will not change in contrast to the changing hour angle of (X). In general, all objects may then have their positions on the celestial background specified by their declinations and by the angles between their meridians and the meridian through ( ). 22 Chapter two Coordinate Systems and The Moon Coordinate The point chosen is the vernal equinox, also referred to as the First Point of Aries, and the angle between it and the intersection of the meridian through a celestial object and the equator is called the right ascension (α) or RA of the object. Right ascension is measured from 0h to 24h or from 0° to 360° along the equator from ( ) eastwards; that is, in the direction opposite to that in which hour angle is measured. This definition again holds for observers in both northern and southern hemispheres [12,14]. Astronomers use the right ascension declination system to catalog star positions accurately. Because of the enormous distances to the stars, their coordinates remain essentially unchanged even when viewed from opposite sides of the earth' s orbit around the Sun. Only few stars are close enough to show a measurable parallax between observations made 6 months apart [8]. 2.1.3. The Ecliptic System This system is especially convenient in studying the movements of the planets, asteroids and in describing the Solar System. The orbits of the planets in the Solar System, except of Pluto, lie within 7° of ecliptic plane. When the Sun is observed over a long period of time, it is found to possess a second motion in addition to its apparent diurnal movement about the Earth. It moves eastwards among the stars at about 1°/day, returning to its original position in one year. Its path is a great circle called the ecliptic which lies in the plane of the Earth’s orbit about the Sun. This great circle is the fundamental reference plane in the ecliptic system of coordinates. It intersects the celestial equator in the vernal and autumnal equinoxes (First Point of Aries and Libra ♎ ) [12]. 23 Chapter two Coordinate Systems and The Moon Coordinate The two quantities specifying the position of an object on the celestial sphere in this system are ecliptic longitude and ecliptic latitude. In figure (2.4) a great circle arc through the pole of the ecliptic (K) and the celestial object (X) meets the ecliptic in the point (D). Then the ecliptic longitude, (λ), is the angle between ( ) and (D), measured from 0° to 360° along the ecliptic in the eastwards direction, which is in the direction in which right ascension increases. The ecliptic latitude, (β), is measured from (D) to (X) along the great circle arc DX, being measured from 0° to 90° north or south of the ecliptic. It should be noted that the north pole of the ecliptic, (K), lies in the hemisphere containing the north celestial pole [12,14]. The point of intersection the celestial equator and the ecliptic plane is often referred to as the ascending node, since an object travelling in the plane of the ecliptic with the direction of increasing right ascension (eastwards) passes through Aries ( ) from southern to northern declinations. By similar reasoning, Libra (♎) is called the descending node [14]. The origins most often used with this system of coordinates are the Earth’s centre and the Sun’s centre since most of the planets move in planes inclined only a few degrees to the ecliptic. This system is particularly useful in considering interplanetary missions [12]. Figure (2.4): Ecliptic coordinates [14]. 24 Chapter two Coordinate Systems and The Moon Coordinate 2.2. Transformation Of One Coordinate System Into Another It is often required to convert from the system to another. This may be achieved by using the equation below: 1- For the transformation from equatorial into ecliptical coordinates (β, λ), the following formulae can be use. sin β = sin δ cos ε – cos δ sin ε sin α (2.1) and sin sin λ cos sin sin cos 2.2 If substitute equation (2.1) in (2.2), we can find formula independent on (β) tan where ( sin cos tan sin cos 2.3 is the obliquity angle 2- For transformation from ecliptical into equatorial coordinates (α, δ), the following formulae was used. tan α = (sin cos ε – tan β sin ε ) / cos (2.4) and sin δ = sin β cos ε + cos β sin ε sin (2.5) 3- Calculation the local horizontal coordinates (A, al) of any sky body from its equatorial coordinate is made using the following formula: tan A = sin H / (cos H sin sin al = sin where ( sin δ + cos – tan δ cos ) cos δ cos H (2.6) (2.7) is the observer latitude. 25 Chapter two Coordinate Systems and The Moon Coordinate 2.3. Date And Julian Date 2.3.1. Computation Of Julian Date (JD) From Calendar Date The irregularities in the present calendar (unequal months, days of the week having different dates from year to year) and the changes from the Julian to the Gregorian calendar make it difficult to compare lengths of time between observations made many years apart. In the observations of variable stars, it is useful to be able to say that the moment of observation occurred so many days and fractions of a day after a definite epoch. The system of Julian Day Numbers was introduced to reduce computational labour in such problems and avoid ambiguity [12,14]. January 1st of the year 4713 BC was chosen as the starting date, time being measured from that epoch (mean noon on January 1st, 4713 BC) by the number of days that have elapsed since then [12,14,19]. The Julian Date is given for every day of the year in The Astronomical Almanac. Tables also exist for finding the Julian Date for any day in any year. Time may also be measured in Julian centuries, each containing exactly 36525 days. Orbital data for artificial Earth satellites are often referred to epochs expressed in Modified Julian Day Numbers in which the zero point in this system is 17·0 November, 1858. Hence [12,14] Modified Julian date = Julian date – 2400000·5 days. The program must then start with a procedure separating the numbers YYYY, MM and DD.dd = day + U.T / 24 and U.T in hours. In what follows, we will suppose that this separation has been performed. 26 Chapter two Coordinate Systems and The Moon Coordinate If MM is greater than 2, take y = YYYY and m = MM If MM = 1 or 2, take (to solve February month problem) y = YYYY – 1 and m = MM + 12. If the number YYYY MM DD is equal or larger than 1582 10 15 (that is, in the Gregorian calendar), calculate AJ = INT ( y / 100 ) BJ = 2 – AJ + INT (AJ / 4) If YYYY.MMDDdd < 1582.1015, it is not necessary to calculate AJ and BJ. The required Julian Day is then [18,19,34,35] JD =INT(365.25 y)+INT(30.6001 (m + 1))+DD.dd+1720994.5+BJ (2.8) We denote by T the number of Julian centuries elapsed since midday of beginning of 1st January 1900 [34,35]: JD 2415020 /36525 2.9 And after year 2000 the following formula can be used [36] JD 2451545 /36525 2.10 2.3.2. Compute Of The Calendar Date From JD The following method is valid for positive as well as for negative years, but not for negative Julian Day numbers [18,19,34,37]. Add 0.5 to the JD, and let (Z) be the integer part, and (CC) the fractional (decimal) part of the result. Z = INT (JD + 0.5) CC = (JD + 0.5) – Z If Z < 2299161, take EC = Z 27 Chapter two Coordinate Systems and The Moon Coordinate If Z is equal to or larger than 2299161, then calculate FC = INT ((Z – 1867216. 25 ) / 36524.25 ) EC = Z + 1 + FC – INT (FC / 4 ) Then calculate GC = EC + 1524 INT I G 122.1 365.25 JC = INT (365.25 IC) INT K G –J 30.6001 The day of the month (with decimals) is then LC = GC – JC – INT (30.6001 KC) + CC The month number (MC) is KC – 1 KC – 13 if KC < 13.5 if KC > 13.5 The year (YC) is IC – 4716 if MC > 2.5 IC – 4715 if MC < 2.5 The day in month is DC = INT (LC) The hours in the day is HC = INT ((LC – DC) 24) The day of the week corresponding to a given date can be obtained as follows. Compute the JD for that date at 0h, add 1.5 and divide the result by 7. The remainder of this division will indicate the weekday, as follows: if the remainder is 0, it is a Sunday, 1 a Monday, 2 a Tuesday, 3 a Wednesday, 4 a Thursday, 5 a Friday and 6 a Saturday. 28 Chapter two Coordinate Systems and The Moon Coordinate 2.4. Calculating The Julian Day For The Crescent Moon We can find the JD for crescent moon using the equation: [18,19,34] JDm = 2415020.75933 + 29.53058868K + 0.0001178T'2 – 0.000000155T'3 + 0.00033 sin (166°.56 + 132°.87T' – 0°.009173T'2 (2.11) These instants are expressed in Ephemeris Time (Julian Ephemeris Days). In the formula above, an integer value of k gives a New Moon, an integer value increased by: 0.25 gives a First Quarter, 0.50 gives a Full Moon, 0.75 gives a Last Quarter An approximate value of K is given by K = (year – 1900) × 12.3685 Where the "year" should be taken with decimals and T' is the time in Julian centuries from 1900 January 0.5, which calculated with a sufficient accuracy from T' = K / 1236.85 To obtain the time of the true phase, the following corrections should be added to the time of the mean phase given by (2.11). The following coefficients are given in decimals of a day, and smaller quantities have been neglected [19,34]. For New and Full Moon : + (0.1734 – 0.000393 T') sin Ms + 0.0021 sin 2Ms – 0.4068 sin Mm + 0.0161 sin 2Mm – 0.0004 sin 3Mm + 0.0104 sin 2Fm – 0.0051 sin (Ms + Mm) – 0.0074 sin (Ms – Mm) + 0.0004 sin (2Fm + Ms) – 0.0004 sin (2Fm – Ms) – 0.0006 sin (2Fm + Mm) + 0.0010 sin (2Fm – Mm) + 0.0005 sin (Ms + 2Mm) 29 Chapter two Coordinate Systems and The Moon Coordinate For First and Last Quarter : + (0.1721 – 0.0004 T') sin Ms + 0.0021 sin 2Ms – 0.6280 sin Mm + 0.0089 sin 2Mm – 0.0004 sin 3Mm + 0.0079 sin 2Fm – 0.0119 sin (Ms + Mm) – 0.0047 sin (Ms – Mm) + 0.0003 sin (2Fm + Ms) – 0.0004 sin (2Fm – Ms) – 0.0006 sin (2Fm + Mm) + 0.0021 sin (2Fm – Mm) + 0.0003 sin (Ms + 2Mm) + 0.0004 sin (Ms – 2Mm) – 0.0003 sin (2Ms + Mm) And, in addition: for First Quarter: + 0.0028 – 0.0004 cos Ms + 0.0003 cos Mm for Last Quarter : – 0.0028 + 0.0004 cos Ms – 0.0003 cos Mm Where: Sun's and Moon's mean anomaly (Ms, Mm) at time JD for year 1900 A.D as a function (K,T') are calculated as [18,19,34]: Ms = 359.2242 + 29.l0535608 K – 0.0000333 T'2 – 0.00000347 T'3 Mm = 306.0253 + 385.81691806 K + 0.0107306 T'2 + 0.00001236 T'3 Moon's argument of latitude as a function (K,T') Fm = 21.2964 + 390.67050646 K – 0.0016528 T'2 – 0.00000239 T'3 Which are expressed in degrees and decimals and may be reduced to the interval (0 – 360) degrees. 2.5. Moon Ecliptical Coordinate : The ecliptical coordinate of the Moon which considered actual coordinate can be computed by an empirical formula. The Moon's longitude is given by [21]: λm = 218.32 + 481267.883T + 6.29 sin (134.9 + 477198.85T) – 1.27 sin (259.2 – 413335.38T) + 0.66 sin (235.7 + 890534.23T) + 0.21 sin (269.9 + 954397.7T) – 0.19 sin (357.5 + 35999.05T) – 0.11 sin (186.6 + 966404.05T) (2.12) 30 Chapter two Coordinate Systems and The Moon Coordinate The Moon's latitude is given by: βm = 5.13 sin (93.3 + 483202.03T) + 0.28 sin (228.2 + 960400.87T) – 0.28 sin (318.3 + 6003.18T) – 0.17 sin (217.6 – 407332.2T) (2.13) Where T is the number of centuries since J2000 which is calculated from equation (2.10). The Moon ecliptical coordinate also calculated using J. Meeus method [34] and the results are in a good agreement with above method. 2.6. Calculating The Moon Distance The Moon's distance from the centre of the Earth can be calculated as the following [36]: Rm= 385000 – 20905 cos l – 3699 cos (2Dd – l) – 2956 cos 2Dd – 570 cos (2l) + 246 cos (2l – 2Dd) – 152 cos (l + l' – 2Dd) (km) (2.14) Where: the Moon's mean anomaly is (l), the Sun's mean anomaly is (l') and the difference between the mean longitudes of the Sun and the Moon is (Dd) , which are functions of Julian centuries (T2000) and calculated as [36]: l = 134°.96292 + 477198°.86753 T l' = 357°.52543 + 35999°.04944 T Dd = 297°.85027 + 445267°.11135 T 2.7. Moon Coordinate Conversion The Moon elliptical coordinates which are computed using the equations (2.12) and (2.13), must be converted to equatorial coordinate using equations (2.4) and (2.5). The equatorial coordinate can be converted to the horizontal coordinate using equations (2.6) and (2.7) to know Moon's position in sky in observer region on Earth surface and to compare with the theoretical model with the observation of the Moon. 31 Chapter two Coordinate Systems and The Moon Coordinate The position components distance in cartesian coordinate can be calculated using equatorial coordinate as the following [38]: Rx = Rm cos δ cos α Ry = Rm cos δ sin α Rz = Rm sin δ The distance of the Moon from the Earth centre can be found as [35]: 2.8. Calculating The Moon Velocity Component The velocity component can be determined from the following formula: ∆ ∆ ∆ Where ∆ = which can be choose 1day or less, and the total velocity is found as [35]: 32 Chapter Three TWO BODY PROBLEM WITHOUT PERTURBATION Chapter three Two Body Problem Without Perturbation 3.1. Introduction Galileo showed that the Ptolemaic theory failed as an adequate description of planetary geocentric phenomena, by his telescope observations. The Copernican System was able to accept the new discoveries regarding the phases of Venus. At that time no one, was able as yet to give any explanation of why the motion of the planets were as observed or why the Moon revolved about the Earth. Half a century had to pass before the explanation was given by Isaac Newton (1642–1727 AD). His work was built on the foundations laid by Tycho Brahe (1546–1601 AD), Johannes Kepler (1571–1630 AD) and Galileo Galilei (1564–1642 AD). Kepler studied the mass of observational data on the planets positions collected by Brahe, formulated the three laws of planetary motion forever associated with his name. They are: (1) The orbit of each planet is an ellipse with the Sun at one focus. (2) For any planet the radius vector sweeps out equal areas in equal times. (3) The cube of the semi-major axis of the planetary orbit is proportional to the square of the planet period of revolution. Kepler’s laws are still very close approximations to the truth. They hold not only for the system of planets moving about the Sun but also for the various systems of satellites moving about their primaries. Only when the outermost retrograde satellites in the Solar System are considered, or close satellites of a non-spherical planet. The mathematical expression of Kepler’s second law can be written as 1 2 1 2 Where (r1, r2) are the distance between two body at different time and (θ1, θ2) are the angular distance at time (t). 34 Chapter three Two Body Problem Without Perturbation But (θ/t) is the angular velocity (ωv) in the limit when (t) tends to zero. Hence, 1 2 1 2 3.1 As shown in figure (3.1.), in order that this law is obeyed, the planet has to move faster when its radius vector is shorter, at perihelion (rp), and slower when it is at aphelion (ra): Where rp = a (1 – e) and ra = a (1 + e) (3.2) Where (a) is a semi-major axis of the orbit [14,39] and, (e) is a eccentricity of the orbit which defined by the relation [14] 3.3 Also [39] a A' b C S A rp ra Figure (3.1.): An orbital ellipse. In the third law, Kepler obtained a relationship between the sizes of planetary orbits and the periods of revolution. Now it happens that the semi-major axis of a planetary orbit is the average size of the radius vector, or the mean distance, so that an alternative form of the third law is to say that the cube of the mean distance of a planet is proportional to the square of its period of revolution. 35 Chapter three Two Body Problem Without Perturbation Hence, if (a) and (T) refer to the semi-major axis and sidereal period of a planet (P) moving about the Sun, then Or 3.4 2 Where (µ) is G(M + m) and (M, m) are the masses of the Sun and planet respectively and (G) is the universal gravitational constant. Let two planets revolve about the Sun in orbits of semi-major axis (a1) and (a2), with periods of revolution (T1) and (T2). Let the masses of the Sun and the two planets be (M, m1 and m2) respectively, then by equation (3.4), 2 2 Where µ1 = G(M + m1) And µ2 = G(M + m2) Hence, Where the mass of the Sun M >> m1 or m2 then Kepler’s third law would have been written as: 36 Chapter three Two Body Problem Without Perturbation Newton’s law of universal gravitation is the basis of celestial mechanics - that branch of astronomy dealing with the orbits of planets and satellites the branch of dynamics that deals with the orbits of space probes and artificial satellites. The law is stated: Every particle of matter in the universe attracts every other particle of matter with a force directly proportional to the product of the masses and inversely proportional to the square of the distance between them. Hence, for two particles separated by a distance (r), is 3.5 Where (F) is the force of attraction, (m1 and m2) are the masses and (G) is the constant of proportionality, often called the universal constant of gravitation, which equal 6.67 6.672 10 ⁄ . 10 . ⁄ [14] or [17,39,40,41]. 3.2. Equations Of Motion The two-body problem, was first stated and solved by Newton. Given at any time the positions and velocities of two massive particles moving under their mutual gravitational force, the masses also being known, provide a means of calculating their positions and velocities for any other time. At any moment, the velocity vector of one of the masses relative to the other, and the line joining them, defines a plane. The gravitational force between the bodies acts along the line joining them. Let the two particles be (P1 and P2) of masses (m1 and m2), and with coordinates (x1, y1) and (x2, y2) with respect to the axes (Ox' and Oy'). The distance (r) is: 37 Chapter three Two Body Problem Without Perturbation The magnitude of the force of gravity (F) is given by equation (3.5) as Consider particle (P1). It is attracted towards (P2), experiencing an acceleration which can be thought of as resolved into components (d2x1/dt2) and (d2y1/dt2) along the (Ox' and Oy') axes respectively. The force (F) can similarly be resolved into components along the (Ox' and Oy') axes. In the case of particle (P1), the force acts from (P1 to P2) so that for (P1), the components are: Y' Y (x2,y2) (x,y) m2 P2 r m1 X P1(x1,y1) O X' Figure (3.2): The two-body problem presented in a rectangular coordinate frame. And respectively, since for (P2) the force acts from P2 to P1. By Newton’s laws first and second ( ), the first particle (P1) written as: (3.6) 38 Chapter three Two Body Problem Without Perturbation It is called differential equations of motion of particle (P1). For the second particle (P2), we have (3.7) By dividing both sides of equations (3.6) and (3.7) on (m1 and m2) one has: (3.8) Subtracting the first equation from the second, one obtains: 0 3.9 If we take a set of axes (P1x, P1y) through (P1) and the origin, with (P1x and P1y) parallel to (Ox' and Oy') respectively, we see that the coordinates (x) and (y) of (P2) with respect to these new axis are given by x = x2 − x1 ; y = y2 − y1. Letting µ = G(m1 + m2) then equation (3.8) may be written as: 0 3.10 By treating second part of equations (3.6) and (3.7) in a similar fashion, we are led to the equation: 0 3.11 In general the equation of motion can be written for the center of mass as [12,42,43]: 0 39 3.12 Chapter three Two Body Problem Without Perturbation 3.3. The Solution Of The Two-Body Problem The solution of (3.12) may be written as (see appendix A) ⁄ 3.13 1 Where (e) is the eccentricity of the orbit and (f) is the true anomaly and (h) is a constant which is twice the rate of description of area by the radius vector, the relation can write as [12,27,43,44,45]: 2 Where (a and b) is the semi major and semi minor axis. Equation (3.13) is the polar equation of a conic section. By obtaining this solution, Newton generalized Kepler’s first law, for a conic section can not only be an ellipse but also a parabola or a hyperbola. 3.4. The Energy Integral If we multiply equations (3.10) by (dx/dt) and (3.11) by (dy/dt) and add, we obtain the relation. 0 3.14 Now 1 2 1 2 Also, r2 = x2 + y2 , so that 2 giving 40 2 Chapter three Two Body Problem Without Perturbation Hence, equation (3.14) may be written as a perfect differential, namely 1 2 1 2 0 Integrating, we obtain [12,14,46] 1 2 3.15 Where (C) is the called energy constant and (V) is the velocity of one mass with respect to the other, and we can benefit from it to determine the shape of the orbit as [27]: C < 0 , the orbit is ellipse C = 0 , the orbit is parabola C > 0 , the orbit is hyperbola The term, , is the kinetic energy per unit mass, energy the planet possesses in its orbit about the Sun by virtue of its speed. The term, (−µ/r) ,is the potential energy, energy the planet possesses by virtue of its distance from the Sun. Equation (3.15) states are that the sum of these two energies is a constant, a reasonable statement since the two-body is an isolated system, no energy being injected or removed from system. In an elliptic orbit, the distance (r) is changing and it shows that there is a continual trade-off between the two energies: when one is increasing, the other is decreasing. If we wish to obtain an expression giving the velocity (V) of the planet, we must interpret the constant (C) [14]. 41 Chapter three Two Body Problem Without Perturbation 3.5. The Velocity Of a Planet In Its Orbit Let (P) be the position of a planet in its elliptical orbit about the Sun (S) at a given time when its velocity is (V) and its radius vector (SP) has distance (r) from focus to point present body (see figure 3.3). Let (VP, VA) be the velocities at perihelion (A) and aphelion (A') respectively. The points (A, A') are the only places in the orbit where the velocity is instantaneously at right angles to the radius vector and where, consequently, we may write V = rωv (3.16) Where (ωv) is the angular velocity. At every point Kepler’s second law holds, namely that r2 ωv = h (3.17) Hence, at perihelion and aphelion only, we have V = h/r For perihelion, VP = h / a(1 – e) For aphelion, VA = h / a(1 + e) So that 1 1 3.18 Now the energy equation (3.15) is: 1 2 P V A' VP r f Ce S A VA Figure (3.3.): The velocity of a planet in an elliptical orbit showing that at perihelion A and aphelion A' the velocity vector is perpendicular to the radius vector. 42 Chapter three Two Body Problem Without Perturbation So that at perihelion, we have 1 2 1 1 2 1 3.19 While at aphelion, 3.20 Subtracting equation (3.20) from equation (3.19) and using relation (3.17) to eliminate (VA), We obtain 1 1 3.21 1 1 3.22 In similar fashion, we obtain Again, equations (3.21) and (3.22) give VAVP = µ/a (3.23) Subtracting equation (3.19) from equation (3.20) and using (3.21) to eliminate (VP), we obtain, after a some reduction, the required relation [12]. 3.24 3.6. The Orbital Element In three-dimensional spaces, it takes three parameters each to describe position and velocity. Therefore, any element set defining an object's orbital motion requires at least six parameters to fully describe dynamics of orbital motion. There are different types of element sets, depending on the use. The Keplerian, or classical, element set is useful for space operations and tells us four parameters about orbits, namely [41]: 43 Chapter three Two Body Problem Without Perturbation • Orbit size • Orbit shape • Orientation - orbit plane in space • Orbit within plane And the following six quantities are called the orbital elements: (figure 3.4.) [10,47] Semi-major axis (a): it is distance between the geometric center of the orbital ellipse and the perigee passing through the focus of center mass. The value of (a) depend on the type of conic, where a=∞ for parabola orbit 0<a<∞ for ellipse orbit -∞ < a < 0 for hyperbola orbit Inclination (i): the angle between the main plane and body's plane. Longitude of the ascending node (Ω): the angle between the vernal equinox vector (γ) and the line of ascending node therefore, it is called right ascension of ascending node. p m f E a ω Ω Earth Equatorial plane i n Moon orbital plane Figure (3.4.): The orbital elements. 44 Chapter three Two Body Problem Without Perturbation Argument of the perigee (ω): the angle from the ascending node to the line between the center and perigee. Time of the perigee passage (tp): it is the time which the body passes in the perigee. Eccentricity (e): the parameter describes how flattened or the ellipse is compared with a circle. Or how elongated the ellipse. It is defined by the relation (3.3) In any way the eccentricity gives the shape of orbit (see figure 3.5.) e=0 for circler orbit 0<e<1 for ellipse orbit e=1 for parabola orbit 1<e<∞ for hyperbola orbit e<1 e=0 e=1 e>1 Figure (3.5.): The shape of the orbit for different values of the eccentricities. Table (3.1) summarizes the Keplerian orbital element set, and orbit geometry and its relationship to the Earth. 45 Chapter three Two Body Problem Without Perturbation Table (3.1) Classical orbital elements [41]. Elements Description Definition Remarks Semi-major Half of the long axis Orbital period and energy Orbit size axis of the ellipse depend on orbit size Ratio of half the Closed orbit 0 ≤ e < 1 Eccentricity Orbit shape focus separation (c) to Open orbit 1 ≤ e the semi-major axis Angle between the orbital plane and Equatorial i = 0 or 180 Orbital equatorial plane, Prograde 0 ≤ i < 90 Inclination planes tilt measured Polar i = 90 counterclockwise at Retrograde 90 ≤ i < 180 the ascending node Right Orbital Angle measured 0 ≤ Ω < 360 ascension of planes rotates eastward from the Undefined when i = 0 or 180 ascending about the vernal equinox to the Equatorial orbit node earth ascending node Angle measured in 0 ≤ ω < 360 Orbit the direction of object Argument of Undefined when i = 0 or 180 orientation in orientation from the perigee Or e = 0 circular orbit orbital plane ascending node to perigee Angle measured in 0 ≤ f < 360 Object True the object motion, Undefined when e = 0 location in its anomaly from perigee to object Circular orbit orbit location 3.7. The Period Of Revolution Of a Planet In Its Orbit Let us suppose the orbit to be circular so that (r = a). Then expression (3.24) becomes V2 = µ/a (3.25) But V = 2π / T Where (T) is the time it takes the planet to describe its circular orbit, which calculated as equation (3.4) [14]. In other way we can find the period of the rotation using mean motion (n), where [45] 46 Chapter three Two Body Problem Without Perturbation 3.26 Hence 2 3.27 If we consider (tp) time of passing the object in perigee, the mean anomaly in any time is: (3.28) And we can calculate the eccentric anomaly for the orbit as [43,46]: (3.29) yw the auxiliary circle a r o E a f xw ae the orbit of body's motion Figure (3.6.): The geometry of an elliptical orbit. This equation is called Kepler equation, although its looks as simple equation but its solved using numerical methods [27]. To find the cartesian coordinate (xw and yw) to the Moon in his orbit, which is illustrated in figure (3.6.) as: 47 Chapter three Two Body Problem Without Perturbation cos cos sin cos sin 1 sin and the displacement radius (r) will be [40,43,44,48]: 1 e cos (3.30) By direct differentiation for (xw and yw) one obtains: cos sin √ sin sin sin 1 cos 1 cos cos sin (3.31) Or V √ sin The conversion of position and velocity of the Moon from this orbital plane to the Earth equatorial plane can be utilized by Gaussian vector (conversion matrix), which content Eular angle [8,27,44,49]. Where R-1 is the inverse of Gauss matrix 48 Chapter three Two Body Problem Without Perturbation The elements are: cos cos Ω sin sin Ω cos cos sin Ω sin cos Ω cos sin sin sin cos Ω cos sin Ω cos sin sin Ω cos cos Ω cos cos sin sin Ω sin cos Ω sin cos Thus Also 49 Chapter three Two Body Problem Without Perturbation 3.8. The Orientation Of The Orbital Plane represents angular momentum of the orbit, and it The vector is vertical vector to the orbital plane which contains position vector velocity vector and . In cartesian coordinates can be calculated as follows [35,36,39,44]: Hence And 3.9. Calculating The Orbital Elements The elliptical orbital elements in general are (i, Ω, ω, a, e, M) can be calculated from the component of position, velocity and angular momentum as follows : a- The inclination (i) of the orbit from the equatorial plane is given by [36,44,49]: tan 3.32 Or as [12,28,35,50]: cos 50 Chapter three b- Two Body Problem Without Perturbation The longitude of ascending node (Ω) is calculated as [28,36,44,49]: tan c- 3.33 The argument of perigee (ω) can be found as [2,36,51]: (3.34) Where ( ) is the true anomaly and ( ) is argument of the latitude [36] tan Or as [35]: cos d- 1 cos Ω sin Ω , sin The semi-major axis (a) of the orbit calculated as: 2 e- 1 sin 3.35 The eccentricity (e) of the orbit is calculated as [44,49]: 3.36 1 Or as: 1 f- √ The eccentric anomaly (E) is calculated as [44,49]: tan 1 3.37 51 Chapter three g- Two Body Problem Without Perturbation The mean anomaly (M) is calculated as [44,49]: 3.38 √ h- The true anomaly ( ) is calculated as [36,44,49,52]: sin √1 cos tan 3.39 Or as: tan 2 1 1 tan 2 3.10.Solution Of Kepler's Equation The equation of Kepler (3.29) has incompletely solution [17,34,53] E = M + e sin E Where (e) is the eccentricity of the orbit, (M) the mean anomaly at a given instant, and (E) the eccentric anomaly. Generally, (e) and (M) are given, and the equation must be solved for (E). The eccentric anomaly (E) is an auxiliary quantity which is needed to find the true anomaly (f). The above equation cannot be solved directly. There are some methods for finding (E), and finally a formula which gives an approximate result. a. First method It uses the angles (M) and (E) in degrees, and multiply (e) by (180/π) to convert from radians into degrees which is denoted by (e0). Kepler's equation is then [12,28,35] E = M + eo sin E sin 52 (3.40) Chapter three Two Body Problem Without Perturbation To solve equation (3.40), an approximate value to (E) should be given. Then the formula will give a better approximation for (E). This is repeated until the required accuracy is obtained. For the first approximation, use E = M We thus have Eo= M El = M + eo sin Eo E2 = M + eo sin El E3 = M + eo sin E2 etc. Ei+1 = M + eo sin En Where El , E2 , E3, etc. are successive and better approximations for the eccentric anomaly (E). This method is simple, and it is accurate when (e) is small. b. Second method When (e) is larger than 0.4 or 0.5, the convergence may be so slow that a better iteration formula should be used a better value (El for E) is [12,28,35]. 1 or 1 Where (Eo) is the last obtained value for (E). In this formula, all quantities are expressed in degrees. It is important to note that the numerator of the fraction contains the eccentricity (eo) defined before, while the denominator contains the ordinary eccentricity (e). Here, again, the process can be repeated as often as is necessary. 53 Chapter three Two Body Problem Without Perturbation c. Third method The formula [34] tan sin cos Gives an approximate value for (E), and is valid only for small values of the eccentricity. d. Fourth method [44,52] Find the root of equation (3.29) sin Find the derivative of equation (3.29) with respect to Ei 1 cos Apply Newton-Raphson method base in approximately ∆ Determine a new (E) from ∆ Continue the repetition until |∆ | where is a small constant appropriately chosen to correspond to the extent of precision desired in the calculation. 54 Chapter Four PERTURBATIONS Chapter four Perturbations 4.1. Introduction Perturbation theory is a very broad subject with applications in many areas of the physical sciences. Indeed, it is almost more a philosophy than a theory. The basic principle is to find a solution to a problem that is similar to the one of interest and then to cast the solution to the target problem in terms of parameters related to the known solution. Usually these parameters are similar to those of the problem with the known solution and differ from them by a small amount. The small amount is known as a perturbation and hence the name perturbation theory [33]. The expression "perturbed motion" implies that there is an unperturbed motion. In Celestial Mechanics the unperturbed motion is the orbital motion of two spherically symmetric bodies represented by the equation of motion (3.11) [28,41]. The constant (µ) is the product of the constant of gravitation and the sum of the masses of the two bodies considered. The numerical value and unite of (µ) thus depends on the concrete problem and on the system of units chosen. The perturbed motion of a celestial body is defined as the solution of an initial value problem as [28,33,47,54,55]: ∆ The term and ∆ 4.1 in equation (4.1) is called "the two-body term", the perturbation term which is the summation of other external force. If the acceleration is a constant, then the solution to the equations of motion will be the solution to the two-body problem. If the perturbation term is considerably smaller than the two-body term [28,33], then. ∆ 56 Chapter four Perturbations Then the differential equation system (4.1) is called "the perturbation equations". Every method for solving this problem is called a "perturbation solution method". In Celestial Mechanics one usually makes the distinction between • General Perturbation Methods, seeking the solution in terms of series of elementary integrable functions. • Special Perturbation Methods, seeking at some stage the solution by the methods of numerical integration. For general perturbation methods, it is mandatory not to use the original equations of motion (4.1) in rectangular coordinates, but to derive differential equations for the osculating orbital elements or for functions thereof. This procedure promises to make the best possible use of the (analytically known) solution of the two-body problem (3.11), because the osculating elements are so-called first integrals of the two-body motion. Special perturbation methods may be applied directly to the initial value problem or to the transformed equations for the osculating elements. Both, general and special perturbation methods provide approximate solutions of the equations of motion [28,41]. 4.2. Orbit Perturbations There are two categories of motion under the influence of perturbative forces: special perturbations and general perturbations. The methods of general perturbations are used to calculate the effect of perturbative forces on the orbital parameters. Analytical integration of series expansions of the perturbative accelerations are carried out to calculate these changes over long periods of time. The methods of special perturbations entail the step-by-step numerical integration of the equations of motion and provide the desired short term solutions for the in orbit position [56]. 57 Chapter four Perturbations These perturbing forces can be classified, based on their effect on an object's Keplerian orbital elements vs time as be illustrated in figure (4.1.). Secular variation represents a linear change with extremely long effect time relative to the period. Short-period variations are periodic with a period of less than or equal to the orbit period. Long-period variations are also periodic, but have a period greater than the orbit period [47,50]. Figure (4.1.): Secular and periodic variations of an orbital element [47,50]. The common forces, which perturb an object's orbit, are listed below and will be discussed in the following sections: Atmospheric drag Non-spherical Earth (Oblateness) Solar Radiation Pressure (SRP) Third body attractions (such as the Sun) Other perturbation forces The above perturbing forces have different effects and intensities, depending on the position of the object in space (its orbital elements according to the appropriate reference frame) and also the position of the object at a particular point in its orbit. 58 Chapter four Perturbations 4.2.1. Atmospheric Drag Although the atmosphere at hundreds kilometers altitude is extremely thin, during the impact between the high speed satellite and the atmosphere particles, momentum transfer happens, accumulates and affects satellites motion greatly. This process generates atmospheric drag which is nonconservative. The derivation of a mathematical model of atmospheric drag is based on the following assumptions [47]: • The momentum of particles hitting at the surface is totally lost to the surface. • The mean thermal motion of the atmosphere is much smaller than the speed of the object as relative to the local atmosphere. • For spinning vehicles, the relative motion between surface elements is much smaller than the speed of the mass center. The magnitude of the drag force can be stated as [12,41,46,56]: 1 2 4.2 The negative sign indicates that the acceleration is in a direction opposite to this unit vector. And (ρ is the atmospheric density, (CD) is the drag coefficient, has a value between 1 and 2. It takes a value near 1 when the mean free path of the atmospheric molecules is small compared with the satellite size, and takes a value close to 2 when the mean free path is large compared with the size of the satellite, (m) is the mass of the satellite, (V) is the velocity of the satellite relative to the atmosphere and (Aa) is the effective surface area which equal Aa = Ao cos ζ . Where (Ao) is the local surface area and (ζ) is the angle between surface normal direction and velocity direction. 59 Chapter four Perturbations Atmospheric drag is the most complex and the most difficult of the important artificial satellite perturbations because the function from of the force law is not known and the atmosphere is variable [41]. Elements such as the atmospheric composition, temperature and density of the atmosphere are strongly related to the solar cycle. The Sun, having an eleven-year Sunspot activity cycle, causes the most uncontrolled re-entry of spacecraft to occur at its peak. This occurs because as the Sunspot activity peaks, it correlates with an increase in atmospheric density, which amplifies the effect of atmospheric drag on a spacecraft. An important factor that determines the intensity of drag perturbation is the spacecraft's coefficient of drag, (CD). The (CD) describes the molecular interaction between the atmosphere and the spacecraft's surfaces. The smaller the coefficient, the more aerodynamic the vehicle and the less intense the effect of atmospheric drag when compared with a spacecraft of a large coefficient of drag. Other factors that describe the level of intensity of drag on a spacecraft are the material and temperature of the vehicle's surfaces, along with their orientation with respect to the oncoming atmospheric particles [50]. Atmospheric drag is an important concern for satellites in the Low Earth Orbit (LEO) region. At altitudes above LEO, drag due to the atmosphere becomes less problematic as its influence decays significantly with increasing altitude. The density of the upper atmosphere can be modeled by a simple analytical equation if the following assumptions are made [50]: • The Earth is spherically symmetric. • The scale height is constant over the altitudes of interest. • There is no time variation in the density. 60 Chapter four Perturbations A simplified useful approximation for the atmospheric density is given by the barometric formula [46,54]: Where (ρo) is the density of the air at the surface of the Earth, (Batm) is called the barometric coefficient, and ( ) a function of the altitude which is in principle different for each layer of the Earth’s atmosphere. 4.2.2. Non-Spherical Gravitational Field Of The Earth Like all planets with high rotational rates, the Earth has developed an equatorial bulge and flattening around its poles. Like an oblate pear with radius in a range from 6357 km to 6378 km. The mean radius of the equator is about 21.4 km longer than that of the poles. The distribution of the mass over the Earth is non-uniform. All these generate extra disturbance force in the velocity and orbital normal direction on the satellite [47]. Unlike a uniformly spherical object, in which the gravitational field depends only on the distance from its center, the Earth's nonuniformity causes gravitational perturbations on a spacecraft to depend on the latitude and longitude of its orbit about the Earth. The variation with latitude is called zonal variation and the dimensionless parameter with the principal effect on spacecraft with an inclined orbit is known as (J2), the second harmonic. The two major effects of (J2) is a regression in the line of nodes and a precession of the line of apsides. The equatorial bulge (Oblateness) of the Earth produces a torque, which rotates the angular momentum vector, causing a regression in the line of nodes. Precession of the line of apsides represents an overall rotation of the orbit within the orbit plane [50]. 61 Chapter four Perturbations In deriving the ideal elliptical Keplerian orbit, it is assumed that the Earth can be modeled as a point mass with a spherical gravitational field. For the purposes of accurate orbit determination, this assumption is no longer valid. The mass of the Earth causes its gravitational field to deviate from the ideal spherical model. A convenient way to account for this variation is to model the Earth's gravitational potential by a spherical harmonics expansion. In this expansion, the value of the J2 zonal coefficient is three orders of magnitude larger than all the other coefficients and thus dominates the gravitational perturbative influences of Earth. It represents the equatorial bulge (Oblateness) of the Earth. If all but this term is neglected and the gradient of the scalar potential function is taken, the vector perturbative acceleration on satellite follows [56]: 3 2 3 3 4 4 2 Where (Re) is the radius of the Earth and (x, y, and z) are the components of the satellite position (r). 4.2.3. Solar Radiation Pressure (SRP) In the space environment, the body is also affected by the solar radiation, which is an effect of electromagnetic radiation, SRP is a force on the satellite due to the momentum flux from the Sun. For most satellites it acts in a direction radially away from the Sun. The magnitude of the resulting acceleration on the satellite is given by [56,57]: a 62 Chapter four Perturbations Where (KS) is a dimensionless constant between 1 and 2 (KS=l: surface perfectly absorbent; KS=2: surface reflects all light), (PS) is the momentum flux from the Sun, (AS) is the cross-sectional area of the satellite perpendicular to the Sun-line and (mS) is the mass of the satellite. Some of the solar radiation is absorbed while the other is reflected. There are two different principles for describing the radiation reflection on a surface, namely specular reflection and diffusive reflection. In specular reflection, the incoming particles have an elastic impact with the surface regulated by the reflection law; in diffusive reflection, atmospheric particles penetrate the satellite surface material, interact with the surface molecules, and finally re-emitted at a number of random angles. This energy transfer process generates the force on the objects [47]. The strongest effects from SRP are observed on spacecraft with large incident surface areas (for example large solar array surfaces) and low mass. The extent of perturbation is also directly linked to the orientation of the orbit. According to the equations below, it is difficult to quantify the level of perturbation a spacecraft receives as a result of SRP because the disturbing force depends on the spacecraft's distance from the Sun, rather than its Earth orbiting altitude [50]. Where is the solar energy flux at the spacecraft and (c) is the speed of light. 63 Chapter four Perturbations 4.2.4. Third Body Attractions The term "third-body" refers to any other body in space besides the Earth which could have a gravitational influence on the Moon. The most significant influences come from the Sun. Planetary gravitational influences are orders of magnitude in different degree depend on position from the Earth or the Moon and its mass. The perturbing acceleration due to the gravitational attraction of a third body can be calculated as follows [56]: Where (µd) is the gravitational parameter of the third body and the definitions of the vectors are given in the following diagram: Moon rM Earth rdm Third body rd Figure (4.2.): Vector definitions for third body attraction. The functions (f) and (q) are: . 2 . 3 1 3 1 ⁄ 64 Chapter four Perturbations Or other form can be expressed on acceleration due to third-body forces as [57]: The main body which has an effect on the Moon motion is the Sun. Although gravitational force decreases as the square of the distance, the Sun's gravity exceeds that of the Earth, because of its far greater mass. We find that the Sun's gravitational attraction is about twice as great as that of the Earth: 2 Where: (M) is Sun's mass 2×1030 kg and (R) is Sun–Moon distance 150 million Km, (m) is Earth's mass distance 6×1024 kg and (r) is Earth–Moon 380 thousand Km The difference between the forces exerted by the Sun on the Moon and on the Earth is about (r / R) times the amount of the force itself, and is thus about 200 times smaller than the force that the Earth exerts on the Moon [16]. The most useful description of the Moon's orbit is therefore the one expressed in a geocentric reference frame. In it, the Moon moves around the Earth in a monthly Keplerian orbit, which is perturbed to a greater or smaller degree by the Sun. The Earth, Moon and Sun attract each other according to Newton’s law of gravitation, all three bodies being taken to be pointmasses. Everything else, the finite sizes of Earth and Moon, tidal effects, the attractions of the planets, etc., may be taken to be small. 65 Chapter four Perturbations 4.3. Other Perturbation Forces There are other perturbation forces, but since they are all significantly smaller than those mentioned then can be neglected during calculation or simulations. The other perturbations include [56]: • relativistic effect • aerodynamic lift • induced Eddy currents in the satellite structure interacting with the Earth's magnetic field • Earth-reflected solar radiation pressure (Albedo) • drag due to solar wind • gravitational effects of Earth tides and ocean tides • precession and nutation of the Earth's • meteorites For more information about satellite's perturbation see [6:3rd section] 66 Chapter Five RESULTS, DISCUSSION, CONCLUSIONS AND FUTURE WORK Chapter five Results, Discussion, Conclusions, and Future Work 5.1. Introduction The program was designed using Quick-Basic language to solve the equations. Two techniques were used to calculate the orbital elements for the Moon: The first one depend on practical formula for the Moon distance and ecliptic coordinate, as in equations (2.12), (2.13) and (2.14) respectively, the Moon position and velocity components used to calculate angular momentum components to find the orbital elements as in equations (3.32) – (3.39). The calculation began from the date of perigee, where determined by the Moon distance (Rm) scanning to find minimum distance, see flowchart (3). The results obtained by this method are illustrated in the following figures as a dished line. The second technique is found using the Moon orbital elements by solving Kepler's equation of motion equation (3.29) using NewtonRaphson method to calculate eccentric anomaly then calculate the position and velocity components, equations (3.30) and (3.31) after that, the position and velocity components are converted from the Moon orbital plane to the geocentric equatorial plane by multiplying them with the inverse of Gaussian matrix, after that using the same equation to calculate the Moon orbital elements as equations (3.32) – (3.36), see flowchart (4). The results obtained by this method are illustrated in the following figures as a solid line. Finally the results of both above methods are compared to find the perturbation effect on the orbital elements (between the Keplerian motion and practical motion) for the Moon's orbit and it's variation with time, see flowchart (5). The results which obtained are illustrated in the following figure for each element for four different months of the year 2010. 68 Chapter five Results, Discussion, Conclusions, and Future Work 5.2. Results and Discussion 1) The distance of the Moon (Rm) from the centre of the Earth found using two techniques for four different months in the year 2010 which are illustrated in figure (5.1). As shown the curves are like sine function and the Moon started from perigee which means the nearest point in its rotation from the Earth and reach to farthest point after about 14 days in apogee of its orbit. In astronomy textbooks it is found that the mean distance from the centre of the Earth to the Moon is 384400 km, and that the eccentricity of the lunar orbit is e = 0.0549. From these values the minimum (perigee) and the maximum (apogee) distances can be deduced between the centre of the two bodies according to equation (3.2) and found as 363296 km and 405504 km, respectively [58]. This is right if we suppose that the orbit is stable but the major axis of the lunar orbit changed direction toward the Sun, see figure (5.5), near these epochs the eccentricity of the lunar orbit reaches a maximum, and the perigee distance of the Moon is much smaller than normal and the apogee distance is larger. When the major axis of the lunar orbit is perpendicular to the direction of the Sun, the eccentricity reaches a minimum; at these epochs, perigee and apogee distances are less extreme. Michelle Chapront - Touze and Jean Chapront, who calculated the perigee and apogee distances of the Moon for the years 1960 to 2040, 81-year period, the perigee distances of the Moon, vary between 356445 km and 370354 km, which gives a spread of 13909 kilometers, during same period the apogee distance varies from 404064 km to 406712 km, a variation of only 2648 km [58]. 69 Chapter five Results, Discussion, Conclusions, and Future Work As for the much longer period of AD 1500 to 2500, during these ten centuries, distances were less than 356425 km, and they grow to larger than 406710 km, during the time interval of ten centuries considered, the extreme distances between the centre of the Earth and the Moon are 356371 km and 406720 km [58]. The distance has not been recorded smaller than the above value in both methods, the smallest distances were 357473 km and 360863 km with and without perturbation. In large distance the recorded value was 407545 km without perturbation in a second method but the large distance was 406177 km with perturbation in a first method which mean it is less than the extreme values. See table (5.1) Table (5.1): The maximum, minimum and average the Moon distance by two techniques for four different months in the year 2010 and the difference between two methods for each month. a b c d Method With out With pert. ∆R With out With pert. ∆R With out With pert. ∆R With out With pert. ∆R RmMax(km) RmMin(km) 404536 362457 404979 362044 443 413 406570 364279 405117 361883 1453 2396 402757 360863 406177 357473 3420 3390 407545 365153 404605 362910 2940 2243 RmAve(km) 383703 384429 726 385632 384808 824 382015 385397 3382 386557 384408 2149 In general the distance of the Moon calculated by the first method is varying around it is value from the second method and some time they are nearly the same value, that means the minimum effect of perturbation. 70 Chapter five Results, Discussion, Conclusions, and Future Work R 4.0E+05 3.9E+05 3.8E+05 3.7E+05 3.9E+05 3.8E+05 3.7E+05 3.6E+05 b R 4.1E+05 4.1E+05 RW R RW 4.0E+05 4.0E+05 The distance R (km) The distance R (km) 2455392 2455388 2455384 2455380 2455376 2455372 2455368 2455364 2455360 2455312 2455308 2455304 2455300 2455296 2455292 2455288 2455284 2455280 The date (day) a 3.9E+05 3.8E+05 3.7E+05 3.6E+05 3.9E+05 3.8E+05 3.7E+05 3.6E+05 2455588 2455584 2455580 2455576 2455572 2455568 2455564 2455560 c 2455556 The date (day) 2455552 2455480 2455476 2455472 2455468 2455464 2455460 2455456 2455452 2455448 2455444 The date (day) RW 4.0E+05 3.6E+05 The date (day) R 4.1E+05 RW The distance R (km) The distance R (km) 4.1E+05 d Figure (5.1): The Moon distance (Rm) variation with date for four different months in the year 2010 using two techniques (with RW and without R). 2) The results of velocity of the Moon (V) is illustrated in figure (5.2) from the two techniques used for the four different months as seen the curves like half of cosine function and the velocity difference with the distance with 180o which mean when the Moon near to the Earth, in perigee, the velocity reach the maximum and vice versa according to second Kepler law. The velocity is more similar when the Earth is near the vernal and autumn equinoxes figure (5.2: a, c) and it was different when the Earth is near the aphelion or perihelion in its orbit around the Sun figure (5.2: b, d). 71 Chapter five Results, Discussion, Conclusions, and Future Work 1.1 1.08 1.06 1.04 1.02 1 0.98 0.96 0.94 VW V 1.1 1.08 1.06 1.04 1.02 1 0.98 0.96 0.94 VW The velocity V (km/s) V 1.1 1.08 1.06 1.04 1.02 1 0.98 0.96 0.94 VW d Figure (5.2): The Moon's velocity (V) variation with date for four different months of the year 2010 using two techniques (with VW and without V). 3) The semi major axis (a) is illustrated in table (5.2) and figure (5.3) using two techniques for the four different months; the plots shows the values of (a) (without perturbation) are constant through month but it different from month to another. The difference between two techniques when the Earth at the vernal and autumnal equinoxes figure (5.3: a, c) is uniform and smaller than the difference when the Earth at the aphelion or perihelion figure (5.3: b, d). For some months the variation of actual semi major axis (a) is like a wave around the fixed value (without perturbation) and for other months the variation are not uniform waves. 72 2455588 2455584 2455580 2455576 2455572 2455568 2455564 2455560 2455556 The date (day) V 2455552 2455480 2455476 2455472 2455468 2455464 2455460 2455456 2455452 2455448 2455444 c 2455392 2455388 2455384 2455380 The velocity V (km/s) VW b 1.1 1.08 1.06 1.04 1.02 1 0.98 0.96 0.94 The date (day) 2455376 a 2455372 2455368 The date day) 2455364 2455360 2455312 2455308 2455304 2455300 2455296 2455292 2455288 2455284 2455280 The date (day) V The velocity V (km/s) The velocity V (km/s) Chapter five Results, Discussion, Conclusions, and Future Work The table (5.2) shows the maximum and minimum values of (a) through the year 2010 are 396531 km and 369713 km that means the perturbation on the (a) element are 11091 km and 7947 km. Table (5.2): The maximum, minimum and average of the semi major axis (a) of the Moon by two techniques for four different months and the difference between them. The semi major axis a (km) a b c d aMax 387645 396531 390172 395075 SMA 4.0E+05 aMin 373782 369713 373878 372920 SMAW 3.9E+05 3.8E+05 aAve 381193 382224 381654 381588 aMax-a 4133 11091 8347 8710 a-aMin 9730 15727 7947 13445 4.1E+05 SMA 3.9E+05 3.8E+05 3.7E+05 SMA SMAW 3.9E+05 3.8E+05 3.7E+05 SMAW 3.9E+05 3.8E+05 3.7E+05 d Figure (5.3): The semi major axis (a) of the Moon in km with date for four different months in the year 2010 using two techniques. 73 2455588 2455584 2455580 2455576 2455572 2455568 2455564 2455560 2455556 The date (day) 2455552 2455480 2455476 2455472 2455468 2455464 2455460 2455456 2455452 2455448 2455444 c SMA 4.0E+05 The semi major axis a (km) The semi major axis a (km) b 4.0E+05 2455392 2455388 2455384 2455380 2455376 2455372 2455368 2455364 2455312 2455308 2455304 2455300 2455296 2455292 2455288 2455284 2455280 The date (day) a The date (day) SMAW 4.0E+05 2455360 3.7E+05 The date (day) ∆a With perturbation The semi major axis a (km) Without perturbation a (km) 383512 385440 381825 386365 Chapter five Results, Discussion, Conclusions, and Future Work 4) Eccentricity (e), as illustrated in figure (5.4), was fixed in all months at a value 0.0549 when the second technique is used and it was varied as shown in table (5.3) when the first technique (with perturbation) is used. The same behavior during all months can be seen. The actual values of (e) are vibrate around the fixed value as nonuniform wave because the effect of the near planets, as well as the Sun’s gravitational pull, which are variable in distance with time period is 31.8 days [12] which know as evection. The evection largest periodic perturbation of the Moon’s longitude, caused by the Sun, displacing it from its mean position by ±1o16'26".4 [7]. Table (5.3) shows the maximum and minimum values of (e) through the year 2010 as 0.093369 and 0.023901 which means that the perturbation value of (e) element are 0.038469 and 0.018285. Table (5.3): The maximum, minimum and average value of actual eccentricity (e) of the Moon orbit for four different months using first technique (perturbed) and the difference each month. a b c d eMax 0.07633 0.093369 0.089551 0.091675 eMin 0.025996 0.036615 0.023901 0.035645 eAve 0.059597 0.061992 0.06071 0.062119 eMax-e 0.02143 0.038469 0.034651 0.036775 e-eMin 0.028904 0.018285 0.030999 0.019255 Eccentricity is at maximum when the major axis of the lunar orbit is directed toward the Sun, see figure (5.5), this occurs at mean intervals of 205.9 days, and it is value can vary between the extremities 0.026 and 0.077 [58], see figure (5.7), or it varies from 0.044 to 0.067 [12] or its varies between 0.0432 and 0.0666 [9]. Present result varied between 0.0239 and 0.0933, see table (5.3), that means extreme the minimum value and exceed the maximum value in the year 2010. 74 Chapter five Results, Discussion, Conclusions, and Future Work ECCW 0.08 0.06 0.04 0.02 0 0.1 ECC 0.06 0.04 0.02 0 ECCW 0.08 0.06 0.04 0.02 0 0.1 ECC ECCW 0.08 0.06 0.04 0.02 0 d Figure (5.4): The eccentricity (e) of the Moon as a function of time for four different months using two techniques for the year 2010. Figure (5.5): When the major axis of the Moon's orbit is aligned with the Earth - Sun line (A), the orbital eccentricity exceeds its mean value. About 103 days later, in B, the two lines are at right angle and the eccentricity reaches a minimum. A new maximum is reached again after another 103 days (C). Sizes and distances are not to scale! [58] 75 2455588 2455584 2455580 2455576 2455572 2455568 2455564 2455560 2455556 The date (day) 2455552 2455480 2455476 2455472 2455468 2455464 2455460 2455456 2455452 2455448 c The eccentricity e (without units) ECC 2455444 The eccentricity e (without units) b 0.1 2455392 2455388 2455384 2455380 2455376 2455372 2455368 2455364 The date (day) a The date (day) ECCW 0.08 2455360 2455312 2455308 2455304 2455300 2455296 2455292 2455288 2455284 The date (day) The eccentricity e (without units) ECC 0.1 2455280 The eccentricity e (without units) Chapter five Results, Discussion, Conclusions, and Future Work Figure (5.6): The effective eccentricity of the lunar trajectory as a function of time; the abscissa gives the time in synodic months, starting with the year 1980 [2]. Figure (5.7): The instantaneous eccentricity of the lunar orbit, 1996 to 1998 [58]. 76 Chapter five Results, Discussion, Conclusions, and Future Work 5) Inclination (i) as illustrated in figure (5.8) there is no change in inclination through each month but it differs in value from month to another when the second technique is used and its variation is clear when the first technique is used. The average value of variation in each month is very close to the value in same month as shown in table (5.4). In both cases the value of inclination decreases with time. As known, the mean inclination on Moon orbit is 5°9' with the ecliptical Earth plane, (inclination oscillates between 4°58' and 5°19') [8,12]. If the trend of the Moon orbit from the Earth equatorial plane is to be calculated, that depend on direction of nodes line, the Moon's ascending node coincides with the vernal equinox direction, the inclination of the Moon's orbit to the equator is a maximum, and when the descending node is at the vernal equinox, the inclination of the Moon's orbit to the equator is a minimum. Thus, the inclination relative to the equator varies between 5o09' + 23o26' (obliquity angle near 2000 AD) [12] = 28o35' or 18o17' [8] , which can be considered as the declination of the Moon. The variation of inclination with time and the difference is called perturbation in the inclination element by the Sun and planet effect on the Moon orbit as gravitational force caused nonuniform curve. From table (5.4) the maximum and minimum of (i) in the year 2010 are found with perturbation 25o.36414 and 24o.13273, and the perturbed terms are not constant but vibrated between 0.07276 and 0.13926. Table (5.4): The maximum, minimum and average values of inclination (i) of the Moon's orbit by first method and inclination (i) by second method for four different months of the year 2010 and the difference between them for each month. a b c d Without perturbation i (deg) 25.24803 25.16704 24.7453 24.40839 ∆i With perturbation iMax 25.36414 25.2398 24.84873 24.40839 iMin 25.10877 24.92401 24.35931 24.13273 iAve 25.29059 25.02852 24.64921 24.21669 iMax-i 0.11611 0.07276 0.10343 0 i-iMin 0.13926 0.24303 0.38599 0.27566 77 Chapter five Results, Discussion, Conclusions, and Future Work IN 25.4 INW The inclination i (deg) The inclination i (deg) 25.6 25.5 25.4 25.3 25.2 25.1 IN 25.3 25.2 25.1 25 24.9 24.8 b 24.9 IN 24.6 INW The inclination i (deg) The inclination i (deg) 2455392 2455388 2455384 2455380 a 2455376 2455372 2455368 The date (day) 2455364 2455312 2455308 2455304 2455300 2455296 2455292 2455288 2455284 2455280 The date (day) 2455360 25 24.8 24.7 24.6 24.5 24.4 IN INW 24.5 24.4 24.3 24.2 24.1 24 24.3 d Figure (5.8): The inclination angle (i) in degree for the Moon with the date using two techniques for four different months in the year 2010. 6) Longitude of ascending node (Ω) are as shown in figure (5.9) where there is no change in longitude of ascending node in each month but it is different in value from month to another when the second technique (without perturbation) was used and it was varied when the first technique (with perturbation) was used which can be seen in the table (5.5) The longitude of ascending node was retrograde with a period of 18.6 years for more accuracy is 6798.3 days [12]. That is the same period of motion of the nodes. The line of nodes is almost stationary when it is directed toward the Sun. This coincides with the maximum value of the orbital inclination, and it is near these epochs that solar and lunar eclipses taken place. 78 2455588 2455584 2455580 2455576 2455572 2455568 2455564 2455560 2455556 c The date (day) 2455552 2455480 2455476 2455472 2455468 2455464 2455460 2455456 2455452 2455448 2455444 The date (day) INW Chapter five Results, Discussion, Conclusions, and Future Work The mean values of (Ω) through the year 2010 with perturbation are varied between (11.3 – 12.8) degrees and its fluctuation continued to increase with date. Table (5.5): The maximum, minimum and average value of longitude of ascending node for the Moon orbit for four different months using two techniques in the year 2010. Without perturbation Ω (deg) 11.24975 12.10544 11.83937 12.91436 ΩMin 11.14856 11.82897 11.7462 12.49302 LOAN LOANW 12 11.8 11.6 11.4 11.2 11 ΩMax– Ω 0.55696 0.49826 0.6706 0.29073 12.7 12.3 12.1 11.9 11.7 b LOAN LOANW 12.4 12.2 12 11.8 11.6 LOANW 13.2 13 12.8 12.6 12.4 d Figure (5.9): The longitude of ascending node (Ω) of the Moon as a function of the date for four different months using two techniques. 79 2455588 2455584 2455580 2455576 2455572 2455568 2455564 2455560 2455556 The date (day) 2455552 2455480 2455476 2455472 2455468 2455464 2455460 2455456 2455452 2455448 2455444 c LOAN 13.4 The longitude of ascending node Ω (deg) 12.6 2455392 2455388 2455384 2455380 2455376 2455372 The date (day) 2455368 The longitude of ascending node Ω (deg) LOANW 12.5 a The date (day) Ω – ΩMin 0.10119 0.27647 0.09317 0.42134 LOAN 2455364 2455312 2455308 2455304 2455300 2455296 2455292 2455288 2455284 2455280 The date (day) ΩAve 11.344863 12.278654 12.018258 12.879839 The longitude of ascending node Ω (deg) ΩMax 11.80671 12.6037 12.50997 13.20509 ∆Ω 2455360 The longitude of ascending node Ω (deg) a b c d With perturbation Chapter five Results, Discussion, Conclusions, and Future Work 7) Eccentric anomaly (E) as illustrated in figure (5.10) where it is started from zero when the Moon in the perigee and reached to 360o at the end of period when Kepler equation of motion was solved using iteration of Newton–Raphson method. When the first technique was used with the true orbit which have small value of eccentricity it is shown that the eccentric anomaly varied up and down the straight line drawn by second technique. In general the variation was increasing when the eccentricity was ECAN ECANW 350 250 150 50 ECAN 250 150 50 150 50 ECAN ECANW 350 250 150 50 d Figure (5.10): The eccentric anomaly (E) in degree of the Moon against the time (JD) using two techniques for four different months in the year 2010. 80 2455588 2455584 2455580 2455576 2455572 2455568 2455564 2455560 The date (day) 2455556 -50 2455552 2455480 2455476 2455472 2455468 2455464 2455460 2455456 2455452 2455448 c The eccentric anomaly E (deg) ECANW 250 2455444 The eccentric anomaly E (deg) ECAN 2455392 2455388 2455384 2455380 b 350 The date (day) 2455376 2455372 2455368 2455364 -50 The date (day) a -50 ECANW 350 2455360 2455312 2455308 2455304 2455300 2455296 2455292 2455288 2455284 The date (day) 2455280 -50 The eccentric anomaly E (deg) The eccentric anomaly E (deg) increasing too. Chapter five Results, Discussion, Conclusions, and Future Work 8) Mean anomaly (M) is as illustrated in figure (5.11) and it is started from zero when the Moon in the perigee and reach to 360o at the end of period because it was depending on the eccentric anomaly in calculation. And when the first technique was used with the true orbit which have small value of eccentricity is shown the mean anomaly (with perturbation) varied up and down, the straight line drawn with second technique (without perturbation). It is very similar with the eccentric anomaly and the difference is very small because the eccentricity is very MEAN MEANW 350 250 150 50 MEAN 250 150 50 b MEAN The mean anomaly M (deg) The mean anomaly M (deg) 2455392 2455388 2455384 2455380 2455376 2455372 2455368 The date (day) 2455364 -50 a 350 250 150 50 MEAN 250 150 50 2455588 2455584 2455580 2455576 2455572 2455568 2455564 2455560 2455556 -50 The date (day) c MEANW 350 2455552 2455480 2455476 2455472 2455468 2455464 2455460 2455456 2455452 2455448 2455444 -50 The date (day) MEANW 350 2455360 2455312 2455308 2455304 2455300 2455296 2455292 2455288 2455284 2455280 -50 The date (day) The mean anomaly M (deg) The mean anomaly M (deg) small (less than 0.1). d Figure (5.11): The mean anomaly (M) in degree of the Moon against the time (JD) using two techniques for four different months in the year 2010. 81 Chapter five Results, Discussion, Conclusions, and Future Work 9) Argument of perigee (ω) as was mentioned in the previous chapter can be obtained from the difference between the argument of latitude and true anomaly as in equation (3.34). This angle was fixed in each month when the second technique without perturbation was used but it was changed from month to another because the line of apsides (line joining perigee and apogee) rotates in the direction of the Moon's orbital motion causing change by 360° in about 8.9 years [8] or 3232.6 days (8.85 years) [12]. The (ω) with first technique vary with time where it was near the sold line in the beginning of the month then it was decreasing near the middle of month then it begun increasing to reach the sold line again at the end of the month as illustrated in figure (5.12). Table (5.6): The maximum, minimum and average values of argument of perigee of the Moon for four different months in the year 2010 using two techniques and the difference between them for each month. a b c d Without perturbation ω (deg) 8.48867 -23.2014 24.36643 -6.31026 With perturbation ωMax 8.48867 -15.1283 25.15403 4.74243 ωMin -49.154 -73.1594 -35.1002 -52.4218 ωAve -16.3165 -46.8597 -4.28069 -19.1418 ∆ω ωMax – ω 0 8.07317 0.7876 11.05269 ω – ωMin 57.64264 49.95796 59.46665 46.11154 82 Chapter five Results, Discussion, Conclusions, and Future Work -40 -50 -60 -70 -80 The date (day) AOP 10 AOPW Theargument of perigee ω (deg) -30 -40 -50 -60 The date (day) d Figure (5.12): The argument of perigee (ω) of the Moon against time for four different months in the year 2010 using two techniques. The period of conjunct month (synodic month) calculated using first technique was started in the year 2010 bigger than mean period and went on decreaseing to less the mean period after the middel of the year and went on increasing to end of the year which exceed the mean value again, as shown in table (5.7). 83 2455588 2455584 2455580 2455576 c 2455572 The date (day) 2455568 -40 -20 2455564 -30 -10 2455560 2455480 2455476 2455472 2455468 2455464 2455460 2455456 2455452 2455448 -20 AOPW 0 2455556 0 AOP 2455552 10 2455444 Theargument of perigee ω (deg) b 20 -10 2455392 -30 a 30 2455388 The date (day) 2455384 -60 AOPW 2455380 -50 2455376 -40 2455372 -30 2455368 2455312 2455308 2455304 2455300 2455296 2455292 2455288 -20 2455284 -10 -20 2455364 0 -10 AOP 2455360 10 2455280 The argument of perigee ω (deg) 0 AOPW The argument of perigee ω (deg) AOP 20 Chapter five Results, Discussion, Conclusions, and Future Work Table (5.7): The period of the conjunct months of the Moon in the year 2010 A.D. No. of month The month length Day Hour Minute Second 1 29.8194 29 19 39 56 2 29.75714 29 18 10 17 3 29.64429 29 15 27 47 4 29.52413 29 12 34 44 5 29.4234 29 10 9 42 6 29.35112 29 8 25 37 7 29.31074 29 7 27 28 8 29.30668 29 7 21 37 9 29.34352 29 8 14 40 10 29.42189 29 10 7 31 11 29.53099 29 12 44 38 12 29.64412 29 15 27 32 Table (5.8): The date and time of the new Moon The date D M Y 15 1 2010 14 2 2010 15 3 2010 14 4 2010 14 5 2010 12 6 2010 11 7 2010 10 8 2010 8 9 2010 7 10 2010 6 11 2010 5 12 2010 My program N S H 11 44 7 51 40 2 1 56 21 43 12 29 4 28 1 9 11 14 46 19 39 7 14 3 51 10 28 31 18 43 51 2 4 39 17 35 [59] H N 7 12 2 52 21 2 12 30 1 5 11 15 19 41 3 9 10 30 18 45 4 52 17 36 [60] H N 7 14 2 54 21 4 12 32 1 7 11 17 19 42 3 10 10 31 18 46 4 54 17 38 [61] H 7 2 21 12 1 11 19 3 10 18 4 17 N 11 51 01 29 4 15 40 8 30 44 52 36 D= day, M= month, Y= year, H= hour, N= minute, S= second Table (5.8) shows the date and time of the new Moon using present program (first technique) which shows a good agreement with other references. This means that the present written program for this research was very accurate to calculate the Hegree date and the actual Moon orbital elements (with perturbation). 84 Chapter five Results, Discussion, Conclusions, and Future Work 5.3. Conclusions: 1) The perturbation term adds some effects on the all orbital elements. 2) The first technique was better than the second one from direction to apply on actual motion. 3) The mean value of the semi-major axis (a) for the Moon was 381665 km and it varied from year to other. 4) The eccentricity (e) of the Moon orbit varied between 0.024 and 0.093 and with mean value equal 0.06. 5) The inclination (i) of the Moon orbit was fluctuated between 24.13 and 25.36 and the inclination preturbation between 0.0737 and 0.385 deg. 6) The longitude of ascending node (Ω) for the Moon orbit was between 11.14 and 13.2 which mean it was perturbed by 0.5 deg. 7) The mean anomaly (M) and eccentric anomaly (E) have small perturbation at all 15 days. 8) The argument of perigee (ω) for the Moon was more influenced than other elements by perturbation. 9) The distance from the Sun was mainly affected by the Moon from other body in the Solar System therefore it was seen the Moon's elements have approximately same shape or same behavior when the Earth near the vernal and autumnal equinoxes or near the aphelion and perihelion. 10) The synodic month is varied steeply between 29.3 and 29.8 day. 11) The present program was suitable to calculate the actual Moon orbital elements and Hegree date with excellent accuracy. 12) The solar radiation pressure and the planets attraction have small but not negligible effects on the Moon’s orbit, and the shape of Earth and Moon themselves contribute to the perturbations. 85 Chapter five Results, Discussion, Conclusions, and Future Work 5.4. Future Work: 1) Studying the secular variation of Moon's orbital elements for long period. 2) Finding the periodic revolution for some elements through 19 year. 3) Improving the programs for the other planet's satellite or for the planets themselves. 4) Using other body attraction technique to calculate the perturbation on the Moon orbital elements. 5) Orbit elements during Lunar and Solar eclipses. 6) Using numerical methods of analysis. 86 REFERENCES Reference Reference Reference [1] Stan Gibilisco, (2003), "Astronomy Demystified", The McGraw-Hill Companies, Inc. [2] Martin C. Gutzwiller, (1998), "Moon-Earth-Sun: The Oldest ThreeBody Problem", Rev. Mod. Phys., Vol. 70, No. 2 [3] W. K. Hartmann, (1972), "Moon And Planets", Bogen & Quigley, Inc., Printed in the United States of America. [4] Tony Buick, (2006), "How To Photograph The Moon And Planets With Your Digital Camera", Springer-Verlag London Limited, Printed in Singapore. [5] John D. 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Freistetter and J. Kurths, (2005), "Chaos And Stability In Planetary Systems", Springer-Verlag Berlin Heidelberg, Printed in The Netherlands. ،" "إﻳﺠ ﺎد ﻣﻌﺎدﻟ ﺔ اﺣﺘﻤﺎﻟﻴ ﺔ رؤﻳ ﺔ اﻟﻬ ﻼل،(٢٠٠٢) ،[ ﺿ ﺤﻰ ﻣﺤﻤ ﻮد ﻣﻨﺼ ﻮر اﻟﻔﻴ ﺎض١٨] . آﻠﻴﺔ اﻟﻌﻠﻮم ﺟﺎﻣﻌﺔ ﺑﻐﺪاد،رﺳﺎﻟﺔ ﻣﺎﺟﺴﺘﻴﺮ "ﺣﺮآ ﺎت اﻟﺸ ﻤﺲ واﻟﻘﻤ ﺮ اﻟﻔﻴﺰﻳﺎﺋﻴ ﺔ،(١٩٩٧) ،[ ﻋﺒ ﺪ اﻟ ﺮﺣﻤﻦ ﺣﺴ ﻴﻦ ﺻ ﺎﻟﺢ اﻟﻤﺤﻤ ﺪي١٩] . آﻠﻴﺔ اﻟﻌﻠﻮم ﺟﺎﻣﻌﺔ ﺑﻐﺪاد، أﻃﺮوﺣﺔ دآﺘﻮراﻩ،"وﺗﻄﺒﻴﻘﺎﺗﻬﺎ ﻟﻠﻤﻮاﻗﻴﺖ اﻹﺳﻼﻣﻴﺔ [20] Monzur Ahmed, (2001), "Moon Calculator Version 6.0 Program And Documentation", www.moonsighting.com/articles. [21] Muhammad Shahid Qureshi, (2005), "Computational Astronomy And The Earliest Visibility Of Lunar Crescent", www.moonsighting .com/articles. "دورات ﻟﺤﻈ ﺔ وﻻدة،(١٩٩٤) ،[ ﺣﻤﻴ ﺪ ﻣﺠ ﻮل اﻟﻨﻌﻴﻤ ﻲ وﻋﺒ ﺪ اﻟ ﺮﺣﻤﻦ ﺣﺴ ﻴﻦ اﻟﻤﺤﻤ ﺪي٢٢] ٢ ﻋ ﺪد٣٥ م، اﻟﻤﺠﻠ ﺔ اﻟﻌﺮاﻗﻴ ﺔ ﻟﻠﻌﻠ ﻮم،"اﻟﻬ ﻼل وﺷ ﺮوط ﺟﺪﻳ ﺪة ﻟﺮؤﻳﺘ ﻪ ﻋﻨ ﺪ ﻏ ﺮوب اﻟﺸ ﻤﺲ .٥٨٠ص [23] Muhammad Sh. Odeh, (2006), "New Criterion For Lunar Crescent Visibility", Experimental Astronomy, 18: pp 39-64 . [24] N. Guessoum and K. Meziane, (2001), "Visibility Of The Thin Lunar Crescent", Journal of Astronomical History and Heritage, 4: pp 1-14. 89 Reference Reference [25] A. H. Sultan, (2007), "First Visibility Of The Lunar Crescent: Beyond Danjon's Limit", The Observatory, 127, No 1: pp 53-59 . [26] Ke Zhang, (2007), "Resonant And Secular Orbital Interactions", PhD thesis, University of Maryland College Park. "ﺗ ﺄﺛﻴﺮ اﺿ ﻄﺮاﺑﻲ آ ﺒﺢ اﻟﻐ ﻼف اﻟﺠ ﻮي،(٢٠٠٣) ،ﻟﻤﻴـ ـﺎء ﻣﺤﻤ ﺪ ﺣﺴ ـﻦ ﻋﻤ ﺎر ﺳﻤﻴﺴ ـﻢ [٢٧] ، رﺳ ﺎﻟﺔ ﻣﺎﺟﺴ ﺘﻴﺮ،" ﻋﻠ ﻰ اﻟﻌﻨﺎﺻ ﺮ اﻟﻤﺪارﻳ ﺔ ﻟﻸﻗﻤ ﺎر اﻟﺼ ﻨﺎﻋﻴﺔ واﻃﺌ ﺔ اﻻرﺗﻔ ﺎعJ2 واﻟﻌﺎﻣ ﻞ .آﻠﻴﺔ اﻟﻌﻠﻮم ﺟﺎﻣﻌﺔ ﺑﻐﺪاد [28] Gerhard Beutler, (2005), "Methods Of Celestial Mechanics Volume I", Springer-Verlag Berlin Heidelberg, Printed in Germany. [29] Jeffrey S. Parker, (2007), "Low-Energy Ballistic Lunar Transfers", PhD thesis, University of Colorado. [30] Nadege Pie, (2008), "Mission Design Concepts For Repeat Groundtrack Orbits And Application To The ICESat Mission", PhD thesis, University of Texas at Austin. [31] Chao, Chia-Chun., (2005), "Applied Orbit Perturbation And Maintenance", American Institute of Aeronautics and Astronautics, The Aerospace Press Segundo, California. [32] Diane Fisher Miller, (1998), "Basics Of Radio Astronomy For The Goldstone-Apple Valley Radio Telescope", California Institute of Technology, Pasadena, California. [33] George W. Collins, (2004), "The Foundations Of Celestial Mechanics", the Pachart Foundation dba Pachart Publishing House. [34] Jean Meeus, (1988), "Astronomical Formulae For Calculation", Fourth Edition, Willmann-Bell. Inc., Printed in the United States of America. [35] Oliver Montenbruck, Translator: A. H. Armstrong, (1989), "Practical Ephemeris Calculations", Springer-Verlag Berlin Heidelberg New York, the United States of America. 90 Reference Reference [36] Oliver Montenbruck and Eberhard Gill, (2001), "Satellite Orbits Models Methods And Applications", Second Edition, Springer-Verlag Berlin Heidelberg, Printed in Germany. [37] Jean Meeus, (1991), "Astronomical Algorithms", Fourth Edition, Willmann-Bell. Inc., Printed in the United States of America. [38] J. R. Salvail and W. Stuiver, (1995), "Solar Sailcraft Motion In SunEarth-Moon Space With Application To Lunar Transfer From Geosynchronous Orbit", Acto Astronautica Vol.35, No.213: pp.215-229, Elwier Science Ltd, Printed in Great Britain. [39] Richard Fitzpatrick, (1996), "Newtonian Dynamics", The University of Texas at Austin. [40] Victor G. Szebehely, (2004), "Adventures In Celestial Mechanics", Second Edition, Wiley-VCH Verlag GmbH & Co. KGaA, Weinheini, Printed in the Federal Republic of German. [41] Ahmed Kadir Izzet Zainal, (2007), "Orbit Determination From Three Angles Observation In The Presence Of Perturbation", PhD thesis, College of Science University of Baghdad. [42] Jan Vrbik, (2006), "Solving Lunar Problem Via Perturbed K–S Equation", New Astronomy 11: pp 366-373. [43] S. Unnikrishna Pillai, Ke Yong Li and Braham Himed, (2008), Space Based Radar Theory & Applications, The McGraw-Hill Companies, Manufactured in the United States of America. "ﺗﺤﺪﻳﺪ ﻣﺪارات اﻷﻗﻤﺎر اﻟﺼ ﻨﺎﻋﻴﺔ واﻃﺌ ﺔ اﻻرﺗﻔ ﺎع،(٢٠٠٣) ،[ ﻓﺮﻳﺪ ﻣﺼﻌﺐ ﻣﻬﺪي اﻟﺪﻟﻴﻤﻲ٤٤] . آﻠﻴﺔ اﻟﻌﻠﻮم ﺟﺎﻣﻌﺔ ﺑﻐﺪاد، رﺳﺎﻟﺔ ﻣﺎﺟﺴﺘﻴﺮ،"ﺑﻄﺮﻳﻘﺔ اﻟﺮﺻﺪ اﻟﺒﺼﺮي [45] Jay McMahon and Daniel Scheeres, (2010), "Secular Orbit Variation Due To Solar Radiation Effects: a detailed model for BYORP", Celest Mech Dyn Astr 106: pp 261-300 . 91 Reference Reference [46] Franz T. Geyling and H. Robert Westerman, (1971), "Introduction To Orbital Mechanics", Bell Telephone Laboratories, Inc., Printed in the United States of America. [47] Ma Jian, (2009), "Formation Flying Of Spacecrafts For Monitoring And Inspection", MSc thesis, University of Wurzburg. [48] Sylvio Ferraz–Mello, (2007), "Canonical Perturbation Theories Degenerate Systems and Resonance", Springer Science+Business Media, LLC. "اﻻﺿ ﻄﺮاﺑﺎت اﻟﻤ ﺆﺛﺮة ﻋﻠ ﻰ ﻣ ﺪارات اﻷﻗﻤ ﺎر،(٢٠٠٢) ،[ أﻧ ﺲ ﺳ ﻠﻤﺎن ﻃ ﻪ اﻟﻬﻴﺘ ﻲ٤٩] . آﻠﻴﺔ اﻟﻌﻠﻮم ﺟﺎﻣﻌﺔ ﺑﻐﺪاد، رﺳﺎﻟﺔ ﻣﺎﺟﺴﺘﻴﺮ،"اﻻﺻﻄﻨﺎﻋﻴﺔ اﻟﻮاﻃﺌﺔ [50] Zoe Parsons, (2006), "Lunar Perturbations Of a Supersynchronous GEO Transfer Orbit In The Early Orbit Phase", MSc thesis, Cranfield University. [51] Imad Ahmad Hussain AL-Hayali, (2003), "Secular And Long Period Variations Of Earth Orbital", MSc thesis, College of Science Baghdad University. [52] Vladimir A. Chobotov, (1996), "Orbital Mechanics", Second Edition, American Institute of Aeronautics and Astronautics, Inc, Virginia. [53] Mauri Valtonen and Hannu Karttunen, (2005), "The Three-Body Problem", Cambridge University Press. [54] Christian Hellström and Seppo Mikkola, (2010), "Explicit Algorithmic Regularization In The Few-Body Problem For VelocityDependent Perturbations", Celest Mech Dyn Astr 106: pp143-156. [55] V. A. 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Curtis, (2005), "Orbital Mechanics For Engineering Students", Printed and bound in Great Britain by Biddles Ltd, King’s Lynn, Norfolk. 93 Appendix A We can rewrite the equation of motion (3.10) as [27,49,62]: µr r r Or µ r Where U (A.1) By taking the cross product for relationship (A.1) with the specific angular momentum h µ r (A.2) Since (A.3) The rela onship (A.2) became µ r (A.4) By recall the vector identity known as the bac −cab rule: A × (B × C) = B (A ∙ C) − C (A ∙ B) Then equa on (A.4) became µ r r . . By substitute U in above equation . . . . Then r (A.5) Then taking the integral for the equation (A.5) r (A.6) By taking dot product for (A.6) with , get . Left side equal . . . , the equation (A.7) became . Since . (A.7) and . . cos (A.8) Where is the angular distance from perigee. Then put these result in (A.8), get cos 1 cos Finally the relation of conic section orbital in polar coordinate is: 1 cos 1 cos Or Where is the semi latus rectum and (e) is eccentricity. Appendix B Flowchart (1) finding the date and time for the crescent Moon. Start INPUT the month M and the year Y M = M + 0.5 Calculate the J.D. for crescent Moon by equa on (2.11) Add correction to the J.D. of the crescent Moon Convert the J.D. to Year, Month, Day, Hour, minute and the name of day PRINT J.D, DATE, TIME END Flowchart (2) calcula ng the Julian date, distance and velocity for the Moon. Start INPUT UT, D, M and Y for new Moon which calculated from flowchart (1) Calculate (J.D) for input data by equa on (2.8) J.D = J.D + 1 Calculate the Julian century (T) by equa on (2.10) Calculate the Earth ‐ Moon distance (Rm) by equa on (2.14) FOR I = 1 TO 29 Calculate the ecliptical coordinate for the Moon λ , β by eq. (2.12) and (2.13) Convert λ , β to equatorial coordinate δ , α by eq. (2.4) and (2.5) Convert δ , α to horizontal coordinate A , al by eq. (2.6) and (2.7) Calculate cartesian coordinate (Rx, Ry, Rz by δ , α and Rm Next FOR I = 1 TO 28 Calculate the velocity components (Vx, Vy, Vz and Vm) PRINT the distance (Rm and Vm with J.D Next END Flowchart (3) calcula ng orbital elements for the Moon using angular momentum which obtained from actual position and velocity according to the first technique. Start FOR I = 1 TO 28 (Rx, Ry, Rz, Vx, Vy, Vz Calculate the angular momentum (h) and components hx, hy and hz using the sec on 3.8 Calculate the inclination (i), longitude of ascending node (Ω), argument of perigee (ω), semi major axis (a), eccentricity (e), eccentric anomaly (E), mean anomaly (M) and true anomaly (f) using the sec on 3.9 PRINT a, e, i, Ω, ω, E, M, f Next END Flowchart (4) calcula ng orbital elements for the Moon by solving Kepler equation according to the second technique. Start Rm(max), Rm(min) from the first technique, e= 0.0549 Calculate mean mo on (n) by equa on (3.26) FOR I = 1 TO 28 Calculate mean anomaly (M) by equa on (3.28) Find the eccentric anomaly (E) by solution Kepler equation (3.29) using Newton-Raphson method Calculate the Moon position (Rx, Ry, R) for Moon in its orbit and Moon velocity (Vx, Vy, V) by equa on (3.30) ‐ (3.31) Convert the components of position and velocity of the Moon from his orbital plane to Earth equatorial plane using Gaussian vectors (conversion matrix), which content Eular angle. Calculate the angular momentum h and components (hx, hy and hz) using the section 3.8 Calculate the inclination (i), longitude of ascending node (Ω), argument of perigee (ω), semi major axis (a), eccentricity (e), by the sec on 3.9 PRINT a, e, i, Ω, ω, E, M, f Next END Flowchart (5) calcula ng perturba on on the Moon orbital elements. Start FOR I = 1 TO 28 Orbital elements from flowchart (3) Orbital elements from flowchart (4) Find the perturbation in the position, velocity and orbital elements ∆R, ∆V, ∆e, ∆i, ∆a, ∆Ω, ∆ω, ∆E, ∆M, ∆f PRINT ∆R, ∆V, ∆e, ∆i, ∆a, ∆Ω, ∆ω, ∆E, ∆M, ∆f Next END اﻟﺨﻼﺻﺔ ﻓﻲ هﺬا اﻟﺒﺤﺚ ﺗﻤﺖ دراﺳﺔ ﻣﻌﺎدﻟﺔ اﻟﺤﺮآﺔ اﻟﻜﺒﻠﺮﻳﺔ ﻟﺤﻞ ﻣﺸﻜﻠﺔ اﻟﺠﺴﻤﻴﻦ ﺑﻮﺟ ﻮد وﺑﻌ ﺪم وﺟﻮد اﻻﺿﻄﺮاﺑﺎت ﻋﻠﻰ ﻣﺪار اﻟﻘﻤﺮ ﺑﺼﻮرة ﺧﺎﺻﺔ آﻨﻤﻮذج ﻟﻤﺪارات اﻟﻘﻄﻊ اﻟﻨﺎﻗﺺ ﺑﻄﺮﻳﻘﺘﻴﻦ : اﻷوﻟﻰ :اﻋﺘﻤﺎدا ﻋﻠﻰ ﻣﻮﻗﻊ اﻟﻘﻤﺮ اﻟﻤﺤﺪد ﻣﻦ اﻟﻤﻌﺎدﻟﺔ اﻟﺘﻄﺒﻴﻘﻴﺔ اﻟﻤﻮﺿﻮﻋﺔ ﻟﺤﺴ ﺎب ﺑﻌ ﺪ اﻟﻘﻤﺮ ﻋﻦ اﻷرض ﺗﺠﺮﻳﺒﻴﺎ .ﺛﻢ ﺣﺴﺎب ﻣﺮآﺒﺎت اﻟﻤﻮﻗﻊ واﻟﺴﺮﻋﺔ ﻻﺳ ﺘﺨﺮاج ﻣﺮآﺒ ﺎت اﻟ ﺰﺧﻢ وﻣ ﻦ ﺛ ﻢ إﻳﺠﺎد اﻟﻌﻨﺎﺻﺮ اﻟﻤﺪارﻳﺔ ﻟﻠﻤﺪار .واﻟﺘﻲ ﺗﻤﺜﻞ اﻟﺤﺮآﺔ ﺑﻮﺟﻮد اﻻﺿﻄﺮاب . اﻟﺜﺎﻧﻴﺔ :اﻋﺘﻤﺎدا ﻋﻠﻰ ﺣﻞ ﻣﻌﺎدﻟﺔ آﺒﻠﺮ ﺑﺎﺳ ﺘﺨﺪام زاوﻳ ﺔ اﻻﻧﺤ ﺮاف اﻟﺤﻘﻴﻘ ﻲ وﺗﻐﻴﺮه ﺎ ﻣ ﻊ اﻟﺰﻣﻦ واﺳﺘﺨﺪاﻣﻬﺎ ﻟﺤﺴﺎب ﻣﺮآﺒﺎت إﺣﺪاﺛﻴﺎت اﻟﻤﻮﻗﻊ واﻟﺴﺮﻋﺔ ﻟﻠﻘﻤﺮ ﻹﻳﺠ ﺎد ﺑﻘﻴ ﺔ اﻟﻌﻨﺎﺻ ﺮ اﻟﻤﺪارﻳ ﺔ ﺑﻌﺪ ان ﺗﻢ ﺗﺤﻮﻳﻠﻬﺎ ﻟﻤﺴﺘﻮي ﻣﺪار اﻷرض ﺑﺎﺳﺘﺨﺪام ﻣﻌﻜﻮس ﻣﺼﻔﻮﻓﺔ آﺎوس .وﻗ ﺪ ﺗ ﻢ اﻋﺘﺒ ﺎر اﻟﺤﺮآ ﺔ ﺣﺮآﺔ آﺒﻠﺮﻳﺔ ﻧﻘﻴﺔ )ﻋﺪم وﺟﻮد اﺿﻄﺮاب(. وﺑﺎﻟﻤﻘﺎرﻧﺔ ﺑﻴﻦ اﻟﻌﻨﺎﺻﺮ اﻟﻤﺤﺴﻮﺑﺔ ﻣﻦ اﻟﻄ ﺮﻳﻘﺘﻴﻦ أﻋ ﻼﻩ ﺗ ﻢ ﺣﺴ ﺎب ﻣﻘ ﺪار اﻻﺿ ﻄﺮاب ﻟﻜ ﻞ ﻋﻨﺼ ﺮ ﻣ ﺪاري وﺗﻐﻴﺮه ﺎ ﻣ ﻊ اﻟ ﺰﻣﻦ ﻷرﺑﻌ ﺔ أﺷ ﻬﺮ ﻣﺨﺘﻠﻔ ﺔ ﻣ ﻦ ﺳ ﻨﺔ .٢٠١٠ﺣﻴ ﺚ ﻟ ﻮﺣﻆ ان اﻻﺿﻄﺮاب ﻳﻮﺛﺮ ﻋﻠﻰ ﺟﻤﻴﻊ اﻟﻌﻨﺎﺻﺮ اﻟﻤﺪارﻳﺔ. ﺑﺎﻹﺿ ﺎﻓﺔ إﻟ ﻰ ﻣﻨﺎﻗﺸ ﺔ ﻣﻮﺿ ﻮع اﻻﺿ ﻄﺮاﺑﺎت ﺑﺼ ﻮرة ﻋﺎﻣ ﺔ واﻟﻤ ﺆﺛﺮة ﻋﻠ ﻰ اﻷﺟﺴ ﺎم اﻟﻤﺪارﻳ ﺔ آﻤﻘﺎوﻣ ﺔ اﻟﻐ ﻼف اﻟﺠ ﻮي وﻋ ﺪم اﻧﺘﻈ ﺎم آﺮوﻳ ﺔ اﻷرض وﺿ ﻐﻂ اﻹﺷ ﻌﺎع اﻟﺸﻤﺴ ﻲ وﺗ ﺄﺛﻴﺮ ﺟﺎذﺑﻴﺔ اﻟﺠﺴﻢ اﻟﺜﺎﻟﺚ واﻟﺬي ﻳﻌﺘﺒﺮ اﻟﻤﺆﺛﺮ اﻟﺮﺋﻴﺴﻲ ﻋﻠﻰ اﻟﻌﻨﺎﺻﺮ اﻟﻤﺪارﻳﺔ ﻟﻠﻘﻤﺮ. أﻳﻀ ﺎ ﺗ ﻢ ﺣﺴ ﺎب وﻗ ﺖ اﻻﻗﺘ ﺮان ﺑ ﻴﻦ اﻟﺸ ﻤﺲ واﻟﻘﻤ ﺮ ﻟﻜ ﻞ ﺷ ﻬﺮ ﻣ ﻦ ﺳ ﻨﺔ ٢٠١٠ ﻟﺘﺤﺪﻳﺪ ﺑﺪاﻳﺔ اﻟﺸﻬﺮ اﻻﻗﺘﺮاﻧﻲ اﻟﻔﻠﻜﻲ واﻟﺬي ﻳﺴﺘﻔﺎد ﻣﻨﻪ ﻟﺘﺤﺪﻳﺪ ﺑﺪاﻳﺎت اﻷﺷﻬﺮ اﻟﻬﺠﺮﻳﺔ . ﺟﺎﻣﻌﺔ ﺑﻐﺪاد آﻠﻴﺔ اﻟﻌﻠﻮم ﻗﺴﻢ ﻋﻠﻮم اﻟﻔﻠﻚ واﻟﻔﻀﺎء ﺣﺴﺎب ﺗﺄﺛﻴﺮ اﻻﺿﻄﺮاﺑﺎت ﻋﻠﻰ اﻟﻌﻨﺎﺻﺮ اﻟﻤﺪارﻳﺔ ﻟﻠﻘﻤﺮ رﺳﺎﻟﺔ ﻣﻘـﺪﻣـﺔ إﻟﻰ ﻗـﺴـﻢ ﻋﻠﻮم اﻟﻔـﻠـﻚ واﻟﻔﻀﺎء آـﻠـﻴـﺔ اﻟـﻌـﻠـﻮم ﺟـﺎﻣـﻌـﺔ ﺑـﻐـﺪاد وهﻲ ﺟﺰء ﻣﻦ ﻣﺘﻄﻠﺒﺎت ﻧﻴﻞ درﺟﺔ ﻣﺎﺟﺴﺘﻴﺮ ﻓﻲ ﻋﻠﻮم اﻟﻔﻠﻚ واﻟﻔﻀﺎء ﻣﻦ ﻗﺒﻞ ﺣﻴﺪر رﺿﺎ ﻋﻠﻲ اﻟﻌﻠﻲ ﺑﻜﺎﻟﻮرﻳﻮس ٢٠٠٤م ﺑﺈﺷﺮاف أ.م.د .ﻋـﺒﺪ اﻟـﺮﺣـﻤـﻦ ﺣﺴـﻴـﻦ ﺻﺎﻟﺢ اﻟﻤﺤﻤﺪي ١٤٣٢هـ ٢٠١١م
