Chapter 2. Orbits and Launching Methods 第二章 衛星軌道與發射方法 亞洲大學 資訊工程學系碩士班 呂克明教授 二○○六年九月二十五日 1/1/2019

Chapter 2. Orbits and Launching Methods (第二章 衛星軌道與發射的方法) Introduction (緒言) Kepler’s Laws (刻卜勒定律) Definitions of Terms for Earth-Orbiting Satellite (繞地軌道名詞之定義) Orbital Elements (軌道元件) Apogee and Perigee Heights (遠地點與近地點之高度) Orbital Perturbations (軌道的擾亂) Homework problems (習題) 1/1/2019

Introduction (緒言) Satellites which orbit the earth follow the same laws that govern the motion of the planets around the sun. Johannes Kepler (刻卜勒,1571-1630) derived three (3) laws describing planetary motion. Sir Isaac Newton (牛頓,1642-1727) derived laws of mechanics and developed the theory of gravitation. Edmond Halley (哈雷,1656-1742) with Newton’s help to predict the next coming Comet Halley. Mark Twain (馬克吐溫, 1835-1910) was born and died in years that Comet Halley came closely to the earth. The more massive of the two bodies is referred to as the primary, the other, the secondary, or satellite. 1/1/2019

Kepler’s Law (刻卜勒定律) First Law: The path followed by a satellite around the primary will be an ellipse. Barycenter (重心的相異根) is always centered on one of the foci. In our specific case, the center of mass coincides with the center of the earth and therefore always at one of the foci. Second Law: For equal time intervals, a satellite will sweep out equal areas in its orbital plane, focused at the barycenter. Third Law: The square of the period time of orbit is proportional to the cube of the mean distance between the two bodies. 1/1/2019

Definitions of Terms for Earth-Orbiting (繞地軌道名詞之定義) Apogee (遠地點): The point farthest from earth. Perigee (近地點): The point of closest approach to earth. Line of apsides (遠近線): The line joining the perigee and apogee through the center of earth. Ascending node (昇交點): The point where the orbit crosses the equatorial plane going from south to north. Descending node (降交點): The point where the orbit crosses the equatorial plane going from north to south. Line of nodes (交點線): The line joining the ascending and descending nodes through the center of earth. Inclination (傾斜角): The angle between the orbital plane and the earth’s equatorial plane. 1/1/2019

Definitions of Terms for Earth-Orbiting (繞地軌道名詞之定義)(continued) Prograde orbit (順行軌道): An orbit in which the satellite moves in the same direction as the earth’s rotation. Retrograde orbit (逆行軌道): An orbit in which the satellite moves in a direction counter to the earth’s rotation. Argument of perigee (近地點輻角): The angle from ascending node to perigee, measured in the orbital plane at the earth’s center, in the direction of satellite motion. Right ascension of the ascending node (昇交點赤經): For the practical determination of an orbit, the longitude and time of crossing of the ascending node are frequently used. An absolute measurement, a fixed reference in space is required. The reference chosen is the first point of Aries, or spring equinox. Mean anomaly (均偏角): An average value of the angular position of the satellite with reference in the perigee. True anomaly (真偏角): The angle from perigee to the satellite position, measured at the earth center. 1/1/2019

Orbit Elements (軌道元件) Keplerian element set: six (6) orbital elements. Semi-major axis (半長軸, a) Eccentricity (離心率, e ) – the eccentricity of Comet Halley is 0.967 Mean anomaly (均偏角,M ) Argument of perigee (近地點輻角, w) Inclination (傾斜角, I) Right ascension of the ascending node (昇交點赤經) Satellite parameters – details from the NASA Bulletins (Table 2.1) Epoch – a reference time. By given the mean motion (rev/day) and using Kepler’s 3rd law to find semi-axis a. 1/1/2019

Apogee and Perigee Heights (遠地點與近地點之高度) Apogee Height: Ra = a (1 + e) Ha = Ra - R Perigee Height: Rp = a (1 – e) Hp = Rp – R Given e=0.0011501, a=7192.3 km, and earth radius, R=6371 km. Ra = 7192.3 (1 + 0.0011501) = 7200.6 km Ha = 7200.6 – 6371 = 829.6 km Rp = 7192.3 (1 - 0.0011501) = 7184.1 km Hp = 7184.1 – 6371 = 813.1 km 1/1/2019

Orbit Perturbations (軌道的擾亂) Keplerian orbit is ideal: The earth is uniform spherical mass and Only force acting is the centrifugal force Perturbations: Sun and moon Atmospheric drag Effects of a non-spherical earth Atmospheric Drag – the drag is greatest at the perigee, the drag acts to reduce the velocity at this point. 1/1/2019

Homework Problems (習題) 2.1 State Kepler’s three laws of planetary motion. 2.6 The orbit for an earth-orbiting satellite orbit has an eccentricity of 0.15 and a semi-axis of 9,000 km. Determine a) its period time; b) Ha; c) Hp. Assume a mean value of earth radius, R=6,371 km. 2.10 Explain what is meant by apogee height and perigee height. Given the Cosmos 1,675 satellite has Ha=39,342 km, Hp=613 km, and earth radius, R=6,371 km. Determine the semi-axis and the eccentricity of its orbit. 2.11 The Aussat 1 geostationary satellite has Ha=35,795 km, Hp=35,779 km, and earth’s equatorial radius, R=6,378 km. Determine the semi-axis and the eccentricity of the satellite’s orbit. 1/1/2019