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Orbits and Launching Methods: Exploring Satellite Motion

This chapter introduces Kepler's Laws and the definitions of terms for earth-orbiting satellites. It explains orbital elements, apogee and perigee heights, orbital perturbations, and provides homework problems.

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Orbits and Launching Methods: Exploring Satellite Motion

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  1. 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

  2. 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)

  3. 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.

  4. 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.

  5. 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.

  6. 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.

  7. 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.

  8. 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

  9. 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.

  10. 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.

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