Transcript Document 7799964

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
5/26/2016
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Chapter 2. Orbits and Launching Methods
(第二章 衛星軌道與發射的方法)
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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)
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Introduction
(緒言)
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Satellites which orbit the earth follow the same laws that govern the
motion of the planets around the sun.
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Johannes Kepler (刻卜勒,1571-1630) derived three (3) laws
describing planetary motion.
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Sir Isaac Newton (牛頓,1642-1727) derived laws of mechanics and
developed the theory of gravitation.
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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.
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The more massive of the two bodies is referred to as the primary, the
other, the secondary, or satellite.
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Kepler’s Law
(刻卜勒定律)
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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.
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Definitions of Terms for Earth-Orbiting
(繞地軌道名詞之定義)
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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.
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Definitions of Terms for Earth-Orbiting
(繞地軌道名詞之定義)(continued)
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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.
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Orbit Elements
(軌道元件)
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Keplerian element set: six (6) orbital elements.
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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)
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Inclination (傾斜角, I)
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Right ascension of the ascending node (昇交點赤經)
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Satellite parameters – details from the NASA Bulletins (Table 2.1)
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Semi-major axis (半長軸, a)
Epoch – a reference time.
By given the mean motion (rev/day) and using Kepler’s 3rd law to
find semi-axis a.
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Apogee and Perigee Heights
(遠地點與近地點之高度)
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Apogee Height:
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Perigee Height:
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Ra = a (1 + e)
Ha = Ra - R
Rp = a (1 – e)
Hp = Rp – R
Given e=0.0011501, a=7192.3 km, and earth radius, R=6371 km.
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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
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Orbit Perturbations
(軌道的擾亂)
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Keplerian orbit is ideal:
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Perturbations:
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The earth is uniform spherical mass and
Only force acting is the centrifugal force
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.
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Homework Problems
(習題)
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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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