Chapter 3. The Geostationary Orbit 第三章 定點衛星通訊軌道 亞洲大學 資訊工程學系碩士班 呂克明教授 二 ○○ 六年十月二日

1. Chapter 3. The Geostationary Orbit 第三章 定點衛星通訊軌道 亞洲大學 資訊工程學系碩士班 呂克明教授 二○○六年十月二日

2. Chapter 3. The Geostationary Orbit(第三章定點衛星通訊軌道) • Introduction (緒言) • Antenna Look Angles (天線視角) • The Polar Mount Antenna (極座標天線) • Limit of Visibility (可見度之極限) • Near Geostationary Orbits (近定點衛星的軌道) • Earth Eclipse of Satellite (衛星的地蝕現象) • Sun Transit Outage (日凌現象) • Launching Orbits (發射軌道) • Homework problems (習題)

3. Introduction(緒言) • Three (3) conditions are required for an orbit to be geostationary: • The satellite must travel eastward at the same rotational speed the earth. • The orbit must be circular. • The inclination of the orbit must be zero. • Given the period P for the geostationary is 23 h, 56 min, 4 s, or 86,164 seconds (the reciprocal of this is 1.00273896 rev/day) and earth’s equatorial radius, R= 6,378 km, the geostationary height can be derived as 35,786 km. • The value is often round up to 36,000 km. • In fact, the gravitational fields of the sun and moon produce a shift of about 0.85 degree per year in inclination. • Also, the earth’s equatorial elliptically causes the satellite to drift eastward along the orbit.

4. Antenna Look Angles(天線視角) • When installing an earth station, we need calculate the antenna look angles. • Definition: The look angles (視角) for the ground station antenna are the azimuth and elevation angles required at the antenna so that it points directly at satellite. • The azimuth angle (方位角), A and elevation angle (俯仰角), El are needed for given the values of latitude and longitude of an earth station antenna. • Azimuth angle for an earth station antenna: Equation 3.10 • Elevation angle for an earth station antenna: Equation 3.12 • Azimuth-elevation angles for an earth station location of given the values of latitude, 48.42 degree N, and longitude, 89.26 degree W. All Ku-band satellites are shown in Fig. 3-4. • Do not worry about the formulas. They are built in all Direct PC and Direct TV antenna installation software.

5. The Polar Mount Antenna(極座標天線) • Where the home antenna has to be steer-able, expense usually precludes (排除) the use of separate (個別的) azimuth and elevation actuators (激勵器, 驅動器). • A single actuator is used which moves the antenna in a circular arc (圓弧). This is known as a polar mount antenna. • The antenna bore sight (瞄準的方向, 準星或槍膛歸零) is normal to the polar axis. Please see Fig. 3.5a. • The dish is tilted at an angle relative to the polar mount until the bore sight is pointing at a satellite position. • The angle of tilt, the declination (傾斜角), is calculated in Equation 3.15.

6. Limits of Visibility(可見度之極限) • There will be east and west limits on the geostationary are visible from any given earth station. • The limits will be set by the geographic coordinates (地理座標) of the earth station and the antenna elevation (天線位置的高度). • The lowest elevation in theory is zero, when the antenna is pointing along the horizontal. • The longitudinal limits can be made by considering an earth at the equator. It is 81.3 degree via Equation 3.16. • The limits of visibility will also depend on the earth station latitude (地球站位置的高度) and a typical minimum value of elevation, 5 degree. • The angle B is shown in Equation 3.19. • Please note that the larger latitude, the smaller angle B.

7. Near Geostationary Orbits(近定點衛星的軌道) • The perturbing forces that cause an orbit to depart from the ideal keplerian orbit are: • the gravitational fields of the moon and the sun • the non-spherical shape of the earth • the solar radiation pressure • The reaction of the satellite itself • Two-lines orbital elements are published at regular intervals. • The period for a geostationary satellite is 23 h, 56 min, 4 s or 86,164 s. The reciprocal of this is 1.00273896 rev/day. • The term geosynchronous satellite (同步衛星) is used in many cases instead of geostationary (定點衛星) to describe these near-geostationary satellites. • Please note that the geosynchronous satellite does not have to be near-geostationary. Why?

8. Earth Eclipse of Satellite (衛星的地蝕現象) • If the earth’s equatorial plane coincided with the plane of the orbit around the sun, geostationary satellites would be eclipsed by the earth once each day (由於同步衛星的軌道面和赤道面在同一平面上，理論上，每一天會發生一次衛星-地球-太陽在一條線上的地球遮掩現象，或稱地蝕現象。) • The equinox (晝夜平分點) and the solstice (至日,最高點,白晝最長和最短之日分別是夏至和冬至) • The spring equinox(春分) is the first day of spring, and the autumnal equinox(秋分) is the first day of autumn. • Eclipse begin 23 days before equinox and end 23 days after equinox. The eclipse lasts about 10 min at the beginning and end of the eclipse period and increases to a maximum duration of about 72 minutes at full eclipse. • During the eclipse, the solar cells do not function, and operating power must be supplied from batteries. (當衛星-地球-太陽在一條線上的最短10分鐘到最長72分鐘的時段內，因為沒有太陽光的充電，衛星操作的能源是從蓄電池來的。這種現象發生的時候，衛星的太陽能電池無法充電,但是衛星依然可以工作。)

9. Sun Transit Outage (日凌現象) • During the equinoxes, the sun comes within the beam-width of the earth station antenna (地球-衛星-太陽在一條線上). • Sun transit outage: When this happens, the sun appears as an extremely noisy source which completely blanks out the signal from the satellite. This effect is termed sun transit outage. (太陽通過發生衛星故障的太陽凌駕現象，或稱日凌現象). • It lasts for short periods each day for about 6 days around the equinoxes (還好僅有春分和秋分前後的六天，衛星不能通訊). • The occurrence and duration of the sun transit outage depends on the latitude of the earth station, a maximum outage time of 10 min being typical.(每一次僅有十分鐘)

10. Launching Orbits (發射軌道) • Low-altitude orbit: Satellite may be directly injected into low-altitude orbits, up to about 200 km altitude, from a launch vehicle. • Hohmann transfer orbit: The transfer orbit is selected to minimize the energy required for transfer, and such orbit is known as a Hohmann transfer orbit that was discovered in 1925 in his book <The Attainability of Celestial Bodies> written by a West German scientist Walter Hohmann (1880-1945), born at a small town Odenwald, 32 years before human entering space (10/4/1957, Russian Sputnik I, Sputnik 是俄語的”衛星”之義). • The Hohmann elliptical orbit is seen to be tangent to the low-altitude orbit at perigee and to the high-altitude orbit at apogee. • Velocity changing in the same plane change the geometry of the orbit but not its inclination. In order to change the inclination, a velocity change is required normal to the orbital plane. • It can be shown that the smallest inclination obtainable at initial launch is equal to the latitude of the launch site. Thus the farther away from the equator a launch site is, the less useful it is, since the satellite has to carry extra fuel to effect a change in inclination.

11. Homework Problems-I(習題) • 3.1 Explain what is meant by the geostationary orbit. How do the geostationary orbit and a geosynchronous orbit differ? • 3.6 An earth station is located at latitude 35 degree N and longitude 100 degree W. Calculate the antenna look angle for a satellite at 67 degree W. • 3.7 An earth station is located at latitude 12 degree S and longitude 52 degree W. Calculate the antenna look angle for a satellite at 70 degree W. • 3.8 An earth station is located at latitude 35 degree N and longitude 65 degree E. Calculate the antenna look angle for a satellite at 19 degree E. • 3.9 An earth station is located at latitude 30 degree S and longitude 130 degree E. Calculate the antenna look angle for a satellite at 156 degree E.

12. Homework Problems- II(習題) • 3.19 Explain what is meant by the earth eclipse of an earth-orbiting satellite. • 3.20 Explain briefly what is meant by sun transit outage.