Precise Orbit Determination of Global Navigation Satellite System of ...

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Chapter 8 Geostationary Orbit Determination And Prediction During Satellite Maneuvers te � ts te � ts te � ts te 2π sin 2( v + ω) ndt = sin 2( v + ) dv = 0 2 ( 1+ ecos v) � ω cos 2( v + ω) ndt = cos 2( v + ) dv = 0 2 ( 1+ ecos v) � ω 2 0 2π sin ( v + ω) 2 ndt = sin ( v + ) dv = 2 ( 1+ ecos v) � ω π 2 0 2π 0 2π 102 (8-43) (8-44) (8-45) cos ( v + ω) 2 � ndt = cos ( v ) dv 2 ( 1+ ecos v) � + ω = π ts 0 From Eq.(8-43) to Eq.(8-46), the solutions of Eq.(8-37) and Eq.(8-38) are given by (8-46) 1 GM i i � 3 ∆ii= πsin 2( Ω 2 3 n sin( i−ε) ri �� 2 3 − v′− ω′ ) − π sin 2( Ω 2 2 � − v′− ω′ ) cos ( i−ε) �� π GM i i = 2 3 n sin( i−ε) r 3 2 [ sin 2( Ω− v′ − ω′ ) −sin 2( Ω − v′ − ω′ ) cos ( i−ε) ] 2 (8-47) i 3π GM i i 2 ∆ΩΩ i =− sin ( − v′ − ω′ ) cos( i−ε) (8-48) 2 3 n r i Considering the effects of the Sun and the Moon on the geostationary satellite respectively and inclinations of the Sun and the Moon with equator as 23.442 o and 28.6 o (Soop, 1994), Eq.(8-47) and Eq.(8-48) can be written as π GM s s 3 2 ο ∆is = [ sin 2( −vs −ω s) −sin 2( −vs −ω s) cos ( −23 .442) 2 3 ] (8-49) ο n sin( −23 .442) r 2 ∆Ω s ∆i m ∆Ω m 3π GM 2 ο =− sin ( −vs −ω s) cos( −23 .442) n = n s s 2 3 rs π G M ο sin( −514) r m m 2 3 m 3π G M 2 ο =− sin ( −vm −ω m) cos( −5 . 14) n m m 2 3 rm s 3 2 2 ο [ sin 2( −vm −ω m) −sin 2( −vm −ω m) cos ( −5. 14) ] (8-50) (8-51) (8-52) According to Eq.(8-49) to Eq.(8-52), the secular variations of the inclination i and the longitude of the ascending node, Ω, of the geostationary satellite due to the Sun and the Moon attractions are drawn in Figure 8-2 and Figure 8-3. In the figures, assuming the Sun and the Moon move along the Sun/Moon orbits from 0° to 360°. For the Sun it takes one year, for the Moon it only needs about one month. From Eq.(8-28) and Eq.(8-29), and the discussion above it is clear that the secular variations of the inclination i and the longitude of the ascending node, Ω , of geostationary satellite are mainly caused by Sun and Moon attractions. Although the secular variations are produced by the J2 term of non-spherical gravitation as well, comparing to the Sun and the Moon, the variations caused by J2 are very small. Therefore the major effects that are considered for GEO satellite maneuvers are the Sun and the Moon attractions.
Chapter 8 Geostationary Orbit Determination And Prediction During Satellite Maneuvers Changes of Omega (arc_sec) Changes of i (arc_sec) 8 6 4 2 0 0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 -2 -4 -6 -8 0 -5 -10 -15 -20 -25 -30 Sun/Moon Positions (deg) Figure 8-2 Secular Variations of Inclination of GEO Satellite due to Sun/Moon Position Changing 0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 Sun/Moon Positions (deg) Figure 8-3 Secular Variations of Longitude of Ascending Node of GEO Satellite due to Sun/Moon Position Changes Figure 8-2 and Figure 8-3 show that the gravitational attractions of the Sun and the Moon pull the geostationary satellites out of equatorial orbit plane, gradually increasing satellite’s orbital inclination. In addition, the noncircular shape of the earth’s equator causes these satellites to be slowly driven to one of two stable equilibrium points along the equator, resulting in an east-west liberation (drifting back and forth) about these points. The secular variation from the Moon attraction is about twice as strong as from the Sun’s attraction. This is because the distance from the Sun is much larger than the Moon and the attraction from the Sun decreases with the cube of the distance from the Sun to the Earth. For the secular variation of inclination i from the Sun attraction, the maximum variations will be reached four times each year, two times for positive changes and two times for negative changes. For the secular variation of inclination i from the Moon attraction, the maximum variations will be reached four times each lunar month, also two times for positive changes and two times for negative changes. For the secular variation of the longitude of the ascending node, Ω, from the Sun attraction, the maximum variations occur at midsummer and midwinter. At the beginning of spring and autumn, the secular variation of Ω is zero. The effect of the lunar perturbation on the orbital inclination is the same as that of the solar perturbation. The variations are maximum twice per lunar month and passes through zero in between. Due to non-spherical geopotential gravitation, the Sun and the Moon attractions, the inclination and longitude of the ascending node of the geostationary satellite will gradually drift away from the original position. 103 Sun Moon Sun Moon
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PRECISE ORBIT DETERMINATION OF GLOB
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Summary GNSS-2 (Global Navigation S
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3.2.2.1 Principle..................
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9.3.3 The Effect of Distribution of
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Chapter 1 Introduction 1.2 Developm
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Chapter 2 Observations of Orbit Det
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Chapter 3 Orbit Tracking System and
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Chapter 4 Major Error Sources of Sa
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Chapter 5 Perturbation Models of IG
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Page 60 and 61: Chapter 5 Perturbation Models of IG
Page 74 and 75: Chapter 6 Algorithms of Orbit Deter
Page 94 and 95: Chapter 7 Orbit Determination Using
Page 104 and 105: Chapter 8 Geostationary Orbit Deter
Page 122 and 123: Chapter 9 Software and Simulation R
Page 152 and 153: Conclusion 144
Page 154 and 155: References Carpino M. (1995): SAEG
Page 156 and 157: References Klobuchar, J. A.(1996):
Page 158 and 159: References Traquilla, J. M. and J.
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Acknowledgments 152
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