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Chapter 8 Geostationary Orbit Determination And Prediction During Satellite Maneuvers = G M ( m m It is known that 1 xmx+ ymy+ zmz − / ) ( x − x) + ( y − y) + ( z −z) ( xm + ym + zm) 2 2 2 2 2 2 3 2 (8-32) m m m 2 2 2 2 2 2 i i i i i i i i | r − r| = ( x − x) + ( y − y) + ( z − z) = r − 2( x x+ y y+ z z) + r and r r i >> , then 1 1 � 2 2( xx i + yy i + zz i ) r � = �1 − 2 + 2 � | ri − r| ri � ri ri � and ri⋅ r xx i + yy i + zz i cosδ = = rr rr thus i 1 1 � �� −2ri⋅r r 2 �� = �1 + �( ) + ( ) 2 �� | ri − r| ri �� ri ri �� i 1 − 2 1 − 2 ⋅ �� ⋅ �� = + + � − � �� + 1 � r � i r 3 rir 2 1 r 2 �1 ( ) ( ) Λ 2 2 � ri �� ri 2 ri 2 ri �� 2 3 1 r r 1 3 2 r 3 5 2 � = �1 + cos δ + 2 ( − + cos δ) + 3 ( − cosδ + cos δ) + Λ � ri � ri ri 2 2 ri 2 2 � R GM i i � 2 3 r 1 3 2 r 3 5 3 � i = �1 + 2 ( − + cos δ) + 3 ( − cosδ + cos δ) + Λ � (8-33) ri � ri 2 2 ri 2 2 � Removing the constant term in Eq.(8-33), Eq.(8-33) becomes R i GM i i � 2 3 r 1 3 2 r 3 5 3 � = � ( − + cos δ) + ( − cosδ + cos δ) + Λ 2 3 � (8-34) ri �� ri 2 2 ri 2 2 �� In the equations above subscript i means s or m, i.e. the Sun and the Moon respectively. The relations between δ and satellite orbit parameters are shown in the following figure. γ O ε δ S/M N i 100 N´ Orbit Plane i - Sat. Figure 8-1 Relations between Satellite and Sun/Moon ε Sun /Moon Orbit Plane Equator In Figure 8-1 ε is the obliquity of the ecliptic. The arc between γ and N ′ is defined as Ω. The arc between satellite and N ′ as v + ω . The arc between γ and Sun/Moon as v′ + ω ′ ,
Chapter 8 Geostationary Orbit Determination And Prediction During Satellite Maneuvers From Figure 8-1, cosδ = cos( v+ ω) cos( Ω− v′− ω′ ) − sin( v+ ω) sin( Ω− v′− ω′ ) cos( i−ε) In Eq.(8-34) if only the first term is considered, Eq.(8-34) becomes R G M r 2 � 1 3 2 2 i = i i − + cos ( v+ ω) cos ( − v′− ω′ ) 3 ri �� 2 2 Ω − 3 cos( v+ ω) cos( Ω− v′ − ω′ )sin( v+ ω)sin( Ω− v′ − ω′ ) cos( i−ε) R i + + − ′− ′ − � 3 2 2 2 sin ( v ω)sin ( Ω v ω ) cos ( i ε) (8-35) 2 �� 2 GM i i a 2 2� 1 1 = ( 1−e) �− 2 ri ri � 2 ( 1+ ecos v) − 2 2 3 cos ( v + ω) 2 + cos ( Ω − v′ − ω ′ ) 2 2 ( 1+ ecos v) + + − ′ − ′ − + − ′ − ′ − + + � 2 3 sin 2( v ω) 3 sin ( v ω) 2 2 sin 2( Ω v ω ) cos( i ε) sin ( Ω v ω ) cos ( i ε) 2 2 � (8-36) 4 ( 1 ecos v) 2 ( 1 ecos v) � Inserting Eq.(8-36) into Eq.(8-1), the following differential equations can be obtained di dt dΩ dt n GM = 2 3 n sin( i−ε) r i i i � 2 3/ 2 � 3 sin 2( v + ω) 2 ( 1−e ) �cos( i−ε) �− cos ( Ω − v′− ω ′ ) 2 �� 2 ( 1+ ecos v) + + − − ′ − ′ − + − ′ − ′ − + + � 3 cos 2( v ω) 3 sin 2( v ω) 2 2 sin 2( Ω v ω ) cos( i ε) sin ( Ω v ω ) cos ( i ε) 2 2 � 2 ( 1 ecos v) 2 ( 1 ecos v) � � 2 3 cos ( v + ω) 3 sin 2( v + ω) −�− sin 2( Ω− v′ − ω ′ ) − cos 2( Ω− v′ − ω′ ) cos( i−ε) 2 2 � 2 ( 1+ ecos v) 2 ( 1+ ecos v) + + − ′− ′ − + � 2 3 sin ( v ω) � 2 sin 2( Ω v ω ) cos ( i ε) 2 �� (8-37) 2 ( 1 ecos v) �� n GM = 2 3 n sin( i−ε) r i i i 2 3/ 2 ( 1− e ) � 3 sin 2( v + ω) � sin 2( Ω − v′ − ω′ )sin( i−ε) 2 � 4 ( 1+ ecos v) + − − ′ − ′ − + � 2 3 sin ( v ω) 2 sin ( Ω v ω ) sin 2( i ε) 2 � (8-38) 2 ( 1 ecos v) � Before solving Eq.(8-37) and Eq.(8-38), the following integrations should be solved first, considering dM r = ( ) dv a − e 2 1 2 1 te � ts te � ts te � ts te � ts 2π 2π sin 2( v + ω) sin 2( v + ω) sin 2( v + ω) r 2 1 2 3/ 2 sin 2( v + ω) ndt = dM ( ) dv ( 1 e ) dv 2 2 2 ( 1 cos ) ( 1 cos ) ( 1 cos ) 2 4 + e v � = + e v � = − + e v a � 1− e ( 1+ ecos v) 0 2π cos 2( v + ω) 2 3/ 2 cos 2( v + ω) ndt = ( 1−e) dv 2 4 ( 1+ ecos v) � ( 1+ ecos v) 2 sin ( v + ω) 2 3/ 2 sin ( v + ω) ndt = ( 1−e) dv 2 4 ( 1+ ecos v) � ( 1+ ecos v) 2 0 2π cos ( v + ω) 2 3/ 2 cos ( v + ω) ndt = ( 1−e) dv 2 4 ( 1+ ecos v) � ( 1+ ecos v) 0 2π 0 2 2 0 101 2π 0 (8-39) (8-40) (8-41) (8-42) Eq. (8-39) to Eq.(8-42) are difficult to integrate, but for geostationary satellite, e ≈ 0 , then Eq.(8-39) to Eq.(8- 42) become
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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 58 and 59: Chapter 5 Perturbation Models of IG
Page 74 and 75: Chapter 6 Algorithms of Orbit Deter
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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):
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References Traquilla, J. M. and J.
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Acknowledgments 152
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