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Chapter 8 Geostationary Orbit Determination And Prediction During Satellite Maneuvers where aei , , , Ω, ω, and M are the semimajor axis of the orbit, eccentricity, orbital inclination, longitude of the ascending node, argument of perigee and mean anomaly, respectively. M = n( t −T ) ,where n is called the mean motion, and T 0 the time of perifocal passage; R is a perturbation function. Clearly, due to the complexity of the perturbation function R in Eq.(8-1), it is impossible to obtain closed solutions, instead, an approximate solution will be obtained. The perturbations on the satellite can be classified as three types, first, the secular variations or the ever-increasing or the decreasing changes from the some epoch; second, the long periodic variations and third the short periodic variations. These can be expressed by (Escobal, 1965, pp. 362). q = q + q&( t − t ) + K cos( 2ω) + K sin( 2v+ 2ω ) (8-2) 0 0 0 1 2 where, v true anomaly In Eq.(8-2) above, the second term is the secular variation that is ever-increasing or decreasing, the third term is the long periodic variation that is related to the argument of perigee ω . ω changes slowly with the period 2π. The last term is short periodic variation, caused by the trigonometric functions of linear combinations of M or v and ω , in which the changes of true anomaly, v, are much more faster than the slow secular variations in the argument of perigee ω . For geostationary satellite maneuver or stationkeeping, the secular variations of Ω , i, are more interesting, because the secular variation makes geostationary satellites drift away from their fixed position and never to return. This is different from the periodic terms. Therefore in the following section only the effects of the secular variation terms are discussed. 8.1.1 Nonspherical Earth Gravitation The geopotential perturbation function can be written as ae R GM P C m S m r n N n = �� n+ 1 nm(sin ϕ)( nm cos λ + nm sin λ ) (8-3) n= 2 m= 0 The secular terms are only related to the coefficients of zonal terms, thus Eq.(8-3) can be expressed as R GM J � 3 2 2 5 J3 2 5 J4 2 4 = 2 ( 1−5sin ϕ) + 3 ( 3−7sin ϕ)sin ϕ − 4 ( 3− 42sin ϕ + 63sin ϕ) �� 2 r 2 r 8 r − − + + − + − + � 3 J5 2 4 1 J6 2 4 6 5 ( 35 210 sin ϕ 231sin ϕ) sin ϕ 6 ( 35 945sin ϕ 3465sin ϕ 3003sin ϕ) Λ (8-4) 8 r 16 r �� where Ji = Ci0 Neglecting the high order terms in Eq.(8-4), considering sinϕ = sinisin( v + ω) , Eq.(8-4) becomes R GM J � 3 2 a 3 1 1 2 1 2 J3 a 4 15 2 3 = 3 ( ) { − sin i+ sin icos 2( v+ ω)} − 4 ( ) {( sin i− )sin( v+ ω) �� 2 a r 3 2 2 a r 8 2 5 2 35 J4 a 5 3 3 2 3 4 2 3 1 2 − sin isin 3( v+ ω)}sin i− ( ) { − sin i+ sin i+ sin i( − sin i)cos 2( v+ ω) 5 8 8 a r 35 7 8 7 2 1 4 � + sin icos 4( v+ ω )} (8-5) 8 �� According to Eq.(8-2), the perturbation function with the secular variations in Eq.(8-5) can be expressed as 96 0
Chapter 8 Geostationary Orbit Determination And Prediction During Satellite Maneuvers R GM J �3 2 a 3�112 � 35 J4 a 5�332 3 4 �� s = � 3 ( ) � − sin i� − 5 ( ) � − sin i+ sin i� �2 a r �3 2 � 8 a r �35 7 8 � � (8-6) � and the periodic variation perturbation function by R GM J � a � � J a � � p = � � i v+ � − � i− v+ − i v+ � i � a r � � a r � � 3 2 3 1 2 3 4 15 2 3 5 2 3 ( ) sin cos 2( ω) 4 ( ) ( sin )sin( ω) sin sin 3( ω) sin 2 2 8 2 8 35 J4 a 5�2 3 1 2 1 4 �� − 5 ( ) �sin i( − sin i)cos 2( v+ ω) + sin icos 4( v+ ω ) � 8 a r � 7 2 8 � � (8-7) � Inserting Eq.(8-6) and Eq.(8-7) into Eq.(8-1), the variations of Ω and i due to perturbations Eq.(8-6) and Eq.(8- 7) can be obtained as dΩ = dt 2 na 1 ∂R 2 1− e sin i ∂i = n 2 1− e �3 J2 a 3 35 J4 a 5�632 �� 2 ( ) ( −cos i) − 4 ( ) � cos i+ sin icos i� �2 a r 8 a r �7 2 � � � (8-8) di = dt 2 na cosi ∂R 2 1 − e sini ∂ω = ncosi �3 J2 a 3 J3 a 4�15 2 3 15 2 � � ( ) { − sinisin 2( v+ ) } − ( ) �( sin i− )cos( v+ ) − sin icos 3( v+ ) � 2 2 ω 3 ω ω 1 − e �2 a r a r � 8 2 8 � 35 J4 a 5�312 1 3 �� + 4 ( ) �2sin i( − sin i)sin 2( v+ ω) + sin isin 4( v+ ω ) � 8 a r � 7 2 2 � � (8-9) � From two-body theory of satellite movement, 2 a( 1− e ) � r = = a( 1−ecos E) � 1+ ecos v � 2 1− e sin E � sin v = � 1− ecos E � � M = E− esin E = n( t−T0) � dM = ndt � � dM = ( 1−ecos E) dE � 2 � a( 1− e )sinvdv sin EdE = � 2 ( 1+ ecos v) � � where n unperturbed two-body mean motion E eccentric anomaly T0 time of perifocal passage then the following equation can be obtained dM r = ( ) dv a − e 2 1 2 1 97 (8-10) (8-11) In order to evaluate Ω and i variations under the influence of nonspherical earth gravitation, Eq.(8-8) and Eq.(8- 9) are integrated over a revolution of geostationary satellite movement. The results are � te te � 1 J a J a ∆Ω s =− �3 2 1 3 35 4 �6 3 2 � 1 5 cos i ndt − � i+ i i� ndt� − e � a � ( ) cos sin cos r a � � � ( ) 2 2 4 1 2 2π 8 7 2 2π r � � ts ts � =− � 2 2 1 3 J � � 2 1 a 3 35 4 �6 3 2 � 1 5 − � + � � 2 2 i − � � 4 � � � 1 e 2 a 2 r 8 7 2 2 � � 0 0 � dM π π J a cos ( ) cosi sin icos i ( ) dM π a π r (8-12)
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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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References Carpino M. (1995): SAEG
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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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