Ω,Λ) X Y Where = Ω Introduction Ex Ecos

ORBIT MANEUVERS OF SATELLITES IN NEAR-EARTH QUASI-CIRCULAR ORBIT

Huang Fuming China Xi’an Satellite Control Center P.O.Box 505, Xi’an 710043, Shaanxi, China

Abstract Perturbation Movement Equations of the Satellite in Quasi-Circular Orbit A near-earth quasi-circular orbit is often used for earth observation satellites. In this paper, the method The orbit perturbations of the satellite include: the that the ground tracks are restricted in a given limit is earth non-spherical gravity perturbation, the atmospheric adopted in order to study the track positioning and the drag perturbation, the luni-solar gravity perturbation, the ground track maintenance of the satellites in the solar radiation pressure perturbation and so on. near-earth quasi-circular orbits. The dynamics models For a quasi-circular orbit, in order to make the orbit of orbit maneuvers are established. On the basis of the perturbation movement equation to avoid having the simplified models, various parameters for orbit singular points, the following standard orbital elements maneuvers are calculated. are adopted:

Key Words: Orbit Maneuver, Dynamics, Satellite. (a,e ,e i,, Ω,λ) x y where = ω Introduction ex ecos

= ω In earth observation missions, the satellite is required e y esin to pass over a same area on the ground periodically, and λ = ω + M the altitudes of the satellite above the ground are Its corresponding orbit perturbation movement required to be almost the same so as to compare the equations are: sequential images of the same area taken from a same angle of view. The near-earth quasi-circular orbit with da 2  P  =  T + ( e sin f ) S  the recurrent and frozen features can meet the above dt n 1 − e 2  r  requirements. In order to ensure the imaging quality and the de 1 esin ω sin uctgi coverage of the earth, the orbit of the satellite should be x = {}2( cos u)T + (sin u)S + W adjusted to the nominal operation orbit in time after the dt na na satellite is put into orbit by a launch vehicle. de 1 e cos ω sin uctgi At the end of the orbit acquisition, the appropriate y = {}2( sin u)T − (cos u)S − W time is selected to adjust the orbit, to complete the track dt na na positioning and to establish the normal operation orbit, di cos u so that the satellite can meet the restriction of the track = W grids in order to ensure the imaging quality and the dt na coverage of the earth. During the lifetime, the satellite may drift from the dΩ sin u = W nominal orbit because of various perturbations. dt na sin i Therefore, the orbital parameters (a, e, ω) in plane should be adjusted in time. So, the errors between the dλ 1 1 r e cos f r  = n −  e sin f 1( + )T + ( + 2 )S  actual tracks and the nominal tracks are not beyond a dt na  2 P 2 P  given limit when the satellite passes over the ground sin uctgi tracks repeatedly. − W na

µ a0 is used to indicate the nominal mean semi-major Here, n = ; S is the acceleration component of the a3 axis, and θ is used to indicate the directional cosine c r gravity perturbation, the atmospheric perturbation and of the modified vector ∆e in plane (e e, ) . When the orbit maneuver thrust in the radial direction in the c x y ∆ θ ω ∆ θ ω π orbit coordinate system, T is in the transversal direction , V>0, c = + E ; when V<0 , c = + E + . If and W is in the normal direction. ∆V and ∆V are the velocity increments generated 1 2 by the first and second tangential pulses respectively,

and θ and θ are the directional cosines of the Dynamics Models for Orbit Maneuvers c1 c2 r r ∆ modified vectors e 1c and ec2 in the plane During the track positioning and the track maintenance, the on-board thruster for orbit maneuvers (e x e, y ) respectively, then is used to generate the transversal thrust T so as to ω 2 change the orbital parameters (a, e, ) in plane  ∆a  a   = 4 0 (∆V 2 + ∆V 2 + 2∆V ∆V )   µ 1 2 1 2 simultaneously. Now, it is assumed that Ft is the  a 0  ∆ constant thrust without errors in attitude, txb is the  ∆ 2 ∆ ∆ 2  a  a0 ∆ ∆ [[[][ θ θ ]]] firing time length of the thruster, and Vx is the ( ec ) =   + 8 V1 V2 cos( c2 - c1 1-)  a  µ velocity increment in the direction T. If the arc of the 0 orbit maneuver is short, the thrust can be considered to be of pulse. Therefore, the change in the parameters of Here, there are three cases: the quasi-circular orbit can be expressed in the following simplified formula: 2  ∆a  2   > ()∆e   c ∆V= F∆ t/ m • a 0 x t xb The two pulses of a same symbol are used to modify a ∆ = ∆ a and ec . a 2a Vx µ θ The position c1 of the first pulse can be selected at

discretion. The parameters ∆V and θ of the a 2 c2 ∆ e = 2 cos( ω + E )∆ V ∆ cx µ x second pulse are related with the parameters V1 and θ c1 of the first pulse. a ∆ = ω + ∆ 2 ecy 2 sin( E ) V x µ  ∆a  2   < ()∆e •   c a 0 The above expressions are the dynamics models used The two pulses of the opposite symbols are used to to control orbital parameters. Here, ∆a, ∆e and cx modify a and ec . ∆ ecy are the target bias. 2 Now, there is a question: How are the least control  ∆a 2 times and the least fuel used to make the orbital   = ()∆e   c ω • a0 parameters a, e and (or a, ex e, y ) meet the Only one pulse can complete the coordinated control requirements simultaneously? Generally, two tangential of a, e and ω. pulses are used for completing the coordinated control of the three parameters. Under special conditions, only one pulse is used. In order to complete the coordinated Mathematical Models for Track Prediction control of the three parameters, the control point is seldom at the apsis. The geographical longitude at the descending node is used to express the position of the nominal track. If the From the above formula, it is obvious that, by nominal track of the descending pass over a specific adjusting the semi-major axis, the track positioning can area is defined to be the first track which is marked as be performed or the track can be restricted within the L . The northern nominal track adjacent to the first given limit. B1 The total longitude bias between the actual track and one is the second track, and the eastern nominal track the nominal track is: adjacent to the first one is the Nth track ( i.e. the last track), then the distance between two adjacent nominal ∆LLL= ∆ + ∆ tracks is 360/N degrees. The longitude of the Kth 2 1 t nominal track at the descending node is: The relationship between the variant ∆p of the period ∆ L = L − (k − )1 ∆L and the variant a of the semi-major axis is: Bk B1 B

a 360 ° ∆p = 3π ∆ a Where, ∆L = . L is transformed within µ B N Bk

(-180 °, +180 °] ( positive in the east longitudes ). & It is assumed that the longitude of an actual track at The relationship between the decay rate a (in ° meter/day ) of the semi-major axis and the decay rate the descending node is L which value is within (-180 , & +180 °], and the nominal track nearest to it is the jth p ( in second/day ) of the period is: track, therefore we obtain 1 µ a&&= p L − L 3π a j = int( B1 + 0.5 ) ∆L B a& and p& are both negative.

Now, using j to replace k and inserted into the calculation formula of , we obtain the LBk Track Positioning and Track Maintenance corresponding longitude LBj of the nominal track. The maneuver sketch of the track positioning and Then, the bias of the actual track relative to the nominal track maintenance is shown in Figure 1. track is gained by means of the following formula:

a ∆ = − L L LBj

Mathematical Models for Track Control

a If T1 is the time of the satellite passing over its certain 0 L descending node, the longitude bias between the actual track and the nominal track is ∆LLL= − in West 1 1 B East degree, the semi-major axis bias between the actual and ∆ = − 东 nominal values is a a a0 in meter, and the decay rate of the semi-major axis is a& in meter/day , then the drift longitude at the descending node after t days is:

a ∆ = − π e ∆ + 1 & 2 in meter L t 3 ( at ta ) a 2 1 2 = −540 ( ∆at + &ta /) a in degree Figure 1: Orbit maneuver for track 2 positioning and track maintenance In terms of the determined orbit and the control errors, University , 1979.4 (in Chinese). the control of the tracks is performed in two steps: in the 3Li Jisheng, Satellite Precision Orbit Determination, first step, a great part of the theoretic adjustment size is XSCC, 1995 (in Chinese). used for control; in the second step, after the first-step maneuver calibration, the remaining adjustment, which is calculated by putting the target on the track drift parabola, is used for control. The in-plane track maintenance maneuver is to change the parameters (a, e and ω ) by means of providing the velocity increment in order to maintain the recurrent and frozen features of the orbit, and to control the bias between the actual track and the nominal track within the given limit in order to guarantee the coverage of the track grids. The main steps for the track maintenance maneuvers are as following: • If the bias between the actual track and the ∆ nominal track is L1 ( in degrees) and the left ∆ * boundary is LL , the calculation formula of the theoretic adjustment size ∆a is as follows: ()∆L* − ∆ L a a& ∆a=() a − a + L 1 0 0 1 270 • In terms of the determined orbit and the control errors, the target semi-major axis deviates ∆()∆a ( in meter) downwards, the actual adjustment size of the ∆ = ∆ − ∆ ∆ semi-major axis is taken as ac a() a . • In terms of the formula given by the dynamics model for orbit maneuver, the velocity increments and their acting phases are calculated. • In terms of the velocity increments and their acting phases, the time to turn on/off the thruster is calculated.

References

1M.M.Pircher, SPOT-1 Spacecraft in Orbit Performances , IAF-98-132. 2Liu Lin, Zhao Deshi, Satellite Orbit Theory, Nanjing