Motion in General Elliptic Orbit with Respect to a Dragging and Precessing Coordinate Frame
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Motion in General Elliptic Orbit with Respect to a Dragging and Precessing Coordinate Frame Jean Albert Kechichian! Abstract The full set of second-order nonlinear differential equations describing the exact motion of a spacecraft subject to drag and oblateness perturbations in general elliptic orbit, relative to a rotating reference frame which drags and precesses exactly as a given spacecraft attached to its center is derived. This attached spacecraft is itself flying a general elliptic orbit and can be considered as the passive or nonmaneuvering vehicle. The unaveraged form of the J 2 acceleration is used for both vehicles leaving this oblateness perturbation position dependent for more exacting calculations. These equations can be effectively put to use in calculating by an iterative scheme, the impulsive rendezvous maneuvers in elliptic orbit around the Earth or those planets that are either atmosphere bearing or have a dominant second zonal harmonic, or both.
Introduction The formationkeeping between two vehicles in low-Earth orbit requires the consideration of both the drag and oblateness accelerations in the accurate design of the proper sequence of maneuvers that bring the spacecraft to proximity once they drift beyond a desired distance. This drift is inevitable when the vehicles have different ballistic coefficients forcing each to experience a different drag acceleration program and therefore relatively diverging trajectories. This differential drag effect will also cause one orbit to shrink faster than the other, thereby creating a differential precession effect due to the J2 harmonic. This oblateness precession will then disrupt the coplanar flight requiring the out-ofplane orbit control of say the active maneuvering vehicle in addition to the in-plane control which brings this vehicle to the vicinity of the passive vehicle for another period of formationkeeping flight. Considering both the drag and J 2 perturbations, we develop the general equations of relative motion for the case where the rotating 1Engineering
Specialist, The Aerospace Corporation, Astrodynamics Department, MS M4-947, P. O. Box 92957, Los Angeles, CA 90009. E-mail: [email protected]. Member AAS.
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frame is attached to the passive vehicle and therefore drags and precesses at the same rate as that vehicle. The motion of the active vehicle is referred to this accelerating frame and it too drags and precesses in its particular orbit. The twoimpulse noncoplanar rendezvous calculations can thus be made easier and an iterative scheme can be devised in order to compute the two Ll V's that achieve a desired rendezvous in a fixed time. The trajectories thus obtained are exact since they are based on numerically integrating the full nonlinear equations of motion without making any approximations. The J 2 acceleration is used in its unaveraged form and is therefore position dependent while the air density model for drag is assumed exponential without loss of generality due to the small variations in altitude for
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