On Periodic Polar Orbits in the Artificial Satellite Problem
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On Periodic Polar Orbits in the Artificial Satellite Problem Martin Lara! Abstract We compute periodic orbits in the Artificial Satellite Theory. We fix our attention on the Earth case where we considered a zonal gravitational model for the potential. In a rotating frame attached to the Earth we have found sixteen polar orbits that are periodic after different number of revolutions. The altitude of each orbit remains almost constant and varies from about 263 km for the sixteen-revolutions orbit to about 35,830 km for the one-revolution orbit. All the orbits are linearly stable and their period is exactly 24 hours.
Introduction One of the problems that confronts mission designers for artificial satellites is the selection of initial conditions for orbits that are approximately periodic. A possible way of choosing initial conditions for periodic orbits of artificial satellites is by computing frozen orbits. These are orbits in which the average orbital elements remain constant. From a theoretical point of view frozen orbits correspond to stable relative equilibria in an averaged form of the zonal problem. A procedure for searching for frozen orbits is presented in Coffey et al. [1] where a combination of analytical, numerical and graphical techniques is applied to a zonal model of the gravity field. Families of frozen orbits are found for all inclinations. These results, obtained from asymptotic approximations by averaging, were later confirmed numerically in Lara et al. [2] for the non-averaged problem, where a frozen orbit was identified with a solution of the zonal problem that is periodic in the rotating meridian plane of the satellite. In this work we go one step further: we compute periodic orbits. Although apparently it has not been realized in the works mentioned above, the polar frozen orbits are exactly periodic in an inertial frame. It becomes evident when the non-averaged problem is formulated in cylindrical coordinates.
'Capitan de Corbeta, Armada Espanola, Real Instituto y Observatorio de la Armada, 11110 San Fernando, Spain. 321
Lara
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A concept that plays an important role on mission planing is "repeating ground track satellites." For a variety of missions, space mission planners search for orbits that retrace their ground tracks on the surface of the Earth. Ideal conditions for these orbits will be those corresponding to orbits that are periodic in a reference frame attached to the rotating Earth. Although a polar frozen orbit is periodic in an inertial frame, it is not necessarily periodic in a frame attached to the rotating Earth. The periodicity will occur only when the intrinsic frequency of the polar frozen orbit is commensurable with the rotation of the planet. To locate polar orbits that are periodic in a rotating frame attached to the Earth we use previous results on frozen orbits. The orbits computed in Lara et al. [2] were grazing frozen orbits but the intrinsic period of the polar one does not satisfy the condition for the periodicity previously stated. Since the period varies with
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