Abstract
The Hill differential equation describes the relative motion of a satellite w.r.t. a circular reference orbit. The deviations in the orbit are caused by a residual acceleration, which is small compared to the effect of the central gravitational field. In this paper, the acceleration of a local mass anomaly in the central body is considered, which rotates w.r.t. the inertial frame with a constant angular velocity. The mass anomaly is modeled by a superposition of radial base functions. The potential and the gradient of each base function are represented in the orbit by the Keplerian elements, the rotation rate of the central body and the parameters of the base function, i.e. the position of its center, the shape and a scaling factor. The inhomogeneous solution of the Hill differential equation for short arcs is found by means of the Laplace transform. A few lower orders of the solution require an additional Laplace transform, to consider the so-called resonance cases. The final deviations are described in a closed and differentiable formula of the Keplerian elements and the parameters of the base function.
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Antoni, M., Keller, W. Closed solution of the Hill differential equation for short arcs and a local mass anomaly in the central body. Celest Mech Dyn Astr 115, 107–121 (2013). https://doi.org/10.1007/s10569-012-9454-7
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DOI: https://doi.org/10.1007/s10569-012-9454-7