Abstract
We give elementary derivations of the gravitational Newtonian potential created by different mass distributions. We begin with several two-dimensional objects such as rectangles, triangles and polygons. We emphasize the presence of two kinds of terms: logarithms and Arc-Tangents. We also show the connection with the well-known logarithmic potential of the wire segment. All these potentials are expressed in closed form.
We also give a few theorems which allow us to reduce several potentials to the potential of a more simple but equivalent mass distribution. For instance, the potential at the vertex of a triangle relates to the potential created by the wire segment at the side opposite to this vertex. On the other hand, the potential at the apex of a pyramid, created by the whole massive pyramid relates to the potential created by an equivalent mass distribution at the base of the pyramid. We use these properties to derive closed-form expressions for the potential created by a polyhedron at inner as well as outer points.
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© 1999 Springer Science+Business Media Dordrecht
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Broucke, R.A. (1999). Closed Form Expressions for Some Gravitational Potentials. In: Steves, B.A., Roy, A.E. (eds) The Dynamics of Small Bodies in the Solar System. NATO ASI Series, vol 522. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-9221-5_32
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DOI: https://doi.org/10.1007/978-94-015-9221-5_32
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