The confinement of most satellite galaxies in the Local Group to thin planes presents a challenge to the theory of hierarchical galaxy clustering. The PAndAS collaboration has identified a particularly thin configuration with kinematic coherence among companions of M31 and there have been long standing claims that the dwarf companions to the Milky Way lie in a plane roughly orthogonal to the disk of our galaxy. This discussion investigates the possible origins of four Local Group planes: the plane similar, but not identical to that identified by PAndAS, an adjacent slightly tilted plane, and two planes near the Milky Way: one with nearer galaxies and the other with more distant ones. Plausible orbits are found by using a combination of Numerical Action methods and a backward in time integration procedure. For M31, M33, IC10, and LeoI, solutions are found that are consistent with measurements of their proper motions. For galaxies in planes, there must be commonalities in their proper motions, and this constraint greatly limits the number of physically plausible solutions. Key to the formation of the planar structures has been the evacuation of the Local Void and consequent build-up of the Local Sheet, a wall of this void. Most of the M31 companion galaxies were born in early-forming filamentary or sheet-like substrata that chased M31 out of the void. M31 is a moving target because of its attraction toward the Milky Way, and the result has been alignments stretched toward our galaxy. In the case of the configuration around the Milky Way, it appears that our galaxy was in a three-way competition for companions with M31 and Centaurus A. Only those within a modest band fell our way. The Milky Ways attraction toward the Virgo Cluster resulted in alignments along the Milky Way-Virgo Cluster line.
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