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As a thought experiment, consider the Earth, as it is with its stratified layers - a dense core with progressively less dense layers on top until you get to the crust and out into the stratified atmosphere. Now take the moon and shrink it down to the size of a softball. Retain the mass of the moon, but now it’s close to a neutron star in density. Hit pause and hold this ultra-dense object directly over the surface of the Earth
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Now start the simulation and let the moon fall. The deepest portion of Earth’s core and the center of the softball-sized moon will quickly displace the less dense materials between them and merge, with the little moon traveling the most distance and the core moving slightly for the merging. There will be some oscillation as the gravity well attains hydrostatic equilibrium again, but the dropped "softball moon" will quickly occupy the core, driving the Earth’s well ever deeper with its added mass. And due to its ultra-density, it will reside at the point of greatest acceleration - the center.
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