The formula is calculating the gravitational flux on the surface of a 3-dimensional sphere, and 3-dimensional spheres have a surface area 4π times their radii.
Saying that this is what the formula intrinsically does, amounts to saying that field lines are more fundamental/"real" than action-at-distance forces on point particles. But in the context of purely Newtonian gravity, both formulations are in fact completely equivalent. (And if you appeal to relativity to justify considering fields more fundamental, then why not better go for simplifying Einstein's equation and including 8π in G?)
Saying that this is what the formula intrinsically does, amounts to saying that field lines are more fundamental/"real" than action-at-distance forces on point particles.
Yep :-). I don't know enough the physics to back that up, but that's what my gut tells me. A more educated version of me might be able to say something "the vocabulary of forces is 'shallow'; the vocabulary of fields is deeper; the vocabulary of group symmetries is deeper still." I certainly do not have the depth of understanding to make that sort of statement wit...
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