For a fixed separation, the gravitational acceleration between two uniform spheres of density $\rho$ is proportional to their radius $r$. But since angular sizes and distances across the celestial sphere are also proportional to separation and $r$, it is straightforward to show that the apparent angular motion across the celestial sphere of two objects released from rest and falling toward each other under gravity, is independent of their size or mass, for a given fixed angular size and separation. In other words, if two moon-size and moon-distance objects are released from rest to fall toward each other in the sky, their motion will appear identical to two much smaller and closer objects that also subtend 30 arc minutes. This seems to be a kind of scale invariance or symmetry, even though inverse square forces are not generally scale invariant. Is there a name for this kind of scale symmetry? The same kind of symmetry also exists when trying to pin down the size and distance to the sun without any more information than its angular diameter, its temperature, and the received solar irradiance at Earth (i.e. the Sun could equally be small and nearby or very large and far).



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