What is the underlying law of nature that determines whether these two
specks in the sky have some kind of influence on a planet millions of
miles away from them?

>From the viewpoint of scientific knowledge, this makes absolutely no
  [Planet Uranus and dwarf planet Pluto in the sky from Earth 2013-04-22]

In addition things like conjunctions where two celestial bodies seem to
line up, is just a function of the perspective of the viewing position;
from a different angle, there is no conjunction, so it is just smoke and
  [Conjunction of Titan and Rhea]

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