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I'm playing around with another idea that probably has been thought of
before, so I thought I'd do a bit of pre-math research...
The "simple" problem: Can we prove that there are other planets in our
solar system using the Doppler Shift of the sun as seen from earth?
The probably-not-simple approach: Work out a 3-to-9 body CoM problem, with
the twitch of a rotating frame of reference so the CoM of the system is
stably viewed. The sun should appear to "wobble" a bit about
that point due
to the gravitational influences of the planets, and if we take the head-on
wobbling portion, that should give us doppler-shifted lines (admittedly of
very small order) that we could observe to confirm the existence
of planets.
Gordon P. Smith