Here I simply wish to have Marlow clarify the problem I'm supposed to work.
Marlow says:
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The condition of the problem specifically says "by applying Newton's laws
in a frame in which Earth and Sun are at rest," so there is no orbiting
going on in that frame at all; however, you may introduce any pseudoforces
necessary to balance the real interaction between Earth & Sun (Newtonian
action-at-a-distance inverse square approximation fully acceptable! No GR
considerations needed.) But no, you cannot make any false assumptions
about the center of mass -- that's a well defined quantity involving
only masses and positions, with no reference at all to velocity or
acceleration.
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and later
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Ground rules as stated; derive an astronomically accurate formula for
either M or M+m by applying Newton's laws relative to a frame of reference
in which Earth and Sun are at rest, using Newtonian formula for
gravitational interaction between Earth and Sun, and any pseudoforces
you need.
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This cleared up several of my questions, but still isn't well posed enough.
Tell the astronomically correct formula? Would deriving
period^2 = (4 pi (earth-sun distance)^3)/G(M+m)
do??
I won't make any assumptions about the center of mass, May I assume a
circular orbit in the inertial frame; i.e. a statics problem in the
non-inertial frame?
I may have a follow up question depending on the answers to these questions