Thus Kepler's 3rd Law is approximately valid because the Sun is much more
massive than any of the planets and therefore
Newton's correction is small. The data Kepler had access to were not good
enough to show this small effect. However,
detailed observations made after Kepler show that Newton's modified form of
Kepler's 3rd Law is in better accord with the data than Kepler's original form.
Two Limiting Cases
We can gain further insight by considering the position of the center of mass
in two limits. First consider the example just addressed, where one mass
is much larger than the other. Then, we see that the center of mass for the
system essentially concides with the center of the massive object:
This is the situation in the Solar System: the Sun is so massive compared with
any of the planets that the center of mass for a Sun-planet pair is always
very near the center of the Sun.
Thus, for all practical purposes the Sun IS almost
(but not quite) motionless at the center of mass for the system, as Kepler
originally thought.
