In deriving Kepler's third law we made an approximation based on the fact that the sun's mass Ms is much greater than that of the planet m. Show that the law should actually read

Γ^2 = [4π^2 / G(Ms + m)]a^3 ,

and hence that the "constant" of proportionality is actually a little different for different planets. Given that the mass of the heaviest planet (Jupiter) is about 2 x 10^27 kg, while Ms is about 2 x 10^30 kg (and some planets have masses several orders of magnitude less than Jupiter), by what percent would you expect the "constant" in Kepler's third law to vary among the planets?