Chapter 8: Problem 15
In deriving Kepler's third law (8.55) we made an approximation based on the fact that the sun's mass \(M_{\mathrm{s}}\) is much greater than that of the planet \(m .\) Show that the law should actually read \(\tau^{2}=\left[4 \pi^{2} / G\left(M_{\mathrm{s}}+m\right)\right] a^{3},\) 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 \(2 \times 10^{27} \mathrm{kg}\), while \(M_{\mathrm{s}}\) is about \(2 \times 10^{30} \mathrm{kg}\) (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?
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Key Concepts
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