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Io, one of the satellites of Jupiter, has an orbital period of 1.769 days and the radius of the orbit is 4.22 × 10^8 m. Show that the mass of Jupiter is about one-thousandth that of the sun.

Q: 8.4  Io, one of the satellites of Jupiter, has an orbital period of  1.769  days and the radius of the orbit is  4.22\times 10^8\hspace{1mm}m .  Show that the mass of Jupiter is about one-thousandth that of the sun. 

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The orbital period in days is   =\ 1.769\times 24 \times 60\times 60\ s 

    Mass is given by :

                                                              M\ =\ \frac{4 \pi ^2 R^3}{GT^2}

Thus the ratio of the mass of Jupiter and mass of the sun is :

                                                \frac{M_s}{M_j} =\ \frac{\frac{4 \pi ^2 R_e^3}{GT_e^2}}{\frac{4 \pi ^2 R_{io}^3}{GT_{io}^2}}

or                                                       =\ \left ( \frac{1.769 \times 24\times 60\times 60}{365.25\times 24\times 60\times 60} \right )^2\times \left ( \frac{1.496\times 10^{11}}{4.22\times 10^8} \right )^3

or                                                      \approx 1045

Thus the mass of Jupiter is nearly one-thousandth that of the sun.

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