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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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