A Saturn year is $29.5$ times the Earth year. How far is the Saturn from the Sun if the Earth is $1.50 \times 10^{8} \mathrm{~km}$ away from the Sun?

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A Saturn year is $29.5$ times the Earth year. How far is the Saturn from the Sun if the Earth is $1.50 \times 10^{8} \mathrm{~km}$ away from the Sun?

According to Kepler’s third law of planetary motion, the following relation can be written,

$T=\sqrt{\frac{4 \pi^{2} r^{3}}{G M}}$

We get $\mathrm{T}^{2} \propto \mathrm{r}^{3}$ from the above equation.

The orbital period and the mean distance of Saturn from the Sun are $\mathrm{T}_{\mathrm{s}}$ and $\mathrm{r}_{\mathrm{s}}$ respectively.

The orbital period and the mean distance of Earth from the Sun are $T_{e}$ and $r_{e}$ respectively.

Time period of Saturn is $T_{s}=29.5 \mathrm{~T}_{\mathrm{e}}$

$\Rightarrow \mathrm{r}_{\mathrm{s}}^{3} / \mathrm{r}_{\mathrm{e}}{ }^{3}=\mathrm{T}_{\mathrm{s}}^{2} / \mathrm{T}_{\mathrm{e}}^{2}$

$r_{s}=r_{e}\left(T_{s} / T_{e}\right)^{2 / 3}$

$=1.5 \times 10^{8} \times 29.5^{2 / 3}$

$=14.3 \times 10^{8} \mathrm{~km}$

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