Two planets, $X$ and $Y$ , revolving in a orbit around star. Planet $X$ moves in an elliptical orbit whose semi-major axis has length $a$ . Planet $Y$ moves in an elliptical orbit whose semi-major axis has a length of $9 a$ . If planet $X$ orbits with a period $T$ , Find out the period of planet $Y$ 's orbit?

Question

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Answer 1

Kepler's third law of planetary motion gives $T^{2} \propto r^{3}$
where $T =$ time period of revolution , $r =$ length of semi major axis of elliptical orbit
by this relation we get , $\frac{T _{X}}{T _{Y}} = \frac{r _{X}^{3}}{r _{Y}^{3}}$
$\frac{T _{X}^{2}}{T _{Y}^{2}} = \frac{a ^{3}}{( 9 a ) ^{3}}$
but given $T_{X} = T$
therefore $\frac{T ^{2}}{T _{Y}^{2}} = \frac{a ^{3}}{7 2 9 a ^{3}}$
or $T_{Y} = 27 T$

Answer 2

Kepler's third law of planetary motion gives $T^{2} \propto r^{3}$

where $T =$ time period of revolution , $r =$ length of semi major axis of elliptical orbit

by this relation we get , $\frac{T _{X}}{T _{Y}} = \frac{r _{X}^{3}}{r _{Y}^{3}}$

$\frac{T _{X}^{2}}{T _{Y}^{2}} = \frac{a ^{3}}{( 9 a ) ^{3}}$

but given $T_{X} = T$

therefore $\frac{T ^{2}}{T _{Y}^{2}} = \frac{a ^{3}}{7 2 9 a ^{3}}$

or $T_{Y} = 27 T$