Question

Two planets X and Y travel counterclockwise in circular orbits about a star, as seen in the figure.



The radii of their orbits are in the ratio 5:3. At some time, they are aligned, as seen in (a), making a straight line with the star. Five years later, planet X has rotated through 86.8°, as seen in (b). By what angle has planet Y rotated through during this time?

Answer 1

You need to use Kepler's Third Law. For two planets, X and Y, orbiting the same star:\[\frac{ T _{X}^{2} }{ R _{X}^{3} }=\frac{ T _{Y}^{2} }{ R _{Y}^{3} }\]Where TX is the orbital period of Planet X; RX is the mean orbital radius for Planet X; TY is the orbital period of Planet Y; and RY is the mean orbital radius for Planet Y.

Answer 2

You can rearrange to get this:\[\frac{ R _{X}^{3} }{ R _{Y}^{3} }=\frac{ T _{X}^{2} }{ T _{Y}^{?} }\]You know that the ratio of RX to Ry is 5:3, so you get:\[\frac{ 5^{3} }{ 3^{3} }=\frac{ T _{X}^{2} }{ T _{Y}^{2} }=4.6296\]That means:

Answer 3

\[\frac{ T _{X} }{ T _{Y} }=\sqrt{4.6296}=2.1517\]This means that Planet X's orbital period is 2.1517 times longer than Planet Y's.

Answer 4

One period is 360°, and Planet X has covered 86.8°. That means that Planet X has covered (86.8/360)=0.2411 of a period.

Answer 5

Planet Y will have covered 2.1517 x 0.2411=0.5188 of its period, which gives 0.5188*360°=186.8°. Planet Y will have rotated through 186.8°.