Question

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 $9a$. If planet $X$ orbits with a period $T$, Find out the period of planet $Y$'s orbit?

• A. $729T$
• B. $27T$
• C. $3T$
• D. $T/3$
• E. $T/27$

Answer 1

The correct option is B $27T$

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 , $\frac{T_X}{T_Y}=\frac{r_X^3}{r_Y^3}$

$\frac{T_X^2}{T_Y^2}=\frac{a^3}{{\left(9a\right)}^3}$

but given $T_X=T$

therefore $\frac{T^2}{T_Y^2}=\frac{a^3}{729a^3}$

or $T_Y=27T$