Question

Three planets orbit a star at distances $a,\;4a,\;9a$. Their orbital periods are proportional to $r^{3/2}$. If the smallest planet has period $T$, after how long will all three be aligned again?
 

• A. $8T$
• B. $27T$
• C. $216T$
• D. $512T$

Hint:

When repeated alignment is required, always compute the LCM of the revolution periods.

Answer 1

The Correct Option is C

Step 1: Use Kepler's law.
Period $\propto r^{3/2}$.
Thus:
$T_1 = T$ (for $a$),
$T_2 = (4a)^{3/2} = 8T$,
$T_3 = (9a)^{3/2} = 27T$.

Step 2: Planets align when LCM of periods occurs.
LCM$(T,\;8T,\;27T) = \text{LCM}(1,8,27) \cdot T$.
LCM$(1,8,27)=216$.

Step 3: Conclusion.
Therefore, planets will align every $216T$.