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Classical Mechanics

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Moderate

Distant Planets

CLMECH-Q86XLN

Astronomers find two planets in orbit around a distant star; the first with a period of $T_1$ and semi-major axis $a_1$, and the second with a different period of $T_2$ and semi-major axis $a_2$.

Which of the following equations could be a correct description of the relationship between these variables?

A

$\frac{a_1}{a_2} = \frac{T_1}{T_2}$

B

$\frac{a_1 T_1}{a_2 T_2} = constant$

C

$T_1 = \Big(\frac{a_1^2 T_2^3}{a_2^2}\Big)^{1/3}$

D

$a_1 = \Big(\frac{T_1^2 a_2^3}{T_2^2}\Big)^{1/3}$