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Classical Mechanics

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Difficult

Double Trouble

CLMECH-J8SAZL

Astronomers discover a star system with two stars, 'A' and 'B' orbiting together. Orbiting just around star A is planet 1, and orbiting just around star B is planet 2. Planet 3 orbits star A and B simultaneously, far enough away from the system that it treats the two stars as being a single, compact object. The image below shows an approximation of the orbits.

If the orbit of planet 1 has a semi-major axis of 3.2 AU and a period of 1.3 years, and the orbit of planet 2 has a semi-major axis of 1.1 AU and a period of 0.32 years. Planet III has a period of 23 years.

What is the semi-major axis of Planet III's orbit?

Note: You may treat the mass of each of the planets as being very small when compared with the mass of the stars. You may find the expression of the gravitational constant G as $1.98 \times 10^{-29} \space AU^{3} kg^{-1} yrs^{-2}$ to be helpful.

A

26 AU

B

44 AU

C

88 AU

D

120 AU