Planetary Opposition: Can You Solve It?

Classical Mechanics Level pending

Consider two planets of perfectly circular and concentric orbits, orbiting around a star. The orbit of the planet closest to the star (Planet Alpha) has an orbital radius of 100 million kilometers and a constant linear speed of 9,513 m/s. The orbit of the other planet (Planet Beta) has an orbital radius of 200 million kilometers and a constant linear speed of 31,710 m/s. The planets are orbiting in opposite directions. If the planets begin in opposition at time $$t=0$$ then when is the next time, in years, when the planets will be in opposition again?

Note that the picture above is to only to serve as an example of what an opposition looks like; a given answer may appear to contradict certain elements in the picture. This does not necessarily mean that an answer is incorrect.

Details and Assumptions:

• The planets' gravitational fields will not interfere with each other
• An opposition is when two planets are aligned, sharing the same angle.

Hint: convert speeds to kilometers per year.

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