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Question: in a distance star system a planet is 9 au...

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In a distance star system a planet is 9 AU (astronomical unit) away from the sun, and has a circular orbit with period 12 earth years. The planet has a moon distanced 0.5 AU away, which circles the planet in a circular orbit also with period 12 earth years. Suppose the orbital planes of the planet around the sun and the moon around the planet coincide, and their motions are both counterclockwise. At midnight March 1 2021, the positions of the sun, planet, and moon are observed to be (0, 0), (9, 0), (9, 0.5) on their common orbital plane.

  1. (a)  Determine the position (xP (t), yP (t)) of the planet, t years after midnight March 1 2021.

  2. (b)  Determine the position (xM (t), yM (t)) of the moon, t years after midnight March 1 2021. Express your

    answer in the form

    xM (t) = A cos(kt + α) + xP (t) yM (t) = A sin(kt + α) + yP (t)

    where A,k > 0 and α (π,π].

  3. (c)  Simplify your expressions for xM (t) and yM (t) to the form

    xM (t) = B cos(kt + β) yM (t) = C sin(kt + γ)

    where B, C, k > 0 and β, γ (π, π], leave your answers in exact form.

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