What is the mass of the planet and its moon using Kepler's 3rd law?

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The discussion focuses on calculating the mass of a planet and its moon using Kepler's 3rd law, given the planet's orbital period around the Sun and the moon's orbital characteristics. The user applies the small angle formula to determine the moon's orbital radius and attempts to use the modified Kepler's law equation to find the combined mass of the planet and moon. Clarifications are sought regarding the relative mass of the moon and the role of the planet's orbital period in the calculations. Key insights emphasize that the planet's mass does not significantly affect the moon's orbit due to the gravitational influences of other celestial bodies. The conversation encourages deriving the equations to better understand the relationships involved.
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Homework Statement


A planet orbits around the Sun with an orbital period of 14.5 years. It is observed it has a moon. At opposition, the moon has an almost circular orbit, and a radius of 7 arcminutes as seen from the telescope. Its orbital period around the planet is 3 days. What is the mass of the planet?

Homework Equations


Newton's ver Kepler's 3rd law: p^2 = [4pi^2/(G*M1+M2)]*a^3
G=6.67x10^-11 m^3/kgs^2

The Attempt at a Solution


I used the small angle formula to get "a";
at oppositon, a=(7")(1deg/60")(1.5x10^8km)(2pi/360deg)= 3.05x10^5km
p=72hrs*(3600sec/hr)
(Mplanet+Mmoon)= [4pi^2*a^3]/[G*p^2]

Am I doing this correctly? How do I find the mass of the moon since it doesn't state the relative mass is small enough to be negligible. Where does the orbital period of the planet come into play?
 
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Check your Keplers 3rd law, the orbit of a planet around another is independent of its own mass. You may have some difficultly accepting then why we don't see the Sun orbiting the Earth once every 100 years or so but remember all of the other planets etc, it's being pulled from many angles resulting in little effect. Whereas in this case it's safe to assume that the only force on the moon is that of the planet.

(Hint : derive it yourself by considering the equations )

\frac{mv^2}{r} = \frac{GMm}{r^2}
v = \frac{2\pi r}{T}
 
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