Determine the mass of Uranus in the masses of the Earth if the period of revolution of the Oberon satellite around Uranus is 13.46 days, its semi-major axis is 5.8 x 10 ^ 8 m.
We will use formula III of the improved Kepler’s law a1 / a2 = T1 (M1 + m1) / T2 (M2 + m2)
where M.1 is the mass of Uranus, m1 is the mass, T1 is the orbital period, and a1 is the semi-major axis of Oberon’s orbit. Since the second body is not indicated, we need to take a body whose parameters we know well, for example, the Moon.
For the Moon, a2 = 3.84 * 10 ^ 5 km, T2 = 27.32 days, m2 is the mass of the Moon, M2 is the mass of the Earth. Since the masses of Oberon and the Moon are small compared to the masses of the Earth and Uranus, they can be neglected, and then the formula will take the form:
a1 / a2 = T1 * M1 / T2 * M2
If the mass of the Earth is taken as a unit M2 = 1, then M1 = a1 / a2 = T1 * M1 / T2 * M2
M1 = (5.8 * 10 ^ 8) ^ 3 27.32 ^ 2 / (3.84 * 10 ^ 8) ^ 3 13.46 ^ 2 = 14; M1 = 14M2, that is, the mass of Uranus is equal to 14 Earth masses.
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