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Kepler's third law, T1^2/T2^2 = r1^3/r2^3, compares the periods and orbital radii of two satellites. Under what condition does this specific form of the law apply?
AIt applies to any two orbiting bodies in the universe
BIt applies only to two small masses orbiting one shared parent body
CIt applies only to perfectly circular orbits of equal radius
DIt applies only to satellites of identical mass
Answer & Solution
Correct answer: B. It applies only to two small masses orbiting one shared parent body
1. Deriving Kepler's third law from Newton's second law and the law of gravitation gives T^2 = (4 pi^2 / GM) r^3, where M is the mass of the parent body being orbited.
2. Taking the ratio of this expression for two satellites cancels the constant 4 pi^2 / G, but the parent mass M only cancels if both satellites orbit the same parent.
3. Because of that cancellation requirement, the simple ratio form T1^2/T2^2 = r1^3/r2^3 is valid only when comparing two small masses orbiting the same large parent body.
4. The law does not require equal radii or equal satellite masses, and it fails if the two bodies orbit different parents, since then M would not cancel out.
_Source: OpenStax College Physics (CC BY 4.0), Ch 6 "Uniform Circular Motion and Gravitation", section 6.6 Satellites and Kepler's Laws: An Argument for Simplicity_
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