Keplette

Meet Keplette, a planet who knows that a bigger orbit means a much longer year.

Johannes Kepler found from Tycho Brahe's observations that a planet's orbital period T and the size of its orbit a are linked: T² is proportional to a³. He published this third law in 1619. Isaac Newton later derived it from his law of gravity and showed what the constant is: T² = 4π²a³/(GM), where M is the mass of the central body.

So doubling the orbit's size makes the year 2^1.5 ≈ 2.8 times longer. In the sticker the outer orbit is 70/44 times wider than the inner one, so its year is (70/44)^1.5 ≈ 2.0 times longer. Mars orbits at about 1.52 times Earth's distance, and its year is about 1.88 Earth years.

Run the law backwards and it weighs things: measure a moon's period and orbit and you get the planet's mass. The same method gives the masses of stars in binaries and of the black hole at the Milky Way's centre.

The simple form neglects the orbiting body's own mass. For two comparable masses, M is replaced by the total.