Why Mercury's year is 88 days.
Mercury orbits the Sun in 88 days; Neptune takes 165 Earth years — roughly 690 times longer. Yet Neptune sits only about 78 times farther from the Sun than Mercury does. Period grows much faster than distance, because it scales with distance to the 1.5 power.
Semi-major axis a
T ∝ a1.5
0.2 AUlog scale · double a · 2.83× the period50 AU
Orbiting body's mass m
~Marslog scale · M fixed at 1 M☉ (the Sun)= the Sun
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Kepler's third law states that the square of a body's orbital period is proportional to the cube of its semi-major axis: T² ∝ a³, or precisely T = 2π√(a³/(G(M+m))), where G is the gravitational constant, M is the central body's mass, and m is the orbiting body's own mass — strictly it's the sum M+m that belongs in the denominator, not M alone. The m slider above lets you drag the orbiting body's mass from planet-sized toward star-sized to see when that matters: for anything actually orbiting the Sun, m is so much smaller than M that it barely moves the answer, which is also why the constants collapse so tidily. The equation above is the full, unit-agnostic law; what makes the numbers below simple is a choice of units — measuring T in Earth years, a in astronomical units (1 AU = Earth's average distance from the Sun), and mass in multiples of the Sun's own mass makes G's numeric value exactly 1, so it drops out of the arithmetic (not the physics). That trick is specific to AU/year/M☉ units; a moon orbiting a planet, or a planet orbiting a different star, would need the same general form worked out in that system's own units. The orbit diagram above is drawn at true linear scale, so Mercury through Mars sit almost on top of the Sun the way they really do — the small inset alongside it re-renders just that inner region at a much larger scale so those four close orbits are still visible individually, rather than compressing all nine planets onto a log scale that flatters the truth. See the Wikipedia article on Kepler's laws of planetary motion for the full derivation.