Physics

Kepler's Third Law is the key here—it connects orbital period to orbital radius through a power-law relationship. Rather than memorizing the law, think about what must be true: as planets orbit farther from the Sun, gravity weakens, so they move more slowly and take longer to complete their orbits. The specific relationship comes from balancing gravitational force with the centripetal force needed for circular motion. Working through that balance reveals how $T$ and $R$ must scale relative to each other.

Kepler's Third Law is the key here—it connects orbital period to orbital radius through a power-law relationship. Rather than memorizing the law, think about what must be true: as planets orbit farther from the Sun, gravity weakens, so they move more slowly and take longer to complete their orbits. The specific relationship comes from balancing gravitational force with the centripetal force needed for circular motion. Working through that balance reveals how $T$ and $R$ must scale relative to each other.