Proving T² ∝ a³ from the law of universal gravitation

Step 1 of 3: Gravity as the centripetal force

Intuition

Like swinging a weight on a string: the tighter the string pulls (larger centripetal force), the faster the mass must whirl to 'hold' its orbit — gravity here plays the role of the invisible string.

\frac{GMm}{a^2} = m\omega^2 a

Analysis

Consider a circular orbit of radius $a$ (the simplest ellipse). In uniform circular motion the centripetal acceleration is $\omega^2 a$ ; the only force is gravity $GMm/a^2$ . Newton's second law in the radial direction gives the equality above — the planet's mass $m$ cancels immediately, meaning the orbit does not depend on the planet's mass.