The Moon is falling, all the time, and missing: it moves sideways fast enough that the ground curves away beneath it as quickly as it drops.

Falling and missing#

Gravity between two masses follows one inverse-square law:

$$\frc{F} = G\frac{Mm}{\pos{r}^2}$$

Circles, ellipses, Kepler's three laws and the length of a year all follow from it. Launch slower than circular speed and the release point becomes the farthest point of an ellipse. Faster, and it becomes the nearest.

Fast close in, slow far out#

That product is the planet's angular momentum per unit mass. Gravity always points at the star, so it exerts no torque about it, and angular momentum stays fixed. Equal areas in equal times, Kepler's second law, is the previous lesson applied to a planet.

A year, and escape#

For a circular orbit, $GMm/r^2 = mv^2/r$ with $v = 2\pi r/T$ gives Kepler's third law. It holds for ellipses too, with $r$ replaced by the semi-major axis, and the planet's own mass cancels:

$$T^2 = \frac{4\pi^2}{GM}r^3$$

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What comes next#

You turned an ellipse into a circle, stretched it to 2 AU, and found the speed that never returns, all from one inverse-square force. Orbits repeat, and so do springs, pendulums, molecules and bridges; next comes the one equation underneath all of them.