Angular Momentum & Orbits

Linear momentum (Chapter 1) is the universe’s ledger for straight-line motion. There is a second ledger for turning, and you have watched it settle accounts a hundred times: the figure skater who folds her arms in and abruptly triples her spin. Nobody pushed her. She traded arm-span for spin-rate at a fixed exchange rate, because the product of the two – angular momentum – is untouchable.

This chapter takes that ledger to the sky: why planets speed up when they swing near the Sun, why one launch speed separates circles from eggs from farewell trajectories, and how Newton convinced the world that the fall of an apple and the orbit of the Moon are one phenomenon with two costumes.

Here is the quantity: mass × speed × distance from the axis. The skater’s arms carry mass at a distance; fold them in and the distance shrinks, so the speed must rise to keep the product fixed. She can redistribute her spin but never shed it – only an outside twist (the ice’s friction, eventually) can touch the total.

Planets play the skater’s trick on a grand scale. A comet on a stretched orbit is “arms out” when far from the Sun – crawling – and “arms in” at closest approach: it whips around perihelion at terrifying speed, for exactly the skater’s reason. Halley’s comet spends decades dawdling beyond Neptune and mere months on its sprint around the Sun.

Johannes Kepler mined twenty years of the best naked-eye data ever taken and found three rules nobody ordered: planets trace ellipses with the Sun off-centre at a focus; they sweep equal areas in equal times; and the square of the year grows as the cube of the orbit’s size. Beautiful, arbitrary-seeming, and unexplained for eighty years.

Newton’s Principia (1687) pulled all three out of a single assumption: every mass pulls every other, weakening as the square of the distance. One force law, three “free” miracles – the moment natural philosophy learned what “explanation” could mean. Try it live below: one slider, one law, and every one of Kepler’s shapes on demand.

The word doing the work in “universal gravitation” is universal. Newton’s scandalous claim was not that things fall – everyone knew that – but that the sky falls too. The Moon, he said, is perpetually falling toward Earth, missing only because it moves sideways fast enough that the ground curves away beneath it. Orbiting is falling, artfully aimed.

And he checked it. The Moon is sixty times farther from Earth’s centre than the apple; inverse-square predicts its fall should be 60² = 3600 times gentler. Compute the Moon’s actual curve-away-from-a-straight-line per second, compare with the apple’s drop – the ratio comes out 3600, on the nose. One law, kitchen to cosmos. In the figure, the launch slider is the whole story: every orbit is a throw that keeps missing.

Now for classical mechanics’ finest hour, which is also its confession. By the 1850s, astronomers could predict planetary motions to breathtaking precision – and Mercury refused to balance. Its ellipse slowly pivots around the Sun, and after every known nudge from every known planet was accounted for, a stubborn leftover remained: the orbit’s closest point drifts an extra 43 arcseconds per century. That is one hundredth of a degree. Per hundred years.

Le Verrier – the man who had found Neptune from a smaller anomaly – tried the same trick and predicted an inner planet, “Vulcan.” Decades of searching: nothing. The discrepancy wasn’t a missing object. It was a missing theory. In 1915 Einstein computed Mercury’s orbit in general relativity, watched the extra 43″ fall out with no tuning, and later said the discovery gave him heart palpitations. You can watch that very precession in the Spacetime Visualizer, simulated in the equations that finally explained it.

Turning has its own conserved currency; orbits are conservation laws drawn in the sky; gravity is universal, and its one measured blemish was the doorbell of the next great theory. One chapter remains – and it saves the deepest idea for last. So far, physics has been local: what happens next depends on pushes happening now. The finale asks a different species of question: of all the paths a thing could take between here and there, why does nature pick that one? The answer has a number, the number has a name, and the name is on the front door of this website.