Why doesn’t the Moon fall down?

A common answer is that the Moon, like the astronauts on the International Space Station, is “beyond gravity”. It isn’t. The station flies about 400 km up, and at that height Earth’s pull is still 89% as strong as it is on the ground. The astronauts float for a different reason: they and their station are falling together, so nothing pushes on their feet. An orbit is a fall that never reaches the ground.

Newton’s cannon

Isaac Newton explained this with a thought experiment. Put a cannon on a very tall mountain, above the air, and fire a ball horizontally. Gravity bends its path down and it lands some way off. Fire faster and it lands farther away. But the Earth is round, so the ground under a fast ball curves away beneath it. Fire fast enough and the ground curves away exactly as fast as the ball falls. The ball never lands; it is in orbit, a path that repeatedly circles a body.

Look at the two numbers under the figure. Whatever the launch speed, the ball falls the same distance in its first second: about 4.9 m at the ground, 3.7 m from the top of the 1,000 km mountain, where gravity is a little weaker. Speed doesn’t change how fast it falls; it changes how far it goes while falling. A circle of radius R drops away below a straight line by about d²/2R over a distance d. Here R is the distance from the Earth’s centre: 6,371 km at the ground. At 7.9 km/s the ball covers 7.9 km in a second, and the ground drops by 7.9²/(2 × 6,371) km = 4.9 m. The fall and the curve match.

That speed is the circular speed. It depends only on how strong gravity is and how far you are from the centre of the Earth:

v_circ = √(GM / r)

Here G is Newton’s gravitational constant, M the Earth’s mass, and r the distance from the Earth’s centre. The product GM for the Earth is known very precisely: 398,600 km³/s². At the surface this gives 7.9 km/s. At the Space Station’s height it gives 7.67 km/s and one lap every 92 minutes, which is what the station really does. Notice what is missing from the formula: the mass of the ball. A feather and a space station at the same height orbit at the same speed.

Fire faster than circular speed and the ball climbs away from the Earth on the far side, slows as it climbs, and falls back to the mountain top: the orbit has stretched into an ellipse. (From a mountain as tall as the one in the figure, a little less than circular speed is already enough: the ball dips on the far side of the Earth but clears the ground. That is an ellipse too, with the mountain top as its highest point.) At √2 times circular speed, 11.2 km/s from the ground, it never comes back. That is the escape speed.

Newton’s Moon test

The Moon is the same kind of cannonball, just much farther away. Newton made this check himself, roughly in the 1660s and carefully in his Principia of 1687. The Moon is about 60 Earth radii away (384,400 km). If gravity weakens with the square of distance, it should pull 60² = 3,600 times more weakly there than at the ground: 9.81 m/s² ÷ 3,600 = 2.72 mm/s². In one second, an acceleration that small makes something fall 1.36 mm.

Now the Moon itself. It travels round a circle of radius 384,400 km once every 27.32 days, a speed of 1.02 km/s. In one second it moves 1.02 km, and its path bends away from a straight line by v²/2r = 1.36 mm. The Moon falls towards the Earth exactly as far each second as an apple would if it were up there. The force that drops apples is the force that steers the Moon.

Launch a planet

The same law holds the planets round the Sun. Johannes Kepler had found, from Tycho Brahe’s observations, that each planet moves on an ellipse with the Sun at one focus, one of the two special points inside an ellipse (the distances from any point on the ellipse to the two foci always add up to the same total). Newton showed that an inverse-square force makes exactly this happen.

The eccentricity measures how stretched an ellipse is: 0 for a circle, close to 1 for a long thin one. Earth’s orbit has 0.017; the Moon’s, 0.055.

The shaded slices show Kepler’s second law. Each slice is swept out in the same time, and each has the same area. Near the Sun the planet moves fast and its slices are short and wide; far out it crawls and they are long and thin. This is the conservation of angular momentum: with a force that points straight at the Sun, nothing can speed up or slow down the planet’s sweeping.

The period depends only on the size of the ellipse. If a is half its longest diameter (the semi-major axis), then T = 2π√(a³/GM): Kepler’s third law. A given launch speed at a given distance fixes the planet’s energy, and the energy fixes a, so swinging the arrow changes the shape but not the period.

Kepler found this third law in 1619 by comparing the planets themselves. Figure 3 plots the real ones: how far each is from the Sun, in astronomical units (AU, the Earth’s average distance, 149.6 million km), against how long its year is. With these units the law becomes simply T² = a³.

What if gravity were different?

Why ellipses, and not some other curve? It comes down to the exact way gravity weakens with distance. Newton’s law says the force falls off as 1/r², the inverse square. Suppose instead it fell off as 1/rⁿ for some other power n.

For any other power the orbit does not close. The closest point creeps round a little each lap and the path draws a rosette. For n below 2 the closest point slips backwards; above 2 it runs ahead; near 3 the orbit becomes violently unstable and the body can spiral into the centre. In 1873 Joseph Bertrand proved that only two force laws make every bound orbit close on itself: the inverse square, and a spring-like force that grows in proportion to distance. The spring gives ellipses too, but with the centre of force at the middle of the ellipse, not at a focus.

Real orbits do creep slightly. Mercury’s closest point to the Sun advances by about 575 arcseconds (0.16°) a century. About 532″ of that comes from the pull of the other planets. The last 43″ puzzled astronomers for half a century, until Einstein’s general relativity explained it in 1915: near the Sun, gravity is very slightly not inverse-square.

So why doesn’t the Moon fall down?

It does. Every second it falls 1.36 mm towards us, and every second it moves 1.02 km sideways, far enough that the Earth has curved away beneath it. The speed an orbit needs depends only on the mass of the body being orbited and the distance from it, never on what is doing the orbiting. And because gravity weakens as exactly the inverse square of distance, the fall closes up into an ellipse with the planet at one focus, repeating for billions of years.