Throw a ball sideways and it curves down to the ground a few metres away. Throw it harder and it lands further away. Isaac Newton asked what happens if you keep going: fire a cannonball from the top of an impossibly tall mountain, faster and faster. His sketch of the answer appears in A Treatise of the System of the World, published in 1728, the year after his death. It is one of the most important drawings in science.
At some speed the ball falls exactly as fast as the curved Earth drops away beneath it. It is still falling, the whole time, but it never gets any closer to the ground. It has gone into orbit. Try it.
Real gravity (GM = 398,600 km³/s²), a round Earth of radius 6,371 km, no air and no spin. The path is the exact solution of Newton's law, drawn to scale.
Play with the speed at a fixed height. Below a certain speed the ball always hits the ground. At exactly the circular speed (about 7.67 km/s at the Space Station's height) the path closes into a perfect circle. Go a bit faster and the circle stretches into an ellipse that swings far out and comes back. At about 1.41 times the circular speed (that's √2), the ellipse never closes: the ball escapes Earth forever.
You can work out the magic speed on the back of an envelope. In its first second of falling, anything drops about 4.9 metres. Earth's surface is curved: if you travel 7.9 km in a straight line along it, the ground falls away by about 4.9 metres. So if you could skim the surface at 7.9 km per second, you would fall 4.9 metres each second and the ground would drop 4.9 metres each second. You'd never land. (In reality, the air would stop you long before. That is why real satellites fly above the atmosphere.)
| Circular speed | Escape speed | Time for one lap | |
|---|---|---|---|
| Skimming the surface (no air) | 7.91 km/s | 11.19 km/s | 84 min |
| Space Station height (408 km) | 7.67 km/s | 10.84 km/s | 92.6 min |
That last column is why astronauts on the Space Station see about 15 or 16 sunrises a day.
A common belief is that astronauts float because there's no gravity in space. At the Space Station's height, gravity is still about 88% as strong as on the ground. The astronauts float because they and their station are falling together, at the same rate, all the time. Nothing pushes up on their feet, so they feel weightless. A diver feels the same thing for a moment after jumping off a high board.
An orbit isn't an escape from gravity. It is a fall that never ends because it keeps missing.
Newton's boldest claim was that the force that drops an apple is the same force that holds the Moon. You can check his idea with three numbers.
The Moon goes round the Earth once every 27.3 days, at an average distance of about 384,400 km. That means it travels about 1.02 km every second, and its straight-line path has to bend towards Earth by an acceleration of about 0.00272 m/s². In one second, that bending pulls it about 1.36 millimetres off a straight line. That's the Moon falling.
Now the prediction. The Moon is about 60.3 Earth radii away. If gravity weakens with the square of distance, it should be 60.3 × 60.3 ≈ 3,640 times weaker there than at the surface: 9.81 ÷ 3,640 ≈ 0.0027 m/s². The two numbers agree to within about 1%, and the small difference comes mostly from simplifications in this back-of-envelope version. The same law, measured with apples, predicts the Moon. Before Newton, many people assumed the heavens followed different rules from the Earth. After this, they didn't.
The interactive shows something else Newton explained. Every orbit drawn by his law is a circle, an ellipse, or (for an escape) an open curve, which is exactly the shape Johannes Kepler had found for planetary orbits from careful observations. Newton's law turned Kepler's rules from a description into a consequence.
And the Moon's orbit isn't fixed. Laser beams fired from Earth at reflectors left by Apollo astronauts show that the Moon is moving away from us by about 3.8 cm a year. The tides it raises on Earth slowly drain energy from Earth's spin and feed it into the Moon's orbit. The days are getting very slightly longer, and the Moon is very slowly spiralling outward.
This is Newton's gravity, and it is good enough to fly spacecraft to other planets. It is not the final word: near very heavy objects or at very high precision, Einstein's general relativity takes over, and the GPS satellites overhead need its corrections every day (see Why Time Is Weirder Than You Think). But for a cannonball, the Space Station and the Moon, the old law is superb.
An orbit is falling while moving sideways fast enough that the ground curves away as fast as you fall: about 7.9 km/s at the surface and 7.67 km/s at the Space Station's height. Faster gives an ellipse, and √2 times faster lets you escape. Astronauts float because they are falling, not because gravity is gone. The Moon falls about 1.36 mm towards us every second, exactly as an inverse-square law tuned on falling apples predicts. That was the moment the sky and the ground turned out to obey the same physics.
Try this: in the simulator, find the slowest speed that just clears the Earth from 100 km up. Then raise the cannon to 2,000 km and see how much less speed you need. Higher orbits are slower.
Sources. Newton's cannonball: De mundi systemate / A Treatise of the System of the World (1728); Wikipedia, "Newton's cannonball", and the Carnegie Science cosmology timeline. ISS figures (about 400–420 km, 7.66 km/s, 90–93 min, 15.5–15.9 orbits a day): NASA JSC "Space Station Orbit Tutorial". The Moon's 3.8 cm/year recession from lunar laser ranging: NASA JPL. Computed in Python and node with GM = 398,600.4418 km³/s² and R = 6,371 km: circular and escape speeds, periods, gravity at 408 km = (6,371 ÷ 6,779)² = 88.3%, the Moon's centripetal acceleration (384,400 km, 27.32-day sidereal month) = 0.00272 m/s² and its 1.36 mm fall per second, and the curvature drop √(2 × 6,371 km × 4.9 m) = 7.9 km. The simulator's analytic paths were checked against a step-by-step numerical integration of Newton's law in node (agreement of impact points and periods to better than 0.1%).