Why an orbit is a fall

Every astronaut you have ever seen floating in the space station is falling. So is the station, and so is the Moon. None of them are “above gravity”. They are falling around the Earth, again and again, and never hitting it.

An orbit is simply a fall that never finishes.

Look

Throw a ball sideways and it curves down to the ground. Throw it harder and it lands farther away. Now imagine throwing it so hard that, by the time it has fallen one metre, the ground beneath it has curved away by one metre too. It is still falling. It just never gets any closer.

On Earth that happens at about 7.8 km/s near the surface (28,000 km/h). Nothing on the ground can move that fast because the air would stop it, so we go above the air first. That is all a rocket does: it lifts a spacecraft above the atmosphere and then pushes it sideways, very hard.

• Slower than orbital speed: the path curves into the ground.
• Exactly orbital speed: a circle.
• Faster: an ellipse, swinging out and back.
• Faster than escape speed (about 11 km/s from low orbit): the path opens up and never returns.

Guess first: how strong is gravity at the height of the International Space Station, compared with the ground?

About 90% as strong. Astronauts float not because gravity is missing, but because they and their station are falling together.

Inside a falling spacecraft everything falls together at the same rate, so nothing presses on anything else. That is why astronauts float. Space agencies call it microgravity.

Understand

Near the Earth, gravity pulls with an acceleration of g = GM/r². Moving in a circle of radius r at speed v needs a pull towards the centre of v²/r. An orbit is where gravity provides exactly that pull:

v²/r = GM/r²  →  v = √(GM / r)

With GM of Earth = 398,600 km³/s² and r = 6,371 + 200 = 6,571 km, v ≈ 7.79 km/s. At the height of Newton’s imaginary 600 km mountain in the film, r = 6,978 km and v = 7.56 km/s.

The time for one lap follows: T = 2πr / v, about 88 minutes at 200 km. Higher orbits are slower and longer: the Moon, 384,400 km away, takes 27.3 days.

Master

Launch horizontally at radius r₀ with speed v and the path is a conic section with Earth’s centre at one focus. The energy per kilogram,

ε = v²/2 − μ/r₀,

decides the shape: ε < 0 is an ellipse (including the circle and the paths that hit the ground, which are ellipses interrupted by the surface), ε = 0 a parabola, ε > 0 a hyperbola. Setting ε = 0 gives escape speed v_esc = √(2μ/r₀), exactly √2 times the circular speed at the same height.

A horizontal launch always puts the launch point at an apsis: the highest point of the orbit if v is below circular speed, the lowest if above. That is why the slower balls in the film fall away from the mountain, and the faster ones come back to it from below.