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Orbit is speed

Rockets and landing · level 4, lesson 2 of 6

A question to start

Some rockets carry tourists past the Kármán line, 100 km up, for a few minutes of floating, then fall back. Others put satellites into orbit at 200 or 400 km. The orbital rockets are enormous by comparison. Why is orbit so much harder than just reaching space?

The idea

Reaching space is about height. Staying there is about speed.

  • To go straight up to 100 km and fall back, a rocket needs to reach about 1.4 km/s.
  • To orbit at 200 km, it must go sideways at about 7.8 km/s, so fast that as it falls, the curved Earth drops away beneath it (Newton's cannon, from Gravity, level 2).

Speed costs much more than height, because the energy of motion grows with the square of the speed. Per kilogram, climbing to 200 km takes about 2 MJ of energy, while going 7.8 km/s takes about 30 MJ: fifteen times as much.

up to space and back:about 1.4 km/sinto orbit: 7.8 km/s sideways

That is why orbital rockets turn over, soon after lift-off, and spend most of their burn speeding up sideways. A rocket going to orbit climbs to its height in the first few minutes and then mostly builds speed along the horizon.

Low orbit Mars Earth
Speed for a low orbit about 3.4 km/s about 7.8 km/s
Time for one lap about 1 h 50 min about 1 h 30 min
Total speed a rocket must add from the ground (with losses) about 4 km/s about 9.4 km/s

On Mars, orbital speed is less than half of Earth's. Because the fuel a rocket needs grows very steeply with the speed it must reach, a rocket from the surface of Mars to orbit can be much smaller for the same payload.

Worked examples

1. Straight up. From \(v^2 = 2gh\) (ignoring air and the small fall in \(g\)): \(v = \sqrt{2 \times 9.8 \times 100{,}000} = 1{,}400\) m/s.

2. Energy per kilogram. Moving at 7,800 m/s: \(\tfrac12 \times 7800^2 = 30{,}400{,}000\) J, about 30 MJ. Lifting to 200 km: about \(9.5 \times 200{,}000 = 1.9\) MJ (using an average \(g\) of 9.5).

Try it

  1. About how fast must something go to orbit the Earth just above the air? To orbit Mars?
  2. Why do rockets going to orbit tip over soon after lift-off?
  3. Is it harder to reach 100 km, or to orbit at 200 km? Why?
  4. How much more energy per kilogram does orbit at 7.8 km/s take than a speed of 3.4 km/s?
Answers
  1. About 7.8 km/s; about 3.4 km/s.
  2. To spend most of the burn speeding up sideways, which is what orbit needs.
  3. Orbiting: it needs far more speed, and energy grows with speed squared.
  4. \((7.8/3.4)^2 = 5.3\) times as much.

Common mistake

"Orbit is a place you go up to." Orbit is a speed you reach. A rocket that climbed to 400 km and stopped would fall straight back down.

Where it leads