Orbits: falling round the Earth

At 400 km up, a satellite must travel about 7.7 km every second to stay in orbit.

Newton's cannon

Isaac Newton described a thought experiment that still explains orbits better than anything. Put a cannon on top of an impossibly tall mountain, above the air, and fire a ball sideways. It falls to the ground some distance away. Fire it faster and it lands farther off. Earth is round, so as the ball goes farther the ground curves away beneath it. At one particular speed the ground curves away exactly as fast as the ball falls, and the ball never lands. It keeps falling round the Earth forever. That is an orbit.

This is also why astronauts float. The International Space Station is only about 400 km up, where gravity is still about 89% as strong as on the ground, (6,371 ÷ 6,771)². The crew float because they and the station are falling together, at the same rate, all the time. Engineers call it free fall, or microgravity.

Earth itself gives every launch a head start. The ground at the equator moves east at 2π × 6,378 km ÷ 23.93 hours ≈ 0.47 km/s, and a rocket standing on the pad already has that speed. Launching eastward adds it to the orbital speed for free, which is why most rockets head east and why launch sites are often placed as close to the equator as a country can manage. At latitude 34° the bonus has shrunk to about 0.39 km/s, and a rocket going into an orbit over the poles gets almost none of it.

How fast is orbital speed?

For a circular orbit the speed comes from one formula: v = √(GM ÷ r), where GM for Earth is 398,600 km³/s² (NASA Earth fact sheet) and r is the distance from Earth's centre, not from the ground. At the surface, r = 6,371 km gives about 7.9 km/s, the speed of Newton's cannonball if there were no air. At 400 km, r = 6,771 km, the speed is 7.67 km/s and one lap takes 2πr ÷ v ≈ 92 minutes. NASA rounds this as the station going round about every 90 minutes, 16 times a day.

Higher orbits are slower and take longer. GPS satellites at about 20,200 km circle the Earth twice a day (GPS.gov), and the formula gives 11.97 hours. Go out to r = 42,164 km, an altitude of 35,786 km, and the period becomes 23 hours 56 minutes, one sidereal day. A satellite there over the equator keeps pace with Earth's turning and seems to hang still in the sky. That is the geostationary orbit used by weather satellites such as Korea's Chollian series and by satellite TV, which is why home dishes can point at one fixed spot.

Escaping for good

Speed up beyond circular speed and the orbit stretches into an ellipse that swings farther out. At √2 times the circular speed, about 41% more, the ellipse opens up completely and the object never comes back. That is escape velocity. From Earth's surface it is √2 × 7.91 ≈ 11.2 km/s; NASA's fact sheet gives 11.186 km/s. Spacecraft bound for Mars or the outer planets often start in a low parking orbit and then fire their engines again to escape Earth. A spacecraft can reach the Moon without exceeding Earth's escape velocity.