Orbital Mechanics

Orbital Mechanics

The planets circle the Sun, the Moon circles the Earth, and thousands of satellites circle both. Not one of them flies off into space, and not one of them falls in. That is stranger than it sounds. A satellite has no engine running. Gravity is pulling it downwards the whole time and never lets up. So why has it not landed?

With a force, and without

Put the two cases side by side and the orbit stops being mysterious:

Label the orbit

Drag each label onto the right feature: the body in the middle, the dashed path, the red arrow pointing inward, and the green arrow along the path.

Is it accelerating?

A satellite travels round a perfect circle at a completely constant speed. Its speedometer would not move all day. Is it accelerating?

• Yes. Acceleration means a change in velocity, and velocity includes direction. The direction is changing every instant, so the satellite is accelerating continuously even though its speed never changes
• No. Acceleration means going faster, and its speed is constant
• Only at the points where it changes direction most sharply
• Only if the orbit is elliptical rather than circular

Naming what holds an orbit

Five terms, and one of them is the source of most confusion in the topic:

Why does it not speed up?

Gravity pulls on the satellite the entire time it is in orbit, and never stops. Yet its speed stays exactly the same. Why does a force that acts non-stop not make it go any faster?

• Because the pull is always at right angles to the direction of travel. A force only changes speed when part of it points along the motion, and here none of it does, so all of it goes into turning
• Because gravity is very weak at that altitude
• Because an equal outward force balances gravity, so the resultant force is zero
• Because there is no air resistance in space to speed it up or slow it down

One speed per radius

For a stable circular orbit at a given radius there is exactly one speed that works. Smaller orbits need higher speeds, which is why Mercury laps the Sun in 88 days and Neptune takes 165 years. What if the speed is wrong for the radius? The orbit is not stable and the radius changes. • Too fast for that radius, and the body climbs into a larger orbit. Fast enough, and it leaves altogether. • Too slow, and it falls into a smaller orbit. If the fall continues it re-enters the atmosphere. Notice what that means: a satellite that loses speed ends up in a lower orbit, where the required speed is higher than it was before. Losing speed moves you to where you need more of it, which is why an orbit that begins to decay does not settle at some new lower level of its own accord.

The satellite that slows

The space station orbits low enough that a very thin trace of atmosphere still drags on it. Left completely alone, what would happen over the following months?

• Drag removes speed, so it drops to a lower orbit. Lower down the air is thicker, so the drag is greater, so it drops faster still, and the process runs away until it re-enters and burns up
• It drops a little and then settles into a new stable orbit at the lower altitude
• It stays at the same height but circles more and more slowly until it stops
• Losing speed lets gravity's grip weaken, so it drifts gradually outwards

Two useful orbits

Geostationary orbit. At about 36 000 km up, directly above the equator, the period is exactly 24 hours, so the satellite goes round in step with the Earth turning beneath it and appears to hang motionless in the sky. A dish on a roof can therefore be bolted in one position and never moved. Used for television and communications. Low Earth orbit. A few hundred kilometres up, with a period of about 90 minutes. Far too fast to stay above one place, and it sweeps over a different strip of the surface on every pass, eventually covering nearly all of it. Used for imaging, weather and the space station. The trade-off. Geostationary is convenient and remote: the signal has to travel 72 000 km there and back, which is a quarter of a second of delay, and that is enough to be noticeable on a phone call. Low orbit is close and fleeting: any one satellite is overhead for minutes, so continuous coverage needs a whole constellation of them.

Which orbit for which job?

True of a stable circular orbit?

A satellite is in a stable circular orbit. Select the THREE statements that are TRUE.

• It is accelerating, even though its speed never changes
• Its kinetic energy stays constant, because gravity acts at right angles to its motion and so does no work on it
• Moved into a larger orbit, it would need to travel more slowly to stay stable
• The forces on it are balanced, since it neither falls nor flies away
• It stays up because it is high enough to be beyond the Earth's gravity
• It needs its engines running continuously to maintain the orbit

Mission control

You are running a satellite in low Earth orbit. Three calls to make.

• A junior colleague says the satellite is weightless because there is no gravity at that altitude. What is the correction?
• You need to move it into a stable orbit with a smaller radius. What must be true of its speed once it gets there?
• A customer asks for a satellite that stays fixed above London. What do you tell them?

How an orbit decays

A low satellite runs out of fuel and can no longer boost itself. Put the stages of what follows into order.

• Thin traces of atmosphere drag on it, taking away a little speed
• Too slow for its radius, it moves into a slightly lower orbit
• Lower down the air is denser, so the drag on it is greater than before
• It loses speed faster, and drops faster, and the process feeds itself
• It meets the atmosphere proper and burns up on re-entry

The orbit rules

A satellite stays in orbit because gravity provides an unbalanced force directed towards the _____ of the orbit. Its velocity points at 90° to that force, so gravity changes its direction without changing its speed, which means the satellite is _____ at all times even at constant speed. For a stable circular orbit, a smaller radius requires a _____ speed. A satellite that appears fixed above one spot on the ground must be in orbit above the _____.