Orbits

Why Astronauts Float

The International Space Station circles about 420 kilometers up, close to the ground for a planet 12,700 kilometers across. Gravity there is nearly nine tenths as strong as at the surface. An astronaut floating past the window has almost all of her weight pulling on her. So why does she float?

Because everything around her is falling in the same way. The station falls, the astronaut falls, and so do the loose spoon and the drop of water, all at the same rate, all missing Earth together. Nothing falls faster than anything else, so nothing presses on anything, and the feeling of weight is gone. The weight you feel right now is the floor pushing up on you, stopping your fall. Take the floor away, or let it fall with you, and the feeling goes with it.

i All falling together, nothing presses on anything ×

Falling together hides a pull, however strong it is. The Sun pulls on the Moon more than twice as hard as Earth does, yet the Moon never leaves Earth for the Sun. Earth falls toward the Sun just as fast, and the two fall together. All that is left to show is the small difference between the Sun’s pulls on the two. Differences of the same kind, from the Moon and, about half as strong, from the Sun, raise the tides.

Nor does the Moon simply go round Earth. Each pulls the other exactly as hard, so the two go round the balance point between them. Earth is 81 times heavier, so that point lies only 4,670 kilometers from Earth’s center, some 1,700 kilometers below its surface, and Earth swings round it in a small circle each month.

Why Ellipses

A ball fired a little faster than the circle needs swings out and back on an ellipse, and real orbits are ellipses too. It shows even in the best observations of the planets made before telescopes. No circle could match their places for Mars to better than eight minutes of arc, about a quarter of the Moon’s width. Johannes Kepler would not throw those eight minutes away, and found two rules instead.

Each planet goes round an ellipse with the Sun at one of its two foci, not at its center. And the line from the Sun to the planet sweeps out equal areas in equal times. Near the Sun the line is short, so the planet must race to sweep its share. Far out the line is long, and a slow crawl sweeps as much.

The second rule is the spin kept, the angular momentum that a turning body keeps. The Sun’s pull points straight at the Sun, and a pull whose line passes through a point cannot change a body’s spin about that point. So equal areas hold for any pull aimed at a fixed center, whatever its strength.

Kepler’s third rule ties the orbits together: the square of a planet’s year grows as the cube of its average distance from the Sun. Jupiter is 5.2 times as far out as Earth, and 5.2 cubed is 141. Its year is 11.86 of ours, and 11.86 squared is 141 too. The rules were found by fitting numbers, but all three follow from one pull that weakens with the square of the distance.

The rules hold for comets too. Three comets seen about 76 years apart turned out to be one comet on a long ellipse. Its return was foretold more than fifty years ahead, and it came back on time. It is Halley’s comet, named for the man who foretold it, and its next swing past the Sun comes in 2061.

To Catch Up, Slow Down

Orbits have a strange rule about speed. The higher the orbit, the slower it goes and the longer it takes. So a push forward does not simply make a craft go faster. It throws the craft up into a higher orbit, where it ends up moving more slowly, and it falls behind where it would have been. Push the space station forward by one meter a second, and half an orbit later it is 3.5 kilometers higher. One orbit later it is 17 kilometers behind.

An early crew in orbit found this out the hard way. Their pilot tried to fly the capsule up to the spent upper stage of the rocket that had launched it. He aimed at it and fired toward it, and it seemed to move away and downward. After 42 percent of the fuel had gone, the attempt was called off.

The way to catch a craft ahead of you is to slow down. Brake, firing against your motion, and you drop into a lower orbit, which takes less time. You lap faster, gain on the other craft, and rise to meet it. In the scene the eye rides with the craft in front, as its crew would see the chase, and the pushes are drawn some fifty times larger than real ones, so that the change of orbit shows. Flown this way, two crewed capsules have met in orbit and come within 30 centimeters of each other.

Drag works the same strange way. The thin air at the station’s height takes energy away, and the station sinks into a lower orbit, speeding up as it sinks.

Fast Enough to Leave

There is a speed at which the ball never comes back. Fired level at about 1.41 times the circle’s speed, the square root of two, its path opens out and never closes. From Earth’s surface that escape speed is 11.2 kilometers a second, about 40,000 kilometers an hour. A coasting body faster than that slows forever but never stops, and Earth’s pull never turns it back.

Escape speed is not a speed you must reach to leave, though it is often put that way. It is the speed needed by something given one shove and then left to coast, as the cannonball is. A rocket that keeps its engine running could climb away at walking pace, given enough fuel, because it keeps pushing all the way out.

The number measures how deep a world’s pull runs. It is 2.4 kilometers a second from the Moon, 59.5 from the cloud tops of Jupiter and 618 from the surface of the Sun. By the same reckoning, a star 500 times as wide as the Sun, and as dense, would need a speed greater than light’s to escape it, so its own light could not leave. That was the first thought of a black hole. The real story of black holes needs Einstein’s gravity rather than Newton’s.

Where Two Pulls Balance

Add a small third body to a pair going round each other, and there are five places where it can keep pace with both, turning with them as if pinned to the pair. They are called Lagrange points. Three lie on the line through the pair. For the Sun and Earth, two of the three in line lie 1.5 million kilometers from Earth. One is toward the Sun, where Earth’s pull weakens the Sun’s. The other is beyond Earth, where the two pulls add up.

Both are working places. Spacecraft have watched the Sun from the near one for decades. The James Webb Space Telescope circles the far one in a wide loop. From there the Sun, Earth and the Moon always lie on the same side, so one sunshield blocks all three, and the telescope stays at around 40 to 50 degrees above absolute zero. These points balance a body like a ball on a saddle: any drift away grows. So Webb fires its thrusters about every six weeks, and its fuel should last some twenty years.

The other two points lie on the orbit itself, 60 degrees ahead of the smaller body and 60 behind, each making an equal-sided triangle with the pair. Seen turning with the pair, they are not valleys that things settle into. They are the tops of hills: a body nudged off one starts to fall away. But seen that way, a moving body is also swung sideways, by the same swing that turns the winds of a hurricane. So the fall away is turned into a circling round the point, and the body stays. This works only while the smaller body has less than about a twenty-fifth of the bigger one’s mass. Jupiter has about a thousandth of the Sun’s.

Jupiter holds two swarms of asteroids there, the Trojans. Nearly 16,000 are known, and there may be hundreds of thousands to a million bigger than a kilometer. The swarm ahead holds about 1.4 times as many as the one behind. How they got there is not settled. The leading account is that they were caught early in the solar system’s history, when the giant planets shifted their orbits, and have stayed ever since. A space probe now on its way flies past the first of them in August 2027.

Tides, and the Moon Moving Away

The Moon’s pull weakens with distance. So it pulls the side of Earth facing it a little harder than Earth’s middle, and the middle a little harder than the far side. Earth as a whole falls toward the Moon at the rate set by its middle. The near seas are pulled a little more than that and heap up toward the Moon. The far seas are pulled a little less and are left behind in a second heap. Earth turns through both each day, which is why most coasts get two high tides a day, about twelve hours and 25 minutes apart.

That is the picture for an Earth covered by an ocean far deeper than any real one, and real seas do not follow it. Earth turns too fast. To stay under the Moon, a heap of water would have to sweep round the equator at 1,600 kilometers an hour, more than twice as fast as a wave can run across an ocean four kilometers deep. And the continents are in the way. Instead the tide sloshes round each ocean basin, timed and sized by its shape. The Bay of Fundy in Canada can rise and fall by 16 meters, and most of the Mediterranean by a few tens of centimeters.

The scene draws the bulges millions of times higher than they are and the Moon far closer, but the overall effect is still that of the simple picture. Earth turns faster than the Moon goes round, and the tide always lags, so on balance the raised water sits a little ahead of the line to the Moon. Its extra pull tugs the Moon forward along its path, and a body tugged forward climbs into a higher, slower orbit, just as a craft pushed forward does. The Moon is moving away. Laser pulses fired at the reflectors left on the Moon by three Apollo crews and two Soviet rovers, and lately at one carried by a commercial lander, time its distance to the millimeter. It grows by 3.8 centimeters a year.

The Moon pulls back on the raised water just as hard, and that brakes Earth’s turning. Eclipses recorded over the last 2,700 years show the day growing longer by about 1.8 milliseconds each century. The tides alone would make it 2.3. Most of the rest is undone as the land rises back after the weight of the ice ages, leaving Earth a little rounder. Corals that grew some 380 million years ago laid down about 400 daily layers in a year, so the days then were shorter. Most of the spin Earth loses is the spin the Moon’s orbit gains.

The Moon has already reached the end of this road. Earth’s pull on the tides it raises in the Moon’s rock long ago slowed the Moon’s spin until it matched its orbit, which is why it always shows us the same face. Earth would one day keep one face toward the Moon too, its day as long as the month, but only after some 50 billion years. The Sun will swell into a red giant in about five billion.

One question here took decades to answer. Suppose the tides had always lagged as much as they do today. Their tug on the Moon grows steeply as it comes closer, so the Moon would have been touching Earth about 1.5 billion years ago, yet its rocks show it is some 4.5 billion years old. The answer is that today’s tides are unusually strong. The Atlantic happens to be close to the size at which its water sloshes in step with the tide, and for most of the past it was not. Recent models of the oceans on a changing Earth fit both today’s rate and the Moon’s age.

Where Newton’s Orbits Stop

Newton’s orbits are astonishingly accurate, and not quite right. Mercury’s ellipse slowly turns, its closest point to the Sun creeping round by about 575 seconds of arc a century. The pulls of the other planets account for all but 43 of them. Neptune had been found by its pull, so the first guess was another unseen planet, inside Mercury’s orbit. No such planet exists. Einstein’s theory of gravity, in which mass curves space and time, gives exactly the missing 43.

Newton’s orbits also last forever, and real ones do not quite. Two neutron stars in close orbit shake spacetime as they go round, and the gravitational waves carry energy away. In one such pair the orbit shrinks by about three and a half meters a year, just as Einstein’s theory predicts.

And three bodies are a different problem from two. Two bodies alone always follow one simple curve forever: an ellipse, or an open curve that leaves. In general, three have no such curve. There is a series that does give the answer, but it closes in so slowly that a useful result would need more terms than a one followed by eight million zeros. Worse, the motion can be chaotic, tiny differences at the start growing until the futures part completely. The solar system is chaotic in that sense, unpredictable in detail beyond some tens of millions of years, yet in nearly every calculation it holds together for billions.

None of this undoes Newton’s picture. Whatever gravity turns out to be underneath, everything in orbit is doing what the cannonball does: falling, all the time, and missing.