"Oh, 'tanstaafl'. Means 'There ain't no such thing as a free lunch.' And isn't," I added, pointing to a FREE LUNCH sign across room, "or these drinks would cost half as much. Was reminding her that anything free costs twice as much in the long run or turns out worthless."
—Robert Heinlein, The Moon is a Harsh Mistress
SURELY ALL of us have at one time or another had it drilled into us that free lunches simply don't exist, as Heinlein had Manuel say in his classic novel. Everything has its associated cost, even if that cost is delayed, or assessed to someone else. So to a certain extent, it's a matter of interpretation, meaning that it's impossible to disprove. You can always come up with some way of looking at a situation so as to establish a kind of cost that one of us is going to have to pay.
Science, too, has its proscriptions on free lunches. There is, for instance, the first law of thermodynamics, which states that energy is always conserved. Even when you apparently create energy out of nothing, at least an equal amount of energy is used up somewhere else. There's no good reason why this has to be true, but it does always seem to be true. There are also conservation laws for things like electric charge, angular momentum, baryon number, and so forth, but energy seems like the most basic one.
That wouldn't be so bad if that were all, since even though it means we can't get a free lunch, at least the cost doesn't go up: the energy doesn't go away. It just moves from one place to another. Alas, there is also a second law of thermodynamics, which states that a quantity called the entropy of any closed system (which we might think of as its evenness) is always increasing. That is, heat always flows spontaneously from a hot component to a cold one, rather than the other way around. If we want it to go the other way, as in a refrigerator, for instance, we have to do work—which involves moving a greater amount of energy in the other direction. (That's why there are heat sinks on a refrigerator.) So not only do we have to pay for our lunch, but the price keeps on going up.
The reason why things work as well as they do is that the Earth isn't a closed system. It gets heat from the Sun, and it gets it in an uneven way, which keeps energy flowing from place to place all around us. It's not energy itself but the flow of energy that allows us to do useful work. And the Sun is so fantastically large that we can expect this ability to be with us for a long time indeed. (Even so, we are increasingly running up against the limits of our current efficiency in extracting work from the available energy.)
In fact, it's particularly in the realm of astronomy that we have such large objects that even though lunches aren't free, they cost so little that they're mostly free.
For a long time, comets were regarded as astronomical pariahs. The stars circled the Earth with admirable constancy, and the motions of the Sun, the Moon, and the other planets, though more complicated, were still predictable in essence, so those objects too were divine, like the stars.
But the comets refused to adhere to the rules of the clockwork universe. They insisted on coming and going every which way, according to no discernible pattern. Humans are very good at seeing patterns, though, even when there are none to see, so rather than conclude that there was no pattern to the comets, we decided that there was a pattern, but it was a mystical pattern, one that might tell us about the future if only we could interpret it correctly. In short, comets were treated as omens from the gods.
That's not to say that every last person felt that way. The Chinese kept a rather prosaic catalogue of cometary observations, and even classified their shapes. Aristotle believed them to be merely atmospheric phenomena, since they were so different from the other celestial bodies. It was only most of humanity that regarded the comets as omens, and almost exclusively bad ones at that.
Thus, for instance, in 1066, a great comet appeared in the skies of England, where Harold II was King. It shone brightly in the skies and remained for weeks, and therefore must be a particularly evil sign. Sure enough, that year, the Normans, under the command of William the Conqueror, sailed to England across the English Channel and made quick work of the English. Poor Harold II himself was killed in the Battle of Hastings, barely half a month after William had crossed the Channel.
To the newly defeated English, that would have confirmed the awful predictive power of the comet. They wouldn't have been likely to consider that to the French, the comet had been a very good sign. In fact, there is some evidence that William took the appearance of the comet as an indication that it was a good time to attack.
Moreover, comets are hardly rare phenomena. These days, with the night sky increasingly invaded by city lights, even prominent comets like Hale-Bopp and Hyakutake find it difficult to pierce the thick dome of light pollution tenting most urban areas, but in pre-industrial times, a few torches or gas lamps were no match for the comets, and a bright one could be seen perhaps once a decade. And comets hang around for weeks, for the most part circling the skies like the rest of the stars. The brighter comets would hang around for an especially long time.
The upshot is that some bad thing is bound to happen during the stay of one comet or another. The prophecy is fulfilled, not because comets aim particularly well, but because there are so many of them and strike over a long period of time.
The beginning of the end of the comet mythology was in 1577, when the Danish astronomer Tycho Brahe (1546–1601) made careful observations of a comet, and found that its apparent position against the stars did not change when it was viewed from different points on the Earth, demonstrating that it was not atmospheric. In fact, the Moon's apparent position changed more than the comet's, indicating that the comet was more distant than the Moon.
This removed some of the mysticism surrounding comets, but not quite all. So what if the comets—and perhaps only some of the comets at that—were astronomical objects, and not atmospheric ones? The fact remained that they didn't obey the rules followed by the other objects in the sky. They were still omens, but celestial rather than earthly.
That part of the mystery would also start with Tycho's work, this time on the motions of the planets. Tycho had been meticulously recording the positions of the planets, with the intention of establishing his model of the universe, one which blended the Earth-centeredness of the ancient Greeks with the Sun-centeredness of Copernicus. As it happened, he didn't succeed as he had thought, but his measurements did permit the German astronomer Johannes Kepler (1571–1630) to conclude that the planets (including the Earth, it turned out) orbited the Sun in elliptical paths. An ellipse can be thought of as a sort of stretched out circle, although in the cases of the planets, the stretching out is very slight indeed. The Earth, for instance, travels in an orbit that is stretched out by only one part in about 7,200. That is to say, the orbit is longer than it is wide, by an amount equal to about 1/7,200 of the width.
This set the stage for Edmond Halley (1656–1742), the English astronomer and statistician. In 1682, newly married, he looked up in the London skies and saw (but did not discover) a comet in the sky. It was not a spectacular comet, by any means. Just two years earlier, there had been a much more impressive comet, one that had enough to it to be named after the year: the Great Comet of 1680 (or in some cases, just "the Great Comet").
This newer comet might not have attracted much interest, therefore, except that Halley happened to look back into the historical records of comets, and found that 76 years earlier, in 1607, and 76 earlier than that, in 1531, there were also observations of notable comets. That alone might be chalked up to coincidence, but Halley also noticed that all three comets seemed to come from the same area in the sky. Knowing of Kepler's findings, he wondered if this comet too might also be orbiting the Sun, in an elliptical path.
Having at his disposal some recent work by his friend Isaac Newton (1642–1727) on the motions of celestial objects, he found that the three comets were indeed one comet, moving in an elliptical orbit—but one that was dramatically more stretched out than any of the planets. The most elliptical orbit held by any planet known in Halley's time was that of Mercury, which is out of round by about one part in 50. The orbit of Pluto, which was discovered almost 250 years after Halley's examination of the comet, is out of round by about one part in 30. At a casual glance, these orbits look circular.
To Halley's surprise, he found the comet's orbit to be enormously stretched out, to the point that its length was almost four times its width. It was distinctly non-circular. All the same, however, he wrote up his findings, and today this comet (which is the same as the comet that inspired the Normans in 1066) is better known as Halley's Comet.
But why? Why should the orbit be so elliptical? If the comets are essentially citizens of the solar system, why should their orbits be so different from those of the planets, which are so much better behaved?
For example, those other rogues of the solar system, the asteroids, have orbits that cross over and under each other, like a swarm of bees. But their orbits, though generally more elliptical than those of the planets, are still more or less circular. The comets still seemed to be strangers in the solar system.
As part of his work on the comet that bears his name, Halley made a prediction: The comet would next appear in 1758, another 76 years after he observed it. He even predicted where in the sky it would first appear, based on what had happened the last three times. He got the right patch of sky, but it was not until the spring of 1759, 16 years after his death, that the comet finally swung again around the Sun.
What had happened to delay Halley's Comet? (Or, to use its modern moniker, 1P/Halley, the 1P signifying the first comet to be identified as a periodically returning one.) What had happened was that the comet did not travel precisely in an elliptical orbit, as prescribed by Kepler's laws. Those laws are only an approximation, which Newton found was precisely valid only when there was a single object in orbit around a star. In all other cases, Newton's law of universal gravitation deviates from Kepler's laws—because the other planets also exert their influence.
By and large, this doesn't matter very much, because the planets travel in mostly circular orbits and therefore never get close enough to one another to create significant disturbances. (Those disturbances are measurable, however, and in the days before interplanetary spacecraft, they were the only way to determine the masses of Mercury and Venus, which have no satellites.) But Halley, on its swing around the outer solar system, temporarily came closer to Jupiter and also Saturn than any other planet does. Working feverishly in an attempt to determine the comet's path around the Sun before it got there, the French mathematician Alexis Clairaut (1713–1765) determined that the pull of those two gas giants was sufficient to slow the comet by three or four months—more or less what was observed.
Why should Halley have been slowed down by those two planets? Why wasn't it sped up? Suppose that there were nothing in the universe besides Halley and Jupiter. Jupiter is so much more massive than Halley that it can be treated like a sun, with Halley acting as a planet. According to Kepler, Halley should then move in an elliptical orbit around Jupiter.
Newton found that that wasn't the only possibility. Halley would travel in an elliptical orbit only if it was moving slowly enough. If it weren't—if it moved faster than Jupiter's escape velocity—it would swing around Jupiter in what is called a hyperbolic path. In such a path, Halley approaches the Jupiter system at a certain speed, swings around it, and then leaves it at exactly the same speed at which it arrived, albeit in a different direction.
That sounds like Halley shouldn't have been either sped up or slowed down by Jupiter, but then, Jupiter and Halley aren't really the only objects in the universe. There is also the Sun. The incoming and outgoing speeds of Halley are equal with respect to Jupiter, but they are not the same with respect to the Sun.
Suppose Halley passes behind Jupiter in its orbit around the Sun. From Jupiter's perspective, Halley comes in at one speed and leaves at the same speed, but its direction is changed. Halley is now travelling more "with" Jupiter than it did before. As a result, its speed with respect to the Sun (and us) is higher than it was before, just as a tennis ball thrown from a moving automobile moves faster, with respect to the ground, if the ball is thrown in the same direction as the automobile, rather than in the opposite direction. On the other hand, if Halley passes in front of Jupiter, it changes its direction so that it is now travelling more "against" Jupiter than it did before, and its speed relative to the Sun and us is lower than it was before.
Since it arrived behind schedule in 1758, Halley must have passed in front of both Jupiter and Saturn before its appearance, slowing it down, and thereby delaying its return.
Now, Halley is not a very recent comet, historically speaking. It has been observed continually for almost two thousand years. Could Jupiter and Saturn really have restricted their influences to the early 18th century? Isn't it much more likely that they had pulled Halley this way and that over many of its orbits? And if Comet Halley, why not numerous other comets? In 1767, for instance, Comet Lexell had its orbit changed by Jupiter from a relatively long one to a short one with a period of just 6 years. Two orbits later, in 1779, it again was deflected by Jupiter and was ejected out of the solar system altogether.
Once a comet interacts with a planet, then, it is likely to continue doing so. Unless it is deflected by some other planet, the comet will return to the scene of the crime, along its new orbit. Sooner or later, the planet will meet it there once more, ready to deflect it into yet another orbit.
What that new orbit looks like depends, of course, on whether the planet slows the comet down or speeds it up. However, the new orbit is not at all likely to be anything close to circular. In order for it to be circular, it would have to be travelling at almost the same speed as Jupiter, since Jupiter is itself travelling in a more or less circular orbit. That would put it in an extended interaction with Jupiter, from which it would almost undoubtedly emerge with a drastically changed path.
Much more likely, the comet would travel significantly faster or slower than Jupiter. Newton showed that if the comet moves slower than Jupiter, it revolves around the Sun in an elliptical orbit, with its most distant point (the aphelion, from Greek words meaning "away from the Sun") near Jupiter's orbit. If, on the other hand, the comet moves faster than Jupiter, it may be ejected entirely out of the solar system. If it survives, though, it also revolves around the Sun in an elliptical orbit, but now the closest point to the Sun (the perihelion, from Greek words meaning "around the Sun") is near Jupiter's orbit.
From this line of reasoning, we might expect there to be a bevy of comets, either with their aphelia or perihelia near Jupiter's orbit, and indeed there is. In fact, the group of comets whose aphelia are near Jupiter's orbit (and whose own orbits are therefore almost entirely within Jupiter's) is collectively called the Jupiter family of comets.
Comets are not the only bodies subject to this kind of acceleration. Any object that travels close to a massive planet, such as Jupiter or Saturn, will also be sped up or slowed down in this fashion—and that includes artificial satellites.
In order to leave the Earth's gravitational influence, a spacecraft must accelerate to speeds of about 10 or 11 km/s (about 25,000 mph). This requires phenomenal amounts of fuel, since the thrust must be used to lift not only the payload, but also whatever fuel is needed for the remainder of the voyage. What's more, that enormous speed is only sufficient to leave the gravitational confines of the Earth. There is, again, also the Sun, and in order to move away from the Sun to explore the outer planets, the spacecraft must be sped up even further, requiring still more fuel.
Not much can be done about this as far as Jupiter itself is concerned, since there are no planets more massive than our own closer than Jupiter, but for exploring the planets beyond Jupiter, the nearest gas giant can be used in a maneuver called a gravity assist. In a gravity assist, the spacecraft (which is typically programmed to examine Jupiter along the way), is sent on a path that brings it just behind Jupiter. As we saw above, this speeds the spacecraft up, relative to the Sun, and gives it enough of a boost to reach Saturn, say, even if it didn't have enough speed to do so before encountering Jupiter.
Perhaps the most spectacular early instance of this was with the Voyager 2 spacecraft, which reached Jupiter in 1979. It was sent behind Jupiter in order to speed it up, allowing it to reach Saturn in 1981. During the encounter with that planet, it was sent behind it, giving it enough of a boost to reach Uranus in 1986, whereupon it was sent behind that planet, accelerating it one more time in order to rendezvous with Neptune, in 1989. Each time, it sent back spectacular photographs of the outer denizens of our solar system, and expanded almost immeasurably our knowledge of the past and present of these planets.
In order for this chain of gravity assists to work, each successive planet had to be at the right place in its orbit, ahead of the preceding planet by an appropriate amount. Too far in front, or behind, and the spacecraft would miss the encounter altogether. By adjusting the distance of the encounters, Voyager control could accommodate a small variety of planetary alignments, but even so, it was estimated that an arrangement permitting such a series of maneuvers would not come again for thousands of years. It was, in short, now or never for the Voyager crew, and they took full advantage of the configuration.
Gravity assists can also be used to slow spacecraft down. There is little friction to speak of in interstellar space, and a spacecraft in a higher, more energetic orbit must be slowed down in order for it to return to the Earth (if it is so designed). After the Genesis explorer collected comet dust in 2002, it had to be braked through a series of planetary encounters before it could be sent down to the Earth, which it did in 2004 (albeit a bit harder than expected, due to a malfunctioning parachute).
But what of the Heinlein's novel, which reminds us that there's no such thing as a free lunch? What is the cost of these gravitational tricks that allow us to save on fuel and gain speed to send spacecraft to the planets—and eventually, perhaps, the stars? As Newton's third law tells us, for every action (or force), there is an equal and opposite reaction. This means that when, for instance, Voyager passed behind Jupiter and Jupiter pulled forward on Voyager, sending it on to Saturn, Voyager also pulled backward on Jupiter. As a result, the giant planet was slowed down ever so slightly in its orbit.
The amount by which Jupiter was slowed down by Voyager can be determined by comparing the gravitational forces on the two bodies (which are equal) to their masses (which are not). Voyager has a mass of perhaps a ton; that of Jupiter is about two billion billion trillion times greater. It was therefore accelerated correspondingly less. The upshot is that billions of years from now, when the Sun expands to a red giant, crisping the Earth and possibly swallowing it, and humanity (assuming it survives) may want to escape to Jupiter's satellites and beyond—they will find Jupiter a few centimeters away from where it would have been, had Voyager 2 never passed by.
That lunch may not be free, but it's sure cheap.
Copyright (c) 2006 Brian Tung