Astronomical Games: April 2002


Can one escape from the Earth's gravity, even in principle?

Oh! I have slipped the surly bonds of earth / And danced the skies on laughter-silvered wings…

—John Gillespie Magee, Jr., "High Flight"

A WHILE AGO, I found myself trapped in a discussion on the nature of gravity. The guy I was arguing with (he never gave his name, so that's what I'll call him) wasn't clear on very many things, but he was adamant about one thing—that there was no such thing as escape velocity. Whatever goes up, as the old saying goes, must come down.

He was aware that gravity decreases as the inverse square, but he insisted that because the force of the Earth's gravity never diminishes to zero (which is true), any object ejected from the Earth not under its own power must return to the Earth (which is not true). I pointed out that there were a few spacecraft (the Pioneer vehicles, for example) that were clearly destined to remain among the stars, even if we waited an arbitrarily long period of time. The guy I was arguing with countered that that was only because there were lots of stars and planets pulling those spacecraft out, and only one Earth to pull them in.

Well, the Sun helps too, but go talk to a brick wall. I tried something else. "What about all the centuries of mathematics that shows that there must be an escape velocity? What are you going to supply to counter that math?"

"Oh, that," he said, dismissively. "I don't trust the mathematics. It's just not understandable."

So that was it. I imagine that sometime in the past, he had been forced into contact with calculus, and found it incapable of suffusing the solid bone constituting his head. And then, with the subtle confidence of the very stupid, he assumed that whatever he couldn't understand was not to be trusted, was not worth the effort of struggling through. Well, I am not going to persist in any conversation where I am putting forth all the energy.

But it set me to thinking: Is there a way to deduce the existence of an escape velocity without resorting to calculus? Not that it would have convinced anyone physiologically resistant to mathematics he can't understand, but I still found the question interesting. And since I found the answer interesting, too, I will explain it here.

For most of human existence, most people would have agreed that there could be no such thing as an escape velocity. If you throw a stone up, it always comes back down. If you throw it a second time, harder than the first, then it stays up longer and goes up higher, but it still comes back down, and comes back harder, to boot.

To be sure, certain animals, such as birds and insects, were able to stay up indefinitely, but they proceeded under their own power, which stones couldn't do, and even they had to come back down to rest, once in a while. And what was most relevant, humans couldn't fly. So there seemed to be no way to escape the Earth's gravity. We were trapped.

On the other hand, not everything was trapped equally. If you let go of a soft, downy feather, it falls slowly and gently to the ground. If you let go of a crumpled patch of fabric, it falls a little faster, and if you drop a stone, it falls faster still. If you drop an anvil (presuming you could pick it up in the first place), it falls so fast that you'd have to be careful to get your feet out of the way. From observations like these came the principle that the heavier something is, the faster and harder it falls.

The Greek philosopher Aristotle (c. 384–322 B.C.) was the first person we know of to set this principle down in writing and attempt an explanation of it. His explanation was based on the notion that everything on the Earth, and even the Earth itself, was composed of differing amounts of the four elements: fire, air, water, and earth (not the Earth, the globe, but an elemental substance called earth). These elements had differing tendencies: Earth and water tended to sink, earth more so than water. Fire, on the other hand, tended to rise. Air, finally, tended to remain where it was, unless forced otherwise.

Anything that was heavy and fell fast and hard must therefore be composed of relatively more earth, and less of the other elements; the lighter objects, which fell slower and gentler, were composed of relatively less earth, and more of the other elements.

But then, in that case, how explain the spherical shape of the Earth? The Earth was known to be roughly spherical as early on as Pythagoras (c. 569–475 B.C.), and he may have gotten the idea from the Babylonians. A century after Aristotle, Eratosthenes (284–192 B.C.) would even measure the Earth's circumference to fairly high precision. But if the Earth was made mostly of earth (as it had to be, if it stayed beneath our feet), and earth wants to be as low down as possible, shouldn't the Earth be as flat as a pancake?

Aristotle was a very clever man: He realized that the answer depended on how you interpreted the words "up" and "down." If we interpret "down" as "toward the center of the universe," then the result is a spherical Earth. The first piece of matter composing the Earth to get to the center of the universe would stay there, since obviously there is no way for that piece to get any more "down" than that. Successive pieces of matter would be unable to dislodge it, but even they would "try" to get as close as possible to the center.

If you think about it for a bit, you can see that this leads very quickly to a spherical Earth. Any really tall mountains or deep valleys will get filled in, because new pieces of matter prefer to go into the deep valleys, where they can be closer to the center, rather than pile up on top of the already unfavorable mountaintops.

Not only does this neat bit of philosophizing explain heavier objects falling harder and the round Earth, it even explains why the Earth is at the center of the universe, which was another self-evident notion. All the celestial objects—the Sun, the Moon, the other planets, and the stars—were made of a fifth element, which we sometimes call quintessence, which was not found on the Earth. The nature of quintessence was not to rise or to fall or to remain in place, but to move around in circles around the center of the universe, which to the Greeks was the most perfect motion imaginable. And since the Earth had to be at the center of the universe, the celestial objects had to circle the Earth (which they obviously did, day after day, night after night). This happens to be dead wrong, but that's another story for another time.

This theory of Aristotle was so successful and so dominating that it persisted down to the time of the Renaissance. It was not so much that scientists in the intervening period were reluctant to challenge Aristotle, so much as that the theory was practically axiomatic, and no one thought it worth the bother of challenging. In much the same way, early dictionaries concentrated on the hard words and neglected the easy ones. It was only later that dictionaries attempted comprehensive coverage and found out just how hard the "easy" words were.

Then, around 1581, the Italian physicist Galileo Galilei (1564–1642) decided, at the tender age of 17, to look into the easy question of falling objects. He had been troubled by a thought experiment that he had thought up. Suppose you have two objects, one heavy, one light. If you drop them at the same time, surely the heavy object should fall faster than the light object. This seemed so obvious that it was hardly worth thinking about.

But what would happen if you connected the objects in some way (say, with a rope)? The heavy object would "try" to fall faster, but would be impeded by the light object, which would "try" to fall slower. The pair of objects, therefore, should fall at a speed somewhere in between the falling speeds of the individual objects.

On the other hand, by connecting the two objects, haven't you made one object that is heavier than either by itself? And shouldn't this combined object fall faster than either object by itself? The more Galileo thought about this, the more he became convinced that something was wrong with Aristotelian physics, and that the only solution to this dilemma was to conclude that all objects, regardless of weight, fell at the same rate.

This thought experiment naturally failed to convince many ecclesiastical scholars, who had grown up with Aristotle and were unwilling to give him up so easily. Nowadays, we think of that attitude as rather stick-in-the-muddish, but actually, it was the right thing to do. What matters in science is not how plausible we think something is, but whether or not it fits the facts. To establish his new law of falling objects, Galileo would have to conduct the proper experiments.

So he did. There is a legend that Galileo dropped two balls from the Leaning Tower of Pisa, but that is almost surely apocryphal. We do know that he rolled balls down an inclined plane. This serves to attenuate the acceleration of the balls due to gravity, making it easier to do a quantitative analysis. This has to be done with some care, since not all the energy provided by gravity goes into the motion of the ball down the inclined plane; some of it goes into its rotation. You have to make sure that the same fraction goes into each ball's rotation; for example, you can't use a solid ball for the heavy object and a hollow ball for the light object, and so forth.

What Gailleo found was that provided he did the experiment with proper attention to all the details, balls of every weight fell with exactly the same speed, in direct contradiction of Aristotle. The reason that a feather falls much slower than an anvil is not because it's much lighter than the anvil, but because it is much more affected by air resistance. Aristotle actually had thought some about air resistance (and its more obvious relative, water resistance), but he didn't think it would actually affect motion to that extent. Galileo's experiment, more than anything else, marked the beginning of the end for Aristotelian physics.

However, whatever Galileo concluded about falling still applied only to terrestrial objects; celestial objects like the Moon clearly were not affected by gravity, since they stayed up in the sky and did not fall. It took another scientific genius, the English physicist Isaac Newton (1642–1727) to recognize that even the Moon, the Sun, and the stars were subject to gravity. He commemorated that fact in his law of universal gravitation (discussed at greater length in "The Grand Illusion"), which is commonly written as

F = Gm1m2/d2

This equation states that the force of gravity between two objects is directly proportional to the product of the masses of the objects, and inversely proportional to the square of the distance between them. It is this inverse square property that makes it possible for the Moon to revolve around the Earth as slowly as it does. The Moon is about 60 times further away from the center of the Earth as we are, standing on the surface, so the Moon is accelerated less than we are by a factor of 602 = 3,600. If the Earth's gravity were constant all the way up to the Moon, it would have to revolve around the Earth about twice a day; if it went any slower than that, it would fall in toward the Earth and be crushed to pieces.

The inverse square law also made it possible for the first time to conceive of an escape velocity, something that isn't possible if the Earth's gravity is constant at all points in the universe. To see that, consider what happens if you throw a ball up. The acceleration due to the Earth's gravity, at the surface of the Earth, is denoted g and its value is about 9.8 meters per second squared. By that, we mean that an object in free fall has its speed (or more precisely, velocity, which includes both speed and direction) changed by 9.8 meters per second, every second—or 9.8 meters per second squared.

For example, suppose we throw a ball straight up at 19.6 meters per second (about 44 mph). After 1 second under the influence of gravity (neglecting the effect of air resistance, which is not really negligible), its velocity is 19.6 meters per second, minus 9.8 meters per second, or 9.8 meters per second (about 22 mph). And after 2 seconds, the velocity of the ball has been reduced another 9.8 meters per second, to 0.0 meters per second. The ball is now (momentarily) motionless.

Immediately, the ball begins to fall downward, which we can represent using negative velocities. After a total of 3 seconds, the velocity of the ball is 0.0 meters per second, minus 9.8 meters per second, or –9.8 meters per second upward, which is the same as 9.8 meters per second downward. And after 4 seconds, its velocity is reduced another 9.8 meters per second to –19.6 meters per second upward, which is equal to 19.6 meters per second downward.

There's no need to confine ourselves to integer numbers of seconds. To find the velocity of the ball 0.5 seconds after we launch it, we simply subtract 0.5 times 9.8 meters per second from 19.6 meters per second, to get 14.7 meters per second (about 33 mph).

This line of reasoning permits us to figure out how high the ball goes, too, before it comes back down. It takes the ball 2 seconds to reach the top of its trajectory, because it starts out at 19.6 meters per second, and it takes 19.6/9.8 = 2 seconds for gravity to consume all that initial velocity. Over those 2 seconds, the ball's velocity decreases uniformly from 19.6 to 0.0 meters per second, so its average velocity in its upward flight must be half of 19.6 meters per second, or 9.8 meters per second. Therefore, its maximum height must be 2 seconds, times 9.8 meters per second, or 19.6 meters (about 64 feet).

Working more generally, if we throw the ball with an initial velocity of v meters per second, it takes (v/9.8) seconds for gravity to consume that initial velocity, and its average velocity over that time must be (v/2) meters per second. Its maximum height, then, must be

hmax = (v/9.8)(v/2) = v2/(2·9.8)

Working still more generally, we can recognize that the 9.8 in this equation is just the acceleration due to the gravity of the Earth. If it were the Moon, we would use 1.6 instead of 9.8. The 9.8 is just a placeholder for acceleration due to gravity, in other words, and we can write, for any acceleration a due to gravity,

hmax = (v/a)(v/2) = v2/(2a)

Equation 2 gives a maximum height for any initial velocity, no matter how large. For example, for an initial velocity of 300,000 km/s, the speed of light, Equation 2 predicts that a beam of light will get as far as about one-half light-year from the Earth before it is slowed down to a standstill and returns to Earth. That's an enormous distance, but it is nevertheless finite, and it shows that if gravity were constant, the Earth would be as good a black hole as any other.

The reason that such a suggestion was never taken seriously is that Newton formulated his inverse square law of universal gravitation about a century before the idea of a black hole was first conceived, by John Mitchell in 1783. By that time, the idea of an escape velocity was well understood (except by people like the guy I was arguing with), and black holes were, by definition, objects whose escape velocity exceeded the speed of light.

But all this was well understood because of calculus, which Newton co-invented. What if we (like the guy I was arguing with) throw our hands up at calculus, and reject its results? What else can we use to convince ourselves that there's such a thing as an escape velocity?

Consider: The acceleration due to the Earth's gravity is equal to 9.8 meters per second squared only because we are separated from the center of the Earth by about 6,375 km (around 3,960 miles), a distance I'll denote as 1 ER, for Earth Radius. If we were higher up, Newton's law tells us that the acceleration would be decreased, and by how much. For example, at a height of 3 ER overhead, the distance from the center of the earth is 1 ER + 3 ER = 4 ER. (I'll refer distances from the Earth's center, from now on.) At this distance, the acceleration is reduced by a factor of 42 = 16, to 9.8/16 = 0.61 meters per second squared. At a distance of 16 ER, the acceleration would be 162 = 256 times smaller, and at a distance of 64 ER, the acceleration would be 642 = 4,096 times smaller, and so on. (I have my own reasons for picking these distances, which I will reveal in a moment.)

Suppose we want to throw a ball from the ground, at 1 ER, to a distance of 4 ER. The initial velocity with which we must throw the ball is difficult to compute, because the Earth's gravity is continuously decreasing from 1 ER to 4 ER, but let's make the simplifying assumption that the Earth's gravity is actually constant at g = 9.8 meters per second squared all the way from 1 ER to 4 ER. If we can throw it to 4 ER under this assumption, we can surely throw it to 4 ER in real life.

Equation 2 then tells us how to compute the initial velocity in order to get an hmax of 4 ER, but I'll spare you the numbers for now and simply represent that initial speed by V. That's the speed necessary for the ball to go a distance of 3 ER, from 1 ER to 4 ER.

Now, let's suppose we throw a second ball, harder than the first, with an initial velocity of V+(V/2). How far does this ball go?

This second ball starts out faster by V/2, and since this takes it up to 4 ER faster than the first ball, it will still have at least that V/2 left, at a point where the first ball petered out. If the Earth's gravity at 4 ER and up were still g = 9.8 meters per second squared, Equation 2 tells us that this remaining velocity would only be enough to get us an additional distance equal to 1/4 as far as the first ball went: 1/4 times 3 = 3/4 ER. But gravity is weaker at 4 ER; as we worked out, it's 16 times weaker at that distance, and continues to decrease as we go upward and outward. However, let's assume again that gravity is constant at g/16, from 4 ER to 16 ER. In that case, Equation 2 tells us that the ball actually goes 16 times further: 3/4 times 16 = 12 ER. Since the ball is already at a distance of 4 ER, its maximum distance from the center of the Earth is (at least) 4 plus 12 = 16 ER.

Next, let's throw a third ball, with an initial velocity of V+(V/2)+(V/4). How far does the third ball go? Again, it must go at least as far as the second ball (16 ER), because it starts out faster, by V/4, and when it gets to 16 ER, it's still going at least V/4 upward. If the gravity were still g at that height, it would only continue upward 1/16 as far as the first ball went (again, 3 ER), or only 3/16 ER, but we noted above that the acceleration due to gravity is down to g/256 at a distance of 16 ER. That acceleration continues to decrease as the inverse square as the distance increases, but once more, let's assume that it's constant at g/256, from 16 ER to 64 ER. If so, Equation 2 tells us that this third ball will continue onward 256 times further than it would if the acceleration were still g, or 3/16 times 256 = 48 ER. Since the ball is already at 16 ER at this point, its maximum distance will be at least 16 plus 48 = 64 ER.

You're ahead of me, I'm sure. We will throw a fourth ball with initial velocity V+(V/2)+(V/4)+(V/8), and it will get to a distance of 64 ER with a remaining velocity of at least V/8. This would ordinarily only get us an extra distance equal to 1/64 as far as the first ball went, or 3/64 ER, but since gravity is now down to g/4,096, the ball actually goes an extra 3/64 times 4,096 = 192 ER, up to a height of 256 ER. Rather than continue ad nauseam in this vein, let's stop here and tabulate how far we've gotten.

V at least 4 ER
V+(V/2) at least 16 ER
V+(V/2)+(V/4) at least 64 ER
V+(V/2)+(V/4)+(V/8) at least 256 ER

Let's take a look first at the right column, the maximum heights. At each step, we see that our maximum height goes up by a factor of 4. In other words, the inverse square law permits us to get to any distance we wish, simply by adding enough increments to the initial velocity. That was true with constant gravity, too, but we already know there that nothing can escape the Earth if its gravity is constant.

If, on the other hand, gravity decreases according to the inverse square of the distance, the increments get smaller by a factor of 2 with each step. And we can get to an infinite distance from the Earth (and escape the Earth's gravity after all, ha!) by adding an infinite number of increments to our initial velocity. And what is the sum of that infinite series of increments? It is simply V+(V/2)+(V/4)+(V/8)+ · · · = 2V. We need only throw our ball twice as hard as we threw the first one, and we will get it to escape the Earth's gravity forever. And without calculus, too.

Actually, the velocities we've been bandying about are overkill, because we've been assuming that the Earth's gravity is constant over large bands of distances: from 1 ER to 4 ER, then from 4 ER to 16 ER, then from 16 ER to 64 ER, and so on. If the Earth's gravity were really equal to g from 1 ER to 4 ER, Equation 2 tells us that we would have to throw our first ball at about V = 20 km/s, in order to get it to a distance of 4 ER. In reality, such a velocity is already enough to escape the Earth's gravity. The true escape velocity is about 11 km/s.

If we imagine that precisely at the escape velocity the ball makes it to an infinite distance from the Earth, then only at that "point" does the Earth's gravity drop to zero. At any finite distance, the Earth's gravity may fall to imperceptibly small levels, but in principle, it is still there. I mention this only because it is occasionally suggested that you can never be completely weightless, that since something is always pulling on you, you always have some weight.

However, that is not true. Although something is always pulling on you, you can still have a weight of zero—you can still be weightless. I'll explain how in my next essay.

Copyright (c) 2002 Brian Tung