gravity

Apart from crystals and broken rocks, not much else in the cosmos naturally comes with sharp angles. While many objects have peculiar shapes, the list of round things is practically endless and ranges from simple soap bubbles to the entire observable universe. Spheres tend to take shape from the action of simple physical laws. So prevalent is this tendency that often we assume something is spherical in a mental experiment just to glean basic insight even when we know that an object is decidedly non-spherical. In short, if you do not understand the spherical case, then you cannot claim to understand the basic physics of the object.

Spheres in nature are made by forces, such as surface tension, that want to make objects smaller in all directions. The surface tension of the liquid that makes a soap bubble squeezes air in all directions. It will, within moments of being formed, enclose the volume of air using the least possible surface area. This makes the strongest possible bubble because the soapy film will not have to be spread any thinner than is absolutely necessary. It can be shown using freshman-level calculus that the one and only shape that has the smallest surface area for an enclosed volume is a perfect sphere. In fact, billions of dollars could be saved annually on packaging materials if all shipping boxes and all packages of food in the supermarket were spheres. For example, the contents of a super-jumbo box of Cheerios would fit easily into a spherical carton that had a four-and-a-half inch radius. But practical matters prevail—nobody wants to chase food down the aisle after it rolls off the shelves.

On an orbiting space station, where everything is weightless, it would be a cinch to make perfect ball bearings. On Earth, one way to make them is to drop molten metal in pre-measured drops into the top of a long shaft. The blob will typically undulate until it settles into the shape of a sphere, but it also needs sufficient time to harden before hitting the bottom. In a weightless environment, you just gently squirt out precise quantities of your metal and you have all the time you need to create perfect beads—they just float there while they cool.

For large cosmic objects, it's energy and gravity that conspire to turn themselves objects into spheres. Gravity is the force that serves to collapse matter in all directions, but gravity does not always win—chemical bonds of solid objects are strong. The Himalayan range in Tibet grew against the force of Earth's gravity because of the resilience of crustal rock. But before you get excited about Earth's mighty mountains, you should know that the spread in height from the deepest undersea trenches to the tallest mountains is about a dozen miles, yet Earth's diameter is nearly eight thousand miles. So contrary to what it looks like to teeny humans crawling on its surface, Earth, as a cosmic object, is remarkably smooth—if you had a gigantic finger, and if you rubbed it across Earth's surface (oceans and all), Earth would feel as smooth as a cue ball. Expensive globes that portray raised portions of Earth's land masses to indicate mountain ranges are grossly exaggerating reality.

Earth's mountains are also puny when compared with some other mountains in the solar system. The largest mountain on Mars, Olympus Mons—65,000 feet tall and nearly 300 miles wide at its base—makes Alaska's Mount McKinley look like a mole-hill. The cosmic mountain-building recipe is simple: the weaker the gravity on the surface of an object, the higher its mountains can reach. Mount Everest is about as tall as a mountain on Earth can grow before the lower rock layers succumb to their own plasticity under the mountain's weight.

If a solid object has a small enough surface gravity, the chemical bonds in its rocks will resist the force of their own weight. When this happens, almost any shape is possible. Two famous celestial non-spheres are Phobos and Deimos, the Idaho-potato-shaped moons of Mars. On thirteen-mile-long Phobos, the bigger of the two moons, a 150-pound person would weigh about four ounces.

In space, surface tension always forces a small blob of liquid to form a sphere. And if the blob has very high mass then it could be composed of almost anything and gravity will ensure that it forms a sphere. Whenever you see a small solid objects that is suspiciously spherical you can assume it formed in a molten state.

Big and massive blobs of gas in the galaxy can coalesce to form near-perfect, gaseous spheres called stars. But if a star finds itself orbiting too close to another object whose gravity is significant, the spherical shape can be distorted as its material gets stripped away. By too close, I mean too close to the object's Roche lobe—named for the mid-nineteenth century astronomer E. Roche, who made detailed studies of the gravity field in the vicinity of double stars. The Roche lobe is an imaginary, dumbell-shaped, bulbous, double-envelope that surrounds any two objects in mutual orbit. If gaseous material from one object passes out of its own envelope, then the material will fall toward the second object. This occurrence is common among binary stars when one of them swells to become a red giant and overfills its Roche lobe. The red giant distorts into a distinctly non-spherical shape that resembles an elongated Hershey's kiss.

The stars of the Milky Way galaxy form a big, flat circle. With a diameter-to-thickness ratio of 1000:1, our galaxy is flatter than the flattest flapjacks ever made. No, it's not a sphere, but it probably began as one. We can understand the flatness by assuming the galaxy was once a big, spherical, slowly rotating ball of collapsing gas. As it swiftly collapsed, it spun faster and faster like a spinning figure skater whose arms are drawn inward to increase the rotation speed. But not unlike a blob of tossed, spinning pizza dough, the galaxy naturally flattened in response to the increasing centrifugal forces that want to spread it apart. Yes, if the Pillsbury Dough Boy were a figure skater, then fast spins would be a high-risk activity. Any stars that happened to be formed within the Milky Way cloud before the collapse maintained large, plunging orbits. The remaining gas, which easily sticks to itself (like a mid-air collision of two hot marshmallows), got pinned at the mid-plane and is responsible for all subsequent generations of stars, including the Sun. The current Milky Way, which is neither collapsing nor expanding, is a gravitationally mature system where one can think of the orbiting stars above and below the disk as the skeletal remains of the original spherical gas cloud.

This general flattening of objects that rotate is why Earth's pole-to-pole diameter is smaller than its diameter at the equator. Not by much: three tenths of one percent—about 26 miles. But Earth is small, mostly solid, and doesn't rotate all that fast. At twenty-four hour per day, anything on Earth's equator is carried at a mere 1,000 miles per hour. Consider the jumbo, fast-rotating, gaseous planet Saturn. Completing a day in just ten hours, its equator revolves at 22,000 miles per hour and its pole-to-pole dimension is a full ten percent flatter than its middle, a difference noticeable even through a small amateur telescope. Flattened spheres are more generally called oblate spheroids, while spheres that are elongated pole-to-pole are called prolate. In everyday life, hamburgers and hot dogs make excellent (although somewhat extreme) examples of each shape. I don't know about you, but the planet Saturn pops into my mind with every bite of a hamburger I take.

We use the effect of centrifugal forces on matter to help calculate the rotation rate of cosmic objects. Consider pulsars. With some rotating at over one hundred thousand RPM, we know that they cannot be made of household ingredients. To picture a pulsar, image the mass of the Sun packed into a ball the size of Manhattan. If that's hard to do, then imagine stuffing about 50-million elephants into a thimble. To reach this density you must merge all electrons and protons into neutrons by compressing all the empty space that atoms enjoy around their nucleus and among their orbiting electrons. What's left is a ball of neutrons with a mind-bogglingly high surface gravity. Existing under such conditions, a mountain range on a neutron star needn't be any taller than the thickness of a sheet of paper for you to exert more energy climbing onto it than a rock climber on Earth would exert ascending a three-thousand-mile-high cliff. For these reasons, and others, we expect pulsars to be the most perfectly shaped spheres in the universe.

For rich clusters of galaxies, where hundreds are moving in beehive fashion around the cluster's center of mass, the overall shape can offer rich astrophysical insight. Some clusters are raggedy, some are stretched along filaments, while others form vast sheets. None of these have settled into a stable shape. Some are so extended that the fifteen billion years of the universe is insufficient time for their constituent galaxies to make one crossing of the cluster. We conclude that the cluster was born that way because the mutual gravitational encounters between and among galaxies has had insufficient time to influence the shape the cluster.

But other systems, such as the beautiful, spherically shaped Coma cluster of galaxies (in the constellation Coma Berenices) tell us immediately that gravity has shaped the cluster into a sphere and, as a consequence, you are as likely to find a galaxy moving in one direction as in any other. Whenever this is true, the cluster cannot be rotating all that fast otherwise we would see some of the flattening that afflicted the Milky Way.

The Coma cluster (like the Milky Way), is also gravitationally mature. In vernacular, such a system is relaxed, which means many things, including the fortuitous fact that the average velocity of galaxies in the cluster serves as an excellent indicator of the total mass, whether or not the total mass of the system is supplied by the objects used to get the average velocity. In other words, gravitationally relaxed systems are excellent probes of non-luminous dark matter. Allow me to make an even stronger statement: were it not for relaxed systems, the ubiquity of dark matter (which may comprise more than ninety percent of all matter in the universe) would have remained undiscovered to this day.

The ultimate sphere is the entire observable universe. In every direction we look, galaxies are observed to recede from us at speeds proportional to their distance—the famous signature of an expanding universe as discovered by Edwin Hubble in 1929. Since nothing can be observed to travel faster than light, there is a distance in every direction from us where the recession velocity for a galaxy equals the speed of light. At this distance and beyond, the light from luminous objects loses all it energy before reaching us. The universe beyond this spherical edge is rendered invisible. And as far as we know, unknowable.

Spheres are indeed fertile theoretical tools that help us gain insight to all manner of astrophysical problems. But one should not be a sphere-zealot. I am reminded of the half-serious joke about how you might increase milk production on a farm: An expert in animal husbandry might say, Consider the role of the cow's diet... An engineer might say, Consider the design of the milking machines... But it's the astrophysicist who says, Consider a spherical cow...!

The first manned spacecraft ever to leave Earth orbit was Apollo 8. This achievement remains one of the most remarkable, yet unheralded firsts of the twentieth century. When that moment arrived, the astronauts fired the third and final stage of their mighty Saturn V rocket, rapidly reaching nearly seven miles per second for the spacecraft and its three occupants. Half the energy to reach the Moon had been expended just to achieve Earth orbit. At about this time, a well-known television news anchor declared that the astronauts had just left Earth's gravity. But the astronauts were on their way to the Moon. And last anybody had checked, the Moon was in orbit around Earth by the action of mutual gravitational forces. So Earth's gravity must extend at least as far as the Moon. Fact is, the force of gravity for any object extends to the infinite reaches of space, even as it grows exponentially weaker.

After the third stage fired, engines were unnecessary, except for any mid-course tuning the trajectory might require to ensure the astronauts did not miss the Moon entirely. For ninety percent of its nearly quarter-million-mile journey, the spacecraft gradually slowed as Earth's gravity continued to tug in the opposite direction. Meanwhile, as the astronauts neared the Moon, its force of gravity grew stronger and stronger. A spot must therefore exist, en route, where the Moon's and Earth's opposing forces of gravity balance precisely. When the command module drifted across that point in space, its speed increased once again as it accelerated toward the Moon.

If gravity were the only force to be reckoned, then this spot would be the only place in the Earth-Moon system where the opposing forces canceled. But Earth and the Moon also orbit a common center of gravity, which lives about a thousand miles beneath Earth's surface along an imaginary line connecting Earth's center and the Moon's center. When things move in circles of any size and at any speed, they create a new force that pushes outward, away from the center of rotation. Your body feels this centrifugal force when you make a sharp turn in your car or when you survive amusement park attractions that turn in circles. In a classic example of these nausea-inducing rides, you stand along the edge of a large circular platter, with your back against a perimeter wall. As the thing spins up, rotating faster and faster, you feel a stronger and stronger force pinning you against the wall. You can't now move. That's when they drop the floor from below your feet and turn the thing sideways and upside down. When I rode one of these as a kid, the force was so great that I could barely move my fingers, they, being stuck to the wall along with the rest of me.

If you actually got sick on such a ride, and turned your head to the side, the vomit would fly off at a tangent. Or it might get stuck to the wall. Worse yet, if you didn't turn your head, it might not make it out of your mouth due to the extreme centrifugal forces acting in the opposite direction. (Come to think of it, I haven't seen this particular ride anywhere lately. I bet they've been outlawed.)

Centrifugal forces arise as the simple consequence of an object's tendency to travel in a straight line after being set in motion, and so are not true forces at all. But you can calculate with them as though they are. When you do, as did the brilliant eighteenth-century French mathematician Joseph Louis Lagrange, you discover spots in the rotating Earth-Moon system where the gravity of Earth, the gravity of the Moon and the centrifugal forces of the rotating system balance. These special locations are known as the points of Lagrange. And there are five of them.

The first point of Lagrange (affectionately called L1) falls between Earth and the Moon, slightly closer to Earth than the point of pure gravitational balance. Any object placed there can orbit the Earth-Moon center of gravity with the same monthly period as the Moon and will appear to be locked in place along the Earth-Moon line. Although all forces cancel there, this first Lagrangian point is a precarious equilibrium. If the object drifts sideways in any direction, the combined effect of the three forces will return it to its former position. But if the object drifts toward or away from Earth ever so slightly, it will irreversibly fall either toward Earth or the Moon, like a barely balanced cart atop a steep hill, a hair's-width away from falling down one side or the other.

The second and third Lagrangian points (L2 and L3) also lie on the Earth-Moon line, but this time L2 lies far beyond the far side of the Moon, while L3 lies far beyond Earth in the opposite direction. Once again, the three forces—Earth's gravity, the Moon's gravity, and the centrifugal force of the rotating system—cancel in concert. And once again, an object placed in either spot can orbit the Earth-Moon center of gravity with the same monthly period as the Moon.

The gravitational hilltops represented by L2 and L3 are much broader than the one represented at L1. So if you find yourself drifting down to Earth or the Moon, only a tiny investment in fuel will bring you right back to where you were.

While L1, L2, and L3 are respectable space places, the award for best Lagrangian points must go to L4 and L5. One of them lives far off to the left of the Earth-Moon centerline while the other is far off to the right, each representing a vertex of an equilateral triangle, with Earth and Moon serving as the other vertices. At L4 and L5, as with their first three siblings, forces are in equilibrium. But unlike the first three Lagrangian points, which enjoy only unstable equilibrium, the equilibria at L4 and L5 are stable; no matter which direction you lean, no matter which direction you drift, the forces prevent you from leaning farther, as though you were in a valley surrounded by hills. For each of the Lagrangian points, if your object is not located exactly where all forces cancel, then its position will oscillate around the point of balance in paths called librations. (Not to be confused with the particular spots on Earth surface where one's mind oscillates from ingested libations.) These librations are equivalent to the back-and-forth rocking a ball would undergo after rolling down a hill and overshooting the bottom.

More than just orbital curiosities, L4 and L5 represent special places where one might build and establish colonies. All you need do is ship to the area raw construction materials (mined not only from Earth, but perhaps from the Moon or an asteroid), leave them there with no risk of drifting away, and return later with more supplies. After all the raw materials were collected in this zero-G environment, you could build an enormous space station—tens of miles across—with very little stress on the construction materials. By rotating the station, the induced centrifugal forces could simulate gravity for its hundreds (or thousands) of residents. The space enthusiasts Keith and Carolyn Henson founded the L5 Society in August 1975 for just that purpose, although the society is best remembered for its resonance with the ideas of Princeton physics professor and space visionary Gerard K. O'Neill, who promoted space habitation in his writings such as the 1976 classic The High Frontier: Human Colonies in Space. The L5 Society was founded on one guiding principle: to disband the Society in a mass meeting at L5, presumably inside a space habitat, thereby declaring their mission accomplished. In April 1987, the L5 Society merged with the National Space Institute to become the National Space Society, which continues today.

The idea of locating a large structure at libration points appeared as early as 1961 in Arthur C. Clarke's novel A Fall of Moondust. Indeed, Clarke was no stranger to special orbits. In 1945, he was the first to calculate, in a four-page, hand-typed memorandum, the location above Earth's surface where a satellite's period exactly matches the 24-hour rotation period of Earth. A satellite with that orbit would hover over Earth's surface and serve as an ideal relay station for radio communications from one part of Earth to another. Today, hundreds of communication satellites do just that. Where is this magical place? Objects in low Earth orbit, such as the Hubble Space Telescope and the International Space Station, take about ninety minutes to circle Earth. Objects at the distance of the Moon take about a month. Logically, an intermediate distance must exist where an orbit of 24-hours can be sustained. That distance lies about 22,000 miles above Earth surface.

Actually, there is nothing unique about the rotating Earth-Moon system. Another set of five Lagrangian points exist for the rotating Sun-Earth system. The Sun-Earth L2 point in particular has become the destination of choice for many scientific satellites. The Sun-Earth Lagrangian points all orbit the Sun-Earth center of gravity once per Earth year. At a million miles from Earth, in the direction opposite that of the Sun, a telescope at L2 will have 24-hours of continuous view of the night sky because Earth has shrunk to the size of the Moon in Earth's sky. For observers in low orbits, such as that of the Hubble Telescope, Earth blocks a significant field of view. The recently launched Microwave Anisotropy Probe (MAP for short), reached L2 for the Sun-Earth system in a couple of months and is now librating there, busily taking data on the cosmic microwave background—the omnipresent signature of the big bang itself. The real estate for the Sun-Earth L2 is even wider than that for the Earth-Moon L2. By saving only ten percent of its total fuel, the MAP satellite has enough to hang around this point of unstable equilibrium for nearly a century.

The next generation space telescope, now being planned by NASA as the follow-on to the Hubble, is also being designed to work at the Sun-Earth L2 point. And there is plenty of room—tens of thousands of square miles—for more satellites to come.

Another Lagrangian-loving NASA satellite, known as Genesis, will librate around the Sun-Earth L1 point. In this case, L1 lies a million miles toward the Sun. For two and a half years, Genesis will face the Sun and collect pristine solar matter, including atomic and molecular particles from the solar wind. The material would then be returned to Earth via a mid-air recovery over Utah and be studied for its composition, which provides a window to the contents of the original solar nebula from which the Sun and planets formed. After leaving L1, the returned sample will do a loop-the-loop around L2 and position its trajectory before it returns to Earth.

Given that L4 and L5 are stable points of equilibrium, one might suppose that space junk would accumulate near them, making it quite hazardous to conduct business there. Lagrange made a prediction that space debris would be found at L4 and L5 for the gravitationally powerful Sun-Jupiter system. A century later, in 1905, the first of the Trojan family of asteroids were discovered. We now know that for L4 and L5 of the Sun-Jupiter system, thousands of asteroids lead and follow Jupiter around the Sun, with periods that equal Jupiter's period around the Sun. As through they were responding to tractor beams, these asteroids are forever tethered in place by the gravitational and centrifugal forces of the Sun-Jupiter system. (These asteroids pose no risk to life on Earth, they, being stuck in the outer solar system, and out of harm's way.) Of course, we expect space junk to accumulate at L4 and L5 of the Sun-Earth system as well as the Earth-Moon system. It does. But not nearly to the extent of the Sun-Jupiter encounter.

As an important side benefit, interplanetary trajectories that begin at Lagrangian points require very little fuel to reach other Lagrangian points or even other planets. Unlike a launch from a planet's surface, where most of your fuel goes to lift you off the ground, launching from a Lagrangian point would resemble the a ship leaving dry-dock—becoming adrift into the ocean with only a minimal investment of fuel. In modern times, instead of thinking about self-sustained Lagrangian colonies of people and farms, we can think of Lagrangian points as gateways to the rest of solar system. From the Sun-Earth Lagrangian points you are half way to Mars; not in distance or time but in the all-important category of fuel consumption.

In one version of our space-faring future, imagine fuel stations at every Lagrangian point in the solar system, where travelers fill up their rocket gas tanks en route to visit friends and relatives elsewhere among the planets. This travel model, however futuristic is reads, is not entirely farfetched. Note that without fueling stations scattered liberally across the United States, your automobile would require the proportions of the Saturn V rocket to drive coast to coast: most of your vehicle's size and mass would be fuel, used primarily to transport the yet-to-be-consumed fuel during your cross-country trip. We don't travel this way on Earth. Perhaps the time is overdue when we no longer travel that way through space.

In nearly all sports that use balls, the balls go ballistic at one time or another. Whether you're playing baseball, cricket, football, golf, jai alai, soccer, tennis, or water polo, a ball gets thrown, smacked, or kicked and then briefly becomes airborne before returning to Earth.

Air resistance affects the trajectory of all these balls, but regardless of what set them in motion or where they might land, their basic paths are described by a simple equation found in Newton's Principia, his seminal 1687 book on motion and gravity. Several years later, Newton interpreted his discoveries for the Latin-literate lay reader in The System of the World, which includes a description of what would happen if you hurled stones horizontally at higher and higher speeds. Newton first notes the obvious: the stones would hit the ground farther and farther away from the release point, eventually landing beyond the horizon. He then reasons that if the speed were high enough, a stone would travel the Earth's entire circumference, never hit the ground, and return to smack you in the back of the head. If you ducked at that instant, the object would continue forever in what is commonly called an orbit. You can't get more ballistic than that.

The speed needed to achieve Low Earth Orbit (affectionately called LEO) is a little less than 18,000 miles per hour sideways, making the round trip about an hour and a half. Had Sputnik 1, the first artificial satellite, or Yury Gagarin, the first human to travel beyond Earth's atmosphere, not reached that speed after being launched, they would never have made it into orbit.

Newton also showed that the gravity exerted by any spherical object acts as though all the object's mass were concentrated at its center. Indeed, anything tossed between two people on the Earth's surface is also in orbit, except that the trajectory happens to intersect the ground. This was as true for Alan B. Shepard's fifteen-minute ride aboard the Mercury spacecraft Freedom 7, in 1961, as it is for a golf drive by Tiger Woods, a home run by Sammy Sosa, or a ball tossed by a child: they have executed what are sensibly called suborbital trajectories. Were the Earth's surface not in the way, all these objects would execute perfect, albeit elongated, orbits around Earth's center. And though the law of gravity doesn't distinguish among these trajectories, NASA does. Shepard's journey was mostly free of air resistance, because it reached an altitude where there's hardly any atmosphere. For that reason alone, the media promptly crowned him America's first space traveler.

Suborbital paths are the trajectories of choice for ballistic missiles. Like a hand grenade that arcs ballistically toward its target after being hurled, a ballistic missile flies only under the action of gravity after being launched. These weapons of mass destruction travel hypersonically, fast enough to traverse half of the Earth's circumference in forty-five minutes before plunging back to the surface at thousands of miles an hour. If a ballistic missile is heavy enough, the thing can do more damage just by falling out of the sky than can the explosion of the conventional bomb it carries on board.

The world's first ballistic missile was the V-2 rocket, designed by a team of German scientists under the leadership of Wernher von Braun and used by the Nazis during the Second World War, primarily against England. As the first object to be launched above Earth's atmosphere, the bullet-shaped, large-finned V-2 (the V stands for Vergeltungswaffen, or Vengeance Weapon) inspired an entire generation of spaceship illustrations. After surrendering to the Allied forces, von Braun was brought to the United States, where in 1958 he directed the launch of Explorer 1, the first U.S. satellite. Shortly thereafter, he was transferred to the newly created National Aeronautics and Space Administration. There he developed the Saturn V, the most powerful rocket ever created, making it possible to fulfill the American dream of landing on the Moon.

While hundreds of artificial satellites orbit Earth, the Earth itself orbits the Sun. In his 1543 magnum opus, De Revolutionibus, Nicolaus Copernicus placed the Sun in the center of the universe and asserted that Earth plus the five known planets—Mercury, Venus, Mars, Jupiter, and Saturn—executed perfect circular orbits around it. Unknown to Copernicus, a circle is an extremely rare shape for an orbit and does not describe the path of any planet in our solar system. The actual shape was deduced by the German mathematician and astronomer Johannes Kepler, who published his calculations in 1609. The first of his laws of planetary motion asserts that planets orbit the Sun in ellipses. An ellipse is a flattened circle, and the degree of flatness is indicated by a numerical quantity called eccentricity, abbreviated e. If e is zero, you get a perfect circle. As e increases from zero to one, your ellipse gets more and more elongated. Of course, the greater your eccentricity, the more likely you are to cross somebody else's orbit. Comets that plunge in from the outer solar system have highly eccentric orbits, whereas the orbits of Earth and Venus closely resemble circles, with very low eccentricities. The most eccentric planet is Pluto, and sure enough, every time it goes around the Sun, it crosses the orbit of Neptune, acting suspiciously like a comet (see my column Pluto's Honor, February 1999).

The most extreme example of an elongated orbit is the famous case of the hole dug all the way to China. Contrary to the expectations of our geographically challenged fellow citizens, China is not opposite the United States on the globe. The Indian Ocean is. To avoid emerging under two miles of water, we should dig from Shelby, Montana, to the isolated Kerguelen Islands.

Now comes the fun part. Jump in. You now accelerate continuously in a weightless, free-fall state until you reach Earth's center—where you vaporize in the fierce heat of the iron core. But let's ignore that complication. You zoom past the center, where the force of gravity is zero, and steadily decelerate until you just reach the other side, at which time you have slowed to zero. Unless a Kerguelian grabs you, though, you will fall back down the hole and repeat the journey indefinitely. Besides making bungee jumpers jealous, you have executed a genuine orbit, taking about an hour and a half—just like that of the space shuttle.

Some orbits are so eccentric that they never loop back around again. At an eccentricity of exactly one you have a parabola, and for eccentricities greater than one the orbit traces a hyperbola. To picture these shapes, aim a flashlight at a nearby wall. The emergent cone of light will form a circle. Now gradually angle the flashlight upward, and you create ellipses of higher and higher eccentricities. When your cone points straight up, the light that still falls on the nearby wall takes the exact shape of a parabola. Tip the flashlight a bit more, and you have made a hyperbola. (Now you have something different to do when you go camping.) Any object with a parabolic or hyperbolic trajectory moves so fast that it will never return. If astronomers ever discover a comet with such an orbit, we will know that it has emerged from the depths of interstellar space and is on a one-time tour through the inner solar system.

Newtonian gravity describes the force of attraction between any two objects anywhere in the universe, no matter where they are found, what they are made of, or how large or small they may be. For example, you can use Newton's law to calculate the past and future behavior of the Earth-Moon system. But add a third object—a third source of gravity—and you severely complicate the system's motions. More generally known as the three-body problem, this ménage à trois yields richly varied trajectories whose tracking generally requires a computer.

Some clever solutions to this problem deserve attention. In one case, called the restricted three-body problem, you simplify things by assuming the third body has so little mass compared with the other two that you can ignore its presence in the equations. With this approximation, you can reliably follow the motions of all three objects in the system. And we're not cheating: many cases like this exist in the real universe. Take the Sun, Jupiter, and one of Jupiter's itty-bitty moons. In another example drawn from the solar system, an entire family of rocks move in stable orbits around the Sun, a half-billion miles ahead of and behind Jupiter. These are the Trojan asteroids, each one locked (as if by sci-fi tractor beams) by the gravity of Jupiter and the Sun.

Another special case of the three-body problem was discovered in recent years. Take three objects of identical mass and have them follow each other in tandem, tracing a figure eight in space. Unlike those automobile racetracks where people go to watch cars smashing into each other at the intersection of two ovals, this setup takes better care of its participants. The forces of gravity require that for all times the system balances at the point of intersection, and, unlike the complicated general three-body problem, all motion occurs in one plane. Alas, this special case is so odd and so rare that there is probably not a single example of it among the hundred billion stars in our galaxy, and perhaps only a few examples in the entire universe, making the figure-eight three-body orbit an astrophysically irrelevant mathematical curiosity.

Beyond one or two other well-behaved cases, the gravitational interaction of three or more objects eventually makes their trajectories go bananas. To see how this happens, simulate Newton's laws of motion and gravity on your computer. Now nudge every object according to the force of attraction between it and every other object in the simulation. Recalculate all forces and repeat. The exercise is not simply academic. The entire solar system is a many-body problem, with asteroids, moons, planets, and the Sun in a state of continuous mutual attraction. Newton worried greatly about this problem, which he could not solve with pen and paper. Fearing the entire solar system was unstable and would eventually crash its planets into the Sun or fling them into interstellar space, he postulated that God might step in every now and then to set things right.

Pierre-Simon de Laplace presented a solution to the many-body problem of the solar system more than a century later, in his magnum opus, Méchanique Céleste. But to do so, he had to invent a new form of mathematics known as perturbation theory. The analysis begins by assuming that there is only one major source of gravity and that all the other forces are minor, though persistent—exactly the situation in our solar system. Laplace then demonstrated analytically that the solar system is indeed stable, and that you don't need new laws of physics to show it.

Or is it? Modern analysis demonstrates that on timescales of hundreds of millions of years—periods much longer than the ones considered by Laplace—planetary orbits are chaotic. That leaves Mercury vulnerable to falling into the Sun, and Pluto vulnerable to getting flung out of the solar system altogether. Worse yet, the solar system might have been born with dozens of other planets, most of them now long lost to interstellar space. And it all started with Copernicus's simple circles.

If you imagine yourself rising above the plane of the solar system, you would see each star in our Sun's neighborhood moving about at relative speeds between ten and twenty kilometers a second. But collectively those stars all orbit the galaxy in wide, nearly circular paths, at speeds in excess of 200 kilometers a second. Most of the hundred billion stars of the Milky Way lie within a broad, flat disk, and, like the orbiting objects in all other spiral galaxies, the clouds, stars, and other constituents of the Milky Way thrive on big, round orbits.

Elliptical galaxies are rounded rather than disk-like, yet the orbits of their constituents are anything but round. Many of their stars follow highly elliptical trajectories, plunging swiftly toward the center from all directions and rising steeply back out, the way comets in our solar system do. Elliptical galaxies take the collective shape of all their stars' orbits, just as a swarm of bees takes the collective shape of all its bees' paths.

If you continue rising now, above the plane of the entire Milky Way, you would see the beautiful Andromeda galaxy, a mere 2.5 million light-years away. It's the spiral galaxy closest to us, and all the currently available data suggest we're on a collision course. As we plunge ever deeper into each other's gravitational embrace, we will become a twisted wreck of strewn stars and colliding gas clouds. Just wait about six or seven billion years. With better measurements of our relative motions, however, astronomers may discover a strong sideways component in addition to the motion that brings us together. If so, the Milky Way and Andromeda will instead swing past each other in an elongated orbital dance.

Whenever you go ballistic, you are in free fall. All of Newton's stones were in free fall toward Earth. The one that achieved orbit was also in free fall toward Earth, but our planet's surface curved out from under it at exactly the same rate as it fell—a consequence of the stone's extraordinary sideways motion. The International Space Station is also in free fall toward Earth. So is the Moon. And, like Newton's stones, they are all maintaining a prodigious sideways motion that prevents them from crashing to the ground. For those objects, as well as for the space shuttle, the wayward wrenches of spacewalking astronauts, and other hardware in LEO, one trip around the planet takes about ninety minutes.

The higher you go, however, the longer the orbital period. At about 22,300 miles up, the orbital period is the same as the Earth's rotation rate. Satellites launched into this orbit are said to be geostationary; they hover over a single spot on the planet, enabling rapid, sustained communication between continents as well as satellite TV. Much higher still, at an altitude of 240,000 miles, is the Moon, which takes 27.3 days to complete its orbit.

A fascinating feature of free fall is the persistent state of weightlessness aboard any craft with such a trajectory. In free fall you and everything around you fall at exactly the same rate. A scale placed between your feet and the floor would also be in free fall. Because nothing is squeezing the scale, it would read zero. For this reason, and no other, astronauts are weightless in space.

But the moment the spacecraft speeds up or begins to rotate or undergoes resistance from the Earth's atmosphere, the free-fall state ends and the astronauts weigh something again. Every science-fiction fan knows that if you rotate your spacecraft at just the right speed, or accelerate your spaceship at the same rate as an object falls to Earth, you will weigh exactly what you weigh on your doctor's scale. You can always simulate Earth gravity during those long, boring space journeys.

Another clever application of Newton's orbital mechanics is the slingshot effect. Space agencies often launch probes from Earth that have too little energy to reach their planetary destinations. Instead the orbital wizards aim the probes along cunning trajectories that swing near a hefty, moving source of gravity, such as Jupiter. By falling toward Jupiter in the same direction as Jupiter moves, a probe can steal some Jovial orbital energy during its flyby and then sling forward like a jai alai ball. If the planetary alignments are right, the probe can perform the same trick as it swings by Saturn, Uranus, or Neptune in turn, stealing more energy with each close encounter. A one-time shot at Jupiter can double a probe's speed through the solar system.

The fastest-moving stars of the galaxy, the ones that give colloquial meaning to going ballistic, are the stars that fly past the supermassive black hole in the center of the Milky Way. A descent towards this black hole (or any black hole) can accelerate a star up to speeds approaching that of light. No other object has the power to do this. If a star's trajectory swings slightly to the side of the hole, executing a near miss, it will avoid getting eaten, but its speed will dramatically increase. Now imagine a few hundred or a few thousand stars engaged in this frenetic activity. Astrophysicists view such stellar gymnastics—detectable in most galaxy centers—as conclusive evidence for the existence of black holes: the black hole's smoking gun.

I've always wanted to live where gravity is so weak that you could throw baseballs into orbit. And it wouldn't be hard. No matter how slow you pitch, there's an asteroid somewhere in the solar system with just the right gravity for you to accomplish this feat. Throw with caution, though. If you throw too fast, e could reach one, and you'd lose the ball for good.