Going ballistic: the many varieties of free fall

Neil deGrasse Tyson

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 whack 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, 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. As a consequence, 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 all 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 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. As the first object to be launched above Earth's atmosphere, the bullet-shaped, large-finned V-2 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 southern 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 Kerguelenian instantly 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--the same amount of time as the space shuttle's.

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 light cone points straight up, the light that still falls on the nearby wall takes the exact shape of a parabola. Tip the flashlight away from the wall 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 manage a trois yields richly varied trajectories whose tracking usually 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--the Sun, Jupiter, and one of Jupiter's itty-bitty moons, for instance. In another case 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 in its solar orbit (as if by Star Trek's 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 race-tracks 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 the system "balances" for all time 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, Newton postulated that God might step in every now and then to set things right.

The eighteenth-century French astronomer and mathematician PierreSimon de Laplace presented a solution to the many-body problem of the solar system more than a century later in his treatise Mecanique Celeste. 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 that prevails 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 this.

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 could somehow rise above the plane of the solar system, you would see each star in our Sun's neighborhood moving to and fro at ten to twenty kilometers a second. But collectively those stars 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.

If you continued rising now, above the plane of the entire Milky Way, you would see the beautiful Andromeda galaxy, a mere 2.3 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, plunging ever deeper into each other's gravitational embrace. Someday 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're going ballistic, you're in free fall. Each of the stones whose trajectory Newton illustrated was 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.

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. Even 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 toward 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.

Astrophysicist Neil deGrasse Tyson is the Frederick P. Rose Director of the Hayden Planetarium in New York City and a visiting research scientist at Princeton University.

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