While hundreds of artificial satellites orbit Earth, 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 1, 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 do so on highly eccentric orbits, whereas the orbits of Earth and Venus closely resemble circles, each 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.
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. A straight path that connects two opposite points on Earth must pass through Earth’s center. What’s opposite the United States? The Indian Ocean. To avoid emerging under two miles of water, we need to learn some geography and dig from Shelby, Montana, through Earth’s center, 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. But unless a Kerguelian grabs you, 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 1, you have a parabola, and for eccentricities greater than 1, the orbit traces a hyperbola. To picture these shapes, aim a flashlight directly at a nearby wall. The emergent cone of light will form a circle of light. Now gradually angle the flashlight upward, and the circle distorts to 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 astrophysicists 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 addressed in Section 2, with 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 one another 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, one can simulate Newton’s laws of motion and gravity on a computer by nudging every object according to the force of attraction between it and every other object in the calculation. 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, as we will see in Section 9, that God might step in every now and then to set things right.
Pierre-Simon Laplace presented a solution to the many-body problem of the solar system more than a century later, in his magnum opus, Traité de mécanique céleste. But to do so, he had to develop 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? As we will see further in Section 6, 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. A situation 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.
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 all maintain 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 90 minutes.
The higher you go, however, the longer the orbital period. As noted earlier, 22,300 miles up, the orbital period is the same as Earth’s rotation rate. Satellites launched to that location are geostationary; they “hover” over a single spot on our planet, enabling rapid, su
stained communication between continents. 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 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 do on your doctor’s scale. So if your aerospace engineers felt so compelled, they could design your spaceship to 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 orbit engineers 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 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, stealing more energy with each close encounter. These are not small boosts; these are big boosts. 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 farthest object visible to the unaided eye is the beautiful Andromeda galaxy, which is the closest spiral galaxy to us. That’s the good news. The bad news is that all available data suggest that the two of us are 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 6 or 7 billion years.
In any case, you could probably sell seats to watch the encounter between Andromeda’s supermassive black hole and ours, as whole galaxies go ballistic.
FOURTEEN
ON BEING DENSE
When I was in the 5th grade, a mischievous classmate asked me the question, “Which weighs more, a ton of feathers or a ton of lead?” No, I was not fooled, but little did I know how useful a critical understanding of density would be to life and the universe. A common way to compute density is, of course, to take the ratio of an object’s mass to its volume. But other types of densities exist, such as the resistance of somebody’s brain to the imparting of common sense or the number of people per square mile who live on an exotic island such as Manhattan.
The range of measured densities within our universe is staggeringly large. We find the highest densities within pulsars, where neutrons are so tightly packed that one thimbleful would weigh about as much as a herd of 50 million elephants. And when a rabbit disappears into “thin air” at a magic show, nobody tells you the thin air already contains over 10,000,000,000,000,000,000,000,000 (ten septillion) atoms per cubic meter. The best laboratory vacuum chambers can pump down to as few as 10,000,000,000 (ten billion) atoms per cubic meter. Interplanetary space gets down to about 10,000,000 (ten million) atoms per cubic meter, while interstellar space is as low as 500,000 atoms per cubic meter. The award for nothingness, however, must be given to the space between galaxies, where it is difficult to find more than a few atoms for every 10 cubic meters.
The range of densities in the universe spans forty-four powers of 10. If one were to classify cosmic objects by density alone, salient features would reveal themselves with remarkable clarity. For example, dense compact objects such as black holes, pulsars, and white dwarf stars all have a high force of gravity at their surfaces and readily accrete matter into a funneling disk. Another example comes from the properties of interstellar gas. Everywhere we look in the Milky Way, and in other galaxies, gas clouds with the greatest density are sites of freshly minted stars. Our detailed understanding of the star formation process remains incomplete, but understandably, nearly all theories of star formation include explicit reference to the changing gas density as clouds collapse to form stars.
OFTEN IN ASTROPHYSICS, especially in the planetary sciences, one can infer the gross composition of an asteroid or a moon simply by knowing its density. How? Many common ingredients in the solar system have densities that are quite distinct from one another. Using the density of liquid water as a measuring unit, frozen water, ammonia, methane, and carbon dioxide (common ingredients in comets) all have a density of less than 1; rocky materials, which are common among the inner planets and asteroids, have densities between 2 and 5; iron, nickel, and several other metals that are common in the cores of planets, and also in asteroids, have densities above 8. Objects with average densities intermediate to these broad groups are normally interpreted as having a mixture of these common ingredients. For Earth we can do a little better: the speed of postearthquake sound waves through Earth’s interior directly relates to the run of Earth’s density from its center to the surface. The best available seismic data give a core density of around 12, dropping to an outer crustal density of around 3. When averaged together, the density of the entire Earth is about 5.5.
Density, mass, and volume (size) come together in the equation for density, so if you measure or infer any two of the quantities then you can compute the third. The planet around the sunlike, naked-eye star 51 Pegasus had its mass and orbit computed directly from the data. A subsequent assumption about whether the planet is gaseous (likely) or rocky (unlikely) allows a basic estimate of the planet’s size.
Often when people claim one substance to be heavier than another, the implicit comparison is one of density, not weight. For example, the simple yet technically ambiguous statement “lead weighs more than feathers” would be understood by nearly everybody to be really a question of density. But this implicit understanding fails in some notable cases. Heavy cream is lighter (less dense) than skim milk, and all seagoing vessels, including the 150,000-ton Queen Mary 2, are lighter (less dense) than water. If these statements were false, then cream and ocean liners would sink to the bottom of the liquids upon which they float.
OTHER DENSITY TIDBITS:
Under the influence of gravity, hot air does not rise simply because it’s hot, but because it’s less dense than the surrounding air. One could similarly declare that cool, denser air sinks, both of which must happen to enable convection in the universe.
Solid water (commonly known as ice) is less dense than liquid water. If the reverse were true, then in the winter, large lakes and rivers would freeze completely, from the bottom to the top, killing all fish. What protects the fish is the floating, less dense, upper layer of ice, which insulates the warmer waters below from the cold winter airs.
On the subject of dead fish, when found belly-up in your fish tank, they are, of course, temporarily less dense than their live counterparts.
Unlike any other known planet, the average density of Saturn is less than that of water. In other words, a scoop of Saturn would float in your bathtub. Knowing this, I have always wa
nted for my bathtub entertainment a rubber Saturn instead of a rubber ducky.
If you feed a black hole, its event horizon (that boundary beyond which light cannot escape) grows in direct proportion to its mass, which means that as a black hole’s mass increases, the average density within its event horizon actually decreases. Meanwhile, as far as we can tell from our equations, the material content of a black hole has collapsed to a single point of near-infinite density at its center.
And behold the greatest mystery of them all: an unopened can of diet Pepsi floats in water while an unopened can of regular Pepsi sinks.
IF YOU WERE to double the number of marbles in a box, their density would, of course, remain the same because both the mass and the volume would double, which in combination has no net effect on the density. But objects exist in the universe whose density relative to mass and volume yields unfamiliar results. If your box contained soft, fluffy down, and you doubled the number of feathers, then ones on the bottom would become flattened. You would have doubled the mass but not the volume, and you would be left with a net increase in density. All squishable things under the influence of their own weight will behave this way. Earth’s atmosphere is no exception: we find half of all its molecules packed into the lowest three miles above Earth’s surface. To astrophysicists, Earth’s atmosphere forms a bad influence on the quality of data, which is why you often hear about us escaping to mountaintops to conduct research, leaving as much of Earth’s atmosphere below us as possible.
Death By Black Hole & Other Cosmic Quandaries Page 12
