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فصل 08

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8.

On Being Round

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. Of all shapes, spheres are favored by 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 the 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. Using freshman-level calculus you can show 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 with a four-and-a-half-inch radius. But practical matters prevail—nobody wants to chase packaged food down the aisle after it rolls off the shelves.

On Earth, one way to make ball bearings is to machine them, or drop molten metal in pre-measured amounts into the top of a long shaft. The blob will typically undulate until it settles into the shape of a sphere, but it needs sufficient time to harden before hitting the bottom. On orbiting space stations, where everything is weightless, you gently squirt out precise quantities of molten metal and you have all the time you need—the beads just float there while they cool, until they harden as perfect spheres, with surface tension doing all the work for you.

For large cosmic objects, energy and gravity conspire to turn 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 Himalayas 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 super-duper, jumbo-gigantic finger, and you dragged 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 landmasses to indicate mountain ranges are gross exaggerations of reality. This is why, in spite of Earth’s mountains and valleys, as well as being slightly flattened from pole to pole, when viewed from space, Earth is indistinguishable from a perfect sphere.

Earth’s mountains are also puny when compared with some other mountains in the solar system. The largest on Mars, Olympus Mons, is 65,000 feet tall and nearly 300 miles wide at its base. It makes Alaska’s Mount McKinley look like a molehill. 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 low 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 a mere four ounces.

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

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 mathematician Edouard Roche, who made detailed studies of gravity fields in the vicinity of double stars. The Roche lobe is a theoretical, dumbbell-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. Moreover, every now and then, one of the two stars is a black hole, whose location is rendered visible by the flaying of its binary companion. The spiraling gas, after having passed from the giant across its Roche lobe, heats to extreme temperatures and is rendered aglow before descending out of sight into the black hole itself.

The stars of the Milky Way galaxy trace a big, flat circle. With a diameter-to-thickness ratio of one thousand to one, our galaxy is flatter than the flattest flapjacks ever made. In fact, its proportions are better represented by a crépe or a tortilla. No, the Milky Way’s disk is 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. During the collapse, the ball spun faster and faster, just as spinning figure skaters do when they draw their arms inward to increase their rotation rate. The galaxy naturally flattened pole-to-pole while the increasing centrifugal forces in the middle prevented collapse at midplane. Yes, if the Pillsbury Doughboy 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 twenty-six miles. But Earth is small, mostly solid, and doesn’t rotate all that fast. At twenty-four hours per day, Earth carries anything on its equator at a mere 1,000 miles per hour. Consider the jumbo, fast-rotating, gaseous planet Saturn. Completing a day in just ten and a half 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 offer insight into the rotation rate of extreme cosmic objects. Consider pulsars. With some rotating at upward of a thousand revolutions per second, we know that they cannot be made of household ingredients, or they would spin themselves apart. In fact, if a pulsar rotated any faster, say 4,500 revolutions per second, its equator would be moving at the speed of light, which tells you that this material is unlike any other. To picture a pulsar, imagine the mass of the Sun packed into a ball the size of Manhattan. If that’s hard to do, then maybe it’s easier if you imagine stuffing about a hundred million elephants into a Chapstick casing. To reach this density, you must compress all the empty space that atoms enjoy around their nucleus and among their orbiting electrons. Doing so will crush nearly all (negatively charged) electrons into (positively charged) protons, creating a ball of (neutrally charged) neutrons with a crazy-high surface gravity. Under such conditions, a neutron star’s mountain range needn’t be any taller than the thickness of a sheet of paper for you to exert more energy climbing it than a rock climber on Earth would exert ascending a three-thousand-mile-high cliff. In short, where gravity is high, the high places tend to fall, filling in the low places—a phenomenon that sounds almost biblical, in preparing the way for the Lord: “Every valley shall be raised up, every mountain and hill made low; the rough ground shall become level, the rugged places a plain” (Isaiah 40:4). That’s a recipe for a sphere if there ever was one. For all these reasons, we expect pulsars to be the most perfectly shaped spheres in the universe.

For rich clusters of galaxies, the overall shape can offer deep astrophysical insight. Some are raggedy. Others are stretched thin in filaments. Yet others form vast sheets. None of these have settled into a stable—spherical—gravitational shape. Some are so extended that the fourteen-billion-year age 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 have had insufficient time to influence the cluster’s shape.

But other systems, such as the beautiful Coma cluster of galaxies, which we met in our chapter on dark matter, tell us immediately that gravity has shaped the cluster into a sphere. 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 flattening, as we do in our own Milky Way.

The Coma cluster, once again like the Milky Way, is also gravitationally mature. In astrophysical vernacular, such systems are said to be “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. It’s for these reasons that gravitationally relaxed systems make 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 may have remained undiscovered to this day.

The sphere to end all spheres—the largest and most perfect of them all—is the entire observable universe. In every direction we look, galaxies recede from us at speeds proportional to their distance. As we saw in the first few chapters, this is the famous signature of an expanding universe, discovered by Edwin Hubble in 1929. When you combine Einstein’s relativity and the velocity of light and the expanding universe and the spatial dilution of mass and energy as a consequence of that expansion, 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, light from all luminous objects loses all its energy before reaching us. The universe beyond this spherical “edge” is thus rendered invisible and, as far as we know, unknowable.

There’s a variation of the ever-popular multiverse idea in which the multiple universes that comprise it are not separate universes entirely, but isolated, non-interacting pockets of space within one continuous fabric of space-time—like multiple ships at sea, far enough away from one another so that their circular horizons do not intersect. As far as any one ship is concerned (without further data), it’s the only ship on the ocean, yet they all share the same body of water.

Spheres are indeed fertile theoretical tools that help us gain insight into all manner of astrophysical problems. But one should not be a sphere-zealot. I am reminded of the half-serious joke about how to 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 . . .”

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