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. 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 [known now as Denali] 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 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, imagine 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 its 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…!”