The Physics of Superheroes: Spectacular Second Edition (8 page)

This is the true meaning of the story of Isaac Newton and the apple. It certainly wasn’t the case that in 1665 Newton saw an apple fall from a tree and suddenly realized that gravity existed, nor did he see an apple fall and immediately write down F = G (m1 × m2)/ (d)
²
. Rather, Newton’s brilliant insight in the seventeenth century was that the exact same force that pulled the apple toward the Earth pulled the moon toward the Earth, thereby connecting the terrestrial with the celestial. In order for the moon to stay in a circular orbit around the Earth, a force has to pull on it in order to constantly change its direction, keeping it in a closed orbit.

Remember Newton’s second law of
F
=
ma
: If there’s no force, there’s no change in the motion. When you tie a string to a bucket and swing it in a horizontal circle, you must continually pull on the string. If the tension in the string doesn’t change, then the bucket stays in uniform circular motion. The tension in the string is not acting in the direction that the bucket is moving; consequently, it can only change its direction but not its speed. The moment you let go of the string, the bucket will fly away from you.

Back to the case of the moon. If there were no gravity, no force acting on it, then the moon would travel in a straight line right past the Earth. If there were gravity but the moon were stationary, then it would be pulled down and crash into our planet. The moon’s distance from the Earth and its speed are such that they exactly balance the gravitational pull, so that it remains in a stable circular orbit. The moon does not fly away from us, because it is pulled by the Earth’s gravity, causing it to “fall” toward the Earth, while its speed is great enough to keep the moon from being pulled any closer to us. The same force that causes the moon to “fall” in a circular orbit around the Earth, and causes the Earth to “fall” in an elliptical orbit around the sun, causes the apple to fall toward the Earth from the tree. And that same gravitational force causes Superman to slow in his ascent once he leaps, until he reaches the top of a tall skyscraper. Once we know that in order to make such a powerful leap, his body had to be adapted to an environment where the acceleration due to gravity is fifteen times greater than on Earth, that same gravitational force informs us about Krypton’s geology.

One consequence of Newton’s law of gravitation—which states that as the distance between two objects increases, the gravitational pull between them becomes weaker by the square of their separation—is that all planets are round. A sphere has a volume that grows with the cube of the radius of the orb, while its surface area increases with the square of the radius. This combination of the square of the radius for the surface area with the inverse square of the gravitational force leads to a sphere being the only stable form that a large gravitational mass can maintain. In fact, to address the astrophysical question of what distinguishes a very large asteroid from a very small planet, one answer is its shape. A small rock that you hold in your hand can have an irregular shape, as its self-gravitational pull is not large enough to deform it into a sphere. However, if the rock were the size of Pluto, then gravity would indeed dominate, and it would be impossible to structure the planetoid so that it had anything other than a spherical profile. Consequently, cubical planets such as the home world of Bizarro must be very small. In fact, the average distance from the center of the Bizarro planet to one of its faces can be no longer than 300 miles, if it is to avoid deforming into a sphere. However, such a small cubical planet would not have sufficient gravity to hold an atmosphere on its surface, and it would be an airless rock. Since we have frequently seen that the sky on the Bizarro world is blue like our own (and shouldn’t it be some other color if it is to hold true to the Bizarro concept?), this would imply that there is indeed air on this cubical planet. We must therefore conclude that a Bizarro planet is not physically possible, no matter how many times we may feel in the course of a day that we have been somehow instantly transported to such a world.

Back to normal spherical planets like Krypton. If the acceleration due to gravity on Krypton g
_K
is fifteen times larger than the acceleration due to gravity on Earth g
_E
, then the ratio of these accelerations is g
_K
/g
_E
= 15. We have just shown that the acceleration due to gravity of a planet is g = Gm/d
²
. The distance d that we’ll use is the Radius R of the planet. The mass of a planet (or of anything for that matter) can be written as the product of its density (the Greek letter ρ is traditionally used to represent density) and its volume, which in this case is the volume of a sphere (since planets are round). Since the gravitational constant G must be the same on Krypton as on Earth, the ratio g
_K
/g
_E
is given by the following simple expression:
where ρ
_K
and R
_K
represent the density and radius of Krypton and ρ
_E
and R
_E
stand for the Earth’s density and radius, respectively. When comparing the acceleration due to gravity on Krypton to that on Earth, all we need to know is the product of the density and radius of each planet. If Krypton is the same size as Earth, then it must be fifteen times denser, or if it has the same density, then it will be fifteen times larger.

Now if, as we have argued at the start of this book, the essence of physics is asking the right questions, then it is as true in physics as it is in life that every answer one obtains leads to more questions. We have determined that in order to account for Superman’s ability to leap 660 feet (the height of a tall building) in a single bound on Earth, the product of the density and radius of his home world of Krypton must have been fifteen times greater than that of Earth. We next ask whether it is possible that the size of Krypton is equal to that of Earth (R
_K
= R
_E
) so that all of the excess gravity of Krypton can be attributed to its being fifteen times denser than Earth. It turns out that if we assume that the laws of physics are the same on Krypton as on Earth (and if we give up on that, then the game is over before we begin and we may as well quit now!), then it is extremely unlikely that Krypton is fifteen times denser than Earth.

We have just made use of the fact that mass is the density multiplied by volume, which is just another way of saying that density is the mass per unit volume of an object. Now, to understand what limits this density, and why we can’t easily make the density of Krypton fifteen times greater than Earth’s, we have to take a quick trip down to the atomic level. Both the total mass of an object and how much volume it takes up are governed by its atoms. The mass of an object is a function of how many atoms it contains. Atoms are composed of protons and neutrons inside a small nucleus, surrounded by lighter electrons. The number of positively charged protons in an atom is balanced by an equal number of negatively charged electrons. Electrons are very light compared to protons or neutrons, which are electrically uncharged particles that weigh slightly more than protons and reside in a nucleus. (We’ll discuss what the neutrons are doing in the nucleus in Chapter 16.) Nearly all the mass of an atom is determined by the protons and neutrons in its nucleus, because electrons are nearly two thousand times lighter than protons.

The size of an atom, on the other hand, is determined by the electrons or, more specifically, their quantum mechanical orbits. The diameter of a nucleus is about one trillionth of a centimeter, while the radius of an atom is calculated by how far from the nucleus one is likely to find an electron, and is about ten thousand times bigger than the nucleus. If the nucleus of an atom were the size of a child’s marble (a diameter of 1cm) and placed in the end zone of a football field, the radius of the electron’s orbit would extend to the opposite end zone, 100 yards away. The spacing between atoms in a solid is governed essentially by the size of the atoms themselves (you can’t normally pack them any closer than their size).

Thus, if quantum mechanics is the same on Krypton as on Earth, the space taken up by a given number of atoms in a rock (for example) will not depend significantly upon which planet the rock resides on. The rock will weigh more on a planet with a larger gravity, but the number of atoms it contains—as well as the spacing between the atoms, both of which determine its mass density—will be independent of which planet the rock finds itself on. Because the number of atoms also determines the mass of the rock, it follows that the density of any given object will be the same, regardless of the planet of origin.

Most solid objects have roughly the same density, at least within a factor of ten. For example, the density of water is 1 gram/cm
³
while the density of lead is 11 gram/cm
³
(a gram is one thousandth of a kilogram). In other words, a cube that measures 1cm on each side would have a mass of 1 gram if composed of water and 11 grams if composed of lead. The higher density of lead is due almost entirely to the fact that a lead atom is ten times more massive than a water molecule. While there is a lot of water on the surface of the Earth, there’s even more solid rock within the planet, so that Earth’s average density is 5 gram/cm
³
. In fact, Earth is the densest planet in our solar system, with Mercury and Venus close behind. Even if Krypton were solid uranium, it would have an average density of 19 gram/cm
³
, which is not even four times larger than Earth’s. In order for Krypton to have a gravity fifteen times greater than Earth’s due to a larger density alone, it would have to have a density fifteen times larger than Earth’s 5 gm/cm
³
—that is, 75 gram/cm
³
—and no normal matter is this dense.

If the density of planet Krypton couldn’t be much greater than Earth’s, perhaps the heavier gravity on Krypton is due to it being a larger planet—one with a radius fifteen times larger than Earth’s. While planets in our own solar system come in all sizes—from Pluto, with a radius one fifth as large as Earth’s, making it just barely bigger than some moons, to Jupiter, with a radius of more than eleven times Earth’s—the geology of the planet is a sensitive function of its size. Planets bigger than Uranus, with a radius four times larger than Earth’s, include Neptune, Saturn, and Jupiter. These planets are gas giants, lacking a solid mantle upon which buildings and cities may be constructed, let alone supporting humanoid life. In fact, if Jupiter were ten times larger, it would be the size of our own sun. In this case, the gravitational pressure at Jupiter’s core would initiate nuclear fusion, the process that causes our sun to shine. So, if Jupiter were just a bit larger, it would no longer be a giant planet but rather a small star.

Big planets are gaseous because if you’re going to build a very big planet, you are going to need a lot of atoms, and when you go to the great celestial stockroom, nearly all of the raw materials available are either hydrogen or helium gas. To be precise, 73 percent of the elemental mass in the universe is hydrogen and 25 percent is helium. Everything else that you would use to make a solid planet—such as carbon, silicon, copper, nitrogen, and so on—comprises only 2 percent of the elemental mass in the known universe. So big planets are almost always gas giants, which tend to have orbits far from a star, where the weaker solar radiation cannot boil away the gaseous surfaces they have accreted. The concentration of heavier elements with which solid planets can form is much lower, so they will tend to be smaller and closer to a star. If these inner solid planets got too large, the gravitational tidal forces
¹³
from their sun would quickly tear them apart. Krypton’s advanced civilization, with scientists capable of constructing a rocket ship, couldn’t arise on a gas giant with a radius fifteen times that of Earth’s.

So, is that it? Is the story of Superman and Krypton, with an Earth-like surface and a gravity fifteen times that of Earth, totally bogus? Not necessarily. Remember that earlier it was stressed that no
normal
matter could be fifteen times denser than matter on Earth. However, astronomers have discovered exotic matter, with exceedingly high densities, formed from the remnants of supernova explosions. As mentioned, when the size of a gaseous planet exceeds a certain threshold, the gravitational compression at its center is so large that the nuclei of different atoms literally fuse together, creating larger nuclei and releasing excess energy in the process. The source of this energy is expressed in Einstein’s famous equation, E = mc
²
or Energy E is equivalent to mass m multiplied by the speed of light c squared. The mass of the fused-product nucleus is actually a tiny bit smaller than that of the two initial separate nuclei. The small difference in mass, when multiplied by the speed of light squared (a very big number) yields a large amount of energy. This energy radiates outward from the star’s center, producing an outward flow that balances the inward attractive gravitational force, keeping the radius of the star stable.