The Physics of Superheroes: Spectacular Second Edition
Page 6
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/cm3 while the density of lead is 11 gram/cm3 (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/cm3. 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/cm3, 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/cm3—that is, 75 gram/cm3—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 forces13 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 = mc2 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.
When all the hydrogen nuclei have been fused into helium nuclei, some of the helium nuclei are in turn fused into carbon nuclei, some of which in turn are compressed to form nitrogen, oxygen, and all of the heavier elements, up to iron. The fusion process speeds up as the star generates heavier and heavier nuclei, so that all of its iron and nickel are created within the last week of the star’s life. As heavier and heavier nuclei are combined, the process becomes less and less efficient, so that the energy released when iron nuclei fuse is insufficient to stably counteract the inward gravitational pressure. At this point gravity wins out, rapidly compressing the material into a much smaller volume. In this brief moment, the pressure at the center of the star is so high that one last gasp of fusion occurs, and heavier elements all the way up to uranium are generated, with a concurrent tremendous release of energy. This last stage in the life of the largest stars is termed the “supernova” phase. With this final blast of energy, the elements that had been synthesized within the star are flung out into space, where gravity may eventually pull them together into clumps that can form planets or other stars. Every single atom in your body, in the chair in which you are sitting, or the paper and ink in Action Comics # 1, was synthesized within a star that died and subsequently expelled its contents. We are all composed of stardust or, if you’re feeling a tad more cynical, solar excrement.
For really big stars, the gravitational pressure at the center is so great that after the supernova phase, there remains a large remnant core for which gravity compresses the protons and electrons into neutrons, which are squeezed until they touch and become a solid, composed of nuclear matter. The remnants of such massive stars are termed “neutron stars,” which are composed solely of neutrons and whose density is exceeded only by that of black holes (which are left over from the death of even bigger stars, whose gravitational attraction is so high that not even light can escape its pull). Compared with the density of lead (11 grams per cubic centimeter), the density of neutron-star material is just a tad higher: one hundred trillion grams per cubic centimeter. That is, a teaspoon of neutron-star material on Earth would weigh more than 100 million tons. This is just the stuff to boost the gravity of Krypton.
If a planet the size of Earth had a small volume of neutron star material within its core, the additional mass would dramatically increase the gravity on the surface of the planet. In fact, it would take only a sphere of neutron star material with a radius of 600 meters (about the length of six football fields) at the center of a planet the size of Earth to create an acceleration due to gravity on its surface of 150 meters/sec2, whereas the acceleration due to gravity on Earth is 10 meters/sec2. So, in order for Krypton’s gravity to have been 15 times greater than Earth’s, it must have had a core of neutron-star matter at its center.
And thus, we see why Krypton was doomed! For such a super-dense core would produce enormous strains on the surface of the planet, making a stable distribution of matter tenuous at best. At some point in the planet’s history, volcanic activity and plate tectonics would result in massive upheavals. Such pre-shock earthquakes would alert an astute scientist that now would be a good time to put their kid into a rocket ship and send him to some other distant planet, preferably one without a neutron-star core.
Let us pause to admire the sc
ientific insight of Jerry Siegel and Joe Shuster. These teenagers from Cleveland, Ohio, either had an understanding of astrophysics and quantum mechanics that exceeded that of many contemporary physics professors in 1938 or they were very lucky guessers. Only five years earlier, the astronomers Walter Baade and Fritz Zwicky had proposed the existence of neutron stars, the definitive evidence of which would not be found until the 1960s. Perhaps if Sheldon Mayer at National Publications had not taken a chance on their Superman strip, Siegel and Shuster might have considered publication in a scientific journal such as the Physical Review, and the history of both science and comic books would be very different today.
Now, in the last chapter I alluded to a question concerning Superman’s acceleration when he lifted off the ground as he leapt a tall building in a single bound. We had calculated that in order to rise to a height of one eighth of a mile, he needed a liftoff speed of 200 ft/sec, or 140 miles per hour. If the time he spent pushing against the ground was one-quarter of a second, then this corresponded to an acceleration of 800 feet/sec2, or twenty-five times the acceleration due to gravity.
Careful readers of the first edition have pointed out that I had overestimated the time that Superman spent jumping. The formula employed to determine his liftoff speed, that is, v2 = 2gh, can also be used to find his acceleration as he leaps. The velocity in this case is still 200 feet/sec, but instead of g, we use his jumping acceleration. While before the height h was the eighth of a mile height of a building, now the height is the difference from his low crouch when he prepares to jump and when he is standing straight up as he first leaves the ground. At most, this distance might be three feet, so whereas earlier we estimated his jumping acceleration to be an incredible 800 feet/sec2, this new calculation finds that his acceleration must be over 6,600 feet/sec2. This corresponds to a time spent jumping of not one quarter of a second, but a much speedier 0.03 seconds. Which makes him quite super, indeed.
Using the longer jumping time of one quarter of a second led to a lower jumping acceleration and a corresponding force of 5,600 pounds that Superman’s legs must supply during this leap. The shorter time of 0.03 seconds and larger jumping acceleration leads to a much larger force of just over 46,000 pounds. The same arguments as employed in Chapter 1 lead us to conclude that the acceleration due to gravity of Krypton gK is an astounding 125 times greater than on Earth, compared with our earlier estimate of gK/gE = 15.
I used the longer time of a quarter second in Chapter 1 for two reasons. Informal experiments in my classes, where I asked students to leap and recorded the time they spent pushing against the ground, seemed consistent with a fraction of a second, and one quarter is roughly the average value obtained. Of course, none of my students were able to leap over a tall building. In Chapter 1, the intent was to explain how one finds an acceleration, knowing the time interval over which a velocity changes. More importantly, I wanted to use a lower jumping acceleration and leg force for the discussion in this chapter. If I had used a leaping time of 0.03 seconds and, upon concluding that the acceleration due to gravity of Krypton must have been 125 times that of Earth, I doubt few would be surprised that it is difficult to construct such a planet. A planet with a gravity fifteen times that of Earth’s seems more plausible, and to show that this is also unlikely requires a discussion of the limits of the density of matter and the geology of large planets. My goal is to use the superhero’s powers to illustrate real-world physics.
Occasionally I have been taken to task by a reader for contending that the acceleration due to gravity on Krypton is “only” fifteen times larger than that of Earth. I would explain my reasons, listing the above arguments. Some fans would protest: “Look, it’s just not realistic that the time spent jumping is as long as a quarter of a second!” To which I responded that we were discussing the exploits of the last son of the imaginary planet Krypton, able to bend steel in his bare hands and change the course of mighty rivers, and the part that they found unrealistic was the time he spent pushing off against the ground while leaping a tall building in a single bound? Few would fail to concede the point at this stage. Not that I mind in the least being corrected and in fact am thrilled that readers are motivated enough to pick up pencil and calculator and keep me honest. That’s the whole point, after all.
3
THE DAY GWEN STACY DIED—IMPULSE AND MOMENTUM
IF A SENATE SUBCOMMITTEE HEARING marked the beginning of the end of the Golden Age of comics, the death blow of the Silver Age was self-inflicted. Viewed from today’s perspective, comics from the Silver Age (from the late 1950s to 1960s) seem suffused with an optimistic outlook and a sunny disposition that borders on the Pollyannaish. The Golden Age characters reinvented by Julius Schwartz and colleagues at DC Comics in the late fifties and early sixties, such as the Flash, Green Lantern, or Green Arrow (an amalgam of Batman and Robin Hood, with a quiver full of gadget arrows such as a “boxing-glove arrow” or a “handcuff arrow,” whose successful application violated several fundamental principles of aerodynamics), carried on the positive outlooks and righteousness of their antecedents, and their plot-driven twelve- or twenty-two-page-long stories did not leave much room for character development. A typical Silver Age hero in a DC comic book would gain superpowers through some implausible mechanism and then decide, as a matter of course, to use said powers to fight crime and better humanity (after first donning, of course, a colorful costume), never questioning this career choice.
The situation was very different with the superheroes populating DC’s main comic-book competitor, Marvel Comics, whose characters such as the Hulk and the X Men lamented that if they didn’t have bad luck, they’d have no luck at all. In 1961, the Marvel Comics (née Timely and then Atlas) Company was on the verge of going out of business. From its Golden Age peak, when it had published the Human Torch, the Sub Mariner, and Captain America comics, the company had fallen to the point where it was barely getting by putting out monster comics, Westerns, funny animal stories, and young-romance stories. This all changed with a golf game between Jack Liebowitz, the head of DC Comics, and Martin Goodman, Marvel Comics’ publisher. Liebowitz boasted of the success DC was having with a particular title, the Justice League of America, that featured a team of superheroes including Wonder Woman, the Flash, Green Lantern, Aquaman, the Martian Manhunter, and others banding together to fight the supervillain du mois. Upon returning to the office, Goodman instructed his editor (and his nephew-in-law, the last remaining full-time employee), Stan Lee, to come up with a comic book featuring a team of superheroes. Marvel was not publishing superhero comics at the time; consequently Lee could not assemble a team book by incorporating characters from other comics, as DC had done. Instead, he and Jack Kirby created a new superhero team from whole cloth. The resulting title, the Fantastic Four, written by Lee and with art by Kirby, became a sales success, and led to Marvel Comics’ reversal of fortune.14
Lee’s and Kirby’s unique contribution was to add character development and distinctive personalities to their comic-book stories. As a way of distinguishing themselves from the heroes in DC comics, the superheroes in Marvel’s stories did not see their superpowers as a blessing, but frequently bemoaned their fate. When cosmic radiation turned Ben Grimm into the large, orange, rock-complexioned Thing in Fantastic Four # 1, he did not revel in his newfound superhuman strength but cursed the fact that he had become a walking brick patio, wanting nothing more than to be restored to his human form. But no character in the Marvel universe complained more about his lot in life than Spider-Man.
In Amazing Fantasy # 15, written by Lee and drawn by Steve Ditko in 1962, young Peter Parker was a thin, nerdy high-school student who suffered endless teasing by the popular jocks at his school. Parker was an orphan, living with his overprotective, elderly relatives, Aunt May and Uncle Ben. Excluded from joining the popular students in an after-school activity, Peter indulged his interest in science by attending a physics lab demonstration on radioactivity. As happened with alarming fr
equency in Silver Age Marvel comics, an accident with radioactivity resulted in the bestowal of superpowers. In this case, a spider was inadvertently irradiated during the demonstration and bit Peter Parker before expiring, leaving him with radioactive spider blood.
Parker found that he had acquired from this spider bite various arachnid attributes, including heightened agility and the ability to adhere to walls. Because a spider can lift several times its own body weight, Peter could now lift several times his own weight, described in the comics as a “proportionate” increase in strength. Peter also gained a “sixth sense” that alerted him to potential dangers, a Spider-Sense, if you will. One can only guess that Stan Lee, stymied in his attempts to kill real spiders in his bathroom, concluded that spiders used ESP to avoid being squished. Presumably, we can thank the protective Comics Code Authority for the fact that Peter did not also gain a spider’s ability to spew organic webbing from his anus, but instead used his knowledge of chemistry and mechanics to construct technological web shooters that he wore on his wrists.15
After a lifetime of ridicule and abuse at the hands of his peers, Peter initially sees his newfound powers as a venue to fame and fortune. After testing his skills in professional wrestling, he creates a colorful blue-and-red costume and mask in order to enter show business. Feeling empowered on the eve of his television debut, he arrogantly refuses to help a security guard stop a fleeing robber, though it would have been easy to do so. However, upon returning home, he learns that gentle Uncle Ben has been slain by an intruder. Capturing the killer using his new powers, Peter discovers to his horror that this was the same robber he could have stopped earlier that day. Belatedly realizing “that with great power there must also come great responsibility,” Peter Parker dedicates himself to fighting crime and righting wrongs as the amazing Spider-Man.