Wizards, Aliens, and Starships: Physics and Math in Fantasy and Science Fiction

by Adler, Charles L.

  • Thermal loss. Molecules in a gas at room temperature or above move with average speeds of a few hundred meters per second, which is about an order of magnitude less than the escape velocity for an Earth-sized planet. However, because this speed represents an average, some of the molecules move a lot faster. In the upper atmosphere, where collisions between molecules are few and far between, molecules moving faster than the escape velocity of the planet can be lost from the atmosphere. This is called the Jeans escape mechanism, after the astrophysicist who first described it. There are other non thermal loss mechanisms, including the fact that chemical reactions in the upper atmosphere can provide the reactants with enough energy to escape into space. Generally speaking, larger planets lose their atmospheres more slowly through this mechanism than do smaller ones.

  • Impacts. Here is where history becomes important. Impacts of planets with large objects (asteroids or comets) can push a lot of the atmosphere into space. Mars’s atmosphere may be thin partly because the planet is close to the asteroid belt and suffered collisions from asteroids over geological time spans.

  • Solar wind. Stellar winds, streams of charged particles from the sun, can strip away planetary atmospheres, particularly for small worlds close to their stars. This may have affected Mars’s atmosphere and almost certainly is responsible for stripping away what little atmosphere Mercury may once have had. In fact, all of the atmospheres of the inner planets are secondary ones, as the primary atmospheres that formed when the planets formed were stripped away by the strong solar winds from the young Sun.

  • Chemical sequestration. The atmospheres can become chemically bonded to the crust of the planet. This is where most of Earth’s carbon is: if all the carbon in the Earth’s crust were liberated into the atmosphere, Earth would have a worse greenhouse effect than Venus.

  The formation of atmospheres is just as complicated, and harder to discuss in detail, as it is due to vulcanism, cometary impacts during the planet’s early history, and, for Earth, the presence of life on the planet (leading to the presence of significant amounts of O2 in the atmosphere).

  14.6.1 Thermal Loss Mechanisms

  The average (rms) speed of a molecule of molar mass m in grams per mole in a gas at temperature T is given by

  whereas the escape velocity of a planet is given by

  Table 14.2

  Relative Properties of Earth and Mars

  Earth

  Mars

  R

  M

  T

  1

  1

  1,000

  0.533

  0.107

  140

  Ve (m/s)

  vrms for H2

  He

  O2

  11,000

  3,500

  2,480

  878

  4,800

  1,860

  930

  328

  where Mp is the mass of the planet measured relative to the mass of the Earth and Rp is the radius, again measured with respect to the Earth’s radius. A rule of thumb derivable from the Jeans escape formula that is used by planetary scientists is that if the rms molecular speed is greater than about one-sixth the escape velocity, thermal loss mechanisms will deplete the atmosphere of that particular molecule over geological periods of time [46, p. 103].

  Table 14.2 shows some comparisons for Mars and Earth. The temperatures were chosen as typical of the top of each planet’s atmosphere. Earth is at a relatively high temperature because of the ozone layer. The absorption of UV light from the Sun puts energy into the atmosphere at that height, while Mars doesn’t have a similar protective layer.

  There are a few points we can glean from table 14.2:

  1. Neither hydrogen nor helium should be present in either atmosphere in appreciable quantities because their average molecular speeds are too high.

  2. However, oxygen shouldn’t be lost because of thermal effects in either atmosphere. In the case of Mars, the low escape velocity is offset by the low atmospheric temperature compared to Earth’s.

  The ratio of the average molecular speeds to the escape velocity is just about the same for molecules in the atmospheres of each planet. We therefore cannot attribute differences in atmospheric composition in either planet to purely thermal effects.

  There is almost no oxygen in the Martian atmosphere because it is bound up in the Martian soil in the form of Fe2O3—rust. Oxygen is so highly reactive that chemical sequestration will bind it unless there is a continual source of it from somewhere else. In the case of Earth, the source of oxygen is the respiration cycle, that is, it is due to life on Earth. Similarly, there is almost no water on Mars because UV light dissociated the water vapor into hydrogen and oxygen; the oxygen was sequestered in the soil, while the hydrogen escaped into space. This shows how complicated atmospheres can be. That Mars has a much more rarified atmosphere than Earth doesn’t seem to owe principally to its mass and size but to a number of complicated factors.

  14.6.2 Impacts

  The total mass of Earth’s atmosphere is roughly 4×1018 kg. If we wanted to get 1% of the total mass of the atmosphere to escape velocity we would need to supply it with an energy of about 2.5×1024 J. Comets or asteroids typically hit Earth at speeds of about 30,000 m/s, that is, at about the orbital speed of Earth. A collision of this energy would require an impact of an object with mass about 5×1015 kg, or, assuming an average density of 5,000 kg/m3, a volume of about 1012 m—the equivalent volume of a cube 10 km on a side.

  This is a very large impactor, about the size of the comet that wiped out the dinosaurs. Such impacts happen to Earth only about once every hundred million years or so. Because of this, we are pretty safe in ignoring this atmospheric loss mechanism for Earth, at least under present Solar System conditions. Comparing Mars to Earth in this manner is interesting. Mars is less massive than Earth, meaning it has a lower escape velocity, and it is closer to the asteroid belt, meaning it will sustain more frequent impacts. Both factors favor atmospheric loss from impacts for Mars over Earth.

  14.6.3 What Is the Range of Sizes for a Habitable Planet?

  From all of this it seems that are no easy criteria with which to establish a lower bound on the size of a planet capable of supporting life. If Mars had an ozone layer, would it have kept its water vapor, leading to higher planetary temperatures from greenhouse warming, or would it have lost its atmosphere faster because it would have developed a thermosphere similar to Earth’s? If we put Mars in Earth’s orbit, would it have kept an atmosphere longer because it didn’t suffer from as many impacts? As a guess, I would say that Mars is close to the lower bound on the mass or radius for a habitable world, as it seems that conditions there are right on the cusp of allowing life. This speculation must be taken with a large grain of salt: because of the interrelation of all of these variables it is hard to give definitive answers.

  How about the upper limit on a habitable planet’s mass? This is similarly hard to estimate. One upper bound is that if a planet retains lighter gases in its atmosphere, it will likely turn into a gas giant planet. However, this depends both on the planetary mass and on its position in the Solar System, as all the gas giant planets formed beyond the “frost line,” outside the orbit of Mars. Perhaps more to the point, all the gas giant planets are thought to have solid cores of about ten times the mass of the Earth; maybe this core mass represents an upper limit. Or maybe not.

  14.7 THE ANNA KARENINA PRINCIPLE AND HABITABLE PLANETS

  All happy families are fundamentally the same; each unhappy family is unhappy in its own way.

  —LEO TOLSTOY, ANNA KARENINA

  Thus reads the famous opening line of Tolstoy’s Anna Karenina novel. Jared Diamond in Guns, Germs, and Steel introduced what he referred to as the “Anna Karenina” principle when reflecting on why, out of all possible animal species on the planet, only a handful had been domesticated by humans. He found that all animals domesticated for food had a number of features in common: they we
re all herbivores, matured quickly, bred in captivity, and had a few other similarities.

  To quote Diamond,

  To be domesticated, a candidate species must possess many different characteristics. Lack of any single required characteristic dooms efforts at domestication, just as it dooms efforts at building a happy marriage.

  As he put it, “For most important things … success actually requires avoiding many separate causes of failure” [65, p. 157].

  Above I introduced Adler’s mantra: “All stars are fundamentally the same; all planets are different from each other.” For planets, I will rephrase slightly:

  Each lifeless planet is different from the others in its own way; all planets with Earth-like life on them will be fundamentally the same.

  This is an extension of the Anna Karenina principle as applied to habitable worlds: all Earth-like planets have a number of similar characteristics, the most obvious being that they fall within a “zone of life,” not too far from or too near their star. The realization of this point has solved a conundrum that has faced scientists for a very long time: if the Earth is an average planet circling an average star in an average galaxy, with nothing special about it, then why haven’t we found life elsewhere in the cosmos yet? Why isn’t the universe teeming with life? Why haven’t the aliens made contact?

  Since the 1960s, two ideas have gradually developed due to advances in planetary science:

  1. The conditions on the different planets are far more diverse than was realized before. Planetary formation seems very chaotic, and planetary history (among other factors) plays a larger role in determining the geological and climatological features of the terrestrial planets than anyone realized.

  2. The conditions required for Earth-like life on a planet are far more restrictive than was thought in the 1960s.

  These two realizations severely reduce the number of possible planets with Earth-like life. Although Earth is in some sense no more special than any other planet, it is special in other ways as a cosmic lottery winner: it got everything right for life to appear on it. The point is that while it is improbable for any one particular person to win the lottery, someone almost always wins it.

  Probability estimates of the number of worlds with life on them, in the fashion of the Drake equation, are meaningless, as we simply don’t have enough data. The criteria I have given so far are pretty solid for any planets with Earth-like life, but in the next section I’ll list a number of criteria for which there is less solid evidence.

  14.8 IMPONDERABLES

  One complication concerning Mars has been discovered by numerical simulations of its rotation. One of the characteristics of Earth’s climate is its long-term stability. This results in part from the fact that the rotational axis of the Earth more or less points in the same direction for long periods of time and doesn’t change dramatically.2 However, because of Mars’s elongated orbit and its lack of a large moon to stabilize it, the orientation of the rotation axis of Mars can change dramatically and chaotically over the course of millions of years. It is believed that this has dramatically changed the characteristics of seasonal change on Mars [135][144]. It isn’t clear whether a stable rotational axis is needed for the evolution of life on a planet, but a stable climate certainly helps, in which case having a large moon might be a requirement for planetary life.

  Another issue is that Earth is currently the only known planet that experiences plate tectonics, a result of both its size and its composition (radioactive decay in the Earth’s core keeps the mantle plastic). Some scientists have speculated that plate movement contributes to evolution because it allows the broad dissemination of plant and animal species, making it harder for individual species to be wiped out by a local catastrophe. Who knows? I am not aware of any science fiction stories that have incorporated the relationship between plate tectonics and evolution thematically; it would be interesting to see if it could be done in any reasonable way.

  Other issues: Most exoplanets are found around stars with high metallicities, that is, around stars that contain more metals than average. (To an astronomer, a metal is any element that is not hydrogen or helium.) It may be that stars with higher metallicity simply contain more of the stuff that planets form from, although the exact details are not entirely clear [100]. More than half of all stars found so far with exoplanets have even higher metallicities than the Sun. What is interesting, however, is that high-metallicity stars are relatively young (population I) stars because the metals were mostly created by higher-order fusion processes in the hearts of the stars; the oldest stars formed at a time when these elements simply didn’t exist [130, pp. 495–497]. Because the higher elements are distributed by supernova explosions, which are more common toward the center of the galaxy than in the spiral arms, there may be a higher occurrence of stars with planets toward the centers of galaxies. However, the higher incidence of supernovas and radiation from supernovas may sterilize life emerging on these planets at distances too close to the galactic center. Thus there may also be a “galactic habitable zone” where life may form, an annulus neither too close to nor too far from the center of the galaxy [100].

  Giant planets on highly eccentric orbits may perturb Earth-like planets out of the life zone because of their gravitational interactions with them, but large planets in the outer system may serve to screen planets from asteroid impacts like the one that eradicated the dinosaurs [247]. Hot Jupiters are thought to form in the outer system but migrate by various processes to close orbits around the star; the migration may disrupt the formation of planets in the zone. So the presence of hot Jupiters (found in over 10% of all exoplanet systems to date) may preclude life from developing, but cold Jupiters on nearly circular orbits in the outer system may be needed for life.

  There is an almost infinite list of considerations which one can go into, especially when we add the question of intelligent life to the mix. I consider this subject in a later chapter. For anyone interested in exploring these ideas further, Brownlee’s book Rare Earth is a good place to start, but there has been a lot of research in this area since the book was published in 2000 [247].

  We have now listed the criteria for a planet to support Earth-like life. In the next chapter I take up the issue of actually finding it out there.

  NOTES

  1. In a prescient piece of writing, Edgar Rice Burroughs in A Princess of Mars wrote that the Martian civilization built an atmosphere plant to combat atmospheric loss from their world. The means by which the atmosphere was replenished (the “ninth ray”) are not particularly scientific, however [43].

  2. Dante Alighieri in 1300 CE understood this, though in a somewhat different way than we do now.

  CHAPTER FIFTEEN

  THE SCIENTIFIC SEARCH FOR SPOCK

  Heaven and earth are large, yet in the whole empty space they are but as a small grain of rice…. It is as if the whole empty space were a tree, and heaven and earth were one of its fruits. Empty space is like a kingdom, and heaven and earth no more than a single individual person in that kingdom. Upon one tree there are many fruits, and in one kingdom many people. How unreasonable it would be to suppose that besides the heaven and earth which we can see there are no other heavens and no other earths.

  —TENG MU, PO-YA CH’IN

  15.1 EXOPLANETS AND EXOPLANTS

  The idea of life in other stellar systems is an old one and a source of speculation at least since the times of the ancient Greeks. However, serious scientific attempts to detect life outside the Solar System dates back only to the 1960s, and at that time were a marginal effort. Although a number of well-known scientists including Carl Sagan and Philip Morrison participated in the search, it was never well funded. It always remained a research sideline even for those people most passionately interested in it.

  This changed significantly in the 1990s, when it moved from the sidelines to a central place in modern astronomy. It is now funded at a rate hundreds of times what it was before then. The reasons have much m
ore to do with developing technology than with the amount of interest in the subject.

  The search for extraterrestrial life has its roots in the 1800s, when physicists and chemists began to realize that life was a physical and chemical process, not something separate from these subjects. As I mentioned in the last chapter, the American astronomer Percival Lowell claimed to have seen canals on Mars through a telescope and speculated they could have been the product of an advanced technological civilization [155, chapter 4]. In The Destinies of the Stars, published about 20 years after Lowell first made his claims, Svante Arrhenius and Joens Elias Fries discussed the limitations to this hypothesis; they pointed out that spectrometers had detected no water vapor in the Martian atmosphere, making it improbable that the canals existed [25, p. 183]. This didn’t stop three generations of science fiction writers, from H. G. Wells and Edgar Rice Burroughs to Robert Heinlein and Ray Bradbury, from using the idea of the ancient dying Martian civilizations in their works [40] [43] [108] [118] [248]. What is of note here is that Arrhenius used state-of-the-art technology, photographic spectrograms of Mars, to refute Lowell’s argument. Science is often driven by available technology, and nowhere is this more true than in the search for life in the universe. Of course, new technology and new discoveries often raise as many questions as they answer: Mariner 9 pretty much ended any serious ideas of an advanced Martian civilization, but it did provide evidence (since confirmed by later probes such as the Mars Rovers) that climate conditions in the past were more favorable for life, Mars having gone through warmer periods when water flowed openly on its surface [163] [193] [208]. This suggested the possibility that instead of life in the form of an advanced technological civilization older than humanity, it may have existed on Mars in a more primitive form, at the bacterial level or as simple plant life.