Wizards, Aliens, and Starships: Physics and Math in Fantasy and Science Fiction
Page 30
Orbital considerations leave only two planets worth considering, Mars and Venus. They are the only two planets close enough to the zone of life in the Solar System. They also have solid surfaces, unlike the outer gas giants. Science fiction writers have also considered the Jovian moons as targets for terraforming. In particular, Robert Heinlein wrote of a large-scale project to make Ganymede, Jupiter’s largest moon, habitable in his excellent juvenile novel, Farmer in the Sky [111]. Although they are well outside the zone of life, Jupiter generates heat because it is slowly collapsing owing to its gravitational self-attraction [130]. As a result the moons Europa and Callisto have liquid water beneath their ice crusts, making them good candidates for life elsewhere in the Solar System. The biggest issue is their lack of an atmosphere because of their low mass (roughly the same as that of Earth’s Moon), and, ironically, their relative warmth, which tends to make planets lose their atmosphere more quickly. Titan, Saturn’s largest moon, has an atmosphere because Saturn is farther from the Sun and hence much colder. Of course, science fiction speculation being what it is, Arthur Clarke in 2010: Odyssey Two imagined the terraforming of Europa by aliens following the forced gravitational collapse of Jupiter, which transformed it into a small star [57].
In this chapter we will consider the terraforming of Mars specifically. This seems to be the easiest planet to consider for several reasons:
• It is essentially within the habitable zone.
• While it is very cold most of the time, its eccentric orbit ensures that at least for part of the Martian year, it warms up enough for liquid water to flow on its surface.
• It is close enough to Earth that we can imagine creating a more or less permanent settlement there.
Why not Venus? The crushing pressure and the extreme heat on the surface of Venus ensure that no one could live there or work there. The terraforming of Venus would be much more difficult.
18.2 CHARACTERISTICS OF MARS
Mars orbits the sun at an average distance of 1.52 AU. Its orbital period can be found from Kepler’s third law:
It spins on its axis with a period of 24 hours, 37 minutes, coincidentally much like Earth’s orbital period, and the inclination of its rotation axis is about 24 degrees, again much like Earth’s.
The atmosphere is about 95% CO2, with trace amounts of argon and other gases, but is very thin. Atmospheric pressure at the surface is less than 1% Earth’s sea-level pressure. There is essentially no free oxygen (O2). As we saw in an earlier chapter, life on Earth is responsible for the large amount of O2 in Earth’s atmosphere: if the respiration cycle didn’t replenish it continuously, it would react to form oxygen compounds. With no life (or very little, at least) on Mars, the life-giving components of the atmosphere are missing. As one writer described the Martian atmosphere, “it’s almost too thin to be considered poisonous.” The big issue with terraforming Mars is putting enough O2 into its atmosphere for it to be breathable.
Table 18.1
Relative Properties of Earth and Mars
Property
Earth
Mars
R
M
T
Ve (m/s)
vrms for H2
He
O2
1
1
1,000
11,000
3,500
2,480
878
0.533
0.107
140
4,800
1,860
930
328
Table 18.2 shows some characteristics of Mars that will be useful in our discussion.
The amount of water on Mars is unknown, but certainly less than on Earth. As discussed in chapter 2, at one point Mars had enough atmosphere to warm it up enough for water to flow freely on its surface [130].
18.3 TEMPERATURE AND THE MARTIAN ATMOSPHERE
The two things needed to turn Mars into a habitable world are (1) increasing the average temperature to above the freezing point of water and (2) increasing and maintaining the O2 content of the atmosphere to breathable levels. The high reactivity of oxygen means that to keep the O2 content at sufficient levels, the respiration cycle on a planetary scale must be created. The most commonly suggested means of doing so is by releasing genetically engineered extremophilic bacteria or plants into the Martian environment. “Extremophilic” refers to life that can survive in Earth’s most challenging enviroments: in extreme heat or cold, under extremely high or low pressures, or in environments toxic to most other living creatures.
Table 18.2
Characteristics of Mars
Property
Value
Orbital distance (AU)
Orbital period (years)
Typical surface temperature (°C)
Atmospheric composition:
CO2 (%)
O2 (%)
N2 (%)
H2O (%)
Solar constant at Mar’s orbit (W/m2)
1.52
1.88
20 (day, equator) to −150 (night, poles)
95
0.13
2.7
0.002
560
Source: Hester et al. [130, p. 218].
Increasing the temperature can be done in only one feasible way, by increasing the concentration of greenhouse gases in Mars’s atmosphere. Again, this was once true of Mars, and maybe could become true again. There is currently only one important greenhouse gas in the Martian atmosphere, CO2; any scheme to increase the average Martian temperature must begin by increasing its concentration. How much CO2 do we need to generate? This is a tricky question to answer, but we can get an approximation by considering that the greenhouse warming of Mars by the CO2 content in its current atmosphere is about 5 K [130]. If we assume that (1) the infrared absorption of the Martian atmosphere is proportional to the partial pressure of CO2 in it; (2) it is independent of the other atmospheric components; and (3) there are no feedback effects once the concentration begins to rise, then we can calculate what we need.
The following is a back-of-the-envelope analysis. We begin with the observation that the temperature is not directly proportional to the atmospheric absorption but instead to its fourth root (see chapter 14). This means that a 1% increase in atmospheric absorption leads to a 0.25% increase in temperature. To make life simple, I’ll assume that we need to increase the mean surface temperature by 80 K (i.e., increase greenhouse warming by a factor of 20), meaning that the partial pressure of CO2 in the Martian atmosphere needs to increase by a factor four times this, or about 80. Since the partial pressure is currently about 6 mbar, it needs to increase to about 480 mbar.
Unfortunately, the situation is not nearly that simple. The absorption due to the infrared bands of CO2 depends on the total atmospheric pressure because of an effect called pressure broadening, and on the column height of the atmosphere, which is higher on Mars than on Earth. Column height also decreases as the surface temperature increases, which means that as we change the Martian temperature, the total absorption will change in a complicated way. In addition, the amount of absorption doesn’t depend simply on the partial pressure but exponentially on it. The simple approximation used above (percent temperature change is one-fourth the percent absorption change) is no longer valid when dealing with a very large temperature change. However, our simple analysis isn’t too far off: Robert M. Zubrin and Christopher P. McKay have a sophisticated model indicating that a partial pressure of 800 mbar will lead to sufficient heating that liquid water can exist for long periods of the Martian year. Their model includes trapping of CO2 by the Martian regolith, the rocky surface layer of the soil [256].
Most plans to terraform Mars begin by using methods to melt the Martian polar caps, which are almost entirely composed of frozen CO2. Here, most of the schemes involve lowering the albedo of the ice caps by placing a layer of some dark material over them and letting the Sun do the rest: for example, a layer of soot has the ad
vantage that the carbon needed to create the soot is already there. Carl Sagan was the first person to suggest doing this, in a paper published in 1973. He estimated that one would need to deposit about 108 metric tons of “a low albedo material such as carbon black” to create runaway melting of the polar caps [208]. Later, Robert Mole speculated that four nuclear bombs exploded over the Martian ice caps would generate enough sooty material to do the trick [167]. It is unknown whether there is enough CO2 in the ice caps to increase the temperature sufficiently, although again it is clear than in the past there was certainly enough atmosphere for this. Unfortunately, this much CO2 in the atmosphere is poisonous; indeed, partial pressures above 10 mbar are poisonous. Therefore, most schemes to terraform Mars hinge on the use of other greenhouse gases such as methane or water vapor. The trickiest part is that because Mars is farther from the Sun, you need a correspondingly larger amount of such gases in the atmosphere to warm it, which is a daunting task. In particular, the amount of water on Mars is almost certainly not enough to provide enough water vapor in the atmosphere for the needed change in temperature.
In the next section I consider the basic physics, chemistry, and estimated costs of producing a breathable atmosphere on Mars. However, before leaving the basic issues involved, let me direct the reader to a few more references that maybe useful if one is writing a story on this subject. Terraforming a planet is an unbelievably complex subject; Martyn Fogg’s book is an indispensable resource for those interested in it [86]. As is appropriate, the largest section of the book is devoted to the case of Mars, which is the most plausible planet of choice in the Solar System. Fogg goes into much more detail than I can here and considers feedback effects, geography and geology, and other tricky issues. The book is unfortunately out of print and hard to find. It should also be considered out of date, given the massive amount we have learned about Mars in the past 17 years. Nevertheles, the book is a clear starting point for those interested in the subject.
A more sophisticated approach to warming of planets by greenhouse gases can be found in a paper by Akira Tomizuka, “Estimation of the Power of Greenhouse Gases on the Basis of Absorption Spectra” [238]. The subject of the paper is the greenhouse warming of the Earth, but it could be used to model the temperature of Mars as well. The study used a sophisticated mathematical/computer model of radiation transport in the atmosphere. It also used high-resolution absorption spectra of greenhouse gases, which most readers probably don’t have access to. Zubrin and McKay give a simpler model for the surface temperature of Mars as a function of partial pressure and the solar constant [256, Eq. 1]. However, it should be treated as an approximation of limited validity only.
18.4 ATMOSPHERIC OXYGEN
The second part of the task of terraforming Mars is to generate oxygen. Most of the oxygen on Mars is bound up in the soil, in the form of rust (Fe2O3) or similar compounds, giving the characteristic red color of the planet. Generating oxygen to breathe means liberating the oxygen from the soil or other places, which will be a time-consuming task. The issue is that of energy: to free the oxygen from the soil, we need to run the following reaction:
For every mole of O2 produced we need to expend a minimum of 494 kJ of energy. Another means of doing this would be through photosynthesis: if we could somehow induce the growth of some sort of vegetation on Mars it would liberate O2 from CO2 through the reaction
This was discussed in detail in chapter 4, where we applied this to growing food and producing the atmosphere for a space station. Photosynthetic energy would be free from the Sun (although it would also require water for the cycle, something of which Mars is in short supply), while the energy to liberate oxygen from the first process would be supplied in some sort of industrial process. One thing is clear, however: some sort of stable respiration cycle would have to be created at some point, or else the newly generated oxygen would simply chemically bond with the Martian “soil” again.
Either process involves a lot of energy. For example, photosynthesis requires 4.77 × 105 J per mole of O2 produced, although plants are not able to utilize all of the energy which falls on them in the form of sunlight. In fact, typical efficiencies are only of order 5 × 10−2%, although some plants can reach efficiencies as high as 1%.
We can do a back-of-the-envelope calculation to estimate the amount of energy it will require to create a breathable atmosphere on Mars. Atmospheric pressure at sea level on Earth is 105 N/m2, of which 25% is oxygen. Assume that we need a mix of roughly one-tenth this pressure as a minimum for the Martian atmosphere, or 2.5 × 103 N/m2. This means that we need to generate approximately 600 kg of oxygen for every square meter of Martian soil. The molecular weight of oxygen is 32 g/mol, or ≈.03 kg/mol, so this represents 600 kg/.03 kg/mol = 2 × 104 moles of oxygen per square meter.
The radius of Mars is only one-third that of Earth, meaning that the surface area is about one-ninth that of Earth, or very roughly 1013 m2. For this area we need to generate approximately 2×1017 moles of oxygen. Through photosynthesis this represents a minimum energy cost of about 1023 J. The energy costs will be similar for any other chemical means used to generate oxygen. Because photosynthesis is very inefficient, we can assume that the true energy cost will likely be two to three orders of magnitude higher, or of order 1025–1026 J. Currently, the world uses approximately 1020 J per year, so this represents about 105–106 years worth of energy use by the entire world. This is clearly an impractical project given current technology and energy resources.
The timescale is likewise daunting. The solar constant at the Martian orbit is 562 W/m2; we have to divide this by four for latitude and day/night variation, so on average each square meter of Martian soil receives 140 W from the Sun. This represents a total insolation of 140 W/m2 × 1013 m2 ≈ 1015 W. Because it will take about 1025–1026 J to generate this much oxygen via photosynthesis, it will take a time of about 1010–1011 s, or 300 to 3,000 years to generate the needed oxygen. This assumes that we can use all of the sunlight which Mars receives, which is doubtful.
This shows the utility of using energy methods for quick feasibility calculations. It’s not a detailed study of the issue, but by making an estimate of the available energy for this system we can state that Mars will not be terraformed in less than about 300 years even if we started today, which we clearly can’t do. My feeling is that the timescale is likely to be much longer, possibly closer to 30,000 years or something like six times the length of recorded human history.
18.5 ECONOMICS
Another way to view the problem is by considering the cost to terraform Mars. As mentioned above, the minimum required energy is 1025 J, or 3 × 1018 kW-hr. (I’m assuming that even if we use sunlight via photosynthesis to power this, we will still have to pay for the energy somehow.) Current energy costs are about $0.1 per kW-hr; if we assume that in the energy-rich future they drop by three orders of magnitude, this represents a total cost of 10−4 $/kW-hr × 3×1018 kW-hr ≈ $3×1014. This is far higher than the current world GDP, and also rests on the assumption that energy costs are going to drop significantly. This is spread over a long time; we’ll be liberal, and estimate that the time to terraform is a mere 1,000 years, meaning an investment of $1011 per year. This amount ($100 billion per year) is less than the United States spends on its military but more than the entire basic science budget.
If we were to look to private industry to pay for this, we need to consider one fundamental issue: how is terraforming Mars going to turn a profit? Damned if I know.… There is nothing, and I mean nothing, that we could manufacture on Mars and ship to Earth that would not be infinitely cheaper to produce here. Anything sent here has to travel across 40 million miles of empty space. This is the weak point of most stories that involve terraforming other planets: the motivation for it is completely lacking. For example, the main motivation given to terraform Ganymede in Farmer in the Sky is to supply food to an overcrowded Earth. This is clearly nonsensical, and (in fact) in one part of the story, one character ev
en does a calculation to show how absurd it is to try to ship items in bulk between the two worlds. The issue at hand was shipping soil from Earth to Ganymede, but the calculations hold equally well for shipping food from Ganymede to Earth. Handling the surplus population is again a loser as an issue: you can’t ship people out fast enough. Maybe we can use the new world to hold our prisoners, as the British did in the eighteenth century, making Mars a new Botany Bay?
Why do I say that we’re going to have to pay for this energy somehow? A lot of writers have written about terraforming a planet as if it were as simple as dropping a load of genetically engineered plants onto the planet and standing back as they grow and generate a breathable atmosphere. It isn’t that easy. As one can see from the discussion here, the atmospheric constituents have to be carefully managed in order to heat the planet up and create the correct mix of a breathable atmosphere. We are barely beginning to understand the processes on Earth that maintain the temperature and atmospheric mix; granted, because Mars will be starting out from nearly a blank slate, these issues will (probably) be simpler than they are on Earth, but humans will have to manage every aspect of them from the ground up. This is a daunting task. Of course, some writers have taken the idea of world-building quite literally. This is the subject of the next chapter.