For main-sequence stars only, the following relation holds:
That is, luminosity increases rapidly with mass, essentially increasing as mass cubed. The power law is an approximation to the true behavior; for very low-mass stars the exponent is somewhat lower, for very high-mass stars the exponent is higher. The fact that luminosity increases so rapidly is why large stars have short lifetimes: a star’s main-sequence lifetime is determined by the ratio of the amount of fuel it has to burn (its mass) and the rate at which it is burning it (its luminosity):
This is an approximate formula. In particular, it doesn’t work well for very low-mass stars. One other thing can be worked out: in these units, along the main sequence,
This isn’t a coincidence. It stems from the fact that the proton-proton cycle operating at the core of the star doesn’t “turn on” until the core reaches a temperature of about 107 K [170].
To convert everything to metric units we need only remember a few numbers. The sun radiates energy at a rate of 3.84×1026 W, so
Lmetric = 3.84 × 1026 W × L.
Similarly,
Rmetric = 6.95 × 108 m × R,
and
Mmetric = 1.99 × 1030 kg × M.
With these formulas we can do a lot. In particular, life emerged on Earth only about 700 million years after the Earth formed. As a wild guess, if we assume that it takes life everywhere in the universe about that long to evolve on a given Earth-like planet, we need to pick a star whose main-sequence lifetime will be longer than about 700 million years, meaning that the O- and B-class stars won’t work, and possibly not A-class stars either. This doesn’t eliminate a lot of them: there are many more smaller, cooler, dimmer, longer-lived stars than there are bigger, brighter, hotter stars. But it gives a point to start from. The smallest class M stars also may not work too well because planets in the zone of life will have to orbit so closely that tidal effects will lock one side of the planet into permanently facing the star, in the same way that the Moon presents only one face to the Earth, making one side much hotter than the other.
Other fun formulas: if the planet is located d astronomical units away from the star, the angular size of the star as seen from the planet is
If the planet has a moon, its angular size as seen from the planet is
where Rm is the radius of the planet’s moon relative to the radius of Earth’s Moon (= 1,737 km) and am is the average distance of the moon from the planet, again relative to the distance of Earth’s Moon from the center of Earth (384,000 km). Kepler’s third law tells us that the length of a year on the planet (i.e., the rotational period around the star) is
where Y is measured in years.
14.3 PLANETARY DISTANCE FROM ITS STAR
Now that we have the star, where do we put the planet? Carbon-based life requires oxygen in the atmosphere and liquid water on the planet’s surface. If the planet is too close to its star, it gets too hot, and water will boil on its surface; too far, and water will freeze. In the Solar System there are only three planetary candidates for Earth-like life based on this criterion, Earth, Venus, and Mars.
A planet both absorbs energy from its star and radiates energy on its own. This fact can be used to figure out the temperature of the planet. There are a few ideas that go into this:
1. Radiation from a star spreads out in all directions; the total amount of light the star radiates away is spread out over a sphere centered on the star. If a planet is at distance r away from its star, the total amount of power emitted by the star is spread over a sphere whose surface area is 4πr2.
2. Some of the light from its star will be absorbed by the planet and some will be reflected away, because of the planetary makeup and the atmosphere of the planet. The total amount of light absorbed will be proportional to the area of the planet.
3. The key point is that the total power absorbed by the planet must equal the total power radiated away (on average); if this weren’t true, the planet wouldn’t maintain an even temperature. Planetary temperature stays relatively constant because if it radiated away less light than it absorbed, its temperature would increase until it radiated away exactly as much as it absorbed. If it radiated away more light, the temperature would decrease until this happened.
We also have to define a concept called the average albedo of the planet. The albedo is the fraction of sunlight that is reflected without being absorbed. Earth’s albedo is about 0.3, meaning that about 30% of the light from the Sun isn’t absorbed by Earth’s surface. This is an average between the oceans (which are relatively absorptive) and the ground and cloud cover (which are relatively reflective), and also changes from point to point and from time to time as well. Our definition is one that is averaged over the surface of the Earth and over time—say, over several years. I’m also assuming that the orbit is essentially circular. With these definitions, for a planet without an atmosphere,
What this means is that there is a zone of life around the star. As a very rough cut, the zone is the region where the mean temperature ranges from 273 K (on the outer edge), that is, where water freezes, to 373 K, where water boils, on the inner edge. The zone doesn’t have hard-and-fast edges because planets have different albedos, and atmosphere plays a role as well. Three points are relevant here:
1. The luminosity of most stars isn’t very near that of our Sun. The range of luminosities goes from about 10−3 to 106—a range over a factor of one billion. This means that the distance of the “zones of life” for different stars will have different values;
2. I call the temperature “T0” because this is the temperature of the planet if it had no atmosphere. Atmosphere plays a major role in determining planetary temperatures; we’ll introduce a simple model for the effect of the atmosphere later in the chapter.
3. This is simply the mean temperature for the planet. The actual temperature will vary quite a bit over time and from point to point on the planet’s surface.
We can rework this to find the inner and outer edges of the zone of life for a given planetary albedo and stellar luminosity. Let di and do be the inner and outer edges of the zone for planets (ignoring the effects of atmosphere):
Again, these are very rough cuts. Because we didn’t include the effects of planetary atmosphere we find that Earth is actually outside the zone, too far from the Sun, in this naive model. However, it serves as a starting point.
The only candidate planets in our solar system potentially within the zone are Venus, Earth, and Mars. In the late 1800s the astronomers Giovanni Schiaparelli and Percival Lowell wondered whether they had detected signs of life on Mars after seeing an intricate network of canals on its surface; Lowell felt the canals could have been the products of an advanced civilization. This work was the inspiration for innumerable works of science fiction, from H. G. Wells’s War of the Worlds to Robert Heinlein’s Red Planet and Stranger in a Strange Land [108][115][248]. However, almost from the beginning critics pointed out that Mars was probably too cold and too arid for life of this kind. The Nobel laureate chemist Svante Arrhenius was one of the first to go over the scientific evidence and conclude that Mars was an unlikely abode for life. He was right. The “canals” were merely the product of low telescope resolution and eye strain. But what Arrhenius took with one hand he gave back with the other; in his book, The Destinies of the Stars (written with Jones Elias Fries), he predicted that the surface of Venus was wet and misty, with conditions similar to Earth’s during its Jurassic period [25]. This prediction led to many a science fiction novel set in the jungles of Venus where dinosaur-like monsters hunted humans through the mud. Heinlein’s novel Podkayne of Mars is perhaps archetypal of these, although S. M. Stirling has written a new alternate-universe series based on this idea. In these novels, Mars and Venus were seeded with life by an unknown alien race. The feel of them is similar to Burroughs’ Barsoom novels [118][229].
But these ideas disappeared from serious science in the 1960s and 1970s with the advent of unmanned probes that flew by and
landed on Mars and Venus. They found Mars a desert with average temperatures around that of Antarctica, which was not very surprising, and Venus a hell whose surface temperatures reached 750 K—hot enough to melt lead! The difference in the temperature of Venus from our ideal model has everything to do with its atmosphere.
14.4 THE GREENHOUSE EFFECT
Certain gases in the atmospheres of planets tend to trap infrared radiation, which is heat radiated away by planetary surfaces; these gases include carbon dioxide (CO2), which makes up 380 parts per million in Earth’s atmosphere and a whopping 98% of the atmosphere of Venus, and methane, which is found in trace amounts in our atmosphere. The most important greenhouse gas, however, is water vapor, which accounts for about 90% of the total for Earth.
The idea behind the green house effect is pretty straightforward: light from the sun is mostly visible light, which the atmosphere is transparent to; about 70% of it reaches the Earth’s surface. However, the Earth radiates away light in the infrared region of the spectrum, so most of it is trapped near the surface. We can see this by considering equation (14.2): because the average temperature of the Earth is about 290, most of the light reradiated from the Earth is at wavelengths near 10 µm, in the infrared region of the spectrum. This is strongly absorbed by the atmosphere, and much of it is reradiated back to the ground. In “building planets,” as Anderson puts it, it is vital to include somehow the effects of the atmosphere in our calculations. Computer models to calculate the temperature typically divide up the atmosphere into vertical layers with different amounts of infrared absorbance in each layer; they also grid the world into cells based on the terrain the cells overlie. However, one can get some good insight into what is going on using a simple model.
Greenhouse gases act as a blanket, so that heat from the planet takes longer to escape into space, leading to an increase in the average temperature not predicted by equation (14.7). Mars has a very thin atmosphere, much thinner than the atmosphere at the top of Earth’s highest mountains, so this greenhouse effect increases its average temperature by only a small amount. Earth has a moderately thick atmosphere, so the greenhouse effect raises its temperature by about 30 K. Venus has a thick atmosphere mostly composed of CO2; it has a runaway greenhouse effect, which raises its surface temperature by over 500 K.
Let’s assume that the atmosphere can be modeled as one layer, not divided up either horizontally or vertically. We can use a simple model developed in Daniel J. Jacob’s book Atmospheric Chemistry to calculate the effect of greenhouse warming on Earth. [129, pp. 128–131]. We assume that the planet has an effective albedo a in the visible region of the spectrum and that the atmosphere as a whole traps a fraction f of the light emitted by the surface in the infrared part of the spectrum. Using this model, the temperature of the surface is given by
As f goes up, T goes up: the more radiation that is trapped by the atmosphere, the warmer the planet will be. Unfortunately, this is as far as we can go using a simple model; predicting f from the atmospheric constituents is a very difficult task. For Earth’s mean temperature of 288 K we need a value f = 0.77 with this model. One can use that as a guide when writing science fiction stories set on other worlds.
The temperature of the atmospheric layer can also be found in this model:
The model is OK for planets with relatively thin atmospheres, like Earth, but breaks down completely for planets with very absorptive atmospheres, like Venus, where we have to use a multilayer model to come close to the truth. The model also ignores convection and the horizontal transport of energy, both important effects when calculating the real temperature.
For the value of f given above, the mean temperature of Earth will remain between freezing and boiling anywhere within 0.6 to 1.1 AU from the sun. This is not to say, however, that the real zone of life is this wide. While planetary temperature depends on atmospheric composition, atmospheric composition also depends on planetary temperature. Venus in many respects is a sister planet to Earth, having about 85% of Earth’s mass and being about 90% of its size, but the surface conditions rival hell’s. Most astronomers think that Venus became the way it is because of a positive feedback effect. Venus started out with a slightly higher insolation than Earth’s, which caused more CO2 to be liberated from the surface of the planet. This led to more warming, leading to more CO2 being liberated, and so on [134]. Earth, being about three-tenths of an astronomical unit farther from the Sun, escaped this fate.
Habitable zones also change over time because of changes in planetary atmospheres and solar irradiance. Over time, the Sun’s luminosity has increased gradually. Shortly after the formation of the Solar System its luminosity was only 70% of what it is now, and it will be about 10% higher than now in about a billion years [47, fig. 11.1]. From this, Kasting, Whitmire, and Reynolds calculated that over the history of the Solar System from 4.5 billion years ago until today, the habitable zone for Earth-like life extended only from 0.95 to 1.15 AU [136].
It is very unlikely that there was ever any Earth-like life on Venus, given conditions there, but Mars is another story. Mars had a thicker atmosphere once, although it didn’t keep it. There is very good geological evidence that liquid water once flowed on the surface of Mars, presumably billions of years ago when its atmosphere was thicker and hence the surface temperature was higher. Where the water is now is anyone’s guess; it may be locked in permafrost beneath the surface of the planet. However, conditions once may have permitted Earth-like life on Mars’s surface, probably at the bacterial level.
Recent science fiction novels have used this idea, notably Greg Bear’s novel Moving Mars [33]. In the novel, a terraforming project to make Mars habitable for humans leads to a rebirth of the older life forms on Mars, which became dormant when conditions on the planet became too inhospitable.
The Kepler mission has just begun finding evidence of planets in habitable zones. As of December 2011, the Kepler mission had identified 54 planet candidates within habitable zones, with one candidate, Kepler 22-b, potentially Earth-like (about 2.4 times Earth’s radius) [1]. The BBC web report is in the form of a press release; one can find a preprint of a paper concerning this planet submitted for publication at the Cornell preprint server [39]. The writers make it very clear that they do not know that the planet is an Earth-like world. The mass could be up to 124 times the mass of the Earth, and it isn’t clear that it is a terrestrial/rocky planet at all. If it is, however, they estimated a value of T0 (as defined above) of 262 K, assuming an albedo of 0.29 (the same as Earth’s), and a temperature of about 295 K if it has an atmosphere with the same properties as Earth’s, which they also state is very unlikely.
14.5 ORBITAL ECCENTRICITY
The habitable zone is a spherical shell surrounding a given star. Its borders are fuzzy because it depends on planetary albedo and atmosphere. Unfortunately, planetary orbits aren’t circular; they are characterized by orbital eccentricities. Even if the average distance from the star, d, is within the zone, a highly eccentric orbit will take a planet out of the zone for large periods of time as it orbits its star.
Let’s say that a planet orbits at an average distance d that is right in the middle of the habitable zone. This “average distance” is really the semi major axis of the planetary ellipse. If the orbit has eccentricity e, then the perihelion and aphelion distances are
If the width of the zone is Δz, then the perihelion and aphelion distances should lie within the zone, meaning that
or,
Earth’s orbital eccentricity is only 0.0167, whereas Δz/d for Earth is approximately 0.2, meaning that the eccentricity is well within the bounds set by this limit: Earth remains well inside the habitable zone for the entirety of its orbit. Other planets have larger eccentricities: Mars’s eccentricity, the largest in the Solar System, is 0.0934. The values for all the planets in the Solar System are relatively small, but eccentricities for exoplanets can be very large, raising the question of whether our Solar System is typical or atypical in
this regard. Ursula K. Le Guin set her novel The Left Hand of Darkness on the ice world of Gethen; owing to a combination of relatively large distance and high orbital eccentricity, the world was frozen for most of its long rotational period [145]. It’s not entirely clear that life could evolve on such a world, although in the story humans have been settled on Gethen from elsewhere.
Exoplanet data show that most exoplanets have average eccentricities higher than those of the planets in the Solar System; it is not clear whether there is a reason for this or whether it is accidental that our system’s planets have low eccentricities. I estimate about 10% of all exoplanets found to date have eccentricities below 0.1. This is ignoring data for hot Jupiters, as tidal friction lowers the eccentricity of their orbits.
14.6 PLANETARY SIZE AND ATMOSPHERIC RETENTION
One other requisite for life (as far as we know) is an atmosphere. It’s interesting to compare Earth, Venus, and Mars in this respect. Mars has a very thin atmosphere that is mostly CO2 (94%, with trace amounts of other gases); Earth has a moderate atmosphere that is 74% N2, 24 % O2, and 2% trace elements; and Venus has a horribly thick atmosphere that is almost entirely CO2.
Mars’s atmosphere was thicker in the past.1 This undoubtedly warmed Mars through to an enhanced greenhouse effect, but the planet lost its atmosphere over several billion years. There are lots of mechanisms by which planets can lose their atmospheres over the course of time. Following are just a few:
Wizards, Aliens, and Starships: Physics and Math in Fantasy and Science Fiction Page 23
