Cosmic Commodities: How much is a new planet worth?

Back in March of 2009, less than a week after the $600-million Kepler planet-hunting spacecraft rode a pillar of fire into orbit on a mission to make history, Greg Laughlin, an astrophysicist at the University of California-Santa Cruz, quietly posted a curious equation on his blog, oklo.org.

This equation's initial purpose, he wrote, was to put meaningful prices on the terrestrial exoplanets that Kepler was bound to discover. But he soon found it could be used equally well to place any planet—even our own—in a context that was simultaneously cosmic and commercial. In essence, you feed Laughlin's equation some key parameters–a planet's mass, its estimated temperature, and the age, type, and apparent brightness of its star–and out pops a number that should, Laughlin says, equate to cold, hard cash.

At the time, the exoplanet Gliese 581 c was thought to be the most Earth-like world known beyond our solar system. The equation said it was worth a measly $160. Mars fared better, priced at $14,000. And Earth? Our planet's value emerged as nearly 5 quadrillion dollars. That's about 100 times Earth's yearly GDP, and perhaps, Laughlin thought, not a bad ballpark estimate for the total economic value of our world and the technological civilization it supports.

At first blush, placing monetary values on mostly unexplored planets for which our sum total of knowledge consists of a few numbers in a database seems like the height of folly, or of hubris. But Laughlin argues that the point of his exercise is simply to measure the perceived potential that we collectively bequeath to a planet. Using his equation, you could begin to define something akin to their market value. And appreciating their dynamic range, the soaring highs and abyssal lows that define both the best and worst of all possible planets, just for a moment you might feel sensations vast, cool, and unsympathetic as you weigh and evaluate the worth of a world.

I spoke with Laughlin last week about his equation, how it works, and whether it deserves to be taken seriously.

Greg Laughlin: I've just always thought that the concept of an "Earth-like planet in the habitable zone" was pretty vaguely defined, and I wanted a metric that I could plug a planet into to see whether its value was high enough to warrant media hype. This is just a way for me to be able to quantify how excited I should be about any particular planet.

GL: Sure. "V" is the output, the total dollar value of the planet that you're evaluating; the "6 x 10⁶" is six million, which is normalizing for a baseline value. That value was obtained by taking the total cost of the Kepler mission—$600 million—and dividing by the number of Earth-like planets—about 100—that they expected to find. That means that society has valued a Kepler-observed Earth-like planet at 6 million dollars of real money. There's nothing arbitrary about this assignation.

After that, you have the age of the star, divided by half a billion years. This term is designed to favor stars that are older. If you're looking for life you don't want to look around stars that are just 100 million years old. You probably should be most interested in stars that are as old or older than the Sun, stars that offer more time for their planets to develop life.

This relates to the next term, the one in parentheses, that's the mass ratio of the planet to the star, raised to the ^1/3rd power. The more massive a star is, the shorter its existence, and the less opportunity it affords life to arise.

The exponent at the end there is giving this a weak dependence, and is the first of several exponents I added that multiply different chunks of the equation. If they're larger than 1, then that is making what's being multiplied more valuable relative to an "average" Earth-like planet found by Kepler. If the exponent is less than 1, then that's making whatever is multiplied less valuable. So the term we're discussing here is meant to favor lower-mass stars, although not by a huge amount.

Besides being better suited for harboring potentially Earth-like planets, lower-mass stars are also much easier to study, so that any promising planets they hold can be characterized. They give you a much better transit signal, and if you're using radial-velocity you're looking at an intrinsically larger signal, as the signal strength depends on the mass of the planet compared to the mass of the star.

GL: Each of those "exp – " terms represents 'e to the negative quantity²'.

Everything in this whole section is basically creating a Gaussian distribution, a bell-curve with the central peak being Earth's mass (M_⊕). This bestows planets that are nearer to Earth's mass with higher values, and devalues planets either less massive or more massive than Earth.

And that's admittedly a rather aggressive function. But I think it's reasonable. For habitability I think one would need to be very strongly biased toward Earth-mass planets, because we really don't know very much at all about planets that lie between Earth and Uranus in mass. None exist in our solar system. We're just barely getting our first pieces of concrete information about these worlds. And to talk about their ability to support habitable environments is premature to say the least.

GL: Right. The "exp – " here is doing the same thing, making a Gaussian that maximizes the value of the Earth, but with the amount of light and heat that the planet is getting from its star rather than the planet's mass.

We only have one example of a habitable planet, the Earth. So this favors planets that are similar in the amount of energy they receive from their star. If a planet is likely to get considerably less—like Mars—or considerably more—like Venus—the equation doesn't treat it well. The equation wants to see planets that get an Earth-like flux of energy from their star.

T_eff is a planet's calculated effective temperature, and 273 is, in kelvins, the melting point of ice at standard atmospheric pressure, and is also fairly close to Earth's average effective temperature. You can calculate an effective temperature basically by considering a spot on the equator of the planet that is painted black, with no overlying atmosphere. It's just a way of calibrating to the Earth. The division by 30 says that, if you're more than 30 degrees away from 273 you're getting a significant devaluation; it's scaling how fast you devalue as you move away from a value defined by the Earth.

GL: Yes. This term places a value on immediacy. Kepler launched in 2009, so that's the baseline. The closer to 2009 that Kepler discovers a planet, the more it's worth. The way to think about this is, if a planet like 51 Pegasi b or HD 209458 b were discovered today, it would barely merit a paper in a major journal, whereas 10 or 15 years ago those planets forged careers and reputations. There is enormous value associated with early discoveries, and that term reflects this. Though perhaps not even as much as it should.

You'll notice the immediacy value has a fall-off of 4 years, during which these planets devalue significantly. And even that might be optimistic—they might go down even faster than that. The first planets of a type are a huge deal, and then subsequent ones rapidly lose excitement among the public as you go further out in time. Discovering an Earth-mass planet in the habitable zone of a Sun-like star in the year 2060 is not going to be a big deal, in all likelihood.

GL: This term is basically saying that the brighter the star appears to you, the more valuable the planet is, though it's weakly dependent. The "V" values for actual astronomical objects range from V=-26.7, for the Noonday Sun as seen from Earth, down to V=+30, for the dimmest galaxies that Hubble Space Telescope can spot. The magnitude scale can be traced back to the ancient Greek system for ranking stars visible to the eye. It's a logarithmic scale, because the human eye has a logarithmic, nonlinear response to light. The difference between a bright value of 0 and the dimmest value of 5 on this scale is a factor of 100 in the actual flux, the energy, arriving on Earth.

This is important because we rely on photons to characterize these exoplanets, and it's just much easier to characterize them and extract their intrinsic value if they're nearby and giving us more photons. So it's basically a proximity term, and other than the $600 million baseline provided by Kepler, it's the most important part of the equation.

GL: Think of it this way: If we're sitting here on Earth, our sun is extraordinarily bright in the sky. The brightness of the Sun thus makes that term enormous if we run this equation for the Earth. If we evaluate this equation for the Earth, we get an answer of about 5 quadrillion dollars. And that's basically the value of all our infrastructure, accumulated through history.

This isn't a man-who-sold-the-Earth kind of statement, but rather it's pinning things down on the other end. Being there. How much is a habitable planet worth if you're actually there? Well, this says it's worth quite a bit.

If, for instance, there is a planet orbiting in the habitable zone of Alpha Centauri B, part of the closest star system in the sky other than our Sun, that planet's worth about $6 billion using this scale. But then if you voyage there, Alpha Centauri B appears brighter, and brighter, and brighter, until it is your Sun in your sky and you're on the planet's surface. So in going there you have this ability to intrinsically increase value. And that's an exciting thing because it ultimately provides a profit motive for perhaps going out and making a go of it with these planets.

This is saying that something that is several billion dollars on Earth, could be, if you go there, a quadrillion-dollar payoff.

It's also interesting to note that the $6 billion evaluation for a promising planet around Alpha Centauri B is very similar to the cost of a direct-imaging mission that could conceivably characterize such a planet. It's not hard to imagine that if you have a really alluring Earth-mass planet in an Earth-like orbit in Alpha Centauri, that even in the current climate there might be the political and public will to launch an ambitious mission to characterize it.

The public would be perhaps willing to spend that much, on the order of billions of dollars. The public would not be willing to spend trillions of dollars to build that space telescope for that task. And the public would certainly be willing to spend more than millions of dollars to do it. So that number is more or less in the ballpark of what is realistically conceivable for NASA, for us, to do.

GL: Well, the formula says Mars is worth nearly $14,000. Which is sobering, but at the same time realistic in the sense that you would last 5 seconds if you showed up there.

And you can apply it to previously discovered exoplanets, giving the benefit of the doubt to radial-velocity planets and their uncertain masses. When you do that, the most valuable and undisputed planet that has been announced as of right now, the end of January, remains Gliese 581 c, which is valued at only I think $160 or so. But the value of the much-discussed candidate planet Gliese 581 g, based on the properties listed in its discovery paper, yields a value of approximately $60,000.

What this is telling us is that despite all of the media excitement that has surrounded the detection of planets supposedly lying in the habitable zone, these aren't the ones to really get excited about yet. These aren't the planets we're looking for. It's going to get a lot better, a lot sooner than people realize. The planets that have been found to date don't really register on this scale as calibrated by Kepler's budget and its projected harvest of Earth-like planets.

This scale is not something that press officers would like in pushing marginal planets into press conferences and so forth. It treats the planets in a fair and objective way. Objects like Gliese 581 c, you can write press releases and say they're habitable, but this formula reveals that they're not very Earth-like. But they are out there. Alpha Centauri aside, some radial-velocity searches could easily yield a planet any day now that would hit $10 million, maybe even $100 million, using this formula.

GL: I will say that an important caveat for this formula is that it considers nothing related to habitability other than the planet's most basic orbital and mass properties, the orbital distance, and the parent star. It's not assigning more value to a planet if it has water or an oxygen-rich atmosphere or anything like that. We're still quite a ways away from being able to properly evaluate what exoplanets are actually like in those terms.

This is strictly a speculative value being assigned. This is rather like putting a value on something like Facebook, which has scant meaningful earnings. It's all in potential. In the dot-com bubble of the late 1990s, you saw things like pets.com and @home.com getting huge valuations. Seeing those valuations and then looking at another internet startup company that had no actual profits, you'd be able to value that company based on the similar valuations that you'd seen for other dot-coms.

That valuation is not a mark of intrinsic worth, but rather a valuation that is in line with prices that are being paid at the time. In that case it was for those stocks, and in this case it's for the outlay you need to make the observations to discover the planets. It's a very real thing that pets.com or @home.com were trading for many dollars per share, and people spent real money for those shares and got them in return. That's exactly the nature of my valuation here.

The question of intrinsic value, which hinges on the rate of occurrence of truly habitable planets, is not addressed at all. That's why I take it seriously and would argue that this is in no way controversial or even particularly forward looking. Given that society is willing to pay for speculative investigations looking for habitable planets, given the amount of money we've been spending, this is how you would value planets that will be coming in.

GL: Yeah. Venus is a great example. It does pretty well in the equation, and actually gets a value of about one and a half quadrillion dollars if you tweak its reflectivity a bit to factor in its bright clouds. This echoes what unfolded for Venus in the first half of the 20th century, when astronomers saw these bright clouds and thought they were water clouds, and that it was really humid and warm on the surface. It gave rise to this idea in the 1930s that Venus was a jungle planet. So you put this in the formula, and it has an explosive valuation. Then you'd show up and face the reality of lead melting on the surface beneath sulfuric-acid clouds, and everyone would want their money back!

If Venus is valued using its actual surface temperature, it's like 10^-12 of a single cent. @home.com was valued on the order of a billion dollars for its market cap, and the stock is now literally worth zero. Venus is unfortunately the @home.com of planets.

It's tragic, amazing, and extraordinary, to think that there was a small window, in 1956, 1957, when it wasn't clear yet that Venus was a strong microwave emitter and thus was inhospitably hot.

The scientific opinion was already going against Venus having a clement surface, but in those years you could still credibly imagine that Venus was a habitable environment, and you had authors like Ray Bradbury writing great stories about it. At the same time, the ability to travel to Venus was completely within our grasp in a way that, shockingly, it may not be now. Think what would have happened, how history would've changed, if Venus had been a quadrillion-dollar world, we'd have had a virgin planet sitting right next door. Things would have unfolded in an extremely different way. we'd be living in a very different time.