Once at a party a good friend parked me in front of her mother, an intelligent and inquisitive woman who had deep connections with one of the country’s leading anti-climate change financiers. As I came through the front door with my wife, I was introduced to her mother with these words, “Doug will explain climate change to you.” She smiled at us both then walked away, leaving me alone with her startled mother.
Earlier I had been invited to teach a class at a small college on the physics of climate change – it was a one stop invitational thing – and our friend was curious what I had talked about. I had said that to me, the question of the Earth’s changing warmth was really a matter of simple physics. If people could understand a few basic things, it wouldn’t seem so political. “You should talk to my mother” she had said then. I had forgotten our discussion when I came through the door. I was as surprised as her mother by her introduction.
I’ve never believed it my mission to convert people from their beliefs, especially religious or quasi-religious ones, but here I was, installed in front of someone who I was told would have none of this nonsense about climate change. But foolish me, I’ve always enjoyed teaching and the challenge of explaining science to people, so I thought I’d give it a shot.
I imagined her mother’s basic argument, if it went this far into science, would run something like this: The earth has evolved for hundreds of millions of years in a certain way to be habitable for man and the rest of the creatures that dwell with us. How could all that possibly have changed in a few decades? I am not buying it.
Or this variation on the same theme:
I’ve heard about the Goldilocks earth idea, that the earth is at the right place in the solar system to have the right amount of heat to support life. Now suddenly we are supposed to believe that it isn’t, and that we’re all doomed? Something is very suspicious with that…
So, to the extent those arguments have the flavor of science – and that was the assumption I was making – I thought I could make a dent. Her mom looked at me intently, possibly curious about what I had to say, at least polite enough to listen for a while. Anyway, I was vain enough to think I could make a difference; and perhaps I could, so I began with the Goldilocks idea, which is entirely true, as far as it goes.
Yes, we do regularly hear about habitability zones around other planets. These are the ‘Goldilocks’ distances from their parent stars, where liquid water can persist, and conditions are temperate enough for potential life. In exoplanet studies, we learn that stability of the parent star is all-important; variability, pulsations, flares, or swellings can be fatal to nascent life. The source of the idea of course is earth’s own location from the sun – our own ideal, archetypical habitable zone – but we rarely hear about why and how earth is in this lovely habitable zone of its own, and how sensitive our place in the solar system and the life on our planet is to just this and not that amount of solar energy.
Outside of the context of exobiology, however, the idea of a “Goldilocks earth” is more broadly thought to suggest conditions on our planet that make life, as we know it, not just possible but comfortable to humans, an oasis for the life we have always known. “Earth is where we are finely tuned to live: it’s not too hot or too cold, but just right—like the porridge in the story of Goldilocks,” in the words of one writer. This notion of course is a thoroughly anthropomorphic one, obviously far broader than determining whether a planet will support liquid water and some form of microbial or higher life. It comes with the idea of our earth as our comfortable home, and that, by whatever means of divine or evolutionary circumstance, it will remain so, perhaps eternally.
I am neither philosopher, climate scientist, nor biologist, but I had assumed this was my guest’s perspective as I began my five-minute, seat-of-the-pants disquisition. My aim in this improv was to present what I would call threshold physics, without any discussion of weather, climate, or any of the myriad details of climate modeling in this location or that. I hoped to give her a simple argument involving nothing more than three physical components: the sun, our distance from the sun, and the fact of the earth’s atmosphere. I’d leave it to her to puzzle out the consequences…
Crucial to the habitability of our planet is of course its fortuitous distance from the sun. Imagine you are outside on a cold night near a roaring campfire. It is the only source of warmth. Stand too close and you are hot; too far away and you are cold. Find the ‘Goldilocks’ spot – the right distance from the fire – where the warmth feels just right to our bodies which reflect, absorb and reradiate heat. Mark that distance in the dirt with a stick in a large circle all around the fire. And that in our analogy is the earth’s just-right, orbital ‘habitable’ distance from the sun, where liquid water and life can abide. It is a place especially right for earth in our solar system given our planet’s particular atmosphere which also reflects, absorbs, and reradiates heat. But there is certainly much more to the story than this.
This is what I told her as my introduction, and she was still listening – a good sign – so I talked on, risking leading her a little deeper into the waters of physics, focusing first on the sun.
The Heat of the Sun
“We can start by measuring the amount of radiation the earth receives from the sun. It is, on average, quite constant. We’ve been measuring the constant from the surface of the earth and from mountaintops tops for over a century. Nowadays we measure it by satellite. Amplify that by the distance between the earth and the sun and we can quantify how much radiation comes from the sun itself. We find that the radiation that emerges from the Sun’s visible atmosphere is about 63 million watts per square meter. From this solar flux, as it’s called, and some mathematics, we can measure the sun’s temperature, which turns out to be about 5,800° K.”
(Had I the leisure and a blackboard, I would have elaborated on a well-known law of physics (the Stefan-Boltzmann law) that relates radiation output to temperature if certain conditions are met. The conditions are that the radiator be a kind of ideal, perfect radiator (known technically by the odd term as an ideal blackbody radiator). From this, and on the assumption that the sun more or less approximates such a perfect radiator, we can take the solar constant (total solar irradiance) at earth’s orbital distance, increase it proportionally by the square of the distance as we travel inward to the sun’s photosphere (as readers recall from my blog on finding the sun’s temperature) where we get the 63 million watts per square meter; then apply the Stefan-Boltzmann equation to estimate the temperature to be 5,772° K. Recognizing the fact that the sun is not in fact a perfect radiator, we hedge a little and call this the Sun’s effective temperature. That is, the temperature the sun would be if it were a perfect radiator.)
The Cool of the Earth
“If we can take the sun’s temperature,” I said, “we can use the same idea in reverse to measure the earth’s temperature. Imagine the 63 million watts per square meter of radiation from the sun’s surface are spreading out in all directions of space and dissipating in an ever-expanding sphere. By the time the radius of that sphere is as large as Earth’s orbit, about 150 million kilometers, the solar flux has dropped to only 1,361 watts per square meter, or about a forty-six thousandth of its original power when it left the sun’s surface.
“If we think of the earth as a ball of rock and water that reflects and absorbs some of the sun’s much-diminished radiation, how much warmth does that leave us?”
How much?
“Well, not much at all. With that distance, there frankly isn’t much solar radiation left to comfortably warm the earth. The earth’s effective temperature actually turns out to be 254°K, which comes out to –18.8° C, or – 1.8° F. This is a surprising result to many.”
Wait, isn’t that way too cold?
“Yes! That temperature is a little cooler than the average annual temperature in the Arctic. It is cooler than the average temperature of the earth’s surface by about 33° C.”
Then why aren’t we colder?
Good question. Here I was telling her how cold the earth should be, based solely on its distance from the sun. That didn’t seem too Goldilocks. I kept chatting above the growing noise of the party. “Let’s focus on two key facts that’ll make it clearer,” I said. “Just some basic principles, super-simplified:
“First fact: If any body has a temperature, it radiates heat. The law of conservation of energy requires that solar radiation received and not reflected by earth must be equal to the radiation emitted by the planet – that is, what is emitted cannot exceed what is absorbed. It can’t keep getting hotter and hotter as it absorbs heat from the sun until we all burn up: it must give up what it takes, until it reaches an equilibrium point where emission equals absorption. And if it is in equilibrium with its surroundings, it should resemble a so-called blackbody, and its effective temperature should be calculable.
“Second fact: Earth does not absorb all the energy it gets from the sun. This accounts partly for its cold blackbody temperature. It reflects a portion of the solar radiation it receives back into space. Oceans, ice, and clouds bounce radiation back up. The reflectivity of a body in the direction of the observer is called its albedo. (Its scattering in all directions is called its Bond albedo.) The Earth absorbs roughly 70% of the sun’s energy and reflects around 30%. And because the Earth is rotating and has an atmosphere, both of which distribute the received heat around the globe, it re-radiates the heat in all directions.”
The Warmth of the Atmosphere
“Getting back to your question, the reason the earth’s surface is warmer than the average temperature of the Arctic is because our thin atmosphere traps radiation below it – it’s the greenhouse effect. The greenhouse thing isn’t just to do with the climate change discussion; it’s why we can live here at all. If our greenhouse-like atmosphere didn’t wrap us up like a warm coat on a cold night, earth would be a very chilly place! It is the atmosphere that makes the earth even remotely habitable to us humans. It is the combination of earth’s distance from the sun and its atmosphere that makes all the difference. If it weren’t for our atmosphere, this would not be a Goldilocks earth!
“We can illustrate this method too by looking at the other planets. At the distance of Mars, for example, our sun’s mighty per-square-meter output of 63 million watts is enfeebled to a mere 589 watts by the time its expanding sphere of radiation has reached the red planet. Such is the effect of the inverse square relation on radiation. Mars’s blackbody temperature is about 210°K. (This is – 63° C, or – 81.5° F.) With about half the diameter of Earth and a tenth of its mass, Mars clings only to the thinnest atmosphere at a tiny fraction (1/150th) of earth’s air pressure. There is therefore no appreciable protective blanket of gas to trap warmth, despite that its thin air is about 95% carbon dioxide, an otherwise ideal “greenhouse gas.” The 210° Kelvin is the mean temperature: actual temperatures vary in the range of 150° K to 310° K (– 123° C to 37° C) with the low extremes being at the poles. Dry ice (carbon dioxide in its solid, frozen state) has a surface temperature of – 78.5°C, so it is not surprising that astronomers should find carbon dioxide frost and ice at the cooler poles, visible as polar ice caps.
“As we go even farther away from the warmth of the sun, the effective temperatures naturally drop further. At Neptune’s distance, the total solar irradiance is a mere 1.51 watts/m2. This is a very small wattage! Its albedo is not much different than that of the Earth, but its effective temperature is 46.6° K. This is – 226.5° C, colder by far than any known temperature hospitable to life and dramatically colder than Earth, which has occasionally approached – 90° C.” (To illustrate this solar system perspective, I've graphed below the planets from Mercury to Neptune and the dwarf planets Ceres (the largest asteroid, at 2.767 AU) and Pluto, in order of their increasing distances from the Sun, against their calculated effective temperatures.)
So our atmosphere has always kept us warm, right? What's so different?
Okay, now we’re getting to the tricky part of the story… . “Remember how I said that the earth has to give up what heat it gets until it reaches thermal equilibrium? That means the earth must re-radiate thermal energy back into space. Well, the sun emits its peak energy at a wavelength of about half a micron (.5 µm) which is in the yellow-green part of the spectrum, and that is the peak solar wavelength the earth gets. But using another law of physics known as Wien’s law, we can also discover earth’s peak emission wavelength. It’s about 11.4 microns (11.4 µm). That is far in the infrared end of the spectrum. So, the earth absorbs at a (relatively) medium wavelength and emits at a much, much longer wavelength, far past the red end of the spectrum.”
Ok. Why is that relevant?
She seemed genuinely interested to know. “It’s important because our air has both water vapor and carbon dioxide, and both strongly absorb radiation in the far-infrared. Received radiation, instead of being re-radiated from earth out into space, is trapped in our atmosphere, keeping it warm.
“Now the earth hasn’t always been such a Goldilocks place. There’ve been ice ages, of course, and great swampy eras with alligators at the poles and all that. Everybody knows this, but the real anxiety is why new heating is supposed to be happening so quickly.”
Yes.
“Measurements of carbon dioxide in the atmosphere taken from Hawaii for decades have shown that the amount of carbon dioxide in the atmosphere is rapidly increasing. It’s not on a geologic-ages sort of timeline. It’s quick. And the more molecules there are up there absorbing infrared radiation from earth, the warmer will be the atmosphere. It’s just the same thing that makes us warm in the first place, only doubling up. Like putting on an extra coat over the coat you’re already wearing. Suddenly you want to step back from the campfire because you’re sweltering. The habitable zone for our accustomed mode of living – what I call the “anthropic habitable zone” – has just moved back a bit from the sun.”
(If I had more time, I would have mentioned that ages ago, purely as a mathematical game and with the fictitious assumption of a circular orbit, I was playing with the question of how far the earth would have to move out from the sun to cool it. I found that if the earth’s temperature rises by about 6 degrees over the next century, it will be as if the earth moved 5% closer to the sun in its orbit. If we could move the earth this distance away from the sun, the effective temperature would drop by about 6.1 degrees K, which is 6.1 degrees C. Now there’s a job for planetary engineers!)
I finished up by saying that unless the earth finds new ways to stop or capture carbon dioxide from going into the air, there’s really nothing that limits the earth from getting warmer. There’s no physical law that says the air can’t get warmer if it traps more heat, though the atmosphere too needs to reach its own thermal equilibrium where absorption equals emission.
I sensed some uneasiness here, but I wanted my closer to be Venus.
“Let’s look at Venus, the classic case. It has almost twice earth’s amount of solar radiation hitting it. But Venus has the highest albedo by far of any of the planets, its clouds reflecting about 3/4ths of the radiation it receives from the sun. So, its blackbody temperature is actually cooler than earth’s at about 227° K. Yet its average surface temperature is 747° Kelvin! Why? Its peak re-radiation wavelength is about 12.8 µm, smack in the strongly absorbing infrared part of the spectrum inhabited by carbon dioxide. And Venus has an atmosphere that is almost 97% carbon dioxide! That's definitely a gas to watch here on earth."
Okay thanks, she said quietly, and left to rejoin the party. Later she thanked me for the discussion, and I believe she was sincere. I apologized for becoming technical, but she said no, she understood the science and was glad to learn about it. She was most patient, and I was grateful she was willing to listen!
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