The units are in wavenumbers, which is the number of waves per centimeter. Longwave radiation is on the left, shortwave on the right.

These curves show longwave energy leaving the surface going toward space. In this figure, the equator radiates at an average of 300K, which is the highest temperature and therefore has the most energy. The rest of the curves have lower energy, showing that temperatures cool as you go northward. But even the very cold poles still radiate quite a lot of infrared energy. The peaks gradually go left — toward longer wavelengths — as temperature decreases, because colder bodies radiate lower energy photons.

This outgoing longwave energy isn’t in the visible spectrum. So from space looking at earth, you wouldn’t be able to see the earth’s blackbody radiation without an infrared camera.

A blackbody is just a theoretical construct. There are no true blackbodies. For anything that radiates, we multiply by an emissivity factor between 0 and 1 to give more realistic numbers. As a gross simplification, the earth’s emissivity is 0.7. Just multiply the blackbody numbers by 0.7 to get a realistic figure for earth’s infrared emissions to space.

Next, the area under the Planck curve represents the total energy radiated by a blackbody at a given temperature across all wavelengths, as described by the Stefan-Boltzmann law. But this is the energy per square centimeter, not for the whole planet. Scientists use this curve to develop models that represent how the atmosphere responds to sunlight hitting the surface. The midlevel latitude of 45 degrees (the middle curve) is roughly the model for Happer and Van Wijngaarden’s figure 1.

Incidentally, it was the shape of this curve as measured that caused Max Planck to realize that energy came in discrete packets and levels, without a smooth transition from one level to another. That realization created the field of quantum mechanics.

Now that we understand the Planck curve, let’s look at the Schwarzschild curve.

The Schwarzschild curve
Sunshine (shortwave photons) strikes the earth, and the earth re-radiates long-wave photons upward in the infra-red. Without greenhouse gases, earth would emit energy directly from its surface according to the curves I just showed. But with greenhouse gases water vapor, carbon dioxide, ozone, methane, and nitrous oxide, the emission spectrum looks much different, often referred to as the Schwarzschild curve. See figure 1 again to remember what it looks like.

To get this spectrum, add up the atmospheric response from each of the greenhouse gases. This breaks it down for each gas:

Figure 3: The radiative response by molecule …

This comes from the paper Atmosphere and Greenhouse Gas Primer, by William van Wijngaarden and Will Happer. It’s a complex representation of very real phenomena in the world, and it is generally misunderstood or written off, even though the IPCC has the same calculations in their reports.

In this essay, I want to explain why this curve accurate, why it’s important, and how it shows that more CO2 can’t have any meaningful effect on surface temperatures. This is a simplification of a very technical topic and is for people interested in the science of radiative transfer in an atmosphere containing greenhouse gases. It’s an addendum to my six-part video series on climate I recommend you watch before you continue. It’s also part of my weekly climate-class curriculum. I’m trying to make it clear without using jargon, but it is only for people truly interested in understanding the details. It’s about a six-minute read. If you prefer, you can listen to Will explain it himself:

Blackbody radiation
We’ll start by just understanding blackbody radiation. Everything is always absorbing photons and radiating photons away (except black holes). The Planck curve describes the radiation profile for any object at a given temperature, whether it’s a marble, a planet, a star, or any group of atoms in solid or liquid form (gases are not blackbodies). The radiation it gives off depends on its temperature, so the earth radiates differently at different latitudes. Here’s a simulation of the Planck curve for seven northern latitudes on an idealized earth with no atmosphere at all.

Figure 2: Blackbody emission (Planck) curves for seven latitudes …

Planck curves for seven latitudes, from the equator (top) to the north pole (bottom)

In this plot, the thousands of dots represent individual molecular responses. You can see the quantum nature of the response. The dots correspond to the jagged lines in the first figure. Photons of different frequencies that collide with one of these asymmetric molecules get absorbed by the molecule, and that energy causes the molecule to jump into different quantum vibrational and rotational modes. The molecule becomes excited by bending, stretching, and rotating in a particular way, according to the particular vibration of its positive and negative charges. This is called radiative absorption. To learn what happens next, watch my video series on the radiative vs thermal model of the atmosphere. My goal here is to continue explaining the Schwarzschild curve.

Happer and van Wijngaarden computed the molecular response to each of thousands of frequencies and made a prediction what the absorption curve would look like at different latitudes (left side, below). They assumed CO2 is at 400 parts per million. Then, they checked their calculations against measured data (right).

Figure 4: Happer and van Wijngaarden’s calculated predictions …

See how close their predictions were? This is science — if your predictions match the data, your theory is right, at least so far. These two scientists are very good at predicting the response of various concentrations of greenhouse-gas molecules to upwelling radiation. Their predictions for all three latitudes match the observed data.

I’ll show the first figure again, because that’s what I’m going to talk about from here on ...

Figure 1: The Schwarzschild curve …

Reading the Schwarzschild curve
Let’s break down the Schwarzschild curve from left (longwave radiation) to right (shortwave radiation).

On the left, in red, we see the response from water vapor to the lowest-energy longwave radiation. Each wavelength of infrared energy (photons) excites water molecules differently, but generally these wavelengths have the right energy to interact with the rotations of the water molecules.

The CO2 section
Next, from about 600 waves per centimeter to about 750 (14-16 microns) we have the response by CO2. At these wavelengths, the bending vibration and rotations of the molecule interact with the radiation. The gray line shows how the curve would look with no CO2, and the magenta line shows the atmosphere’s response to 50 parts per million CO2. The white area between those two lines represents the “greenhouse” effect of those 50 PPM CO2. That’s a lot of absorption! (To learn what happens after those molecules absorb the radiation and energize, see my six-part video series.)

Several papers show that this gap represents roughly 33 degrees C of temperature at the surface, which means that without those 50 parts per million of CO2, earth would not be habitable for life as we know it (nevermind the fact that almost all life on earth is made of carbon).

Earth has probably always had at least 150 parts per million CO2, even as it was forming. There may have been some epochs before plants arrived (called “snowball earth”) when there was less CO2, but we don’t really know. Certainly for the last 400 million years, there has always been at least 200 PPM CO2, and at times possibly as much as 10,000 PPM, which would be one percent of the atmosphere.

This absorption by CO2 slows down the energy’s escape to space and causes it to warm the air before it leaves. It also causes some of this heat to reradiate back down toward the surface. This is sort of like a blanket and sort of not like a blanket. Again, this is explained in my video series.

Note the gray line has space above it to the blue line. That represents water vapor’s absorption in these wavelengths. So even though CO2 does quite a bit of the greenhouse-gas work, water vapor is also absorbing radiation in that same range.

After the magenta line, we see the bright green line represents 100 PPM and the cyan line represents 200 PPM. There’s just a bit more greenhouse effect at those concentrations, but not much. This takes place in the “wings,” where CO2 molecules that have already absorbed a photon of longwave radiation and become excited then absorb an additional photon and go into a more complex vibrational pattern. These events are statistically very rare. While they may technically do a bit of greenhouse warming, the effect is going to be very small — possible to calculate but impossible to measure.

Today, at the black line, the amount of CO2 in the atmosphere doesn’t change the temperature much more than if we had half that amount or twice that amount, and that’s what this curve shows. To see the effect of doubling CO2 from here, you need to look at the difference between the black line and the red line, which I’ll quantify in a minute.

The atmospheric window
Next section, we see that there is almost no greenhouse effect from any molecule. This is the “atmospheric window,” which lets photons in the range of 8 to 13 microns escape to space unimpeded. Quite a lot of energy coming from the ground and oceans escapes directly to space this way, with no delay on the way up.

The ozone contribution
Ozone lives up in the stratosphere, from about 10 to 50 miles up. It’s a dynamic layer that sloshes around as the planet spins. It has its own set of oscillations and waves, and it is not affected by hair spray or refrigerants or any other gases humans “emit.” But it can and does absorb outgoing radiation and re-radiate it in all directions, including down. You can see the total greenhouse effect from ozone — it’s represented by the area between the black/red curve and the blue curve above. On a Watts-per-square-meter basis, it turns out to be about four percent of the greenhouse effect.

The stratosphere
The stratosphere also contains water vapor. It’s not ice up there, because a) the extremely low pressure and b) individual water molecules can’t form ice if they are separated. That water vapor also has an effect on photons across the wavelength spectrum, just as it does in the troposphere. In fact, the Hunga Tonga eruption added about ten percent more water vapor (individual molecules) to the stratosphere in 2022. According to Javier Vinos, the greenhouse effect from that volcanic eruption has had a dramatic effect on surface temperatures.

Methane and nitrous oxide
Next there’s a small section where methane and nitrous oxide interact with photons in a very narrow bandwidth. But if you look back at figure 3, they absorb energy in the same basic wavelengths as water vapor does. As Will Happer explains:

For a uniform temperature increase, the forcing increase ∆F = 0.23 W /m2 after 50 years that would result if methane concentrations continued to rise at the rate of the previous 10 years as shown in Fig. 9, would cause a surface-temperature increase of ∆T = ∆F/(dZ/dT) = 0.05 C.

Whatever you’ve heard about methane, this is what you should remember:

• It’s in very small concentrations measured in parts per billion

• It only absorbs in a very narrow range of wavelengths

• Water is also active in those wavelengths

• It breaks down in about nine years

• Adding ten or twenty times more methane than we have now would have a negligible effect on temperature.

Here is William van Wijngaarden’s presentation on methane.

The shortwave tail
On the right side, water vapor is the only greenhouse gas that interacts with photons with frequencies of 1300/cm and higher. This is a considerable part of the greenhouse effect. Water vapor absorbs a small number longwave photons at almost all frequencies, while CO2 absorbs a large number of photons within a much narrower range. Together, they are the two biggest contributors to the greenhouse effect. This explains why overnight temperatures in the tropics are mild, while deserts are usually very cold at night.

The gap
The area under each curve represents the amount of energy going from the surface to space. The gap between these two curves represents the greenhouse effect. The area under the Planck curve is the amount of energy the earth would emit with no atmosphere, it comes to 394 Watts per square meter. The area under the black (400 PPM) Schwarzschild curve is 277 Watts per square meter, which is less than what would be emitted with no greenhouse gases.

Yet we know the energy isn’t “trapped” in the atmosphere — all the energy coming in from the sun generally balances the energy being sent back to space. Because the greenhouse molecules are so effective at absorbing longwave photon energy, a very small amount of energy rises via radiation. But at higher elevations, when the water vapor evaporates or the air molecules collide with greenhouse molecules, the process is reversed and the energy is dethermalized — that is, it goes from kinetic molecular energy into longwave photons that eventually make their way to space. This takes place far above the earth’s surface, where it is much colder, so the radiation is weaker. The photons emitted from the top of the atmosphere are similar to those emitted at the North Pole — there are a lot of them, but they are much lower energy than those coming off the surface lower down.

At 800 PPM, the area under the curve is 274 Watts per square meter, and this is a theoretical figure that everyone — including the IPCC — agrees with. So there’s a difference of three Watts per square meter between 400 PPM and 800 PPM CO2.

The IPCC says this would be catastrophic and would end life on earth. Actually, no, the Summary for Policymakers says that. The science report is much less hyperbolic. I’ll come back to temperature in a minute.

Look at the bottom row of figure 4 — at the poles, there is more energy under the Schwarzschild curve in the region where CO2 absorbs. More CO2 actually cools the atmosphere at the poles. This is because at the poles there is a temperature inversion nine months out of the year, so the surface is colder than the air above, and that causes a positive lapse rate that CO2 accentuates.

The gap between the Schwarzschild curve and the Planck curve represents the amount of energy stored in the atmosphere before it goes to space. Any atmosphere with greenhouse gases will have this gap.

Temperature
As I explained earlier and you can see in figure 1, the difference between today’s greenhouse effect and the effect if you double CO2 amounts to about three Watts per square meter — less than one percent of the total of the energy represented by the Schwarzschild curve. This is why you can’t see the black line very much - it’s directly underneath the red line. There’s virtually no difference between the two scenarios.

Let’s think about that for a second. Double CO2 and the atmosphere might retain about three watts per square meter more energy at the surface. This is about the same amount of energy your wireless router gives off, spread over a square meter. How much would that really heat the atmosphere at the surface?

Again, Will Happer calculates:

The forcing increase ∆F = 2.2 W /m2 after 50 years, if carbon dioxide concentrations continued to rise at the rate of the previous 10 years, would cause a surface-temperature increase of ∆T = ∆F/(dZ/dT) = 0.59 C.

This is a model, a calculation. The IPCC takes extreme liberty with this process and predicts 3-4 degrees Celcius, or whatever gets them the most attention in the press, and their models magically agree. Yet the IPCC models have been consistently too hot since they started predicting future temperature. I honestly think they come up with their numbers in a PR meeting with no scientists present.

Remember that three Watts per square meter is for doubling the CO2 concentration in our atmosphere. Doubling. In reality, how long would that take? I don’t think it could be done in 50 years, I think it would most likely take 80 to 100 years, and the resulting increased area of plant growth would be spectacular.

How much warming have humans already caused by adding 50 PPM to the atmosphere in the last 80 years or so? We can’t measure it. The IPCC relies on attribution studies that require carefully constructed computer models to show that humans have already caused a huge amount of warming, but no one has ever measured it.

The IPCC says this three Watts per square meter that may or may not happen over the next 100 years will cause all kinds of never-seen-before “tipping points” with the climate system, but there are mountains of data showing otherwise. Plus, temperatures have gone up very suddenly many times in the past, while CO2 was unchanged. It’s much more likely that CO2 from human activities simply has no significant effect on temperature. In reality, nobody knows. The earth’s climate system is too complex to predict, but almost all feedbacks in the climate system work to reduce, rather than accentuate, any changes. Clouds are a big unknown in all models. More heat in the atmosphere tends to cause more clouds, which then cool the surface and reduce the temperature, sending more incoming sunlight back to space. But they also insulate the troposphere overnight. We are decades from being able to model this with any degree of precision. The earth is constantly getting rid of excess heat and has done so effectively for billions of years.

Why should we believe this?
People say this analysis is wrong. The people who say that don’t understand it. Happer and van Wijngaarden have already shown that they are very skilled at predicting the Schwarzschild curve for a given latitude and CO2 concentration under clear skies. If they can get the 400 PPM scenario to within one percent, they must also be able to calculate the 800 PPM scenario to the same degree of accuracy. There is every reason to believe these scientists are capable of making good predictions.

Conclusion
Figure 1, by Will Happer and William van Wijngaarden, shows that the first 50 parts per million of CO2 in our atmosphere have a huge effect on temperature, the next 350 parts per million have a possibly measurable but very small effect, and doubling CO2 from here would have such a small effect I don’t think it could be measured. We can’t predict the response of clouds to this small amount of warming — they could easily cancel it out.

The earth’s climate system is complex, and most feedbacks are negative — that is, when something goes toward out of balance, the atmosphere has a tendency to send it back into balance. There are ice ages and tropical warm periods on earth, and CO2 rises and falls, but these two things are not correlated. CO2 does not drive temperature today, nor has it ever. Earth has always benefitted from the first 50 parts per million of CO2, which are always hard at work keeping the earth habitable, and that’s pretty much the whole story. This is not only the science that Al Gore doesn’t understand, it’s the science he doesn’t want to understand.

For the final word, watch this presentation by Javier Vinos:

If you’ve read this far, you’ll be interested in my climate class. We have fantastic lectures and discussions every Wednesday.