# 11+12 =23, or How I know the Greenhouse Effect is real

Note: This essay is somewhat mathematical.

Recently, someone asked me, “Do you think the Greenhouse Effect is real?” I found myself answering, “I don’t ‘think’ it’s real. At this point, I know beyond any shadow of a doubt that it’s real. As real as 11+12 = 23.”

I think it’s about time that I spell out how I know it.

It’s a combination of:

• Trust in logic, science, math, and the coming together of multiple lines of understanding and evidence
• A few foundational principles of physics
• Rigorous math
• A few simple bits of data
• Personal experience

I don’t have much to say about the first point right now. But, I’d like to unpack the other four.

Before doing that, it is useful to distinguish between the “Greenhouse Effect” (GHE) and the “Enhanced Greenhouse Effect.” The GHE refers to an effect which involves materials in the atmosphere which can absorb longwave radiation increasing the average surface temperature of a planet. The “Enhanced Greenhouse Effect” refers to an effect which involves adding longwave-absorbing materials to the atmosphere changing the planetary temperature. This essay is about the former, but not the latter.

This essay is about my certainty that the GHE exists and plays a critical role in setting the average surface temperature of planets like Earth.

This essay does NOT address the issue of post-industrial changes in Earth’s climate or the role of the GHE in those changes. That is a much more complex topic. What I’m saying is certain is the fundamental effect that the GHE has on a planet’s temperature.

## A few foundational principles of physics

I’ve come to understand that, conceptually, the atmospheric Greenhouse Effect (GHE), depends on only a few very basic principles of physics. Those principles might be expressed as:

1. Energy is conserved
2. Temperature is a measure of the amount of thermal energy inside something
3. Hotter things shed more heat
4. Energy imbalance leads to temperature change
5. Temperature shifts towards the temperature that balances energy in and out
6. Matter emits thermal electromagnetic radiation at a rate determined by its temperature
7. Emitting radiation is the only way to cool into a vacuum
8. Electromagnetic radiation travels unless something gets in the way of it doing so

That list will be self-explanatory to some people. However, in case you’d like more details, the principles are explained more fully at the end of this essay (in Appendix A).

Given the reality of those 8 principles, the GHE has to be real as well.

If I wanted to help you understand the reality of the GHE, I should probably offer some sort of relatable lesson, to explain how those principles inevitably lead to certain conclusions.

I’ve tried to do that elsewhere.

But I won’t do that here, because this essay isn’t about you. It’s about how I know what I know about the GHE.

## Rigorous math

Unlike most people, I love and trust mathematics.

In mathematics, properly done, there is extreme clarity. Mathematical assertions are correct or incorrect, or their correctness is unknown. It’s possible to objectively identify which of those categories an assertion is in. Anyone who knows the language of math sufficiently well, and is willing and able to work through the arguments, will come to an identical conclusion about whether an assertion is correct or not.

And, while mathematics is in many ways its own world, it connects in amazing ways to the world that we live in. Many of the most basic things about how our world functions seem to be described perfectly by laws formulated in the language of mathematics.

When people try to reason about how something is likely to work, and they do it by simple verbal reasoning, without any math involved, most people are lucky if they’re right 80 percent of the time. (They’ll usually be convinced they’re right much more often than that, but if you carefully check—they’re usually wrong far more often than they realize.)

Yet, in realms of science and engineering that rely heavily on math, the rate of success, and the accuracy of predictions, can sometimes be almost inconceivable high.

In physics, there is at least one result that was predicted accurately to within 1 part in 100 trillion (1 in 1014).1I’m talking about a prediction from the field of Quantum Electrodynamics. How accurate is that? Let’s consider an analogy. What if we lived in a world where a certain natural process placed one object in San Antonio, Texas and another in Rome, Italy, cities which are around 5,900 miles / 9,400 kilometers apart. What if people who understood the mathematical laws that govern how nature works could predict the distance between those two objects to within the thickness of a human hair? That would be amazing, right? Yet, the real prediction I’m talking about was a thousand times more accurate than that.

The amazing technologies that have shaped our modern world are built on top of deep foundations of amazingly accurate mathematical understandings about many aspects of how the world works.

There are certainly a lot of things about the world that aren’t very predictable. But, some of the things that are predictable are astonishingly predictable, with complete reliability.

When I talked about the foundational principles of physics that I listed above, I did so using casual informal language. But, the heart of physics, everything that we understand about how the world works, is deeply rooted in mathematics. In many ways, mathematics seems to be the language and operating system of nature.

There are some things about how the world works that science has established to be true with a level of certainty that is beyond what most people can imagination.

And those certainties are expressed mathematically. When you work with those certainties and simply rearrange them, in ways that anyone who fully understands the languages of physics and mathematics, then one can be equally certain about some other things.

That is a large part of why I know that the Greenhouse effect is real. I can take a few principles of physics which science has made us quite sure of, and do some manipulations which might seem like scary math to some people, but to a mathematician or physicist are unimaginably simple and boring and obviously correct.

And those sort of obviously correct mathematical manipulations, relying only on principles of which scientists are sure of, lead directly to the conclusion that, of course the Greenhouse effect is real. It couldn’t possibly not be.

It’s as simple as 11 + 12 = 23.

So, now I’m going to show you a formula. I know that is likely to lose a lot of you. But, to me, it’s a thing of beauty. It says the GHE is real. Where the formula comes from is described in an appendix to this essay. To those who don’t love math, it might seem complicated and suspect. But, to someone who really gets the math and physics, it offers clarity in a way that is obviously correct.

So, fair warning, there is math ahead.

### Physical quantities with a role in the formula

There are a number of quantities that are relevant to this discussion:

• : The temperature of the surface of the planet.
• : Global Mean Surface Temperature — the value of as averaged over the surface of the planet.
• : Absorbed solar irradiance flux (energy per unit time per unit area)2In principle this term includes heating from all sources. On some planets, this term includes a significant geothermal heating term. However, on Earth, the geothermal heat flux is about 0.09 W/m2, which is negligible in comparison with the absorbed solar flux of 240 W/m2.
• : Outgoing Longwave Radiation flux — the flux of thermal radiation emitted to space from the top of the atmosphere (TOA). Longwave radiation is radiation with a wavelength longer than 4 microns.
• : Surface-emissions Longwave Radiation flux — the flux of thermal radiation emitted by the surface.
• : the Stefan-Boltzmann constant — used in calculating the relationship between the temperature and the flux of emitted thermal radiation.
• : the emissivity of the surface — a dimensionless number between zero and one which is used in calculating the thermal radiation flux emitted by the surface; it is a characteristic of the material doing the emitting.

I’ll also refer to these quantities:

(1)

(2)

is a dimensionless quantity between zero and one. is a dimensionless quantity greater than or equal to zero. I will explain the significance of these quantities below.

### Formula for the equilibrium Global Mean Surface Temperature of a planet

One of the core principles I’ve mentioned is “Temperature shifts towards the temperature that balances energy in and out.” This temperature where energy in and out balance is called the equilibrium temperature. If things are changing, the actual temperature may not have entirely caught up with the equilibrium temperature, but the actual temperature will always be shifting towards the equilibrium temperature.

It can be shown (see Appendix B) that the equilibrium value of the Global Mean Surface Temperature is given exactly by this formula:

(3)

I’ll call the above equation the Exact Formula for equilibrium Global Mean Surface Temperature (EF-E-GMST).

The factors , , and are defined as:

(4)

(5)

(6)

### Interpreting the Formula

The reason I’m referring to the formula for as “exact”, is that it involves no approximations or assumptions of any kind. It is exactly equivalent to the Stefan-Boltzmann law, which has been part of the foundations of modern physics and thermal engineering for over a century.3Despite its central role, for purposes of this essay, the Stefan-Boltzmann law can be regarded as simply a definition of emissivity; in that sense the SB law is a tautology which inherently must be true. The foundational physics associated with the SB law is really the principle that emissivity can’t be larger than one; however, the analysis presented here does not rely on that principle.

What does EF-E-GMST tell us?

It says the equilibrium Global Mean Surface Temperature is made up of three factors:

• The “idealized planet” temperature, — This is the surface temperature a planet would have if there were no temperature variations and the atmosphere was transparent to longwave radiation, so that .
• The temperature-variation temperature-reduction factor, — This less than or equal to 1. It reflects the fact that when temperature varies, the average temperature can be lower while still achieving the same whole-planet emission rate needed to balance the rate of incoming energy. If the surface temperature was uniform, then would be 1; but given temperature variations, .
• The longwave temperature boost factor or Greenhouse temperature boost factor, — This is greater than or equal to 1. It reflects the fact that , i.e., the power of thermal radiation emitted to space is less than or equal to the power of thermal radiation emitted by the surface.4There are exotic scenarios in which, for a hypothetical planet, perhaps it wouldn’t be true that OLR ≤ SLR, in which case there might be a negative Greenhouse effect which would lower the average surface temperature. However, the planets in our solar system with well-defined surfaces do not satisfy the relevant hypothetical conditions for that to happen. Earth and Venus both have OLR < SLR and Mg > 1. If the atmosphere was transparent to longwave thermal radiation, then all thermal radiation emitted by the surface would reach space, so that the rate of power being emitted to space would be . Since a planet only cools itself via thermal radiation emissions to space, the rate of emissions to space is the planet’s rate of cooling. So, with an atmosphere transparent to LW, a planet’s cooling rate wold be . But, in practice the cooling rate is . On Earth, is less than . Thus, the non-LW-transparent atmosphere reduces Earth’s net rate of cooling to space, for a given temperature. To compensate for this reduced cooling efficiency, the surface temperature needs to be higher to generate sufficient cooling. The factor describes the amount by which this effect increases GMST.

The names chosen for the factors above are my own, and don’t reflect any standardized usage.

At this point, I will go ahead and name that the quantity is known as the normalized Greenhouse effect, for which I’ve seen others use the symbol .5IPCC 2021 AR6 WG1 Full Report, p. 968 There is a related quantity, , defined as:

(7)

is the measurable quantity which climate scientists call the Greenhouse effect.

Both and would necessarily be zero if the atmosphere was transparent to longwave radiation. Thus, their nonzero values is a direct consequence of the presence in the atmosphere of substances in the atmosphere that absorb, emit, scatter, or reflect longwave radiation.

You might be puzzled by how EF-E-GMST could be an exact formula for Earth’s average surface temperature when the formula makes no mention of convection, latent heat transport, advection, lapse rate, cloud cover, water vapor concentration, ocean circulation, or any of the many other factors that play obviously significant roles in Earth’s climate.

The solution to that puzzle comes from realizing that those factors do influence the equilibrium GMST, but the way that they to so is through influencing the quantities that are in the formula. Those various phenomena can influence:

• — absorbed sunlight, which can be influenced, e.g., by changing cloud-cover or snow-cover
• — the emissivity of the surface, e.g., by influencing vegetation growth and ice-sheet extent
• and — the non-uniformity of global temperatures, which can be influenced, e.g., by changing the rate of heat transport between the tropics and the poles.
• , , and — the Greenhouse effect, which can be influenced, e.g., by changing cloud cover, water vapor concentration and distribution, or lapse rate.6The Greenhouse effect depends on the lapse rate. The Greenhouse effect is larger when the temperature drops more quickly with altitude, and smaller when the temperature drops more slowly with altitude; the Greenhouse effect would actually vanish if the atmospheric temperature was equal at all altitudes.

Thus, many of the factors that we intuitively think ought to impact GMST likely do impact it. However, that impact is indirect, insofar as those factors affect the four factors above, and those four factor determine the equilibrium .

## What the data tell us

Let’s plug some numbers for Earth into these formulas and see what they tell us.

Here are estimates for key numbers:

• = 240 W/m2
• = 239 W/m2
• = 398 W/m2
• 0.935
• = 0.0187 ± 0.00037This is based on my analysis of CERES data

The first three numbers are recent versions of values that have been widely circulated.8IPCC 2021 AR6 WG1 Full Report, p. 934

Note that the values for emissivity and could be combined if one treated emissivity as effectively being about 0.95.

The above values lead to:

• = 0.40
• = 159 W/m2
• = 1.14
• = 0.9954
• 293K / 20℃ / 68℉
• 259K / -14℃ / 7℉
• 258K / -15℃ / 5℉ — Temperature when is included but GHE impact is absent
• 35K / 35℃ / 63℉ — Temperature change attributable to GHE

When I look at those numbers, I notice:

• There is a substantial Greenhouse effect, insofar as = 0.40 is well into its allowed range, , and very much not even close to the value = 0 that would would have if the atmosphere was transparent to LW radiation.
• The temperature with temperature-variation accounted for but the impact of the GHE absent is slighly lower than the “idealized planet” temperature, as expected.
• The temperature change attributable to the Greenhouse effect is slightly larger than the commonly-reported number of around 33℃, because the GHE is not achieving our actual temperature relative to the “idealized planet” temperature, but relative to the even lower temperature associated with temperature variations.

## Sanity checks

The EF-E-GMST formula definitely indicates that if an atmosphere absorbs thermal radiation (so that ), then that absorption will make a planet’s surface warmer on average.

And the specific data for Earth indicates a fairly significant effect () on Earth.

But, are there other things we can look at, to sanity-check those conclusions?

### Is the value of for surface emissions unreasonably large?

Some people question the value of average surface emissions, = 398 W/m2; some are incredulous that this value could be larger than the 160 W/m2 value for the average rate at which the Earth’s surface absorbs power from sunlight. However, 398 W/m2 is not simply a made-up number. And, to agree that “Earth has a Greenhouse Effect,” all we need to agree on is that is greater than 239 W/m2, so that . Let’s consider this idea that “should” be much smaller than the quoted number:

• Surface thermal radiation emissions are routinely measured. Whether or not those measurements are done using your favorite technique everywhere, the measurements aren’t remotely consistent with the idea that the average could be anywhere near as small as 240 W/m2 (an emission rate characteristic of objects as cold as -18℃ / -1℉).
• According to Planck’s law, an emission rate of 398 W/m2 is characteristic of any and all objects with an emissivity of 0.94 and a temperature of 21℃ / 69℉. The climate science numbers simply reflect the “Earth’s surface” emitting as much thermal radiation as the objects that make up that surface.
• Human skin, at a temperature around 35℃ / 95℉ and with an emissivity around 0.97 emits around 496 W/m2. That might sound like a lot, but if the person is in a room with a temperature of 27℃ / 80℉ and an emissivity of 0.5, then they are also being absorbing 446 W/m2 that was radiated from the walls around them. The net effect is that, if a body has a surface area of 2 m2 and is naked, that amounts to the human body experiencing a radiant heat loss of about 99 W to the environment. So, the math all checks out as eminently reasonable.

### Are outgoing longwave radiation emissions really smaller than surface thermal radiation emissions?

I enjoy looking at measurements of the spectrum of Earth’s longwave emissions to space.

Figure 1 shows the spectrum of longwave thermal emissions measured looking down over Fort Sumner, New Mexico on June 7, 2005 from a balloon at an altitude of 35 km.

• Note that wave number as referred to on the x-axis is frequency divided by the speed of light.
• The chart includes the spectrum of emissions expected from a surface emitting thermal radiation at a temperature of 320K / 47℃ / 116℉. While that seems rather hot, historical weather data is available for that date and location to verify the temperature. Those records make a surface temperature of 320K seem entirely plausible.9The chart suggests a surface temperature of 320K / 47℃ / 116℉. Historical data from climate.gov indicates that on June 7, 2005 in Fort Sumner, NM:
• at one measurement station, the air temperature range was 13.9-30.6℃ / 57-87℉;bare ground there at a depth of 10 cm had a soil temperature range of range 23.3-33.3℃ / 74-92℉;at another measurement station the maximum air temperature was hotter than at the first by 4.4℃ / 8℉, reaching a maximum of 35℃ / 95℉; extrapolation from the first site suggests that soil 10cm down likely reached a peak temperature around 37.8℃ / 100℉.
Given:
• the semi-arid climate and clear-sky conditions,the 16.7℃ and 10℃ temperature swings for air and for soil 10cm down imply that temperature 10cm down would significantly lag the surface temperature,the 4pm time of the measurement
the indicated surface temperature seems plausible. From personal experience, the surface can get very hot in NM!
• The observed spectrum of radiation emitted to space seems consistent with a surface temperature of 320K. (Given that there are spectral regions where emissions reach the level of the 320 K black-body emissions curve, e.g., between 800-1000 cm-1, it wouldn’t be plausible for the surface to be any cooler than 320K).
• The enormous dip around a wavenumber of 670 cm-1 (corresponding to a 15-micron wavelength) is consistent with the known absorption characteristics of CO2.
• The lowered emissions to the left and right of the chart are consistent with the known absorption characteristics of water vapor.
• An analogous chart of satellite measurements over the Sahara desert shows that computer modeling aligns reasonably well with the observed spectrum.
• Note that under clear-sky (no cloud) conditions, the global average increases from 240 W/m2 to around 267 W/m2,10IPCC 2021 AR6 WG1 Full Report, p. 934. meaning that drops from around 0.40 to around 0.33. The balloon measurements were made under clear-sky conditions, so we would expect that the emissions reaching the the balloon (on the edge of space) would be roughly 33% less than the surface emissions leaving the surface.

Looking at the size of the gap between the emissions expected from a surface at 320K and the measured emissions to space, it seems entirely plausible that around 33% of surface emissions aren’t making it to space.

For me, the pieces of the picture fit together in a very compelling way.

## Personal experience

I know from personal experience that, for a given heating rate, the temperature is warmer when you’re better insulated, i.e., cooling is less efficient. So, it makes sense that things getting in the way of thermal radiation emissions to space could lead to higher temperatures.

I know from personal experience that it’s warmer on cloudy nights than it is on clear nights (even though clouds are cold). So, I have direct experience that things in the sky that intercept thermal radiation help keep things warm.

And, I have personal experience that, whenever I work through physics principles and mathematical analyses related to the Greenhouse effect and related phenomena, it consistently makes sense, and is consistent with with everything else I’ve learned about science and engineering. The physics and math together tell a coherent, consistent story, which makes sense of what is observed.

## Conclusions

The details of how climate functions are often complicated.

But, when viewed broadly, the Greenhouse effect is fairly simple. The presence of material in the atmosphere that interact with longwave thermal radiation leads to the amount of thermal radiation that reaches space being less that the amount of thermal radiation that leaves the surface. To the extent that that happens, it leads to a higher equilibrium global average surface temperature.

This follows, as inevitably as night follows day, from basic principles of physics, with the help of a bit of rigorous mathematical analysis.

The longer I’ve worked with all this, the simpler and more obvious and certain it seems.

I know the Greenhouse effect is a real phenomenon. And the publicly available data indicates that the Greenhouse effect plays an enormous role in keeping the Earth warm.

## APPENDIXES

### Appendix A: Principles of physics

In this Appendix, I’ll explain in a bit more detail what I mean by each of the 8 principles that I named at the beginning of the essay.

#### 1. Conservation of energy

Energy is conserved. That means it doesn’t appear or disappear―though energy might change location or form. For example:

• When matter emits thermal radiation, the power (energy per unit time) carried by the electromagnetic radiation drains the thermal energy inside that matter at an equal rate.11Thermal energy includes both kinetic energy, associated with the motion of particles, and potential energy, associated with interactions between particles. When atoms are organized into molecules, the kinetic energy can be thought of either as (a) the energies of individual atoms moving, or as (b) the energies of molecules moving, rotating, and vibrating. Potential energy includes both energy associated with the gas, liquid, or solid phase of the matter and the energy of chemical bonds.
• When matter absorbs electromagnetic radiation, the power carried by that radiation is converted to thermal energy inside that matter.12More specifically, overall, the power of absorbed radiation is almost always converted to internal kinetic energy—either directly or indirectly. Here is a detailed explanation, if it’s needed:

The “indirect” case occurs when some of the absorbed power is used to alter the internal chemical potential (through a phase change or chemical change) or produces electricity (as in the case of human-made photovoltaic cells). In steady state, the rate at which the chemical potential is increased or electricity is generated is generally balanced by a subsequent process which converts those forms of energy back into internal kinetic energy at an equal rate. There are a number of such balanced cycles, including:
• the water evaporation/condensation cycle
• the UV-absorbing oxygen-ozone cycle
• photosynthesis and biomass metabolism/decay
• human photovoltaic electricity production and use
These cycles do not alter the principle that, as a net effect over the long run, the power of absorbed radiation is almost always converted to internal kinetic energy.

The “almost always” qualifier is relevant when radiation absorption leads net melting of ice; in that case, some of the thermal energy is in the form of the internal chemical potential energy of the phase transition, but the principle that the power of absorbed radiation is converted to internal thermal energy still applies.

#### 2. Temperature is a measure of the amount of thermal energy inside something

Temperature is a measure of the amount of thermal energy in matter, in much the same way as the level of the water in a lake is a measure of the amount of water in the lake.13Specifically, temperature is a measure of the amount of the internal kinetic energy, which is a component of thermal energy.

Although the space surrounding Earth doesn’t contain significant amounts of matter, space does contain energy, in the form of the electromagnetic radiation which pervades the universe.14This is the so-called microwave background radiation which originated shortly after the “Big Bang.” Because of that radiation, when thinking about radiation transferring heat from hot to cold, space can be regarded as having a temperature of 3 Kelvin / -270℃ / -454℉.

#### 3. Hotter things shed more heat

As the temperature of matter rises, relative to the temperature of what surrounds it, the rate at which heat flows from that matter to its surroundings increases.

#### 4. Energy imbalance leads to temperature change

What happens when the rate of energy entering into an object does not equal the rate of energy coming out of the object?

It can be helpful to think about the issue in the abstract. In Figure 2, an “Energy Gain Mechanism” sends energy IN to an object of interest (which has temperature ) and an “Energy Loss Mechanism” removes energy OUT from the object of interest.

Because of energy conservation, if the rate of energy flowing into something and the rate of energy flowing out are not equal, then:

• the difference between the rates of energy flowing in and energy flowing out must be the rate at which the amount of thermal energy inside changes.

In the form of an equation, this looks like:15Strictly speaking, the “energy inside” which changes in response to an energy imbalance can at times include non-thermal forms of energy (such as kinetic energy associated with advection, or gravitational potential energy when the atmosphere expands due to a temperature rise). However, unless those other forms of energy can store energy in ever increasing amounts, or be supply energy in endless amounts (which is not the case for any relevant form of energy I can think of), they can’t alter the ultimate need for energy balance. Those other forms of energy take in or give out energy only on a transient basis. Over the long term, those other forms of energy storage stop affecting the energy balance equation. Those non-thermal forms of energy don’t alter the ultimate requirement that the rates of energy coming in and going out must balance for temperature to stabilize.

(8)

Since temperature is a measure of the amount of thermal energy inside:

• When energy flows in faster than energy flows out, temperature increases.
• When energy flows in slower than energy flows out, temperature decreases.16There is an exception to these rules when a phase change happens, as when something melts, freezes, evaporates of condenses. During a phase change, the temperature change pauses until the phase change is completed; then the temperature increase or decrease continues.

#### 5. Temperature shifts towards the temperature that balances energy in and out

A previous principle was that “hotter things shed more heat.” In terms of our abstract energy flow diagram, that means the the rate of “energy out” (as extracted by the energy loss mechanism) will increase as the object’s temperature increases.

How will the rate of “energy in” vary with the object’s temperature? That depends on the nature of the energy gain mechanism. It that mechanism is involves some hot object supplying heat, then as the temperature of the object of interest increases, the rate of “energy in” will decrease. However, if the energy gain mechanism is an electric heater, then the rate of “energy in” might not depend on temperature at all.

Regardless of the particular energy gain and energy loss mechanism, we can plot how “energy in” and “energy out” vary as a function of the object temperature, , as I’ve illustrated in Figure 3.

Regardless of exactly what the “energy in” and “energy out” curves look like, in all real systems, there is a temperature at which those two curves cross. I’ll refer to that temperature where energy “in” and “out” balance as the equilibrium temperature. In Figure 3, the equilibrium temperature is labeled .

If the object’s temperature, , is cooler than (to the left of) the equilibrium temperature, , then energy flows in faster than it flows out, there is net warming, and the object’s temperature rises. If is cooler than (to the right of) the equilibrium temperature, , then energy flows in slower than it flows out, there is net cooling, and the object’s temperature falls. In either case, shifts towards . If we wait long enough and nothing else changes, eventually will equal .

If something changes (as shown in Figure 4), then the balance point may shift from to another temperature, . If that happens, then will start shifting towards the new equilibrium temperature, .

The general principle is:

• An object’s temperature always shifts towards the temperature at which the rates “energy in” and “energy out” are equal, i.e., towards the equilibrium temperature.

#### 6. Matter emits thermal electromagnetic radiation at a rate determined by its temperature

The Stefan-Boltzmann law (SB) tells us that matter at a temperature, , emits electromagnetic radiation at a rate:

(9)

where the Stefan-Boltzmann constant, , is a fixed constant, and the emissivity, , is a dimensionless quantity characteristic of the type of matter doing the emitting.

This electromagnetic radiation is often called “thermal” radiation, because it’s emitted just because matter is at a non-zero temperature, and the process effectively converts thermal energy into radiation.

There are multiple reasons to be willing to rely on the SB law:

• The SB law was formulated in 1884 and has been verified and relied on in countless ways in scientific and industrial applications on a continual basis for well over a century. It has long-been a foundational principle of both physics and thermal engineering.
• Even if it weren’t so well-established and validated, accepting the SB law shouldn’t be a problem for anyone. That’s because the SB law is, in a way, simply the definition of the quantity called emissivity, . One can always make the SB law yield a correct result, simply by choosing the value of appropriately (and allowing to vary with the temperature ).

The actual physical law isn’t the SB law itself, but the principle that the power of the emitted radiation is a function of temperature, and the principle that is never greater than 1. However, in my analysis, I don’t rely on the principle that emissivity isn’t greater than 1.

So, while it might look like I’m relying on the SB law, the only principle of physics that I am actually relying on is that matter emits thermal electromagnetic radiation at a rate determined by its temperature.

#### 7. Emitting radiation is the only way to cool into a vacuum

Heat can flow from someplace warm to someplace cold via three heat transfer mechanisms: conduction, convection, and radiation. The only one of these mechanisms that does not require the presence of matter is radiation. So, it’s clear that emitting radiation is the only way for a body surrounded by a vacuum to cool itself.

#### 8. Electromagnetic radiation travels unless something gets in the way of it doing so

If you send electromagnetic radiation through a medium that is transparent to the radiation (i.e., doesn’t have anything it that absorbs, emits, scatters, or reflects that radiation), then, by virtue of energy conservation and what it means for something to be “transparent”, the power (energy per unit time) of the radiation that exits the medium will be the same as the power that entered.

#### Afterthoughts about these principles

These principles seem to me to be pretty straightforward. They don’t seem hard to understand. And, correctly understood, they’re valid. The essential physics has been well-established for over a century, has been tested in countless ways, and is routinely applied in contexts having nothing to do with climate science.

### Appendix B: Deriving the formula for equilibrium global mean surface temperature

#### Thermal radiation emitted by Earth’s surface

The Earth’s surface (including land and oceans) emits thermal electromagnetic radiation in accordance with the Stefan-Boltzmann law:

(10)

where is the power per unit area (W/m2) of surface longwave radiation emissions17In climate science, longwave (LW) radiation is defined as electromagnetic radiation with a wavelength longer than 4 microns; and shortwave (SW) radiation is defined as radiation with a wavelength shorter than 4 microns The definitions are designed so that, in essence, all sunlight is SW and all thermal radiation from Earth’s surface and atmosphere is LW., is the emissivity of the surface, and is its temperature.

At least, that’s how it works for a particular portion of the Earth’s surface. We’d really like to know the total thermal emission rate for Earth’s entire surface. To end up with workable units, it’s helpful to divide the total by the surface area of Earth, which puts things in terms of the average value per unit area. I’ve done the math for that elsewhere. The result is, that for the whole Earth, the SB equation becomes:

(11)

where is the global average surface emissions, is the global average emissivity (weighted by ), and is the global average (or mean) surface temperature. The new factor, , is an adjustment that accounts for the effect of temperature variations (such as the poles being colder than the tropics):

(12)

is always greater than or equal to 0, and I call it the temperature-variation emissions boost factor.

#### Thermal radiation emitted by Earth as a whole

There is thermal electromagnetic radiation emitted to space from the top of Earth’s atmosphere (TOA). I’ll use the symbol , for outgoing longwave radiation, to denote the power per unit area emitted to space.

Let’s define an empirical quantity as:

(13)

That definition makes it possible to express the outgoing longwave radiation emitted to space as:

(14)

That’s the rate of “energy out” for Earth.

#### Energy received by Earth

Earth’s land surface, oceans, and atmosphere receive energy at a rate I will denote as .

For all practical purposes, is the the rate at which sunlight is absorbed by Earth’s surface and atmosphere (around 240 W/m2). There is also a negligible contribution associated with geothermal energy rising from Earth’s core (around 0.09 W/m2).

#### Energy balance equation for Earth as a whole

As I mentioned above, one of the scientific principles I’m relying on is “temperature shifts towards the temperature that balances energy in and out.”

(15)

(16)

#### Formula for equilibrium global mean surface temperature

Solving the energy balance equation (“energy in” equals “energy out”) for the temperature yields the “Exact Formula for Equilibrium Global Mean Surface Temperature” (EF-E-GMST), exactly as I presented it earlier in this essay.

A very closely-related formula is explored in my technical essay Quantifying the Greenhouse Effect. A minor difference is that:

• This essay references the absorbed sunlight, , in order to calculate the equilibrium value of .
• Quantifying the Greenhouse Effect does not reference absorbed sunlight or the idea of an equilibrium temperature. Instead, it focuses on the relationship between the outgoing longwave radiation flux, , at the top of the atmosphere, and the actual value of .

### Footnotes

• 1
I’m talking about a prediction from the field of Quantum Electrodynamics.
• 2
In principle this term includes heating from all sources. On some planets, this term includes a significant geothermal heating term. However, on Earth, the geothermal heat flux is about 0.09 W/m2, which is negligible in comparison with the absorbed solar flux of 240 W/m2.
• 3
Despite its central role, for purposes of this essay, the Stefan-Boltzmann law can be regarded as simply a definition of emissivity; in that sense the SB law is a tautology which inherently must be true. The foundational physics associated with the SB law is really the principle that emissivity can’t be larger than one; however, the analysis presented here does not rely on that principle.
• 4
There are exotic scenarios in which, for a hypothetical planet, perhaps it wouldn’t be true that OLR ≤ SLR, in which case there might be a negative Greenhouse effect which would lower the average surface temperature. However, the planets in our solar system with well-defined surfaces do not satisfy the relevant hypothetical conditions for that to happen. Earth and Venus both have OLR < SLR and Mg > 1.
• 5
IPCC 2021 AR6 WG1 Full Report, p. 968
• 6
The Greenhouse effect depends on the lapse rate. The Greenhouse effect is larger when the temperature drops more quickly with altitude, and smaller when the temperature drops more slowly with altitude; the Greenhouse effect would actually vanish if the atmospheric temperature was equal at all altitudes.
• 7
This is based on my analysis of CERES data
• 8
IPCC 2021 AR6 WG1 Full Report, p. 934
• 9
The chart suggests a surface temperature of 320K / 47℃ / 116℉. Historical data from climate.gov indicates that on June 7, 2005 in Fort Sumner, NM:
• at one measurement station, the air temperature range was 13.9-30.6℃ / 57-87℉;bare ground there at a depth of 10 cm had a soil temperature range of range 23.3-33.3℃ / 74-92℉;at another measurement station the maximum air temperature was hotter than at the first by 4.4℃ / 8℉, reaching a maximum of 35℃ / 95℉; extrapolation from the first site suggests that soil 10cm down likely reached a peak temperature around 37.8℃ / 100℉.
Given:
• the semi-arid climate and clear-sky conditions,the 16.7℃ and 10℃ temperature swings for air and for soil 10cm down imply that temperature 10cm down would significantly lag the surface temperature,the 4pm time of the measurement
the indicated surface temperature seems plausible. From personal experience, the surface can get very hot in NM!
• 10
IPCC 2021 AR6 WG1 Full Report, p. 934.
• 11
Thermal energy includes both kinetic energy, associated with the motion of particles, and potential energy, associated with interactions between particles. When atoms are organized into molecules, the kinetic energy can be thought of either as (a) the energies of individual atoms moving, or as (b) the energies of molecules moving, rotating, and vibrating. Potential energy includes both energy associated with the gas, liquid, or solid phase of the matter and the energy of chemical bonds.
• 12
More specifically, overall, the power of absorbed radiation is almost always converted to internal kinetic energy—either directly or indirectly. Here is a detailed explanation, if it’s needed:

The “indirect” case occurs when some of the absorbed power is used to alter the internal chemical potential (through a phase change or chemical change) or produces electricity (as in the case of human-made photovoltaic cells). In steady state, the rate at which the chemical potential is increased or electricity is generated is generally balanced by a subsequent process which converts those forms of energy back into internal kinetic energy at an equal rate. There are a number of such balanced cycles, including:
• the water evaporation/condensation cycle
• the UV-absorbing oxygen-ozone cycle
• photosynthesis and biomass metabolism/decay
• human photovoltaic electricity production and use
These cycles do not alter the principle that, as a net effect over the long run, the power of absorbed radiation is almost always converted to internal kinetic energy.

The “almost always” qualifier is relevant when radiation absorption leads net melting of ice; in that case, some of the thermal energy is in the form of the internal chemical potential energy of the phase transition, but the principle that the power of absorbed radiation is converted to internal thermal energy still applies.
• 13
Specifically, temperature is a measure of the amount of the internal kinetic energy, which is a component of thermal energy.
• 14
This is the so-called microwave background radiation which originated shortly after the “Big Bang.”
• 15
Strictly speaking, the “energy inside” which changes in response to an energy imbalance can at times include non-thermal forms of energy (such as kinetic energy associated with advection, or gravitational potential energy when the atmosphere expands due to a temperature rise). However, unless those other forms of energy can store energy in ever increasing amounts, or be supply energy in endless amounts (which is not the case for any relevant form of energy I can think of), they can’t alter the ultimate need for energy balance. Those other forms of energy take in or give out energy only on a transient basis. Over the long term, those other forms of energy storage stop affecting the energy balance equation. Those non-thermal forms of energy don’t alter the ultimate requirement that the rates of energy coming in and going out must balance for temperature to stabilize.
• 16
There is an exception to these rules when a phase change happens, as when something melts, freezes, evaporates of condenses. During a phase change, the temperature change pauses until the phase change is completed; then the temperature increase or decrease continues.
• 17
In climate science, longwave (LW) radiation is defined as electromagnetic radiation with a wavelength longer than 4 microns; and shortwave (SW) radiation is defined as radiation with a wavelength shorter than 4 microns The definitions are designed so that, in essence, all sunlight is SW and all thermal radiation from Earth’s surface and atmosphere is LW.
• 1
I’m talking about a prediction from the field of Quantum Electrodynamics.
• 2
In principle this term includes heating from all sources. On some planets, this term includes a significant geothermal heating term. However, on Earth, the geothermal heat flux is about 0.09 W/m2, which is negligible in comparison with the absorbed solar flux of 240 W/m2.
• 3
Despite its central role, for purposes of this essay, the Stefan-Boltzmann law can be regarded as simply a definition of emissivity; in that sense the SB law is a tautology which inherently must be true. The foundational physics associated with the SB law is really the principle that emissivity can’t be larger than one; however, the analysis presented here does not rely on that principle.
• 4
There are exotic scenarios in which, for a hypothetical planet, perhaps it wouldn’t be true that OLR ≤ SLR, in which case there might be a negative Greenhouse effect which would lower the average surface temperature. However, the planets in our solar system with well-defined surfaces do not satisfy the relevant hypothetical conditions for that to happen. Earth and Venus both have OLR < SLR and Mg > 1.
• 5
IPCC 2021 AR6 WG1 Full Report, p. 968
• 6
The Greenhouse effect depends on the lapse rate. The Greenhouse effect is larger when the temperature drops more quickly with altitude, and smaller when the temperature drops more slowly with altitude; the Greenhouse effect would actually vanish if the atmospheric temperature was equal at all altitudes.
• 7
This is based on my analysis of CERES data
• 8
IPCC 2021 AR6 WG1 Full Report, p. 934
• 9
The chart suggests a surface temperature of 320K / 47℃ / 116℉. Historical data from climate.gov indicates that on June 7, 2005 in Fort Sumner, NM:
• at one measurement station, the air temperature range was 13.9-30.6℃ / 57-87℉;bare ground there at a depth of 10 cm had a soil temperature range of range 23.3-33.3℃ / 74-92℉;at another measurement station the maximum air temperature was hotter than at the first by 4.4℃ / 8℉, reaching a maximum of 35℃ / 95℉; extrapolation from the first site suggests that soil 10cm down likely reached a peak temperature around 37.8℃ / 100℉.
Given:
• the semi-arid climate and clear-sky conditions,the 16.7℃ and 10℃ temperature swings for air and for soil 10cm down imply that temperature 10cm down would significantly lag the surface temperature,the 4pm time of the measurement
the indicated surface temperature seems plausible. From personal experience, the surface can get very hot in NM!
• 10
IPCC 2021 AR6 WG1 Full Report, p. 934.
• 11
Thermal energy includes both kinetic energy, associated with the motion of particles, and potential energy, associated with interactions between particles. When atoms are organized into molecules, the kinetic energy can be thought of either as (a) the energies of individual atoms moving, or as (b) the energies of molecules moving, rotating, and vibrating. Potential energy includes both energy associated with the gas, liquid, or solid phase of the matter and the energy of chemical bonds.
• 12
More specifically, overall, the power of absorbed radiation is almost always converted to internal kinetic energy—either directly or indirectly. Here is a detailed explanation, if it’s needed:

The “indirect” case occurs when some of the absorbed power is used to alter the internal chemical potential (through a phase change or chemical change) or produces electricity (as in the case of human-made photovoltaic cells). In steady state, the rate at which the chemical potential is increased or electricity is generated is generally balanced by a subsequent process which converts those forms of energy back into internal kinetic energy at an equal rate. There are a number of such balanced cycles, including:
• the water evaporation/condensation cycle
• the UV-absorbing oxygen-ozone cycle
• photosynthesis and biomass metabolism/decay
• human photovoltaic electricity production and use
These cycles do not alter the principle that, as a net effect over the long run, the power of absorbed radiation is almost always converted to internal kinetic energy.

The “almost always” qualifier is relevant when radiation absorption leads net melting of ice; in that case, some of the thermal energy is in the form of the internal chemical potential energy of the phase transition, but the principle that the power of absorbed radiation is converted to internal thermal energy still applies.
• 13
Specifically, temperature is a measure of the amount of the internal kinetic energy, which is a component of thermal energy.
• 14
This is the so-called microwave background radiation which originated shortly after the “Big Bang.”
• 15
Strictly speaking, the “energy inside” which changes in response to an energy imbalance can at times include non-thermal forms of energy (such as kinetic energy associated with advection, or gravitational potential energy when the atmosphere expands due to a temperature rise). However, unless those other forms of energy can store energy in ever increasing amounts, or be supply energy in endless amounts (which is not the case for any relevant form of energy I can think of), they can’t alter the ultimate need for energy balance. Those other forms of energy take in or give out energy only on a transient basis. Over the long term, those other forms of energy storage stop affecting the energy balance equation. Those non-thermal forms of energy don’t alter the ultimate requirement that the rates of energy coming in and going out must balance for temperature to stabilize.
• 16
There is an exception to these rules when a phase change happens, as when something melts, freezes, evaporates of condenses. During a phase change, the temperature change pauses until the phase change is completed; then the temperature increase or decrease continues.
• 17
In climate science, longwave (LW) radiation is defined as electromagnetic radiation with a wavelength longer than 4 microns; and shortwave (SW) radiation is defined as radiation with a wavelength shorter than 4 microns The definitions are designed so that, in essence, all sunlight is SW and all thermal radiation from Earth’s surface and atmosphere is LW.