# Mathematical Proof of the Greenhouse Effect

by Bob Wentworth, Ph.D. (Applied Physics) | June 4, 2021

I am sometimes shocked by the number of climate change skeptics who are certain that the “Greenhouse Effect” (GHE) isn’t real.

As a physicist, I’m as certain of the reality of the Greenhouse Effect as I am that 1 + 1 = 2.

The GHE depends on physical principles that have been well-known and well-tested for 137 years. There really should be no question as to its reality, among anyone who knows and respects science.

Note that being certain about the GHE being real is different than being certain about Anthropogenic Global Warming (AGW), the hypothesis that human-caused increases in the concentrations of “greenhouse gases” in the atmosphere are causing highly problematic changes in the Earth’s climate.

AGW is a far more complex phenomenon than the GHE alone. One can be skeptical about AGW while totally accepting the reality of the GHE.

I know many readers are deeply skeptical about AGW. I encourage you to consider finding a way to honor your beliefs without denying the reality of the GHE.

Based on everything that’s known about physics, denying that the GHE is real seems to me to be just as wrong-headed as insisting that the Earth is flat. (Any Flat-Earthers here?)

Today, I’m going to do something that will likely be pointless, with regard to its ability to change anyone’s mind. But, for the record, I want to offer it anyway.

I’m going to offer a mathematical proof of the reality of the Greenhouse Effect.

I expect that skepticism about mathematics is likely to be common among folks who deny the reality of the GHE.

Oh, well. So be it.

* * *

There are various ways that the idea of the “Greenhouse Effect” might be expressed. Today, I’d like to focus on a formulation of the GHE that is simple and rigorously provable:

Suppose a planet (or object) absorbs shortwave (SW) radiant energy from the Sun (or another source of illumination), and loses energy by emitting longwave (LW) radiation into space at a known average rate.

Then, it follows that there is a maximum average temperature that the surface of the planet (or object) can have, unless there are materials capable of absorbing (or reflecting) LW radiation between it and space.

If the average surface temperature of the planet (or object) is higher than this limit, then that can only happen because of the presence of LW-absorbing (or reflecting) materials between the planetary surface (or object surface) and space.

When the average temperature of a planetary surface is higher than the temperature limit that would be possible in the absence of LW-absorbing materials in the atmosphere, this is called the “Greenhouse Effect” (GHE).

* * *

This result can be proven if one accepts a single principle of physics:

1. The rate at which LW radiation is emitted by the surface of the planet (or an object) is given by the Stefan-Boltzmann Law, Mₛ = ??⋅T⁴, where ? is the emissivity of the surface, ? is the Stefan-Boltzmann constant, and T is the temperature of the surface. (This quantity Mₛ is technically called the radiant exitance from the surface, and is measured in W/m².)

The Stefan-Boltzmann law was deduced based on experimental evidence in 1879, and was derived theoretically in 1884. This law has been a key part of the foundations of physics for 137 years, and has been verified countless times, in countless ways.

The reality and nuances of this law are as well-known and well-tested as anything in physics.

* * *

I will divide the proof into two parts. First, I’ll prove that there is a limit to how high the average surface temperature can be in the absence of LW-absorbing (or reflecting) materials. Then, I’ll show that LW-absorbing (or reflecting) materials create the possibility of the average surface temperature being higher.

Let’s define a few terms:

• T is the temperature of the surface of the planet (or object).
• Mₛ is the radiant exitance from the surface of the planet (or object). The subscript “s” is for “surface.”
• Mₜ is the radiant exitance into space from the top of the atmosphere of the planet (or from the materials associated with the object). The subscript “t” is for “top-of-atmosphere (TOA).”

Each of these quantities, T, Mₛ and Mₜ quantities may vary over the surface of the planet (or object) and vary in time as well.

I will use the notation ⟨X⟩ to denote the average of a quantity X over the surface of the planet (or object) and over some defined period of time.

Thus, the average values of surface temperature, surface radiant exitance, and TOA radiant exitance are ⟨T⟩, ⟨Mₛ⟩ and ⟨Mₜ⟩, respectively.

Let’s average each side of the Stefan-Boltzmann Law:

⟨Mₛ⟩ = ??⋅⟨T⁴⟩

This is the point where we come to the only fancy math in the entire proof.

There is a mathematical law, first proven in 1884, called Hölder’s Inequality. The general formulation of this inequality is rather abstract, and might be scary to a non-mathematician. However, what the inequality says regarding the current problem is very simple. Hölder’s Inequality says it will always be the case that:

⟨T⟩⁴ ≤ ⟨T⁴⟩

In other words, the fourth power of the average surface temperature is always less than or equal to the average of the fourth power of the surface temperature.

It turns out that ⟨T⟩⁴ = ⟨T⁴⟩ if T is uniform over the surface and uniform in time. To the extent that there are variations in T over the surface or in time, then this leads to ⟨T⟩⁴ < ⟨T⁴⟩.

(One of the reasons the surface of the Moon is so cold on average (197 K) is that its surface temperature varies by large amounts between locations and over time. This leads to ⟨T⟩⁴ being much smaller than ⟨T⁴⟩, which leads to a lower average temperature than would be possible if the temperature was more uniform.)

Combining the inequality with the equation preceding it, one finds:

⟨T⟩⁴ ≤ ⟨Mₛ⟩/??

In other words, if you know the average radiation emitted by the surface, then there is an upper limit to how hot the surface could be on average.

Let’s consider the case where there are no LW-absorbing (or reflecting) materials in the atmosphere of the planet (or in between the object and space).

It should be clear that in this situation, Mₜ = Mₛ. The rate at which radiant energy reaches space must be identical to the rate at which radiant energy leaves the surface, if there is nothing to absorb or reflect that radiation.

So, in this situation,

⟨T⟩⁴ ≤ ⟨Mₜ⟩/??

We can re-write this as

T ≤ Tₑ

where the radiative effective temperature Tₑ is given by

Tₑ⁴ = ⟨Mₜ⟩/??        [equation 1]

In other words, if you know how much radiation is emitted at the top of the atmosphere, and if you know there are no LW-absorbing (or reflecting) materials in the atmosphere, then you can calculate the radiative effective temperature Tₑ and you can be certain that the average temperature of the surface will not be larger than this value.

* * *

Often, the “Greenhouse Effect” (GHE) is expressed in relation to the insolation, or the rate of energy being absorbed by the planet. Under an assumption of “radiative balance,” the average insolation is equal to the ⟨Mₜ⟩, the average rate at which LW radiant energy is emitted into space.

However, there can be small discrepancies between the average insolation and the rate of energy being emitted into space. And, some people who don’t trust climate science dispute the assumption of radiative balance.

So, I’m choosing to offer a formulation of the GHE which is valid even in the absence of radiative balance between the rates of energy being received and emitted by the planet (or object).

If you know the rate at which LW radiant energy is being emitted by the planet (or object), then there is a limit to how warm the planet can be without LW-absorbing (or reflecting) materials.

* * *

What happens if there are materials present that absorb (or reflect) some of the LW radiation emitted by the surface, before it can get to space?

This creates the possibility that the rate of LW radiation being emitted to space could be different than the rate of LW radiation being emitted from the surface. In other words, such materials create the possibility that Mₛ ≠ Mₜ.

Let’s define the “LW enhancement” ∆M as ∆M = (Mₛ − Mₜ).

On Earth, ∆M is generally positive. More LW radiation is emitted by the surface than reaches space. This is possible only because of the presence of materials in Earth’s atmosphere which absorb (or reflect) LW radiation.

(In Earth’s atmosphere, there is more LW absorption than reflection. However, some reflection of LW radiation does occur in the form of LW scattering by aerosols and clouds. For purposes of this analysis, “reflection” and “scattering” are interchangeable concepts.)

If we go back to the inequality above that was expressed in terms of ⟨Mₛ⟩, and apply the definition of LW enhancement, we can rewrite the inequality as

⟨T⟩⁴ ≤ ⟨Mₜ⟩/?? + ⟨∆M⟩/??

Applying the definition of the effective radiative temperature Tₑ we can further rewrite the inequality as:

⟨T⟩⁴ ≤ Tₑ⁴ + ⟨∆M⟩/??      [equation 2]

Equations 1 and 2 together offer a formal expression of the “Greenhouse Effect” (GHE).

What do these equations say? They say that:

1. Given the average LW radiant exitance at the top of the atmosphere, you can calculate a radiative effective temperature Tₑ.  (To the extent that radiative balance applies, one could alternatively use the average absorbed insolation to calculate Tₑ.)
2. In the absence of materials in the atmosphere that absorb (or reflect) LW radiation, it would be impossible for the average temperature of the planet to exceed Tₑ.
3. If there are LW-absorbing (or reflecting) materials in the atmosphere, then this creates the possibility of the average surface temperature being higher than Tₑ.
4. How much higher than Tₑ the average surface temperature could be is determined by how much the average LW surface radiant exitance ⟨Mₛ⟩ exceeds the average LW TOA radiant exitance being emitted to space ⟨Mₜ⟩.

In this formulation, the GHE refers to the phenomenon of LW-absorbing (or reflecting) materials making it possible for the average surface temperature to be higher than would otherwise be possible.

I’ve shown that a single principle of physics (the Stefan-Boltzmann Law) sets a limit on how high the average surface temperature can be, and says that this limit can be increased if and only if there are LW-absorbing (or reflecting) materials present in the atmosphere.

* * *

How does this apply to Earth?

Earth’s atmosphere includes LW-absorbing-or-scattering materials such as water (in the vapor, liquid and solid phases), aerosols, carbon dioxide, methane, nitrous oxide, ozone, and fluorinated gases.

Equations 1 and 2 allow us to assess whether the LW-absorbing (or LW-scattering) properties of these materials are essential to accounting for the Earth’s average surface temperature.

Let’s put in some numbers. I’ll use poster data from NASA averaged over a 10-year period. (The results wouldn’t be much different if another data source was used.) That data indicates an average LW TOA radiant exitance ⟨Mₜ⟩ = 239.9 W/m².

(The absorbed SW insolation is given as 240.4 W/m², which is almost, but not quite, in balance with the LW TOA radiant exitance. This imbalance is evidence that Earth was not in steady-state, but experienced a net warming over the decade of measurement.)

The data indicates an average LW enhancement ⟨∆M⟩ = 158.3 W/m².  As a reminder, the LW enhancement ⟨∆M⟩ isn’t a measure of “back-radiation.” It’s a measure of how much more LW radiation leaves the surface than reaches space.

If we assume an average surface emissivity ? = 0.94, then equations 1 and 2 lead to:

Tₑ = 259 K (-14℃)

⟨T⟩ ≤ 294 K (21℃)

In other words:

1. If there were no LW-absorbing (or LW-scattering) materials in Earth’s atmosphere, and it emitted the same average LW radiant exitance (upwelling LW radiation) to space (which would be expected in steady-state if the absorbed insolation was held constant), then the average surface temperature could not be warmer than Tₑ = 259 K (-14℃).
2. Given that Earth’s atmosphere does include LW-absorbing and LW-scattering materials which allow there to be more LW radiation emitted by the surface than what reaches space, the average surface of the Earth can be no higher than 294 K (21℃).

Given that the average surface temperature of the Earth is typically estimated to be about 288 K (15℃), this satisfies the constraint of being no higher than 294 K (21℃).

According to equation 1 and this particular data set, the surface of the Earth is 29℃ warmer than it could possibly be, given the same average LW TOA radiant exitance, if there were no LW-absorbing (or scattering) materials in the atmosphere.

(The more typically quoted figure of 33℃ would result if one assumed an emissivity ? = 1.)

This result demonstrates that the presence of LW-absorbing and LW-scattering materials in the atmosphere is mathematically essential to explaining at least 29℃ of the Earth’s current temperature, provided only that one accepts the Stefan-Boltzmann Law.

* * *

Note that this result (that LW-absorbing materials are needed to enable the Earth to be as warm as it is) is entirely independent of any details of what happens in the atmosphere and ocean.

Convection, heat engines, ocean currents, thermal storage, turbulence, atmospheric pressure—none of these make the slightest difference to the basic conclusion.

No matter what physical processes happen on Earth, its average surface temperature would be need to be colder, if it were not for the presence of LW-absorbing materials in the atmosphere.

* * *

* * *

* * *

APPENDIX 1: “Proof” in the Context of Science

The term “proof” is generally reserved for mathematics, and is not used in science. In science, one doesn’t “prove” things; one offers evidence that confirms or disconfirms the predictive accuracy of a hypothesis or theory.

So, what do I mean when I say I’m “proving” the GHE?

Technically, I proved that the GHE is mathematically an inherent consequence of the Stefan-Boltzmann Law.

The reality of the GHE effect is equivalent to the reality of the Stefan-Boltzmann Law.

The offered “proof” implies that any evidence confirming the Stefan-Boltzmann Law should also be considered to be evidence confirming the GHE.

There has been enormous evidence over 137 years confirming the predictive accuracy of the Stefan-Boltzmann Law. It is a key component in the foundations of physics.

APPENDIX 2: Does the GHE Offer More Specific Predictions?

Some readers may feel frustrated that the GHE, as I’ve formulated it, doesn’t offer any specific predictions for what surface temperatures should result from LW-absorbing (or reflecting) materials being present in the atmosphere.

Maybe you take issue with the results of climate models and you want to refute the predictions that arise from “assuming the GHE exists.”

Maybe it would be nice to be able to identify “the part of these models that is the GHE” so that that part can be separately tested.

I think this sort of thinking reflects a misunderstanding of the nature of the GHE.

The GHE is not a specific process. It’s an emergent phenomenon that arises from the basic laws of physics.

Modelers do not “add the GHE” to their models. They build climate models using the established laws of physics, with some model components being addressed empirically. (How well models reflect the basic laws of physics may vary.)

The GHE simply arises when one takes the laws of physics into account. It’s not something separate that one adds to a model.

There are no specific predictions that the GHE alone gives rise to. There are only the predictions that arise from the laws of physics. Sometimes, some aspect of these predictions may be attributed, after the fact, to the “Greenhouse Effect.”

But, the GHE is not a separate theory. It’s an observation of the consequences of the fundamental theories that form the foundations of modern physics.

APPENDIX 3: But How Does the GHE Work?

There are a variety of ways of talking about the GHE.

Some approaches focus on explaining how LW radiation absorbing-and-emitting gases can raise the surface temperature. People engaging with such explanations often get mired down in disputing details.

In this essay, I’m taking a different approach. What I’ve offered here makes no attempt to explain how LW-absorbing (or scattering) materials can raise the average surface temperature.

Instead, I’m offering an analysis that simply says, if a planetary surface exceeds a certain average temperature, Tₑ, then it’s certain that LW-absorbing (or scattering) materials must play an essential role in whatever process causes this warming to happen.

While the approach in this essay doesn’t offer any explanation of “how,” it arguably makes up for that by being so ridiculously simple that there would appear to be no legitimate loopholes for disputing it.

If you follow the logic offered here, it should be clear that the GHE is real.

Once one has accepted the GHE as real, I imagine there might be more motivation to work through and understand the explanations offered elsewhere about how the GHE works. Without being committed to trying to prove the GHE wrong, it is likely to be easier to understand how works.

(Do I expect that anyone will follow this path? Probably not. Yet, I’ve done what I can to offer the opportunity.)

APPENDIX 4: Variations in Emissivity

An astute reader might notice that the analysis above did not account for variations in the emissivity, ?. If one takes this into account, the key equations become:

Tₑ⁴ = ⟨Mₜ/?⟩/?

⟨T⟩⁴ ≤ Tₑ⁴ + ⟨∆M/?⟩/?

This refinement to the result doesn’t change the basic conclusion.

A majority of the Earth’s surface is ocean with an emissivity of about 0.96.   Emissivity on land is mostly greater than 0.9, though it sometimes dips lower. Suppose we conservatively estimate 67% of the planet to be open ocean with an emissivity of 0.96, estimate that 80% of land has an emissivity of at least 0.85, and the remainder has an emissivity of at least 0.6.

This would lead to an effective emissivity, for purposes of calculating Tₑ, of about ?ₑ ⪆ 1/(0.67/0.96 + 0.264/0.85 + 0.066/0.6) = 0.89. While this is a crude calculation (and ignores the need to weight in proportion to the TOA radiant exitance), it represents an approximate “worst case”; the actual effective emissivity will be higher than this.

An effective emissivity of 0.89 would lead to Tₑ = 263 K (-11℃). This is still about 26℃ colder than Earth’s observed average surface temperature.