In 2017, Ned Nikolov and Karl Zeller (N&Z) published a paper on their work to identify an empirical model for planetary temperatures (*“New Insights on the Physical Nature of the Atmospheric Greenhouse Effect Deduced from an Empirical Planetary Temperature Model”*).[1]

The empirical model they present offers a formula for Global Mean Annual near-surface Temperature (GMAT) which depends on atmospheric effects only through the average near-surface atmospheric pressure, **P.** The model was developed via a curve-fitting process and matches the temperatures of six planetary bodies.

The model does not depend on any measure of atmospheric greenhouse gases. This is surprising, insofar as mainstream science says that greenhouse gases account for why planets with atmospheres are warmer than the planet would be in the absence of an atmosphere. N&Z offer hypotheses that purport to explain how something other than greenhouse gas absorption and re-radiation of longwave radiation could be responsible for atmospheric warming of planets.

The findings of this paper have been received with enthusiasm by some of the people who reject the idea that greenhouse gases warm planets.

What does this result mean? *Has the greenhouse effect based on longwave absorption and emission been disproven, and has an alternative explanation for planetary warming been established?*

**No, definitely not.**

- Nikolov and Zeller perceived a problem that wasn’t really there, a discrepancy that they believed to be present in standard models of atmospheric warming.
- They set out to solve the perceived problem empirically, through a curve fitting process, largely ignoring knowing physics.
- They discovered a correlation relating temperature to pressure. They interpreted this relationship as significant and unique—but I show that it’s not significant or unique. (In this answer, I offer an equally precise and significant correlation relating temperature to amounts of greenhouse gases.)
- They interpreted that correlation as implying causation, offering their formula as a new “natural law.” However, that “law” is definitely wrong. I show this using simple logic and core principles of physics.
- The discrepancies that N&Z believed to exist in the standard theory were based on a flawed temperature calculation they had done and a failure to understand how energy recirculation can increase power levels. If they had had a good conversation with the right person, they could have cleared up what was puzzling them years ago, and might have saved themselves a lot of work.

So, as I interpret things, it seems to me like a sad story. I’ve spent an intense week or so unraveling what to make of their paper. I’ve learned quite a bit along the way. I share what I’ve learned in this answer.

I want to fully address the issues raised by N&Z’s paper. If you’re interested in the topic and are willing to stick with me, I hope that you will learn something.

**Contents** Described

This is a long essay. I’ll offer an outline to orient you, in case you want to just sample what interests you.

**Correlation**

Correlation doesn’t imply causation.- I
**s N&Z’s Formula for Temperature Based on Pressure Significant and Unique?**

I demonstrate that N&Z’s formula isn’t special, by showing that formulas I developed, which depend on the amount of greenhouse gases present, work just as well as N&Z’s formula in predicting planetary temperatures. **Connection to Physics: Conservation of Energy**

I talk about the physics that determines temperature.**Connection to Physics: Hypothesis that Pressure Determines Temperature**

I offer a simple “big picture” argument that demonstrates that the idea of pressure causing planets to be warmer can’t possibly be true.

For those interested, I also explain why the existence of an “adiabatic lapse rate” doesn’t mean that adiabatic processes can heat a planet.**Connection to Physics: Hypothesis that Radiative Properties do not affect Temperature**

I offer a simple “big picture” argument to demonstrate that core principles of physics say it must be true that greenhouse gases warm planets.**What Led Nikolov and Zeller Down this Path: Temperature Gap**

I identify where N&Z went wrong in believing the atmosphere needs to account for 90ºC of warming on Earth, not just 33ºC.**What Led Nikolov and Zeller Down this Path: Power Levels**

I explain how it can make sense that the surfaces of Earth and Venus receive more radiant power from greenhouse gas back-radiation than they do from the Sun.**Conclusions****APPENDIX: TECHNICAL DETAILS**A1. Details of Formulas for Atmospheric Temperature Enhancement

A2. Assessing the Significance of these Models

A3. A Nuance About Energy Flows vs. Heat Flows

A4. Analysis Indicating Rotation Rate Does Matter

Related Information from Others

## Content Links

- Contents Described
- 1. Correlation
- 2. Is N&Z’s Formula for Temperature Based on Pressure Significant and Unique?
- 3. Connection to Physics: Conservation of Energy
- 4. Connection to Physics: Hypothesis that Pressure Determines Temperature
- 6. What Led Nikolov and Zeller Down this Path: Temperature Gap
- 7. What Led Nikolov and Zeller Down this Path: Power Levels
- 8. Conclusions
- 9. APPENDIX: TECHNICAL DETAILS
- Related Information from Others
- Footnotes

**1. Correlation**

The model of N&Z offers a formula that matches the planetary temperature (GMAT) values N&Z are using for six celestial bodies (Venus, Earth, Moon, Mars, Titan, Triton) to within 1.5ºC.

This demonstrates a correlation between the value of their formula, and the temperatures of these bodies.

It is a widely appreciated principle that **correlation does not imply causation**.

It is easy to find countless examples of correlations that are purely coincidental, with no underlying causal connection involved.[2][3][4][5]

See, for example, the chart above, which shows a close correlation between arcade revenues and the number of computer science doctorates awarded. There is strong correlation. But, the idea of causation, while amusing, is ultimately implausible.

At most, correlation can suggest the possibility of causation. When considering the possibility of a causal connection, it’s important to look at both (a) the case for believing the correlation is statistically significant, and (b) whether or not there are any underlying factors that could justify or rule out a causal connection.

**2. Is N&Z’s Formula for Temperature Based on Pressure Significant and Unique?**

N&Z’s formula for calculating planetary temperature (GMAT) from the near-surface atmospheric pressure, P. They write (p. 17):

“Our analysis revealed that the equilibrium global surface temperatures of rocky planets with tangible atmospheres and a negligible geothermal surface heating can reliably be estimated across a wide range of atmospheric compositions and radiative regimes using only two forcing variables: TOA [top of atmosphere] solar irradiance and total surface atmospheric pressure… the relative atmospheric thermal enhancement (RATE) defined as a ratio of the planet’s actual global surface temperature to the temperature it would have had in the absence of atmosphere is fully explicable by the surface air pressure alone… At the same time, greenhouse-gas concentrations and/or partial pressures did not show any meaningful relationship to surface temperatures across a broad span of planetary environments considered in our study.”

N&Z appear to conclude that (a) the correlation with a function of pressure that they discovered offers a uniquely good fit to temperature data, and further infer that (b) this correlation must imply causation. They conclude that pressure causes temperature, greenhouse gases don’t.

All parts of this conclusion are demonstrably wrong. In this section, I’ll address the error in believing that the correlation they found is significant and unique.

There is little reason to believe that the correlation N&Z discovered is as significant as they imagine it to be.

One argument for this lack-of-great-significance goes as follows. N&Z assumed a particular functional form with 4 tunable parameters (regression coefficients), tried curve fitting using that functional form with a variety of variables, and found one variable and set of parameters that did a good job of fitting 6 data points. However, one of these data points (the Moon) had been arranged to be well-fit by the model almost regardless of what parameter values were chosen. So, N&Z were effectively fitting only 5 data points. And, for the particular variable they ended up selecting, two data points (Earth and Titan) ended up being nearly indistinguishable. So, to achieve a good fit, they essentially only needed to tune their 4-parameter model to fit 4 data points. Because they chose a functional form that can’t conform to all data patterns, it wasn’t automatic that they would succeed. But, neither was finding a good fit a miracle, or even very surprising.

As the previously mentioned examples of spurious correlations illustrate, it’s not at all unheard of to stumble across things that seem to line up, even though it’s purely chance.

I can make the argument that what N&Z found was just coincidence, and in the Appendix I apply a statistical test that supports this conclusion. Yet, you still might not believe me.

So, let’s demonstrate that there is more than one way of getting lucky at “predicting” planetary temperatures.

N&Z tried to get a version of their model to work using greenhouse gas data instead of pressure. But, it didn’t work. Their attempts at curve fitting used a particular functional form. Yet, there is no reason whatsoever to believe that the functional form that NZ chose is the only way that nature could work.

So, I tried a different functional form, and used it to see if I could match planetary temperatures using measures of the amount of greenhouse gases present.

I used the data supplied in N&Z’s 2017 paper.

After just a few hours of experimentation, I was able to come up with a few closely related formulas, each of which “predicts” planetary temperatures as well as N&Z’s formula does. My formulas work by considering the three greenhouse gases CO₂, CH₄, and H₂O individually, rather than lumping them together as N&Z had done.

Let’s look at the results.

Like N&Z, I’m focusing on how warm the planetary surface temperature, T, is relative to the temperature that would be predicted (by N&Z) if there was no atmosphere, Tₙₐ.

The above chart shows actual temperature data, the results of N&Z’s 4-parameter model based on pressure, and the results of 3 variations of my model based on the amounts of the various greenhouse gases. Model GH6 has 6 tunable parameters (regression coefficients). Models GH4a and GH4b each have only 4 tunable parameters, like NZ’s model.

Like N&Z’s choice of functions, the functional form I used is somewhat “rigid” so that it can’t just automatically fit any set of data.

(Note that there is every reason to believe that the underlying physics would need quite a few parameters per gas type to describe how each gas affects things. So, reducing the number of parameters to 4 total parameters addressing 3 distinct gases surely vastly oversimplifies the physics. But, I wanted to prove a point by using as few parameters as N&Z had.)

Details of each model are provided towards the end of this essay, in the Appendix.

In the above figure, all the models seem to fit the data pretty well. The exceptions are that models GH4a and GH4b are a bit off for Triton, and the NZ4 model is a bit off for Titan.

We can see the accuracy of the models more clearly by plotting the residual errors.

In the above chart, I’ve normalized the residual errors (difference between the model prediction and the observed value) relative to the uncertainty (standard deviation) in the observed value.

I used the uncertainty values offered by N&Z. But, note that I believe N&Z have significantly underestimated some of the actual uncertainties in their data. They repeatedly write things like “there is no consensus value” and then proceed to offer a value for which they assign an uncertainty only slightly larger than the uncertainty for a body for which we have consensus and vastly more data. So, please take the uncertainty values with a big “grain of salt.”

Examining the residual errors in the chart, we see that all three models involving greenhouse gases have much smaller errors than N&Z’s pressure-based model, for 4 of the 6 celestial bodies. The 6-parameter GH6 model has much smaller errors than NZ’s model for 5 of 6 bodies (i.e., for every body with a non-negligible atmosphere).

All the models do very well overall, at matching the observed temperatures.

* * *

What does this mean?

It demonstrates that **the formula N&Z discovered does not constitute a uniquely-good correlation to planetary temperatures.**

It further demonstrates that **comparably good correlation can be achieved using formulas that rely on measures of greenhouse gases**, rather than on overall pressure.

* * *

So, which of these formulas do I believe reflects the reality of how planetary temperatures are determined?

None of them. **They are all almost certainly simply chance correlations.**

The real physics that determines planetary temperatures is known, and is more complex than could plausibly be accurately represented by such simple formulas.

(Among other things, different celestial bodies are at very different temperatures, so which absorption bands of a given greenhouse gas interact with thermal radiation is likely to be different for different bodies. As a result, there is little reason to expect that the effect of a given gas would be numerically similar for different bodies. Another complication is that N&Z’s calculation of temperatures in the absence of an atmosphere appears to be wrong, as discussed in section 6 on this answer; that casts doubt on the significance of any calculation of “atmospheric temperature enhancement” relative to the results of that calculation.)

What we can say with certainty, if we examine the physics, is that *pressure alone does not and cannot raise planetary temperatures*, and *greenhouse gases can and do raise temperatures.* I will justify these assertions in what follows.

**3. Connection to Physics: Conservation of Energy**

N&Z apparently believe that the ability of greenhouse gases to warm planets is merely an unproven “hypothesis.” They write (p. 16):

“The hypothesis that a freely convective atmosphere could retain (trap) radiant heat due its opacity has remained undisputed since its introduction in the early 1800s even though it was based on a theoretical conjecture that has never been proven experimentally.”

Such a stance seems to be common among people who deny the possibility of anthropogenic global warming (AGW).

But, **such a stance is sustainable only if one obstinately refuses to think clearly about conservation of energy, and the relationship between energy and temperature.**

What determines the temperature of an object?

Consider an object whose temperature, T, is to be determined. Energy flows into the object at rate P_{in}(energy/time) and leaves the object at rate P_{out}. In equilibrium, it must be the case that P_{in}=P_{out}. If the energy flowing in and the energy leaving is not the same, then by conservation of energy, the total energy content of the object will change, and the object will not be in “steady state.”

How a balance is achieved, in which the power-in equals the power-out, is a consequence of the way that thermal energy content affects temperature which in turn affects energy flows. More total energy usually means a higher temperature. (The exception is when some mechanism is storing energy non-thermally. But, this exception is irrelevant if the stored energy is in steady state, since in that case non-thermal energy storage does not affect the net flow of thermal energy.)

A hotter object loses more energy, in the form of heat flows (whether that is through radiation, conduction, convection, or latent heat flow). The temperature of the object adjusts until it reaches a point where P_{in}=P_{out} and that is what sets the temperature of the object.

So, the balance of power flows is what fundamentally determines temperature.

(Details of interest only to some people: If an object doesn’t have a single, uniform temperature, that adds some complications. People who deny the reality of planetary warming due to the absorption and re-radiation of longwave radiation sometimes obsess about these complications, as if that might offer them a way “out” from reaching the conclusion they want to avoid. But, those complications do not change the big picture. The balance of heat flows determines temperatures. And it can be very instructive to look at situations where there is a single uniform temperature in an object, because, in such cases, it becomes blatantly obvious that the sort of warming effects (which some people suspect are not real) must, in fact, be real.)

In general, tracking power flows can be complicated. But, there are also ways in which it can be pretty simple.

With regard to planetary temperatures, there are two places where it is relatively simple to look at power flows:

- At the interface between the surface of the planet and the atmosphere.
- At the interface between the atmosphere and space.

Considering energy balance at either of these interfaces makes it clear that greenhouse gases *must* raise planetary temperatures.

It’s not simply a “hypothesis” or a “conjecture.” It follows directly from the fundamental physics involved, including conservation of energy and the way that energy content affects temperature which affects power flows.

I’ll spell this out in detail in subsequent sections (especially section 5).

**4. Connection to Physics: Hypothesis that Pressure Determines Temperature**

N&Z seem to hypothesize that what they found is not just a correlation, but a true physical phenomenon in which pressure is responsible for planetary warming. They write (p. 12):

“The pressure-induced thermal enhancement at a planetary level portrayed in [a plot of the model vs. actual temperatures] and accurately quantified by [the model formula] is analogous to a compression heating, but not fully identical to an adiabatic process. The latter is usually characterized by a limited duration and oftentimes only applies to finite-size parcels of air moving vertically through the atmosphere. [The model for temperature increase via pressure], on the other hand, describes a surface thermal effect that is global in scope and permanent in nature as long as an atmospheric mass is present within the planet’s gravitational field. Hence, the planetary RATE [“Relative Atmospheric Thermal Enhancement”] could be understood as a net result of countless simultaneous adiabatic processes continuously operating in the free atmosphere…. [The model] describes an emergent macro-level property of planetary atmospheres representing the net result of myriad process interactions within real climate systems.”

I might paraphrase that as follows:

It’s known that adiabatic compression of gases results in heating. But, that’s only a temporary, localized effect. We hypothesize that lots of these temporary effects could happen in a way that adds up to a permanent, global heating effect.

Unfortunately, it is **fundamentally impossible that this hypothesis could be true. **What is being hypothesized would be a sort of perpetual motion machine, creating heat out of nothing.

We can see this either at a “big picture” level, or by considering the specific mechanisms that N&Z hypothesize.

* * *

To see the big picture, assume that what N&Z says is true: that an atmosphere without any greenhouse gases would warm a planet.

Suppose our planet initially has no atmosphere, and has a temperature T_{na}. The surface of the planet is receiving energy from the Sun, P_{in}. This is balanced by energy being radiated to space with power P_{out}, where P_{out}=P_{in.} Now we add an atmosphere which is transparent to electromagnetic radiation, and allow the situation to come to a new equilibrium. If N&Z’s pressure-warms-planets hypothesis is true, the temperature rises to a new steady state temperature T > T_{na}. But, according the Stefan-Botzmann law, the power radiated is proportional to 𝜀𝜎T^{4}. So, the power radiated by the surface has increased. Because the atmosphere is assumed to be transparent to thermal radiation, all of what was radiated by the surface will reach space, so P_{out} will be larger than it was. Now, P_{out} > P_{in}. This means that the system is not in equilibrium, and T is not the steady-state temperature of the planet. The power flow imbalance will cause the surface of the planet to cool down, contradicting the hypothesis that the atmosphere had created a new, higher, steady-state temperature for the surface of the planet.[6]

The only way around this logic is if the atmosphere spontaneously creates energy out of nothing, and uses this something-from-nothing energy to increase the energy flow P_{in} to the surface. However, energy conservation rules out this possibility.

Therefore, **an atmosphere without greenhouse gases cannot warm a planet.**

It’s actually slightly worse than that. An atmosphere transparent to radiation won’t alter radiative energy flows. But, it will introduce additional heat flows away from the surface. In particular, convection and possibly latent heat flow (from evaporation of volatile compounds) will increase 𝑃𝑜𝑢𝑡Pout from the planetary surface. Therefore, **an atmosphere transparent to thermal radiation will tend to** (at least slightly) **cool the planetary surface.**

* * *

*Those for whom the big picture explanation was sufficient might want to skip to the next section (on “Radiative Properties”).*

We can also see the non-viability of the pressure-causes-heating hypothesis by looking at the sort of atmospheric processes N&Z hypothesize could lead to heating.

Although adiabatic processes associated with convection do produce localized heating and cooling, there is *zero net heating* effect in the atmosphere as a whole. There are two equivalent ways to see that this must be true:

- When packets of air move downward through the atmosphere, they compress and get warmer. When packets of air move upward through the atmosphere, they expand and get cooler. Yet,
*the amount of air moving downward and upward is always exactly equal*, so that zero net heat is added to the atmosphere. - Adiabatic cooling and heating of air as it changes altitude can be thought of as a trade-off between gravitational potential energy and thermal energy. As gravitational potential energy decreases, heat increases, and vice versa. If we look at the gravitational potential energy of the atmosphere as a whole, we know that
*the total gravitational potential energy is not changing*since the total mass and density vs. altitude profile of the atmosphere are stable. Consequently, by conservation of energy, there can be no net heating of the atmosphere via adiabatic processes. (Even if the atmospheric mass or density profile changed, this would produce at most a transient effect, not an ongoing infusion of energy.)

So, **adiabatic processes (associated with convection) add no net energy or net heat to the system.**

But what about the “adiabatic lapse rate”?[7] We know, after all, that adiabatic processes lead to the atmosphere being warmer at low altitudes and cooler at high altitudes (within the part of the atmosphere where convective mixing is strong).

Doesn’t that prove that adiabatic processes can affect temperature?

Yes and no. Adiabatic processes affect *relative temperature (at different altitudes)*, but *not absolute (overall) temperature.*

In the troposphere (the lower part of the atmosphere where convection is active), upward and downward moving air flows experience adiabatic cooling and heating in ways that help establish a particular rate of temperature change vs. altitude within the atmosphere, i.e., they affect the lapse rate.

However, these processes do not set the absolute temperature. The surface of the planet could get 2ºC cooler or warmer, and everything in the troposphere would also get about 2ºC cooler or warmer. The lapse rate only sets relative temperatures at different altitudes, not the baseline temperature. Adiabatic heating and cooling can’t affect the baseline temperature because these processes produce no net heat overall.

In particular, overall, adiabatic heating and cooling is associated with zero net heat transfer to the planetary surface (aside from the associated convective transfer of heat away from the surface—which cools the surface, rather than warming it). Adiabatic processes can and do transfer heat between different parts of the surface, effectively transferring heat from high altitude locations like mountains to low altitude locations like valleys. But, these effects balance out globally.

The temperature of a planetary surface is, from first principles, determined by the temperature at which energy received and energy departing that surface are in balance. Adiabatic processes provide no net thermal energy to heat the surface, and the associated convection cools the surface.

N&Z also suggest (p. 13):

“Pressure as a force per unit area directly impacts the internal kinetic energy and temperature of a system in accordance with thermodynamic principles inferred from the Gas Law; hence, air pressure might be the actual physical causative factor controlling a planet’s surface temperature…”

The Ideal Gas Law specifies a relationship between pressure, volume, amount of gas, and temperature.[8] The Gas Law does not mean that pressure “causes” temperature. **For a given pressure, a gas may have any temperature.** The Gas Law tells us that **pressure affects temperature only insofar as changes in pressure can alter temperature**, as during adiabatic expansion or compression. Yet, we have already established that adiabatic cooling and heating introduce no net heat globally. So, invoking the Gas Law adds nothing to the plausibility of pressure determining planetary temperatures.

**The idea that atmospheric pressure could increase planetary temperature is incompatible with conservation of energy.** The hypothesis would require energy to appear out of nowhere on an ongoing basis.

**5. Connection to Physics: Hypothesis that Radiative Properties do not affect Temperature**

N&Z hypothesize that the correlation they have found means that the radiation absorption and emission characteristics of the atmosphere do not affect a planet’s temperature. They write (p. 13):

“air pressure might be the actual physical causative factor controlling a planet’s surface temperature rather than the atmospheric infrared optical depth, which merely correlates with temperature due to its co-dependence on pressure. Based on evidence discussed earlier, we argue that [this] is the most likely reason for the poor predictive skill of greenhouse gases with respect to planetary GMATs [Global Mean Annual near-surface Temperatures] revealed in our study.”

N&Z offer some vague philosophical reasons (related to dimensional analysis and the significance of pressure) as to why they find this hypothesis plausible. Philosophical reasoning can occasionally be helpful in sorting out science. But, it’s often unreliable. I don’t personally find N&Z’s philosophical reasoning compelling.

The main substantive claim N&Z make relates to their assertion that climate models do not properly account for the interaction of different forms of heat transport within the lower atmosphere (i.e., the troposphere). They argue that the effects of convection dominates other forms of heat transport here, so that (p. 14):

“a correctly coupled convective-radiative system will render the surface temperature insensitive to variations in the atmospheric infrared optical depth”

This argument references a sliver of truth, but reaches a false conclusion.

Convection and latent heat flux (evaporation, transport of vapor, and condensation) are very important aspects of heat flow within the troposphere. At the interface between the surface and the atmosphere these effects are important, but radiation is not negligible. Quantitative estimates that I’ve seen suggest that for heat flow from the Earth’s surface, 52% is latent heat flux (evaporation, vapor transport, condensation), 33% is net radiative heat flux, and 15% is sensible heat flux (convective movement of warm air).

However, that’s not the whole story. In the stratosphere, there is a temperature inversion that prevents convection. Such a temperature inversion in the upper atmosphere is typical of planets with atmospheres.[9] The lack of convection in this region largely eliminates sensible heat flux, and greatly reduces latent heat flux (since convection is a significant mechanism for transporting water vapor).

Radiative heat transport becomes much more important to heat transport within the upper atmosphere. And, **how efficiently heat flows through the upper atmosphere and out to space affects the temperature of the lower atmosphere and the planetary surface.**

Yet, it’s very easy to get lost in details. Qualitative reasoning about these details and their implications is unlikely to lead to accurate conclusions. Is there some “big picture” way we can bypass all these details to establish whether or not longwave radiation absorption and re-emission is important in determining global temperature? I believe there is.

Let’s consider energy balance at the interface between the atmosphere and space.

The chart below shows the spectrum of longwave radiation leaving the Earth as measured from above the atmosphere.[10]

The red curve shows the predicted spectrum of a black-body at 294 K. This is essentially the spectrum of the radiation we expect to be emitted by the Earth’s surface. (The actual emission spectrum of the Earth’s surface would have some small irregularities since it is a “grey-body” rather than a “black-body.” But since the surface of the Earth has an average emissivity of 0.94, we would expect the spectrum of radiation from the surface to be pretty close to what is shown by the red curve.[11])

The black curve with blue underneath shows how much longwave radiation the Earth actually emits. One can distinctly see the places in the spectrum where absorption by H₂O and CO₂ and other greenhouse gases reduces the amount of longwave radiation that reaches space.

Now, consider this: in equilibrium, the energy flux of shortwave radiation from the Sun that is absorbed by the Earth (averaged over a day), P_{in}, equals the energy flux of longwave radiation leaving the Earth, P_{out}. (This must be the case if the total thermal energy of the Earth, and the average temperature of the Earth, are to be relatively stable.)

**What would happen if suddenly we magically turned off the longwave absorption and re-emission properties of all the greenhouse gases?**

The dips in the above chart due to greenhouse gas absorption would fill in. The thermal radiation emitted by the Earth’s surface would all reach space, and the radiated spectrum would be much closer to what is shown by the red curve. Most importantly, the total flux of power being radiated to space, P_{out}, would increase.

But, wait! Radiant power received and radiant power lost were in balance before. If the power being lost, P_{out}, increases, that means things are now imbalanced: more energy is being lost than is being received, i.e., P_{out} > P_{in}.

That means the total thermal energy of the Earth must decrease, and the Earth would become cooler.

So what have we shown? **If you turned off the longwave absorption properties of greenhouse gases, this would inevitably lead to the Earth becoming cooler! **Conversely, this makes it clear that the Earth is warmer with greenhouse gases than it would be without them.

**This argument shows that greenhouse gases increase the temperature of the Earth. The argument depends only on: fundamental principles of physics, the observable fact that greenhouse gases produce dips in the spectrum of radiation emitted into space, and simple logic.**

*This argument is completely independent of the details of what happens inside the atmosphere.*

(This simple and definitive argument was suggested to me by David Borojevic. It seems obvious in retrospect.)

**The hypothesis that the absorption and re-emission of longwave radiation by greenhouse gases does not lead to planetary warming is simply not consistent with straightforward measurements and core principles of physics.**

**6. What Led Nikolov and Zeller Down this Path: Temperature Gap**

I think I’ve made a strong case that Nikolov and Zeller came to some seriously flawed conclusions.

Yet, in reading their paper, I get a sense of mathematical competence and an understanding of some of the relevant physics. So, what led them down a path that I think has ultimately not been fruitful?

If you’ll bear with me, just a little bit longer, I think it is instructive to examine how N&Z explain what led them to the work presented in their 2017 paper. They write (p. 1):

“In a recent study Volokin and ReLlez demonstrated that the strength of Earth’s atmospheric Greenhouse Effect (GE) is about 90 K instead of 33 K as presently assumed by most researchers. The new estimate corrected a long-standing mathematical error in the application of the Stefan–Boltzmann (SB) radiation law to a sphere pertaining to Hölder’s inequality between integrals.”

The 2014 paper nominally written by Volokin and Rellez[12] was in fact written by Nikolov and Zeller using pseudonyms, as they admitted in a 2016 published errata to that paper.[13] In their 2014 paper, N&Z predict that the global mean average temperature (GMAT) of Earth without an atmosphere should be 197.0 Kelvin, 90 K lower than the observed global average of 287.4 K.

That 197 K temperature for an airless Earth is much lower than what other researchers calculate. A. P. Smith (2008) does a similar calculation, and notes that the surface of the Earth experiences relatively small daily temperature variations as a result of “rapid rotation and the high heat capacity of water covering most of the surface.” Because of these factors, Smith’s calculations indicate that an airless Earth that somehow still had the heat retention associated with oceans would have a GMAT only modestly lower than the 254 K effective radiative temperature of the planet.[14]

Roy Spencer (2016) did a calculation showing that decreasing the Moon’s rotational period from 29.5 (Earth) days to 1 day would increase its predicted GMAT by 56 K. His model of an airless Earth predicts a temperature of 251 K.[15]

It appears that N&Z insisted on modeling an Earth with a rotation rate and heat retention properties similar to that of the Moon, without accounting for the high heat capacity of the Earth’s oceans and the Earth’s crust.

While they considered the affect of rotation rate, they dismissed the significance of this, concluding (p. 17) “ω [angular velocity] cannot affect ηₑ [heat retention] and the average surface temperature of a planet.” This result is incompatible with the conclusions reached independently by both Smith and Spencer. I haven’t delved into just where N&Z’s analysis of the impact of rotation rate went wrong, but it’s clear that it did. (In Appendix A4, I offer a sketch of a proof that planetary rotation rate affects the average surface temperature.)

N&Z (p. 14) dismiss the heat conductivity of the oceans because an airless planet wouldn’t have oceans. They also (p. 15) dismiss the heat conductivity of the Earth’s crust, because air and moisture in the soil greatly contribute to this conductivity. While it might be fair to say that these effects would not be present on an airless version of Earth, it is not fair to dismiss the relevance of these effects in explaining the 90 K temperature gap between an Earth with and without an atmosphere, and to insist that purely atmospheric effects should be able to account for this full temperature difference.

The 90 K discrepancy between N&Z’s calculations and observed temperatures would seem to be a result of a combination of N&Z incorrectly dismissing the importance of rotation rate and not accounting for the heat capacity of the oceans and soil. Contrary to N&Z’s assertion, there is no need for the impact of the atmosphere alone to account for this.

“Since the current greenhouse theory strives to explain GE [Greenhouse Effect] solely through a retention (trapping) of outgoing long-wavelength (LW) radiation by atmospheric gases, a thermal enhancement of 90 K creates a logical conundrum, since satellite observations constrain the global atmospheric LW absorption to 155–158 W m⁻². Such a flux might only explain a surface warming up to 35 K. Hence, more than 60% of Earth’s 90 K atmospheric effect appears to remain inexplicable in the context of the current theory.”

There is no “logical conundrum.” What N&Z interpret as an “inexplicable” gap in temperatures appears to be readily explained by Earth’s rotation rate and the heat capacity of the oceans.

* * *

Note that, if N&Z’s calculation of planetary temperature with “no atmosphere” is wrong, as seems to be the case, this calls into question the meaningfulness of any of the calculations of the atmospheric enhancement factor (whether it be via their formula or mine).

**7. What Led Nikolov and Zeller Down this Path: Power Levels**

N&Z continue (p. 1):

“Furthermore, satellite- and surface-based radiation measurements have shown that the lower troposphere emits 42-44% more radiation towards the surface (i.e. 341-346 W m⁻²) than the net shortwave flux delivered to the Earth-atmosphere system by the Sun (i.e. 240 W m⁻²). In other words, the lower troposphere contains significantly more kinetic energy than expected from solar heating alone, a conclusion also supported by the new 90 K GE estimate. A similar but more extreme situation is observed on Venus as well, where the atmospheric downwelling LW radiation near the surface (>15,000 W m⁻²) exceeds the total absorbed solar flux (65–150 W m⁻²) by a factor of 100 or more. The radiative greenhouse theory cannot explain this apparent paradox considering the fact that infrared-absorbing gases such as CO₂, water vapor and methane only re-radiate available LW emissions and do not constitute significant heat storage or a net source of additional energy to the system. This raises a fundamental question about the origin of the observed energy surplus in the lower troposphere of terrestrial planets with respect to the solar input. The above inconsistencies between theory and observations prompted us to take a new look at the mechanisms controlling the atmospheric thermal effect.”

N&Z are correct that the back-radiation from the atmosphere to the surface is observed to have an energy flux higher than that of sunlight, modestly higher in the case of Earth, and dramatically higher in the case of Venus.

Yet, this fact does not present a “paradox” or “inconsistencies” so much as it indicates a way in which N&Z’s intuition about how things work is failing them. I suspect this passage reveals a central aspect of what confuses many people who don’t believe that warming by greenhouse gases can be real.

To make sense of what N&Z find puzzling here, it might be helpful to consider what happens inside a reflective cavity.

The diagram above illustrates a type of reflective cavity. This diagram could relate to light or sound or something else. The mathematics are the same. The cavity is made up by a perfect reflector on the left, and a partial reflector on the right. The perfect reflector reflects 100% of the energy it receives. The partial reflector reflects a fraction 𝛼 of the energy it receives, and transmits the remaining fraction (1−𝛼). (Note that it doesn’t matter if what is happening is genuine reflection, or absorption and re-emission of the energy, so long as it happens quantitatively in the amounts described.)

Each time energy hits the partial reflector, some energy is reflected and some is transmitted through. How this might work for several passes of the energy back and forth is illustrated in the figure. For clarity, I’ve drawn things as if the energy is moving diagonally. However, I’d really like you to imagine that the energy is moving back and forth on a single line.

I’ve labeled the various flows of energy with the associated power level. In principle, the energy bounces back and forth an infinite number of times, growing a bit weaker with each pass. The series of power levels forms what is known as a “geometric series” and it’s well known now to add up such a series.[16] I’ve performed the addition for you.

It turns out that, if you direct power P at the reflective cavity, the amount of power reflected or emitted from the cavity (the sum on the right of the diagram) is also P. This makes sense, because nothing in the cavity absorbs or destroys energy. It needs to go somewhere. Consequently, the power that enters the diagram towards the left and the power that leaves the diagram towards the right are equal.

What is more curious is that the power that strikes the perfect reflector on the left also adds up to P. The amount of power inside the cavity is just as high as the power outside the cavity. Yet, the amount of power that enters the cavity is only (1−𝛼)⋅P. Suppose 𝛼 is 0.99 so that most of the energy gets reflected. Then the power that enters the cavity is only 0.01⋅P. Yet, the total power flowing in each direction inside the cavity is P, which is 100 times higher than the amount of power that enters the cavity!

How is that possible? It’s the magic of energy recirculation, with energy being reflected back and forth inside a reflective cavity. Each reflection increases the total amount of power in the cavity, without ever violating conservation of energy. Each unit of energy “visits” a given place in the reflective cavity multiple times. Power is a measure of how often a unit of energy “visits,” not a measure of how much total energy exists in the system. This may be a bit counter-intuitive, but power really does work that way.

As a rough analogy, imagine that you throw one ping-pong ball at a wall every 10 seconds, and count how many bounces you hear. Then imagine putting two walls very close to one another. Again, you throw a ping-pong ball into the gap between walls and count how many bounces you hear. There will be many, many more.

You might have experienced this sort of multiplication-of-power directly, if you’ve ever been inside a room where the walls strongly reflect sound. Any sound made seems much louder than it ordinarily would. The walls aren’t super-great reflectors, so you probably don’t get an amplification by a factor of 100. But, even so, the sounds are made louder due to the reflections.

Now, let’s relate this to what happens with the atmospheric greenhouse effect. Let’s draw a slightly modified version of the previous diagram.

The difference in this case is that the “partial reflector” is assumed to allow all shortwave radiation through, and be a partial reflector only for longwave radiation.

The “perfect reflector” is assumed to be an ideal “black-body” radiator.[17] It absorbs all energy it receives, converts it to heat, then radiates an equal amount of energy in the form of longwave thermal radiation.

So, shortwave energy from the Sun passes through the “partial reflector” without any power being lost. When this is absorbed by the “perfect reflector,” it heats that reflector and causes an equal amount of power to be emitted as longwave radiation. This longwave radiation is partly reflected and partly transmitted by the partial reflector.

Once again, in principle power goes back and forth an infinite number of times. I’ve labeled the amount of power present on each pass. I’ve also added up the infinite sums.

As before, we see that the total power leaving the system is equal to the power arriving, since no energy is created or destroyed.

We also see that the total power incident on the “perfect reflector” or “black body” is P/(1−𝛼). If 𝛼 is close to 1, this power can be enormous. If 𝛼=0.99, then the total power incident on the “black body” would be 100 times the power of the Sun. (This is similar to what happens on Venus.)

Yet, no energy is being created or destroyed. It’s simply bouncing back and forth in a way that creates a localized intensity that is much higher than the intensity of the original source.

(No, this doesn’t violate the *Second Law of Thermodynamics*.[18] Such an arrangement can never lead to the temperature of a “black body” inside a reflective cavity exceeding the temperature of the Sun. If the black-body temperature approached that of the sun, its emitted radiation wavelength would shift to become shorter and the “partial reflector” would no longer reflect that radiation. This places an upper limit on how much warming could occur.)

Let’s redraw the diagram, to help complete the analogy to the atmospheric greenhouse effect.

This includes the same energy flows from the prior diagram. But, I’ve added up all the reflections, and now show only the total flows in each direction. Also, I’ve re-labelled the “perfect reflector” and “partial reflector” as “surface of planet” and “atmosphere,” respectively. All the math is exactly as it was before. The thickness of the lines are drawn to reflect the magnitude of each energy flow for a value of 𝛼=0.6.

It might still seem odd that the radiant energy flows between the surface and the atmosphere are larger than the energy flow from the Sun. It might be reassuring if we look at heat flows, instead of radiant energy flows. Recall that heat flow is defined as the *net* energy flow.

The diagram above shows the same situation, but shows heat flows, rather than radiant energy flows. The heat flow between the surface and the atmosphere is the net energy when energy flux radiated by the surface, P/(1−𝛼), has the back-radiation flux 𝛼P/(1−𝛼) subtracted from it, yielding a net heat flux, P.

The situation might seem surprising when we look at radiant fluxes, which can be larger than the flux received by the Sun. However, when we look at net heat flow, it’s clear that energy is being conserved.

The above diagrams reflect how the atmospheric greenhouse effect would works, at a mathematically level. This is true even though greenhouse gases do not technically reflection radiation, and even though they produce “back-radiation” as a cumulative effect of many layers of gas, not as something that happens at a single surface. Nonetheless, mathematically, the net effect is analogous to what happens in a reflective cavity.

Although it’s surprising to people not familiar with energy recirculation or reflective cavities, **there is no “paradox” or “inconsistency” involved in down-welling long-wave radiation in the atmosphere having more power than sunlight.**

So, we see that “puzzle” that Nikolov & Zeller set out to solve in looking for a new explanation for planetary temperatures was no puzzle at all. It was merely a matter of well-understood physics that can be surprising if you haven’t had it explained to you or been able to think it through on your own.

I feel a bit sad they N&Z have apparently invested years of work in a project that would seemingly have been unnecessary, if they had been willing and able to have a conversation about what it was that they didn’t yet understand.

**8. Conclusions**

As you can see, I’ve spent quite a bit of time unpacking how to think about the claims Nikolov and Zeller made. I could have just shared my conclusions (which haven’t changed much over the week I’ve been working on this answer). But, why would you believe me? I wanted to really spell things out in detail, to support mutual learning.

Nikolov and Zeller reached conclusions which I believe clearly do not relate to the underlying physics in the ways they’ve hypothesized. Atmospheric pressure alone cannot possibly warm planets. Gases which absorb and re-radiate longwave radiation must warm planets, based on logic which is hard to dispute if you pay attention to some core principles of physics. Yes, there are things about how longwave radiation warms planets that might seem surprising. But, it all makes sense if you’re willing to take the time to work it through.

I’ve enjoyed developing this answer. I hope you get something out of it. Let me know if you got value from it.

Thanks, and be well.

**9. APPENDIX: TECHNICAL DETAILS**

**A1. Details of Formulas for Atmospheric Temperature Enhancement**

As described above, I wanted to see if I could identify formulas to match observed planetary temperatures using greenhouse gas data.

N&Z’s experiments with curve fitting demonstrated, and my own experience confirmed, that simply looking at the total amount of all greenhouse gases seems difficult to relate simply to the thermal effects observed. I thought about why that might be. Looking at the real physics of the atmospheric greenhouse effect shows that different gases affect shortwave radiation in different (but sometimes overlapping) bands of wavelength, and the effect is non-linear in the concentration of each gas.[19] It seemed plausible that, to fit the data, a formula would need to reflect these dynamics, at least qualitatively.

My formula is based on the following ideas:

- Different greenhouse gases affect different segments of the electromagnetic spectrum, with different radiative properties. Thus, the contribution of each greenhouse gas must be addressed separately.
- To assess radiative impact, it seems likely to matter how much of a given gas radiation needs to pass through between the planetary surface and space. So, I chose as my metric for the amount of a particular gas x the number of moles of gas in a column of gas extending from the surface out to space, which I denote by Uₓ.

(Technical details: This is computed as Uₓ/Aᵣ = L⋅ρₓ/Mₓ, where Aᵣ is a reference area, ρₓ and Mₓ are the near-surface density and molar mass of gas x, and L is the nominal scale height of the atmosphere, given by L = P/(g⋅ρ) where g is the surface gravity, P and ρ are the total atmospheric pressure and density at the surface, and g is the gravitational acceleration. The density of gas x is calculated as ρₓ = ρ (0.01⋅Cₓ)⋅(Mₓ/M), similar to N&Z equation 9. To render Uₓ dimensionless, I use a reference area Aᵣ = 1 m².) - For each gas, I assumed that when there is little gas, its influence increases linearly with Uₓ, but as the amount of gas gets large, the influence shifts towards increasing logarithmically with Uₓ. To achieve this result, I assumed that the contribution of gas x takes the form aₓ⋅ln(1 + Uₓ/bₓ).

The goal is to compute values for what N&Z refer to as the “relative atmospheric thermal enhancement” (RATE), E = T/Tₙₐ, the ratio of actual surface temperature, T, to N&Z’s interpretation of what the temperature would be without an atmosphere, Tₙₐ.

My models have the form:

E = 1 + a꜀ₒ₂⋅ln(1 + U꜀ₒ₂/b꜀ₒ₂) + a꜀ₕ₄⋅ln(1 + U꜀ₕ₄/b꜀ₕ₄) + aₕ₂ₒ⋅ln(1 + Uₕ₂ₒ/bₕ₂ₒ)

(This formula is unlikely to accurately represent the physics involved. But, it is somewhat inspired by the underlying physics, at least as much as is N&Z’s formula.)

Associated with this general form, I’ve identified three specific models:

**GH6:**a꜀ₒ₂=2.47461964e-01, b꜀ₒ₂=3.46821712e+03, a꜀ₕ₄=2.52997123e-02, b꜀ₕ₄=1.49966410e-03, aₕ₂ₒ=1.81685678e-01, bₕ₂ₒ=7.97199109e+01**GH4a:**a꜀ₒ₂=2.47085039e-01, ꜀ₕ₄=1.16558785e-01, aₕ₂ₒ=1.99513528e+00, b꜀ₒ₂=b꜀ₕ₄=bₕ₂ₒ=3.42189402e+03**GH4b:**a꜀ₒ₂=a꜀ₕ₄=aₕ₂ₒ=2.47283033e-01, b꜀ₒ₂=3.44616690e+03, b꜀ₕ₄=3.36453603e+04, bₕ₂ₒ=1.67913332e+02

Model GH6 has 6 tunable parameters (regression coefficients), while models GH4a and GH4b each have 4.

For comparison, N&Z’s 4-parameter model has the following form (N&Z equation 11):

E = exp[a⋅(P/Pᵣ)ᵇ + c⋅(P/Pᵣ)ᵈ]

where

**NZ4:**a=0.174205, b=0.150263, c=1.83121e-5, d=1.04193

* * *

(You can skip this paragraph if you’re not trying to reproduce my numerical work. Based on the data offered by N&Z, the moles per square meter for the greenhouse gases on each celestial body are given by: Venus U꜀ₒ₂=2.33129328e+07; Earth U꜀ₒ₂=1.39185086e+02, Uₕ₂ₒ=8.62947532e+02; Moon none; Mars U꜀ₒ₂=4.05755574e+03, Uₕ₂ₒ=8.93922268e-01; Titan U꜀ₕ₄=1.93756806e+05; Triton U꜀ₕ₄=4.38436244e-02. Note: In the course of doing my numerical work, I discovered some inconsistencies in N&Z Table 5 (p. 10) with regard to the greenhouse gas parameters on Mars. The greenhouse gas partial pressure and density were given as 667.7 Pa and 0.018 kg/m³, but should have been 653.5 Pa and 0.0188 kg/m³. It is unlikely that these errors altered N&Z’s overall conclusions.)

**A2. Assessing the Significance of these Models**

The chi-squared (𝛘²) metric can be used to compare goodness-of-fit. Chi-squared is the sum of the squares of the difference between prediction and observation, with these differences normalized by the uncertainty in the observation. Small values of 𝛘² generally indicate a better fit.[20]

The value of this metric for these models is: GH6 𝛘²=0.1, NZ4 𝛘²=5.9, GH4a 𝛘²=9.7, GH4b 𝛘²=9.7.

It’s clear that model GH6 does by far the best job of matching the data. NZ4 fits the data somewhat better than GH4a and GH4b, but the difference is not dramatic.

Note that, for the GH6 model, slight variations in the formula or the variables involved led to a poor fit signified by an enormous 𝛘² value. So, it’s not the case that having 6 tunable parameters automatically guaranteed the excellent fit that was eventually found.

* * *

The value of 𝛘² can also be used to assess the significance of the fit.

(Such an assessment of significance is rigorously correct only when fitting with curves like polynomials that are very flexible in fitting data. But, even so, applying this statistical test is more objective than simply intuitively guessing whether the fit is significant.)

In order to assess significance, one divides 𝛘² by 𝞶, the the number of “degrees of freedom” involved in the fit. The number of degrees of freedom is the amount by which the number of independent data points exceeds the number of tuning parameters.[21]

How many degrees of freedom are associated with the fit for each model? There are six celestial bodies that have been considered. That seems like six data points, but it’s not really. All the formulas have been designed to automatically fit the Moon for any likely values of the fitting parameters. So, there are effectively five data points. In the case of the NZ4 model, the data points for Earth and Titan happen to be nearly identical, so do not constitute independent data points for fitting purposes; there are effectively only four data points.

So, the number of “degrees of freedom” for 𝛘² are NZ4 𝞶=0, GH6 𝞶=0, GH4a 𝞶=1, GH4b 𝞶=1.

The associated normalized 𝛘² values are NZ4 𝛘²/𝞶=∞, GH6 𝛘²/𝞶=∞, GH4a 𝛘²/𝞶=9.7, GH4b 𝛘²/𝞶=9.7. For a curve fit to be regarded as significant, one generally looks for 𝛘²/𝞶 to be not too much larger than 1.

These normalized 𝛘² values suggest that there is little significance to these curve fits.

**This statistical test suggests that the “good fit” of these models is likely simply the result of chance.**

The interpretation that the fits reflect chance is reinforced by the observation that different models depending on unrelated variables (pressure vs. greenhouse gas amounts) fit the data equally well.

**A3. A Nuance About Energy Flows vs. Heat Flows**

An astute and suspicious reader might notice that in section 4 I argued “The only way around this logic [that pressure cannot increase temperature] is if the atmosphere spontaneously creates energy out of nothing, and uses this something-from-nothing energy to increase the energy flow 𝑃𝑖𝑛Pin to the surface”; yet in section 7 I argued that radiative recirculation of energy could increase the power flow to the planetary surface, without any new energy being introduced. Am I sneakily applying a double-standard to different arguments?

No, I’m not. It’s just that, to get to rigorously correct answers, one needs to pay careful attention to whether one is reasoning from the perspective of energy flows, or from the perspective of heat flows.

One can analyze a thermodynamic system from a perspective of e*nergy flows*, or from a perspective of *heat flows* (which are *net* energy flows), or from a perspective of *mixed energy flows and heat flows*. Slightly different rules apply, depending on which perspective one uses.

The argument in section 4 was rooted in the heat-flow perspective, while the discussion in section 7 was rooted in the energy-flow perspective.

Analyses from within in either perspective are valid. But, if you don’t track which perspective is being talked about, it’s easy to get confused, make incorrect arguments, or see things as contradictory when they’re not.

At a level of radiative energy flows (used in section 7), it is possible for energy recirculation to multiply energy flow power levels even though no net energy has been added to the system.

However, when one looks at radiative effects from a heat perspective, energy recirculation does not create higher heat flux levels; to the contrary, radiative energy recirculation *reduces* the net radiative heat flux flowing from warmer planetary surface to the cooler atmosphere.

One can increase the temperature of a planetary surface either by increasing the energy returned to the surface (in the energy flow perspective) or by decreasing the rate of heat leaving (in the heat flow perspective). The presence of atmospheric gases that absorb and re-emit longwave radiation has these warming effects (in whichever perspective you choose).

In the case of an atmosphere transparent to radiation, we know, in the heat-flow perspective (used in section 4), that the atmosphere (1) has no effect on radiative heat flow, and (2) adds additional heat flow away from the surface, in the forms of convection and latent heat flow (evaporation of volatiles). It follows that a transparent atmosphere will cool the planet.

It’s within this heat-flow perspective where we know that, if no energy is being generated in the atmosphere, then a transparent atmosphere cannot reverse the net heat flow between the surface and the atmosphere, as would be needed for the transparent atmosphere to increase the temperature of the surface.

So, the arguments I’ve made are correct and consistent with one another. But, at some points an argument implicitly relates to either the energy-flow perspective or the heat-flow perspective, and it can be confusing if one doesn’t realize this.

### A4. Analysis Indicating Rotation Rate Does Matter

As described in Section 7, N&Z’s analysis is built on top of their 2014 analysis (as Volokin & Rellez), in which they derived a formula for the temperature of a planet without an atmosphere, a formula which does not depend on the planetary rotation rate.

It seems clear that that conclusion of rotation-rate independence can’t possibly be right. I’ll present some arguments that I hope will make that clear.

Consider two extreme cases: planet A always keeps the same face to the Sun, and planet B rotates so fast that there’s barely time for the temperature to change between day and night. The diurnal temperature variations on planet A will necessarily be much larger than the variations on planet B. Yet, That means if the average thermal emissions, i.e., the average value of T⁴, is the same on both planets, then T will be lower on planet A than on planet B.

I can also sketch a more formal version of the argument for a less extreme difference in rotation rates.

The argument relates to an airless planet without horizontal heat transport along the surface, and so focuses on one location the temperatures at the surface and deeper under the surface where temperature variations are smoothed out. (If an illustration would help, see Figure 7 in Vasavada (2012).)

[SETUP]

- Consider the Moon, but with a rotation rate of our choice.
- Assume we are considering a fixed point on the Moon’s surface, at coordinates (𝜃, φ).
- Assume the rotation rate is either 𝟂₁ or 𝟂₂ where 𝟂₁ < 𝟂₂/10.
- Assume that the mean solar irradiance at that location is S.
- Let T₁(z,t) and T₂(z,t) be the temperatures associate with rotation rate 𝟂₁ or 𝟂₂ as a function of depth under the surface, z, and time, t.
- Let z=0 denote the surface, and z=d denote a depth where the temperature T₁(d,t) and T₂(d,t) do not vary over a diurnal period; i.e., each is a constant with respect to t.
- Let ⟨f(t)⟩ denote the average of f(t) over a diurnal period.
- Define the effective radiative temperature Teff by the relationship 𝜀𝛔⋅Teff⁴ = S.

[ASSUMPTION 1] Suppose that heat conduction is given by Newton’s cooling law, so that the heat transfer rate between the surface and the deeper regolith goes as dQ/dt = -h A⋅(T(0,t) – T(d,t)). [I know this isn’t entirely accurate, but let’s see what this assumption leads to.]

- [CLAIM 1] Given that 𝟂₁ < 𝟂₂/10, 𝟂₁ is a much slower rotation rate than 𝟂₂. We expect that the temperature swings in T₁(0,t) will consequently be significantly larger than the temperature swings in T₂(0,t).
- [CLAIM 2] We assume energy balance, such that the solar irradiance absorbed over a day balances the thermal radiation emitted by the surface. Therefore, S / 𝜀𝛔 = Teff⁴ = ⟨T₁(0,t)⁴⟩ = ⟨T₂(0,t)⁴⟩.
- [CLAIM 3] Because of CLAIM 1 and CLAIM 2, we expect that ⟨T₁(0,t)>
**<**⟨T₂(0,t)⟩**<**Teff (This example illustrates an analogous situation, albeit with spatial variations instead of temporal variations.) - [CLAIM 4] Given ASSUMPTION 1, dQ/dt = -h A⋅(T₁(0,t) – T₁(d,t)). However, we know that over a diurnal period, in steady-state, dQ/dt must integrate to zero. Or equivalently, ⟨dQ/dt⟩=0.
- [CLAIM 5] It follows from CLAIM 4 that ⟨T₁(0,t)⟩ – ⟨T₁(d,t)⟩ = 0. It follows that ⟨T₁(0,t)⟩ = ⟨T₁(d,t)⟩.
- [CLAIM 6] A result analogous to CLAIM 5 must also be true for rotation rate 𝟂₂. In particular, ⟨T₂(0,t)⟩ = ⟨T₂(d,t)⟩.
- [CLAIM 7] Combining CLAIM 3 with CLAIM 4 and CLAIM 5, it follows that ⟨T₁(0,t)⟩ = ⟨T₁(d,t)⟩ is less than ⟨T₂(0,t)⟩ = ⟨T₂(d,t)⟩.
- [CLAIM 8] Restating CLAIM 7, it has been shown that different rotation rates 𝟂₁ and 𝟂₂ lead to different mean temperatures ⟨T₁(0,t)⟩ = ⟨T₁(d,t)⟩ and ⟨T₂(0,t)⟩ = ⟨T₂(d,t)⟩, respectively.

QED.

The above argument is quite straightforward, providing one understands that greater non-uniformity of temperature leads to a greater value of ⟨T⁴⟩ – ⟨T⟩⁴, which justifies CLAIM 3.

I think the above argument establishes the erroneousness of the assertion that “[the planet spin rate] ω cannot affect… the average surface temperature of a planet.” N&Z’s belief that average temperature doesn’t depend on rotation rate is wrong.

(N&Z’s calculation (2014, p. 16-17) which leads them to the conclusion of rotation-independence takes the form of their trying to analytically relate their ad hoc heat storage parameter η to actual heat conduction. The argument is somewhat convoluted and appears to me to contain serious logical errors. But, as yet, it hasn’t risen to high enough in priority for me to sort out which of the “bad smells” in their argument is the core problem.)

**Related Information from Others**

- An earlier version (2011) of N&Z’s paper is available.[22]
- Willis Eschenbach (2012) critiqued the 2011 version of N&Z’s paper.[23][24] Subsequently, N&Z responded (2012) to aspects of that critique.[25]
- Roy Spencer responding to N&Z in 2011[26] and also in 2018[27]. N&Z responded (2019) to the latter.[28] Spencer noted (2016) large errors in N&Z’s calculations of the Earth’s temperature in the absence of an atmosphere.[29]
- Richard Larson (2018) critiqued aspects of N&Z’s work.[30]
- Douglas Cotton (2020) published a refutation of N&Z.[31]

## Footnotes

[1] https://www.omicsonline.org/open-access/New-Insights-on-the-Physical-Nature-of-the-Atmospheric-Greenhouse-Effect-Deduced-from-an-Empirical-Planetary-Temperature-Model.pdf

[2] 15 Insane Things That Correlate With Each Other

[3] 5 brilliant graphs that teach correlation vs. causality

[4] The 10 Most Bizarre Correlations

[9] The Structure of Planetary Atmospheres

[10] https://www.giss.nasa.gov/research/briefs/schmidt_05/

[11] What is Earth’s surface emissivity?

[12] https://tallbloke.files.wordpress.com/2016/09/ge_magnitude_volokin_rellez_2193-1801-3-723_springerplus_2014.pdf

[13] Erratum to: On the average temperature of airless spherical bodies and the magnitude of Earth’s atmospheric thermal effect

[14] Proof of the Atmospheric Greenhouse Effect

[15] http://www.drroyspencer.com/2016/09/errors-in-estimating-earths-no-atmosphere-average-temperature/

[16] Geometric series – Wikipedia

[18] Second law of thermodynamics – Wikipedia

[19] Dependence of Earth’s Thermal Radiation on Five Most Abundant Greenhouse Gases

[20] Chi-square distribution – Wikipedia

[21] http://maxwell.ucsc.edu/~drip/133/ch4.pdf

[22] Unified Theory of Climate

[24] The Mystery of Equation 8

[25] Nikolov & Zeller: Reply to Eschenbach

[26] Why Atmospheric Pressure Cannot Explain the Elevated Surface Temperature of the Earth

[27] http://www.drroyspencer.com/2018/12/giving-credit-to-willis-eschenbach/

[28] Nikolov & Zeller: Reply to Dr Roy Spencer’s blog article.

[29] http://www.drroyspencer.com/2016/09/errors-in-estimating-earths-no-atmosphere-average-temperature/

[30] Why Nikolov and Zeller are wrong

[31] Refutation-of-Nikolov-and-Zeller-universal-theory-of-climate

*[This work was originally published as an Answer on Quora, albeit without Appendix A4 which offers an argument that average temperature depends on planetary rotation rate.]*