By Markus Ott
I assume that most readers are aware that the 33°C overall greenhouse effect (GHE) is the product of an arithmetically and physically incorrect calculation. It inappropriately averages the insolation evenly over the entire surface of the Earth, as if it were flat and not a rotating sphere, and then uses this average of approximately 240W/m2 to calculate the mean surface temperature for an earth without an atmosphere via the Stefan-Boltzmann law, see page 97 in AR4 here. This physically incorrect mean-value calculation produces a mean surface temperature for an earth without an atmosphere of -19°C. Based on this incorrect result an overall GHE of 33°C (14 – -19) is postulated to explain the measured mean near surface temperature of about 14°C.
Here I present a model by Uli Weber that produces a surprisingly good estimate for the mean near surface temperature of the earth simply by fixing the problems in the arithmetic of the calculation described above.
To avoid this incorrect mean-value formation, a model is developed, that allows the irradiated power to be calculated at every point on the planet’s surface. Using the Stefan-Boltzmann law one then establishes a surface temperature distribution function T(ϴ) that assigns a temperature to each point on the planet’s surface. With this surface temperature distribution function T(ϴ), one can then calculate the mean surface temperature of the planet “practically error-free.”
This sounds quite hypothetical. But thanks to the Lunar Diviner Experiment, we have detailed data on the surface temperature of the Moon, that can be used to calculate the surface temperature distribution function T(ϴ) described above for the Moon.
With this data, our moon becomes the ideal model for a celestial body without an atmosphere.
When analysing the temperature data from the Lunar Diviner Mission, it turns out, that on the sunny side of the Moon, the surface temperatures are very well described by the Stefan-Boltzmann law when the angle of incidence of the solar irradiation is taken into account (Figure 1).
According to Williams et al. 2017, the surface temperature distribution function T(ϴ) is as follows.
where ϴ = solar elevation angle, α = 0.11 (albedo of the moon), and S0 = 1368W/m2 (insolation or solar constant). σ is the Stefan-Boltzmann constant.
Near the poles, the solar elevation angle ϴ (Teta) approaches 90°. This means that the cosϴ approaches zero and very little power is radiated. The surface there is very cold. At the equator, when the sun is at its highest point and ϴ = 0°, cosϴ = 1. There the full power S0 reaches the surface and (1-α)S0 is absorbed. The highest surface temperature is measured there.
Concentric circles with the same angle of incidence and thus the same temperature are thus formed around the point of the sun’s highest point (Figure 2).
The shadow side of the moon is not considered further. It simply cools down overnight and is heated up again the next day.
The moon rotates about 28 times slower than the earth. This means that the shady side has much more time to cool down than the night side of the Earth. After sunrise, however, the lunar surface also has more time to heat up. The diurnal variation in temperature is therefore more symmetrical than on Earth. The daily maximum is reached at the highest point of the sun (Figure 3).
If one compares the diurnal variation of the lunar surface temperature with corresponding terrestrial measurements (Figure 4), one finds that the temperature variations measured on the Earth’s surface differ from the lunar values primarily in that the maximum temperature is reached only after the solar maximum (noon) and that the difference between day and night is smaller. These differences are mainly due to the fact, that the Earth rotates about 28 times faster than the Moon and that the heat capacity of the Earth’s atmosphere and surface slows down the heating and cooling of the Earth’s surface.
Despite these differences, the diurnal variation of the Earth’s surface temperature is more similar to the diurnal variation of the Moon’s surface temperature than to the homogenous -19°C surface temperature assumed in the 33°C-GHE-model. Therefore, let’s take a look at what comes out when the surface temperature distribution function of the Moon is transferred to Earth.
When transferring the “moon model” to earthly conditions, we simply replace the albedo of the moon with the albedo of the earth. This means that although we calculate without an atmosphere, the reflection of sunlight on clouds and water surfaces is still taken into account.
with ϴ = solar elevation angle, α = 0.3 (albedo of the earth) S0 =1368W/m2 (insolation or solar constant). σ is the Stefan-Boltzmann constant.
This temperature distribution function T(ϴ) divides the sunny side of the earth into concentric circles with the same surface temperature (Figure 5).
Uli Weber describes a numerical solution that calculates the global average temperature via this distribution function. The calculation effort is manageable. Anyone can do the calculation themselves using Excel. The result is 14.03°C and is surprisingly close to the global average temperature of about 15°C “measured” by the IPCC.
A detailed description of the calculation can be found here: “Anmerkungen zur hemisphärischen Mittelwertbildung mit dem Stefan-Boltzmann-Gesetz,“ (Notes on hemispheric averaging using the Stefan-Boltzmann law), Uli Weber.
Because it is not particularly difficult in this case, I will also demonstrate a “proper” integral solution for Uli Weber’s hemispherical approach.
Here, the concentric ring with the same angle of solar irradiation becomes an infinitesimally narrow area element dA (see Figure 6).
The area-averaged mean temperature of the sunlit hemisphere Tm is calculated by forming the area integral of the surface temperature distribution function T(ϴ) over the sunlit hemisphere and then dividing by the area of the hemisphere.
With an area-averaged temperature of Tm ≈ 15°C, the integral solution provides an average temperature of approximately 15°C as specified by the IPCC and determined by weather station measurements.
Result: If you place sunshine on only one side of the Earth, as in real life, and use the results of the Lunar Diviner experiment to calculate the average surface temperature of the Earth, the “computed” overall atmospheric greenhouse effect disappears.
Why does Uli Weber’s hemispheric Stefan Boltzmann approach work so well?
In addition to his “Hemisphärischer Stefan-Boltzmann-Ansatz” for calculating the mean near-surface Earth temperature”, Uli Weber has published a series of articles explaining this model on the EIKE website.
Here I try to break down his detailed explanation to the absolute essentials.
The “measured” mean near-surface temperature of the earth (NST = Near Surface Temperature ≈ 15°C) is the result of averaging, in which a time- and area-averaged average is formed from the temperature data of a measurement network spanning the globe. This mean value is of course heavily dependent on the method used to calculate the mean value. This means that it makes little sense to haggle over half degrees here. Further information on the determination method can be found here.
Before we look at why Weber’s model is able to calculate the near-surface mean Earth temperature so well, let’s look at how well the temperature distribution function describes the lunar surface temperature over the course of a day-night cycle. For this purpose, we compare the diurnal variation of the surface temperature measured at the landing site of Apollo 15 with the values calculated for the same location.
In Figure 7, the measured daily variation of the lunar surface temperature (in blue) and that calculated using the distribution function (in red) are superimposed.
It is noticeable that the measured and calculated values agree very well during the lunar day. During the night, however, there are considerable deviations between the measured and calculated surface temperature. This deviation of approx. 77°C (-196°C measured instead of -273°C calculated) is due to the fact that the calculation is based on the assumption that the surface temperature is caused exclusively by solar radiation and that the part of the moon not exposed to the sun must therefore have a temperature of 0K (-273°C). However, as the surface of the moon warmed up during the day and stores some heat overnight, the theoretical value of -273°C is never reached.
This 77°C deviation appears far less dramatic if we calculate the heat flux from the lunar surface, which corresponds to the measured -196°C.
This results in a heat or radiation flux of approx. 2W/m2. Compared to the total energy fluxes occurring during a day-night cycle, this is negligible.
It can therefore be seen that the temperature distribution function T(θ), due to the low heat capacity of the lunar surface, describes the daily temperature variation on the moon pretty well.
If the temperature distribution function is used to calculate the ground-level Earth temperature for a vertical solar incidence (θ = 0), a value of almost 90°C results.
with θ = 0° and α = 0.7 results in
Which equals ~ 360K or 87°C
Almost nowhere on Earth is such a high temperature reached near the ground.
Similarly, the -273°C calculated for the night side is never reached.
The temperature distribution function T(θ) calculates the maximum achievable temperature for each location, assuming that the power radiated by the sun is immediately and completely converted into IR radiation. Uli Weber calls this value the Stefan-Boltzmann temperature equivalent to distinguish it from the actual measurable temperature.
On Earth, such a direct and complete conversion into IR radiation is not possible (due to thermalization, evaporation, …). Here, most of the absorbed solar radiation is converted into sensible heat. This heat drives global circulation and charges the very large heat reservoirs in the climate system. The heat content of the global circulation is approximately 4 orders of magnitude greater than the energy absorbed from the sun in one day.
Heat content of the oceans ≈ 50,000 days of solar radiation
or
Heat content of the oceans ≈ 100,000 times the heat required to prevent a nighttime cooling as observed on the moon (source).
The calculated extreme temperature values are therefore never reached and the night side cooling can be neglected. For a more detailed explanation of why 87°C is not reachable on Earth’s surface due to evaporation and circulation see Sud, et al.
This results in a surprisingly simple convection model. The heat input takes place via solar radiation. The irradiation occurs with S0 (solar constant 1368W/m2) over a circular area with the size πR2 (R = earth radius). Approx. 30% of this radiation is reflected by the earth back into space (albedo α ≈ 0.3). This results in a heat input of approx. 957W/m2 over an area of πR2.
This heat input is converted into sensible heat, distributed over the entire Earth by means of global circulation and then radiated back into space over the entire surface of the Earth (4πR)2.
Similar to the “33°C greenhouse effect model”, this results in an average outgoing long wave radiation of approx. 240W/m2. Assuming thermal equilibrium, this results in:
Irradiation = radiation
However, in contrast to the “33°C greenhouse effect model,” the average outgoing long wave radiation is not used to calculate the average surface temperature, a procedure that ignores the rules of arithmetic. In Uli Weber’s model, the “temperature genesis” takes place directly and locally on the site of irradiation, before the mean value is formed.
At the respective irradiation point, the solar radiation is almost completely converted into sensible heat. This conversion into sensible heat means that the irradiated power (P) or the associated heat quantity (Q) becomes a linear function of the temperature.
“Locally”, for a very small surface element A, the following then applies (with t =time, C=heat capacity of the surface element A, σ = Boltzmann constant):
or
or
After being converted into sensible heat, the radiated power becomes part of the global circulation, ensuring proper “arithmetically correct” averaging.
This is shown in the following graphic, which was taken from Uli Weber’s article “Der hemisphärische Stefan-Boltzmann-Ansatz ist kein reines Strahlungsmodell – Teil 1” (31 December 2023, Figure 8).
The takeaway from this short and relatively simple analysis is that the presence of atmosphere and oceans makes Earth warmer than a comparable planet without atmosphere and oceans. But this higher mean surface temperature is not caused by back radiation from greenhouse gases. The higher mean surface temperature is the result of the enormous heat capacity of the oceans and atmosphere and their ability to distribute the absorbed heat over the planet.
Andy May lightly edited this post for clarity.
Download the bibliography here.
Uli’s writings on this subject go back to 2017, see the bottom of this post for a list and links.
“Result: If you place sunshine on only one side of the Earth, as in real life, and use the results of the Lunar Diviner experiment to calculate the average surface temperature of the Earth, the “computed” overall atmospheric greenhouse effect disappears.”
Bravo!
Finally a reasonable model
In contrast to Mr. Ott’s awkward claims, Weber’s numerical model is based on the inadequate assumption of a radiation balance on a local scale. This assumption is generally an invalid approximation of the energy (flux) budget. Obviously, Mr. Ott has no adequate knowledge regarding the scientific literature in this matter.
Note that Weber, a college dropout, ignored all relevant astronomical aspects like the elliptical orbit of the Earth-Moon barycenter, the obliquity of the ecliptic compared to the celestial equator, the rotation of the Earth and the Moon in the radiation field of the Sun and, hence, either the geocentric declination or the selenographic declination of the Sun . Weber’s knowledge in the matter is rather poor. He offers only Paleolithic knowledge.
1) In the case of the system Earth-atmosphere, as considered by Weber, a local radiation balance does not generally exist on a local scale, neither at the top of the atmosphere nor at the Earth’s surface. Thus, Weber’s results only fulfill the criteria of esoteric garbage. There are many field campaigns in which energy flux budgets have been determined, including, of course, the flux densities of sensible and latent heat and the soil heat flux density, completely ignored by Weber. He also completely ignored the role of vegetation. Weber’s target illustrated in Figure 5 means that the area between 115°E and 65°W would always be in darkness. Mr. Ott should realize that the Earth ist not tidally locked to the Sun like the Moon to the Earth.
2) Also in the case of the Moon, the assumption of a radiation balance on the local scale is an invalid approximation of the energy (flux) budget. It is well-known for more than 7 decades that for some local time hours around solar noon, the assumptions of a local radiation balance provides useful results, but for more than the half of a synodic month (usually called a lunation), this assumption provides, again, esoteric garbage, as already documented by Wesselink (1948).
Because of Weber’s poor knowledge he also confused the sidereal month with the required synodic month, which is about 2.21 days (mean solar days) longer. Thus, Weber’s results illustrated in Figure 7, are wrong, too. And he knows it.
The quote of Williams et al. (2017) is based on horse-trading. Williams et al. (2017) wrote in their subsection “4.4. Minimum and maximum temperatures”:
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“Daytime temperatures on the Moon are approximately in radiative equilibrium. For slowly rotating bodies with low thermal inertias like the Moon, heat diffusion models predict surface temperatures at the equator within ∼1 K of radiative equilibrium between local time hours 8 and 16 (i.e. incidence angles < 60°) ( Vasavada et al., 2012; Bandfield et al., 2015 ). Maximum temperatures therefore occur at noon and will depend on the albedo while being sensitive to the orbital and celestial geometry. The minimum temperatures will occur just prior to local sunrise and are dependent on the thermophysical properties of the near-surface.
Maximum and minimum global surface temperature maps are shown in Fig. 18 . The mean temperature at the equator is 215.5 K with an average maximum of 392.3 K and average minimum of 94.3 K ( Fig. 19 ), representing an average change in temperature of ∼300 K. Average maximum and minimum temperatures in the polar regions (poleward of 85 °) are 202 K and 50 K respectively with a mean average temperature 104 K. Mean maximum temperatures in the south polar region are ∼11 K warmer than the north polar region, however the average minimum temperatures are the same at both poles. This discrepancy is likely due to differences in the distribution and configuration of the topography which is the dominant control of polar temperatures on the Moon. The south polar topography is more rugged, displaying a larger range of elevations ( Smith et al., 2010 ). The maximum solar declination of 1.54 °results in surfaces that are permanently shadowed down to roughly 60 °latitude ( McGovern et al., 2013; Siegler et al., 2015 ). Though a larger surface fraction of the south polar region is in permanent shadow compared to the north polar region, the larger topography range responsible for this results in generally more favorable illumination conditions for equator facing slopes than in the north ( Mazarico et al., 2011b )."
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Weber's numerical model provides a zonal mean of the surface temperature of 161,4 K for Moon's equator. This value is 54.1 K lower that the value mentioned by Williams et al. (2017). Furthermore, Weber's numerical model provides a zonal average of 77 K for a latitude of 85°N and 56 K for the north pole. His model provides similar results for 85°S and the south pole.
In the case of the landing site of the Apollo-15 mission the differences between the observed zonal mean of the surface temperature and the result provided by Weber's numerical model is 56 K. Since July 14, 2017, Weber knows that his results are completely worthless.
Using the assumption of a local radiation balance, a local albedo of 0.3 and a so-called solar constant of 1367 W/m^2 , Gerlich and Tscheuschner (2009) obtained for the global average of the surface temperature of an Earth in the absence of an atmosphere of about 144 K. Weber always claimed that the results of Gerlich and Tscheuschner (2009) must be corrected by a factor of 2. Webers claim is completely wrong. The formula derived by Gerlich and Tscheuschner would provide for the Moon a global average for the surface temperature of about 151 K. Based on Weber's claim, it would provide a surface temperature of 302 K. However, based on the 24 datasets of the Diviner Lunar Radiometer Experiment (see Williams et al. (2017), the global average of the bolometric temperature is 201,1 K plus/minus 0,6 K.
Consequently, Weber's model results fulfill only the criteria of estorec garbage.
References quoted by me:
Gerlich, G. and Tscheuschner, R.D. (2009) Falsification of the Atmospheric CO2 Greenhouse Effects within the Frame of Physics. International Journal of Modern Physics B, 23, 275-364.
Wesselink, A.J. (1948) Heat Conductivity and Nature of the Lunar Surface Material. Bulletin of the Astronomical Institutes of the Netherlands, 10, 351-363
Williams, J.P., Paige, D.A., Greenhagen, B.T. and Sefton-Nash, E. (2017) The Global Surface Temperatures of the Moon as Measured by the Diviner Lunar Radiometer Experiment. Icarus , 283, 300-325.
Gerhard,
While the arguments you make are generally correct, they are not reason to personally criticize either Markus or Uli. The model is only meant to show that the typical “flat earth” model used in the literature is incorrect because it assumes the whole surface of the earth is under the sun 24 hours per day.
The post clearly states that it does not propose a precise model of radiation in versus radiation out. Such a model as you describe does not exist, rather than criticize, you should write your own. Whether Weber finished college or not is irrelevant. Please be more restrained and respectful in the future or I will be forced to delete your comments.
I have sent this to Marcus in case he decides to reply.
Uli Weber’s model is also discussed at EIKE : https://eike-klima-energie.eu/2019/09/11/anmerkungen-zur-hemisphaerischen-mittelwertbildung-mit-dem-stefan-boltzmann-gesetz/ ; there’s a white paper (in German), https://eike-klima-energie.eu/2019/09/11/anmerkungen-zur-hemisphaerischen-mittelwertbildung-mit-dem-stefan-boltzmann-gesetz/?print=pdf, and there are 2 videos published at Tom Nelson’s YouTube channel after Markus Ott’s booklet ; which also discuss the Lunar Diviner experiment (which examined an average temperature for the moon). Finally in Ott’s booklet: https://tomn.substack.com/api/v1/file/71a8adc6-7aa4-42e4-88eb-caa6d0f04057.pdf
I have one complaint. The original GHE idea assumes all outgoing radiation originates at sea level (or at the surface), and neglects to tell us that about 50% of surface cooling is due to evaporative cooling, a mechanism which depends on convection and the enthalpy of vaporization of water EVW. The water vapor mostly condenses in the troposphere; so this energy (EVW), is released there, then radiated from clouds. So the original S-B calculation of a +33C pretended this never happened. Uli Weber’s model would need to be modified to account for evaporative cooling too.
But this is minor complaint – given the original IPCC S-B model ignored evaporative cooling to get its +33C GHE, its OK, by me for Andy to ignore it too to show why their +33C GHE is an artifact of a bad model.
The various discussions of radiation in versus radiation out of Earth are interesting academic exercises, but mostly beside the point in my opinion. The real issue is that the Earth is a rotating sphere that absorbs energy. While the total energy striking Earth is nearly constant, the storage of energy on Earth’s surface and atmosphere varies constantly. This is mostly due to clouds, but other factors matter as well. I discuss one small part of this problem here:
https://andymaypetrophysicist.com/2025/03/12/the-earth-without-greenhouse-gases/
Clouds are discussed here:
https://andymaypetrophysicist.com/2024/12/17/climate-models-clouds-olr-and-ecs/
The main problem is the large changes in cloud cover that occur over time