The temperature of the earth’s surface is often explained using the “Greenhouse Effect”. However, having refuted the “Greenhouse Effect”, we may wonder if it was necessary in the first place. The earth orbits the sun in the vacuum of space. There is no aether as Fourier, Tyndall and Arrhenius believed. Moreover, there is no heat capacity or thermal conductivity in space. The only way for heat to escape the planet is by emission to space. That makes the temperature of the absorbing mass of the earth a question of radiative heat transfer. Hereafter, I will refer to the that portion of the earth’s mass which absorbs solar radiation as the “solarsphere” because the atmosphere does not include the surface layer warmed by the sun on a day to day basis and there is no other term to encompass both. The method of calculation is to treat the solarsphere as an absorbing body subject to incident radiation from the sun.
Given the solar constant of 1368 Wm-2 (Fröhlich & Brusa, 1981) and the fact that the cross-sectional area of solar radiation incident upon the earth is roughly one quarter of the earth’s surface area, it is unsurprising to observe that authors such as Kiehl & Trenberth (1997) arrive at 342 Wm-2 as the mean quantity of solar radiation that falls on the entire surface of the earth. Using this, we may calculate the expected geographical and altitudinal mean temperature of the earth’s solarsphere.
Wm = σT4
T4 = Wm/σ
T = {Wm/σ}0.25
Given Wm = 342:
T = {342/0.000000056704}0.25 = 278.7ºK = 5.5ºC
This figure, is an average or mean temperature for all times, latitudes, and altitudes of the the earth’s solarsphere. Just as the balance point or centre of gravity is found at the centre of mass, this average temperature may be found at the centre of heat capacity. In materials of similar heat capacity, this can be found near the centre of mass. Thus, in order to determine how well our 5.5ºC result -calculated above- corresponds to observed reality, we must first determine the average observed temperature at the barometric median in the part of the earth penetrated by solar energy.
From the diagrams supplied by Vallier-Talbot (2007, pp. 25-26), we may roughly determine the centre of mass for a one square metre column extending from two metres below the surface to 50 kilometres above the surface. Soils and clays amount to roughly 2 tons per cubic metre, with the atmospheric column having to weigh 10 tons in order to yield a mean barometric pressure of roughly 1000 hectopascals at the surface. The total column weighs 14 tons with the centre of gravity corresponding to the barometric median at 700 hPa. Referring once again to Vallier-Talbot (2007, p. 26) we may determine that on average, this pressure corresponds to an elevation of roughly a mile or 1600m above the surface. Given the observed average atmospheric thermal gradient of -7ºC with every 1000m of elevation above the surface (Vallier-Talbot, 2007, p. 25), we may calculate the average absorbing mass temperature as it occurs at the altitude of the barometric mean for our absorbing column. No doubt you’ve worked out that the temperature drop over a tropospheric ascent is 11ºC per mile, and we all know that the average surface temperature is 15ºC (Arrhenius, 1896, p. 239; Burroughs, 2007, p. 124). Notwithstanding 100 years of apparently constant mean temperature from Arrhenius to Burroughs, we may determine that the observed temperature at the altitude corresponding to the centre of absorbing mass is 4ºC or 277ºK. This, via the reasoning above, extends to an observed average absorbing mass temperature for planet earth of 4ºC or 277ºK. This is slightly cooler than the mean absorbing mass temperature calculated above from the solar constant (278.7ºK, 5.5ºC) even if we do allow for 0.5º warming over the last century. However, if we were to consider the impact of convective cooling, I think we can agree that the temperature we derive from the Stefan-Boltzmann equation is well within the tolerance we must allow for such tests.
Adding the tropospheric thermal gradient of 11ºC per mile we got from Vallier-Talbot (2007) above, our temperature (278.7ºK, 5.5ºC), calculated from the Stefan-Boltzmann Equation using the Solar Constant, yields a calculated surface temperature of around 16.5ºC. The fact that this is warmer than the observed mean surface temperatures of Arrhenius and Burroughs (15ºC) leaves no room for such dubious free energy mechanisms as Arrhenius’ “Greenhouse Effect”. The surface temperature of the earth can be much more simply explained without resorting to such complex and unverifiable entities as radiative amplification and power recycling via backradiation of the “Greenhouse Effect”. Absorptivity of any of the parts can vary, but that only alters the overall emissivity, which in turn leaves unchanged, the gross power flowing though the system. Once equilibrium is reached it is only the power flowing through a thermally isolated system that controls and maintains mean temperature. This is because comtinuing and ongoing power is required to offset the amount of heat that is lost spontaneously and continuously due to emission of radiation.
Our calculation of mean surface temperature without the “Greenhouse Effect” above (16.5±0.5ºC corresponding to 16-17ºC) is made without considering the effect of carbon dioxide. According to Arrhenius (1906a, translated by Gerlich & Tscheuschner, 2009, pp. 56-57) the observed temperature should be 20.9ºC higher than that yielded by a calculation such as this, owing to the carbon dioxide in the atmosphere. The observed surface temperature of 15ºC (Arrhenius, 1896; Burroughs, 2007) is actually 1-2ºC lower than the calculated mean surface temperature of 16-17ºC. The lower atmosphere will always be warmer than the upper atmosphere because higher material density in the lower atmosphere dictates a much higher thermal conductivity, absorption and density of heat. In contact with an opaque surface warmed by the bulk of the heat absorbed from the sun, it is not difficult to explain why the surface is so much warmer than the altitude corresponding to the centre of mass in the solarsphere. Moreover, the Ideal Gas Law (PV = nRT) dictates that the temperature of a gas containing a given amount of heat invariably increases with pressure. As the highest atmospheric pressure is at the surface, it makes sense that the higher temperature is there, especially if obstruction to radiative outflow decreases with altitude.
Turning our attention to the example of Langley’s greenhouse experiment on Pike’s Peak in Colorado (mentioned by Arrhenius, 1906b), we may be tempted to ask how it is that a greenhouse can reach such high temperatures. Qualitatively, we may attribute the difference between the 15ºC mean surface temperature and the 113ºC observed in Langley’s greenhouse to the fact that noon-time radiation at the surface is three to four times as intense as the mean radiation over the whole of the earth’s surface. Repeating our calculation method, this time for the midday conditions of a greenhouse:
T = {Wm/σ}0.25
Given Wm = 1368:
T = {1368/0.000000056704}0.25 = 394.1ºK = 121.0ºC
As you can see, our application of the Stefan-Boltzmann Equation predicts that incident Solar radiation at 1368 Wm-2 should produce a maximum daytime temperature of 394.1ºK or 121.0ºC in a greenhouse fully protected from heat losses to conduction. Although Langley’s temperature is lower by eight degrees, it is near enough and, allowing for conductive heat loss, remains a testament to the insulating effectiveness of double glazing.
What is demonstrated in the above examples, is the fact that surface temperature and the temperature in a greenhouse can be explained without resorting to the extraneous entity called the “Greenhouse Effect”. This is significant in light of Ockham’s Razor, which states:
Entia non sunt multiplicanda praeter necessitatem.
This reads in English as:
Entities are not to be multiplied beyond necessity.
Although the terminology may seem unfamiliar in light of 20th century usage, if we look at the words for what they mean we can, nonetheless, understand this statement. This suggests, in modern palance, that it is simply not valid to hypothesise beyond what is strictly necessary to explain the material evidence we possess. A hypothesis that does go beyond the support of material evidence violates this principle in that the evidence is already explained by a simpler theory. This is one of the most fundamental and definitive principles of science.
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