INT1Task 3

An Investigation of Averaged Radiation and the Properties of CO2

Abstract

The methodology used when calculating the effective temperature of rotating planetary bodies treats the average input over an entire rotation as a valid approximation of the overall energy received. The resulting temperature calculations are assumed to be accurate enough to be used as working inputs for climatological models or the theories they are derived from. Research and examination of radiation laws raises doubts about the viability of such averaged inputs, and this reasoning is used to inform and test a hypothesis.

Background

The purpose of this paper is not to go into a full derivation of the physics behind the absorption and emission of electromagnetic radiation. Planck, Stefan, Boltzmann, Einstein, and Wien were all better suited to such an examination, and the experimental apparatus needed to plumb the realms of quantum physics further is not easily acquired. The work of Stefan and Boltzmann is accepted herein as a valid description of the ideal case where emissivity is equal to unity, also known as a black body.

The discovery in 1905 by Einstein[1] that emission spectra are quantized is not his most famous finding, of course, however it is the one for which he received a Nobel Prize. While his work with other scientists in Brussels is not as well known, one of his fellow Nobel Laureates Max Planck initially rejected, but was later convinced by Einstein who had expanded on Planck's earlier work on what is

now known as the Planck Law of black-body radiation. Planck[2] himself was seeking a resolution to the problems plaguing the earlier work of Wien, which correctly predicted radiative properties at high frequencies, but did not work at shorter ones.

Later Planck would say that it was in an “act of despair” that he revised his work using the statistical work of Boltzmann[3]. Though he did not realize it at the time, what he considered a mere assumption about the discrete nature of emission which he used in his formulations could arguably be considered the introduction of quantum physics onto the scene. Einstein expanded upon that earlier “merely formal assumption” and wound up proving it to be true. Boltzmann himself actually suggested the possibility of discrete energy levels in a physical system, though he is more well known for his statistical work. In this case we are specifically concerned with a particular law named after his teacher Jozef Stefan and himself, deduced experimentally by the former, and theoretically with thermodynamics by the latter. The Stefan-Boltzmann law states that a black body will emit a certain amount of radiation per unit area per unit time. One can show that by integrating the area underneath a curve given by Planck's Law you can get the total power radiated over the emission spectrum according to Stefan-Boltzmann. Unfortunately this case can only be an approximation because black bodies are not real objects. This approximation remains useful nonetheless, as the ratio of energy emitted by a body at a given temperature and the theoretical black body emissions at the same temperature forms a dimensionless value known as emissivity.

For a body with known emissivity and temperature, one can calculate the amount of radiation it should emit. Alternatively one can take the energy absorbed by a surface and determine at what temperature thermal equilibrium would be reached, that being the (effective) temperature at which radiative input equals radiative output. However one must be careful when making these calculations and never assume that an observed level of emission must necessarily correlate to a particular

temperature.

This brings us to the question at the beginning of the paper, wherein the globally averaged input from the Sun is calculated to be 340 W/m^2. This value is taken to be the power available at the top of the atmosphere, with the albedo—or reflectivity—giving the final averaged input of approximately 240 W/m^2 at the surface.

As Kirchoff[4] shows, absorption equals emission for a body in thermal equilibrium, so the total radiation emitted to space by the planet should also be 240 W/m^2. Taking that value as an input for the S-B Law one finds what is known as the “effective emission temperature” of the planet, 255 K. Observations at the surface, though, find that the planet averages closer to 288 K, some 33 K warmer than the value expected from the emissions alone.

At this point it is difficult to see why this would not be a reasonable point to start determining what processes could be responsible for this 33 K difference, and indeed one can show that an input of 240 W/m^2 is not sufficient to raise the global average temperature above 255 K. It would seem that a way for the atmosphere to produce a higher surface temperature is needed, and at this point the arguments for what is commonly called the “greenhouse effect” are normally introduced.

This is where an error may have been made which appears to partially if not completely undermine the need for proposing a greenhouse effect at all. The value of 240 W/m^2 for emissions is acceptable, and it corresponds to observations. It is not acceptable to treat temperature as though it is determined by emissions, though, much less a black body value of 255 K. The assumption of unity emissivity is easily discarded as unphysical, and as such it can be shown that the 255 K value is a minimum temperature, not an actual one.

Taking the 288 K average temperature and the 240 W/m^2 output, one finds that a surface with emissivity equal to 0.6 would have the observed properties. That alone is not sufficient to falsify the claimed 33 K greenhouse effect, as arguments can be made without the above errors that support emissivity values between 0.6 and 0.98. This paper will not go into those arguments, because there is another issue more readily addressed at this time.

Does the 240 W/m^2 averaged insolation value properly describe the system?

It would do no good to continue working out whether the observed average global temperature and emissions can be supported physically without first determining whether or not the averaged input value is a solid assumption to build a description of the system around.

The justification given essentially states that the temperatures would be the same whether one uses the averaged value or not, because over a long enough interval any given point on the surface will receive that much power. Much like the subtle error induced by treating temperature as decided by emissions, the error here is difficult to identify when presented with the standard model of the climate.

The difficulty lies in how the discussion is framed, the language used and explanations put forth all tend to assume these methods can be taken as a given, and it is easy to get sidetracked into examinations of other portions of the subject when discussing it with others. Without another planet and star handy to test these ideas, it is difficult to provide a physical justification for alternative explanations and descriptions of the processes involved.

Hypothesis

A body which receives radiation that varies from 100% to 0% and back to 100% power over a given period of time will reach a higher peak and average temperature than a body which receives constant radiation at 50% power over the same period of time.

I started out by asking whether or not there was a paradox induced by assuming that averaging the power received over a period of intermittent illumination would result in the same temperature distribution regardless of whether or not the power output was averaged. Naive calculations suggested that the averaged power would produce a lower temperature, as suggested by the theoretical work of Stefan, Boltzmann, and Planck.

There is a bit of complexity introduced by calculating the temperatures as though the surface was radiating into a vacuum, rather than radiating into an atmosphere. The calculations which assume a vacuum wind up with far greater variation in temperatures, while the effect of an atmosphere appears to reduce those variations dramatically, regardless of composition.

Taking this into effect I hypothesized that the variation would be weighted towards the higher temperatures in such a manner as to raise the averages of the full power-half duration datasets above those of the half power-full duration datasets.

Experimental Setup and Methodology

You will need:

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Three outdoor thermometers

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Three boxes, at least two insulated styrafoam with removable lids is ideal and a platform to place them a safe distance below the light source if necessary

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A light source capable of illuminating the boxes evenly from above Two sets of light bulbs, one at half power, one at full power, (120 W and 240 W for example)

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Two containers which will fit inside of the large boxes and hold water, open at the top Two black trash bags large enough to line the two boxes placed directly below the light source Two flat objects which can support the thermometers and shade them from direct illumination or contact with the sides or bottom of the box, lids from the two containers should suffice

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Two sheets of clear plastic large enough to cover the lids A tool to cut a window into the box lids Tape to fasten the plastic onto the lids

Arrangement and Procedure

Place the black trash bags into the two boxes which will receive direct illumination and smooth the air out until the bags are mostly flush with the interior surface of the boxes. Place them a safe distance from the light source (no closer than 2 feet if using 240 W, one foot + another inch for every 20 W should be sufficient) and after filling the containers over half way with water, place them inside the boxes where they will be directly illuminated. Cut windows into the box lids large enough for the water filled containers to be illuminated, then fasten the plastic over the windows, one sealed completely, the other with gaps sufficient to allow convection.

Place the third box and thermometer near the two boxes to be illuminated, but not where it will be directly warmed by the light source. This will be a control for the air temperature in the room. Attach the other two thermometers to the flat surfaces, the container lids will suffice, and prop them up so they are neither directly illuminated nor directly in contact with the sides or bottom of the box. Use a water bath to prime all three thermometers to the same temperature, 70 to 72 Fahrenheit (295 K) is what I used for a baseline. Half Power Dataset Place the thermometers in position below the light source at half power, 120 W from two 60 W bulbs in my case. After recording the starting temperature, turn the light on for the first interval. I recorded the temperature at the 30 and 60 minute marks for all three thermometers, then reset them to

the baseline value. Place the thermometers back into their original boxes, and put the lids on, noting which box is sealed to prevent convection and which is not, then turn the light source on at half power again for the same duration, and make measurements at the same intervals, 30 and 60 minutes in my case. Full Power Dataset Repeat the above steps with the higher output bulbs, 240 W from four 60 W bulbs in my case. Reset the thermometers to the baseline, place them into the boxes with the lids off, and turn the light on for the first interval. I recorded the temperature at the 30 minute mark and turned the light off, then returned at the 60 minute mark to record the end temperature. Reset the thermometers to the baseline value, place them back into their box, then fasten the lids in the same arrangement as the half power dataset. Turn the light on for the second interval, record the temperature at the 30 minute mark and turn the light off, then return at the 60 minute mark and record the final temperatures.

Discussion and Results

I performed several variations on this basic experiment, to eliminate errors and examine the influence of different variables on the results. In this case the independent variables are the power radiated by the light bulbs, the time they are left on, and the configuration of the contents of the box. Each of which is only changed one at a time to ensure repeatability and sound results. The dependent variable is the air temperature inside of the box as measured by a thermometer. Controlled variables considered included albedo, convection, and the output of the light bulbs. I sought to eliminate error in this case by using the same type of bulbs from the same box, and thus the same manufacturing batch. Similarly I made efforts to prevent variations in the amount of light reflected by the environment, equipment, and boxes. Additionally I ran simultaneous comparisons with one vented and one unvented box to examine the changes introduced by convection or the lack thereof. For each data set I varied

certain aspects of these variables in a controlled fashion to determine whether or not they would invalidate my results. These variations include several versions of the following, many reversed and taken as single and twinned box datasets:

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Bare boxes with the thermometers resting on the containers but directly illuminated Same as the above but with bags containing high CO2 concentrations (large freezer bags with a bottle of soda inside, as much air squeezed out as possible before sealing the bags, then shake the bottle up and open it inside of the bag slowly enough that the bag inflates, but not so quickly that the soda escapes from the bottle, reseal the bottle, then place it inside the bare boxes so the inflated portion of the bag is above the thermometer

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Bare boxes with CO2 inflated bags over the thermometers and windowed lids as described above

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Bare boxes with the thermometers resting on the containers as before but inserted into freezer bags mostly filled with water

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Same as last but with the lids in place Just the water filled bags and thermometers on the platform without the boxes Several of the above setups for twice as long, 2 hours illuminated at half power and 1 hour illuminated at full power + 1 hour with no illumination

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The setup with the CO2 inflated bags and lids was also ran backwards, 1 hour at full power and 30 minutes at half power + 30 minutes with no illumination

After taking several datasets I pinned down what I think are the important variables and could not think of any other important variations which could invalidate my findings. It is worth noting an interesting secondary result that I obtained after taking the initial no lid and lidded data sets. Upon

consideration of earlier variations I decided to see if I could replicate a “greenhouse effect” and compare it directly to an actual greenhouse scenario.

To do so I took the same setup just described and added a single CO2 inflated bag and tray of water to the box with the vented lid, and a bag hand fluffed to approximate normal atmospheric makeup and emptied the water from the tray in the box with the unvented lid. Upon seeing the results I decided to reset the entire experiment and perform several instrument swaps to make sure it was not experimental error. After doing so I can say that the effect observed was a result of the combined CO2+water+convection arrangement rather than a fluke, and that data is included below for comparison with a final comment on the implications.

Data

Unvented and Vented lids, no CO2/air bags, both trays with water:

No Lids, 120 W (60x2) 7:15 PM 7:45 PM 8:15 PM Average Lids, 120 W (60x2) 8:20 PM 8:50 PM 9:20 PM Average

Left 72 F (295 K) 75 F (297 K) 77 F (298 K) 74.67 F (296.85 K) No Convection 72 F (295 K) 80 F (299 K) 82 F (300 K) 78 F (298.71 K)

Right 72 F (295 K) 75 F (297 K) 77 F (298 K) 74.67 F (296.85 K) Convection 72 F (295 K) 80 F (299 K) 82 F (300 K) 78 F (298.71 K)

Control 72 F (295 K) 72 F (295 K) 73 F (296 K) 72.3 F (295.56 K) Control 72 F (295 K) 73 F (296 K) 73 F (296 K) 72.67 F (295.74 K)

No Lids, 240 W (60x4) 10:15 PM 10:45 PM 11:15 PM Average Lids, 240 W (60x4) 11:20 PM 11:50 PM 12:20 AM Average

Left 72 F (295 K) 85 F (302 K) 77 F (298 K) 78 F (298.71 K) No Convection 72 F (295 K) 85 F (302 K) 79 F (299 K) 78.67 F (299.08 K)

Right 72 F (295 K) 85 F (302 K) 77 F (298 K) 78 F (298.71 K) Convection 72 F (295 K) 84 F (302 K) 78 F (298 K) 78 F (298.71 K)

Control 72 F (295 K) 73 F (296 K) 72 F (295 K) 72.3 F (295.56 K) Control 72 F (295 K) 73 F (296 K) 73 F (296 K) 72.67 F (295.74 K)

Unvented lid+air and no water, vented lid+CO2 and water

GHE test 240 W (60x4) 9:00 PM 9:30 PM 10:00 PM Average GHE test 120 W (60x2) 10:15 PM 10:45 PM 11:15 PM Average

Dry/Air/No Vent 72 F (295 K) 84 F (302 K) 79 F (299 K) 78.33 F (298.89 K) Dry/Air/No Vent 72 F (295 K) 79 F (299 K) 80 F (300 K) 77 F (298.15 K)

Wet/CO2/Vent 72 F (295 K) 80 F (300 K) 76 F (297 K) 76 F (297.59K) Wet/CO2/Vent 72 F (295 K) 77 F (298 K) 78 F (299 K) 75.67 F (297.41 K)

Control 72 F (295 K) 73 F (295 K) 72 F (296 K) 72.3 F (295.56 K) Control 72 F (295 K) 73 F (296 K) 74 F (295 K) 73 F (295 K)

Final Thoughts

I was a bit surprised to find how close some of the half power runs averages got to the full power runs, but the peaks and averages of the full power runs were never lower than the half power runs. They were either similar or clearly higher often enough that my hypothesis is not falsified, though I expected a more significant difference than observed overall. Additional research and experimentation is needed to examine the properties of surfaces exposed to varied illumination with various properties, though the results were within the range considered most probable by my understanding.

On the other hand, I did not expect the magnitude of the difference between the CO2/Water/Vented and Air/Dry/Unvented runs, and was actually certain I had made some mistake. Upon reseting and controlling for instrument error, experimenter error, and possible external variations I am confident that this is an actual result worth sharing. The presence of a CO2 filled bag and water from which vapor can rise was expected to produce a higher temperature than the other runs, and my unformed hypothesis was essentially that the presence of these elements should help mimic the hothouse-greenhouse type properties of the non-convecting box. In fact the opposite result was found, adding CO2 and water vapor appeared to enhance the ability of the box to cool beyond that due to convection alone. This is very interesting, and merits further investigation, but is beyond the scope of this paper as it was a secondary outcome.

The difficulties which arise when trying to isolate the effects of varying the power radiated were troublesome at first. Noticing that I could balance the radiation reaching the boxes and only vary the amount emitted by the light bulbs allowed me to design an experiment which I could be confident was only testing the effect or effects while my hypothesis relied upon.

An important thing here is being able to replicate results, and ensuring you are not introducing a bias through experimental error. By keeping track of everything that I changed and providing the

reasoning behind my choice of variables, it should be possible for others to replicate the experiment and either confirm my results, or identify where I had made an error in my methodology. Science which can not be replicated is arguably no longer science, so it is better to be too careful when recording data and methodology than not.

As I was able to reproduce the same results for a range of controlled variables, I am confident enough for others to test my hypothesis, though as always a single set of results can not be taken as validation of said hypothesis by itself. It is merely a lack of falsification at this point, though the confirmation of the predicted temperature profiles does lend support to the hypothesis which would then be that much stronger of a result if replicated by the work of another individual performing this experiment besides myself.

References

[1] Einstein, Albert (1905a), "Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt (On a Heuristic Viewpoint Concerning the Production and Transformation of Light)"

[2] Planck, M. (1900a), "Über eine Verbesserung der Wienschen Spektralgleichung". Verhandlungen der Deutschen Physikalischen Gesellschaft 2: 202–204. Translated in ter Haar, D. (1967). "On an Improvement of Wien's Equation for the Spectrum"

[3] Boltzmann, Ludwig (1995). "Conclusions". In Blackmore, John T.. Ludwig Boltzmann: His Later Life and Philosophy, 1900-1906. 2. Springer. pp. 206–207. ISBN 978-0-7923-3464-4.

[4] Kirchhoff, G. (1860). "Ueber das Verhältniss zwischen dem Emissionsvermögen und dem Absorptionsvermögen der Körper für Wärme and Licht". Annalen der Physik und Chemie (Leipzig) 109: 275–301. Translated by Guthrie, F. as Kirchhoff, G. (1860). "On the relation between the radiating and absorbing powers of different bodies for light and heat". Philosophical Magazine Series 4, volume 20: 1–21.