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Your foot is likely at a temperature of around 80°F. So, according to the known formulas for Black Body Radiation, it radiates heat away centered on a frequency (color) that is far in the infrared range of light. A bare foot easily does this radiation, and the foot quickly becomes cold if the air temperature is extremely cold, as the foot is able to radiate large amounts of heat.
Wearing a sock interferes with this radiation. Specifically, the sock is selected to be made of a material which ABSORBS most of that infrared radiation. As a result, the sock itself warms up, and then starts doing its own infrared radiation. Radiation is not selective regarding what direction it goes, so the radiation from the sock is essentially split in two, half being radiated outward and half being radiated back inward toward the foot.
Note that this has two major effects. Only half of the original amount of radiative heat losses occur AND the foot receives an additional supply of heat energy! The foot is warmer as a result, which is the whole point of the sock in the first place!
Actually, this single-sock situation is a little more complicated than this. The new additional heat sent back to the foot from the sock allows the temperature of the foot to rise. Again, due to the known radiation equations (specifically the Stefan-Boltzmann equation), the warmer foot will now radiate MORE heat than it previously had done at the lower skin temperature. We can consider this to be a second iteration of the radiation audit now. The skin has risen in temperature, sending more radiation to the sock, so the sock becomes slightly warmer, so the sock radiates more heat both outward and inward. This now warmer sock therefore results in slightly more radiation being sent back to the foot, so now the foot becomes slightly warmer yet. And this happens again and again, each time involving slightly greater amounts of heat going back and forth between the foot and the sock.
Have you noticed that it is NOT IMMEDIATE that your feet are toasty warm when you first put on socks? That it seems to take a minute or two before your feet become nicely warm? That is essentially because of these many iterations of the foot and sock sending radiation back and forth. Eventually, they come to a situation which we call Equilibrium, where the radiation sent each direction is such that no further changes in temperature (of foot or sock) is necessary.
For a single sock, which intercepts ALL the radiation that the foot tries to radiate away, there is a mathematical process of Summation which can calculate the total effect of this infinite number of iterations. It turns out that the maximum net effect is where the foot receives a total heat supply of exactly TWICE what it had initially created on its own. The Stefan-Boltzmann Equation can then be used to calculate the increase in the temperature of the foot (which sock manufacturers obviously never do!) It obviously results in the comfort that we all insist on!
Notice that the end result of this is that the total outward radiation is HALF of the DOUBLED (Summed) radiation of the ultimately warmed sock, in other words, actually the same NET amount of heat loss that the foot's blood supply was providing all along, BUT with the difference of that the skin temperature was now much warmer than originally true.
A second sequence of iterations occur, involving the radiation between the first and second socks. The important result here is that the FIRST sock is now not only receiving heat from the foot, but now also from re-radiation from the second sock!
If you had a hundred pairs of socks on, that second sock would eventually get warmed up to virtually the same temperature that the first sock is at (again, receiving re-radiated infrared heat from the next sock out, etc.) In that case, the first sock could eventually be receiving TWICE the original radiation (from the warmed skin of the foot) and ALSO a lot of radiation received from the second sock. So now the first sock is even warmer, so it re-radiates even greater amounts of heat back to the skin of the foot, again causing all the iterations of additional heating.
With a hundred pairs of socks on, the net effect is that the ORIGINAL source of heat radiation from the foot essentially gets MULTIPLIED MANY TIMES, by all this re-radiation back and forth between the various layers of socks. The very outermost sock would have to radiate away (outward) an amount of radiation exactly equal to what the foot was creating, for an eventual Equilibrium. But that would be true because even it was warmed enough to actually be radiating TWICE that amount, half outward and half back inward to the next sock in. And that next sock in would be warmer still, in order to provide that increased radiation that the outermost sock had to be receiving. And so on.
The net effect of these multiple layers of socks is that the foot can be impressively warm or even hot, even in the most intensely cold environment. NOT from any external source of heating, but simply due to this complex handling of the radiation between the many layers of socks. There is nearly no practical limit to just how hot the skin of the foot can be made in this way! If too many socks are worn, then the person can have severe problems of hyperthermia, over-heating. Again, the important thing to note here is that this hyperthermia is NOT due to any external heat source, but entirely due to heat provided by the foot itself.
We know of some historical data points regarding our atmosphere and planet. From around the year 1000 AD to 1800 AD, we know that the carbon dioxide concentration in our atmosphere was relatively constant near 280 ppmv (parts per million by volume). We know that the average Earth temperature was around 56.75°F during that period. This provides one reliable data point for us here.
We also know that in the extreme distant past, maybe four billion years ago, the Earth did not have any carbon dioxide in the atmosphere (0 ppmv). The distance to the Sun was still as it is now, and the Sun was radiating at essentially the same rate as now, so by using the Stefan-Boltzmann Equation, we can calculate the average surface temperature of the Earth at that time. The exact number depends on a variable called the Albedo, how much solar energy is immediately REFLECTED away to space. The value of the Albedo for the early Earth is not known, but different estimates range from about 30% to about 35%, relatively comparable to now. If different values of the Albedo are used in the Stefan-Boltzmann Equation for the early Earth, values for the average Earth temperature then are in the range of -4°F to -9°F. The first of these two values is commonly assumed, although I personally suspect that the second is likely more accurate. The following discussion uses the popularly accepted value, although the differences regarding future temperature projections are never more than about one degree different.
So we have a second reliable data point, at 0 ppmv and -4°F.
There is a third data point which is considered nearly as reliable as those two. The Earth has experienced Ice Ages at a number of times in its history. It is believed that the average Earth temperature never dropped below 32°F, because with all water freezing, all life might have been exterminated, so it is generally accepted that the Ice Ages were times when the average Earth temperature dropped to around 35°F. Several research projects where ice cores have been drilled and brought up, find tiny air bubbles inside the ice, and the air in those bubbles is carefully studied to determine their composition. As a result of many such Research Projects, it is pretty well known that the carbon dioxide concentration during several Ice Ages has been around 188 ppmv. This then provides a third reliable data point.
We can look at these three data points on a graph.
We can join those data points with a line.
It seems clear that we have essentially a straight line. However, nearly all current claims of analysis of this data are corrupted by an apparent problem with the data! This graph SEEMS to indicate that the CURRENT data point should be at around 390 ppmv and 81°F, but all spokespeople constantly repeat that the current average Earth temperature is only measured to be around 59°F. Therefore, they insist that the graph must be extended as suggested here:
This is obviously a silly idea!
It would be wonderful if it could be true, but for the graph to suddenly make such a drastic bend, without any obvious cause, is ridiculous!
IF the Earth's atmosphere behaves as being composed of Multiple Layers (akin to the many layers of socks analogy above) then the graph would extend in essentially a straight line, at least throughout this graph and far beyond it before ever bending downward.
The explanation for this matter (of not yet SEEING major temperature rise) is actually quite simple. The Earth is so massive, and its Crustal rocks are so poor at heat conduction, that the Earth requires some amount of time in order to heat or cool. The three initial data points were each from very long term situations, where the Earth had had plenty of time to get into an Equilibrium situation, of getting to the appropriate temperature for Radiative Equilibrium. The CURRENT temperature readings cannot be correct, because they do not include that time lag factor.
In fact, in a related Research Project several years ago, we had analyzed the thermal conductivity of the Crustal rocks and determined that there must be around a 140-year time lag of temperature changes in the Earth's surface temperature after any stimulus that affects it. Therefore, the increase in temperature that is now observed (58.75°F), is only a couple degrees F above that of the previous 800 years (56.75°F), because it only reflects the increase in carbon dioxide in the atmosphere which was in the very early part of the Industrial Revolution, around 1860. This reasoning indicates that, if a Multiple Layer atmosphere is in effect, then around the year 2140, we should see the average Earth temperature finally get to Equilibrium (82°F) with current carbon dioxide concentrations (390 ppmv), as shown in the following chart.
However, IF the Earth's atmosphere is behaving as a Single Layer, then the Iteration analysis indicates that the Earth's surface could never get warmer than that which would radiate exactly twice the incoming Solar radiation. Standard calculations then would indicate that the average Earth temperature could never exceed around 76.6°F.
Between billions of years of nature and a few decades of human efforts, the current opacity of the Earth's atmosphere to infrared radiation is around 85.4%. This opacity results in the current average Earth temperature of 58.75°F. These numbers will both significantly rise due to our continued burning of fossil fuels.
Both Horizontal and Vertical Scales Expanded
These last two graphs represent the limiting cases, and the reality is probably somewhere between the two. What is important to note is that the Multiple Layer graph shows a current Equilibrium average Earth temperature of around 81°F (for the current CO2 concentration of around 390 ppmv), while the Single Layer graph shows a current Equilibrium average Earth temperature of around 68°F.
BOTH of these values are in extreme excess of the 2°C or 3.6°F absolute maximum rise that political leaders are claiming to be able to guarantee! This temperature rise is due to PREVIOUS burning of fossil fuels! Even if ALL fossil fuel burning was ended, world-wide, immediately, we certainly face a CERTAIN increase in average Earth temperature of between 22°F (or 12°C) and 9°F (or 5°C) in coming decades!
For leaders to claim to offer guarantees of a maximum of 2°C temperature rise as a result of ADDITIONAL FUTURE burning of fossil fuels, it simply demonstrates their ignorance about the actual reality of the situation.
NO ONE can guarantee a coming temperature rise of less than the 5°C or 12°C noted just above, but those numbers will be INCREASED due to future burning of fossil fuels!
It is astounding that a recent news report announced that ONE-FOURTH OF ALL FOSSIL FUELS EVER BURNED, WERE BURNED IN THIS SINGLE DECADE (2000-2009).
Absolutely NO ONE is getting the message!
Governments and businesses fully intend to burn far MORE fossil fuels in this coming decade, especially since China, India and many other countries are now also doing this activity to drive their economies.
Profit is EVERYTHING! Even at the expense of exterminating all of mankind and all animals! It is all really amazing.
No one yet knows which of these two Models applies to the behavior of the Earth's atmosphere.
http://mb-soft.com/index.html
C Johnson, Theoretical Physicist, Physics Degree from Univ of Chicago