Category Archives: Weather

Global warming takes a 15 year rest

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I have long thought that global climate change was chaotic, rather than steadily warming. Global temperatures show self-similar (fractal) variation with time and long-term cycles; they also show strange attractors generally states including ice ages and El Niño events. These are sudden rests of the global temperature pattern, classic symptoms of chaos. The standard models of global warming is does not predict El Niño and other chaotic events, and thus are fundamentally wrong. The models assume that a steady amount of sun heat reaches the earth, while a decreasing amount leaves, held in by increasing amounts of man-produced CO2 (carbon dioxide) in the atmosphere. These models are “tweaked” to match the observed temperature to the CO2 content of the atmosphere from 1930 to about 2004. In the movie “An Inconvenient Truth” Al Gore uses these models to predict massive arctic melting leading to a 20 foot rise in sea levels by 2100. To the embarrassment of Al Gore, and the relief of everyone else, though CO2 concentrations continue to rise, global warming took a 15 year break starting shortly before the movie came out, and the sea level is, more-or-less where it was except for temporary changes during periodic El Niño cycles.

Global temperature variation Fifteen years and four El Niño cycles, with little obvious change. Most models predict .25°C/decade.

Fifteen years of global temperature variation to June 2013; 4 El Niños but no sign of a long-term change.

Hans von Storch, a German expert on global warming, told the German newspaper, der Spiegel: “We’re facing a puzzle. Recent CO2 emissions have actually risen even more steeply than we feared. As a result, according to most climate models, we should have seen temperatures rise by around 0.25 degrees Celsius (0.45 degrees Fahrenheit) over the past 10 years. That hasn’t happened. [Further], according to the models, the Mediterranean region will grow drier all year round. At the moment, however, there is actually more rain there in the fall months than there used to be. We will need to observe further developments closely in the coming years.”

Aside from the lack of warming for the last 15 years, von Storch mentions that there has been no increase in severe weather. You might find that surprising given the news reports; still it’s so. Storms are caused by temperature and humidity differences, and these have not changed. (Click here to see why tornadoes lift stuff up).

At this point, I should mention that the majority of global warming experts do not see a problem with the 15 year pause. Global temperatures have been rising unsteadily since 1900, and even von Storch expects this trend to continue — sooner or later. I do see a problem, though, highlighted by the various chaotic changes that are left out of the models. A source of the chaos, and a fundamental problem with the models could be with how they treat the effects of water vapor. When uncondensed, water vapor acts as a very strong thermal blanket; it allows the sun’s light in, but prevents the heat energy from radiating out. CO2 behaves the same way, but weaker (there’s less of it).

More water vapor enters the air as the planet warms, and this should amplify the CO2 -caused run-away heating except for one thing. Every now and again, the water vapor condenses into clouds, and then (sometimes) falls as rain or show. Clouds and snow reflect the incoming sunlight, and this leads to global cooling. Rain and snow drive water vapor from the air, and this leads to accelerated global cooling. To the extent that clouds are chaotic, and out of man’s control, the global climate should be chaotic too. So far, no one has a very good global model for cloud formation, or for rain and snowfall, but it’s well accepted that these phenomena are chaotic and self-similar (each part of a cloud looks like the whole). Clouds may also admit “the butterfly effect” where a butterfly in China can cause a hurricane in New Jersey if it flaps at the right time.

For those wishing to examine the longer-range view, here’s a thermal history of central England since 1659, Oliver Cromwell’s time. At this scale, each peak is an El Niño. There is a lot of chaotic noise, but you can also notice either a 280 year periodicity (lat peak around 1720), or a 100 year temperature rise beginning about 1900.

Global warming; Central England Since 1659; From http://www.climate4you.com

It is not clear that the cycle is human-caused,but my hope is that it is. My sense is that the last 100 years of global warming has been a good thing; for agriculture and trade it’s far better than an ice age. If we caused it with our  CO2, we could continue to use CO2 to just balance the natural tendency toward another ice age. If it’s chaotic, as I suspect, such optimism is probably misplaced. It is very hard to get a chaotic system out of its behavior. The evidence that we’ve never moved an El Niño out of its normal period of every 3 to 7 years (expect another this year or next). If so, we should expect another ice age within the next few centuries.

Global temperatures measured from the antarctic ice showing stable, cyclic chaos and self-similarity.

Global temperatures measured from the antarctic ice showing 4 Ice ages.

Just as clouds cool the earth, you can cool your building too by painting the roof white. If you are interested in more weather-related posts, here’s why the sky is blue on earth, and why the sky on Mars is yellow.

Robert E. Buxbaum July 27, 2013 (mostly my business makes hydrogen generators and I consult on hydrogen).

What’s Holding Gilroy on the Roof

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We recently put a sculpture on our roof: Gilroy, or “Mr Hydrogen.” It’s a larger version of a creepy face sculpture I’d made some moths ago. Like it, and my saber-toothed tiger, the eyes follow you. A worry about this version: is there enough keeping it from blowing down on the cars? Anyone who puts up a large structure must address this worry, but I’m a professional engineer with a PhD from Princeton, so my answer is a bit different from most.

Gilroy (Mr Hydrogen) sculpture on roof of REB Research & Consulting. The eyes follow you.

Gilroy (Mr Hydrogen) sculpture on roof of REB Research & Consulting. The eyes follow you. Aim is that it should withstand 50 mph winds.

The main force on most any structure is the wind (the pyramids are classic exceptions). Wind force is generally proportional to the exposed area and to the wind-speed squared: something called form-drag or quadratic drag. Since force is related to wind-speed, I start with some good statistics for wind speed, shown in the figure below for Detroit where we are.

The highest Detroit wind speeds are typically only 16 mph, but every few years the winds are seen to reach 23 mph. These are low relative to many locations: Detroit has does not get hurricanes and rarely gets tornadoes. Despite this, I’ve decided to brace the sculpture to withstand winds of 50 mph, or 22.3 m/s. On the unlikely chance there is a tornado, I figure there would be so much other flotsam that I would not have to answer about losing my head. (For why Detroit does not get hurricanes or tornadoes, see here. If you want to know why tornadoes lift things, see here).

The maximum area Gilroy presents is 1.5 m2. The wind force is calculated by multiplying this area by the kinetic energy loss per second 1/2ρv2, times a form factor.  F= (Area)*ƒ* 1/2ρv2, where ρ is the density of air, 1.29Kg/m3, and v is velocity, 22.3 m/s. The form factor, ƒ, is about 1.25 for this shape: ƒ is found to be 1.15 for a flat plane, and 1.1 to 1.3 a rough sphere or ski-jumper. F = 1.5*1.25* (1/2 *1.29*22.32) = 603 Nt = 134 lb.; pressure is this divided by area. Since the weight is only about 40 lbs, I find I have to tie down the sculpture. I’ve done that with a 150 lb rope, tying it to a steel vent pipe.

Wind speed for Detroit month by month. Used to calculate the force. From http://weatherspark.com/averages/30042/Detroit-Michigan-United-States

Wind speed for Detroit month by month. Used to calculate the force. From http://weatherspark.com/averages/30042/Detroit-Michigan-United-States

It is possible that there’s a viscous lift force too, but it is likely to be small given the blunt shape and the flow Reynolds number: 3190. There is also the worry that Gilroy might fall apart from vibration. Gilroy is made of 3/4″ plywood, treated for outdoor use and then painted, but the plywood is held together with 25 steel screws 4″ long x 1/4″ OD. Screws like this will easily hold 134 lbs of steady wind force, but a vibrating wind will cause fatigue in the metal (bend a wire often enough and it falls apart). I figure I can leave Gilroy up for a year or so without worry, but will then go up to replace the screws and check if I have to bring him/ it down.

In the meantime, I’ll want to add a sign under the sculpture: “REB Research, home of Mr Hydrogen” I want to keep things surreal, but want to be safe and make sales.

by Robert E. Buxbaum, June 21, 2013

Paint your factory roof white

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Standing on the flat roof of my lab / factory building, I notice that virtually all of my neighbors’ roofs are black, covered by tar or bitumen. My roof was black too until three weeks ago; the roof was too hot to touch when I’d gone up to patch a leak. That’s not quite egg-frying hot, but I came to believe my repair would last longer if the roof stayed cooler. So, after sealing the leak with tar and bitumen, we added an aluminized over-layer from Ace hardware. The roof is cooler now than before, and I notice a major drop in air conditioner load and use.

My analysis of our roof coating follows; it’s for Detroit, but you can modify it for your location. Sunlight hits the earth carrying 1300 W/m2. Some 300W/m2 scatters as blue light (for why so much scatters, and why the sky is blue, see here). The rest, 1000 W/m2 or 308 Btu/ft2hr, comes through or reflects off clouds on a cloudy day and hits buildings at an angle determined by latitude, time of day, and season of the year.

Detroit is at 42° North latitude so my roof shows an angle of 42° to the sun at noon in mid spring. In summer, the angle is 20°, and in winter about 63°. The sun sinks lower on the horizon through the day, e.g. at two hours before or after noon in mid spring the angle is 51°. On a clear day, with a perfectly black roof, the heating is 308 Btu/ft2hr times the cosine of the angle.

To calculate our average roof heating, I integrated this heat over the full day’s angles using Euler’s method, and included the scatter from clouds plus an absorption factor for the blackness of the roof. The figure below shows the cloud cover for Detroit.

Average cloud cover for Detroit, month by month.

Average cloud cover for Detroit, month by month; the black line is the median cloud cover. On January 1, it is strongly overcast 60% of the time, and hardly ever clear; the median is about 98%. From http://weatherspark.com/averages/30042/Detroit-Michigan-United-States

Based on this and an assumed light absorption factor of σ = .9 for tar and σ = .2 after aluminum. I calculate an average of 105 Btu/ft2hr heating during the summer for the original black roof, and 23 Btu/ft2hr after aluminizing. Our roof is still warm, but it’s no longer hot. While most of the absorbed heat leaves the roof by black body radiation or convection, enough enters my lab through 6″ of insulation to cause me to use a lot of air conditioning. I calculate the heat entering this way from the roof temperature. In the summer, an aluminum coat is a clear winner.

Detroit High and Low Temperatures Over the ear

High and Low Temperatures For Detroit, Month by Month. From http://weatherspark.com/averages/30042/Detroit-Michigan-United-States

Detroit has a cold winter too, and these are months where I’d benefit from solar heat. I find it’s so cloudy in winter that, even with a black roof, I got less than 5 Btu/ft2hr. Aluminizing reduced this heat to 1.2 Btu/ft2hr, but it also reduces the black-body radiation leaving at night. I should find that I use less heat in winter, but perhaps more in late spring and early fall. I won’t know the details till next year, but that’s the calculation.

The REB Research laboratory is located at 12851 Capital St., Oak Park, MI 48237. We specialize in hydrogen separations and membrane reactors. By Dr. Robert Buxbaum, June 16, 2013

Chaos, Stocks, and Global Warming

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Two weeks ago, I discussed black-body radiation and showed how you calculate the rate of radiative heat transfer from any object. Based on this, I claimed that basal metabolism (the rate of calorie burning for people at rest) was really proportional to surface area, not weight as in most charts. I also claimed that it should be near-impossible to lose weight through exercise, and went on to explain why we cover the hot parts of our hydrogen purifiers and hydrogen generators in aluminum foil.

I’d previously discussed chaos and posted a chart of the earth’s temperature over the last 600,000 years. I’d now like to combine these discussions to give some personal (R. E. Buxbaum) thoughts on global warming.

Black-body radiation differs from normal heat transfer in that the rate is proportional to emissivity and is very sensitive to temperature. We can expect the rate of heat transfer from the sun to earth will follow these rules, and that the rate from the earth will behave similarly.

That the earth is getting warmer is seen as proof that the carbon dioxide we produce is considered proof that we are changing the earth’s emissivity so that we absorb more of the sun’s radiation while emitting less (relatively), but things are not so simple. Carbon dioxide should, indeed promote terrestrial heating, but a hotter earth should have more clouds and these clouds should reflect solar radiation, while allowing the earth’s heat to radiate into space. Also, this model would suggest slow, gradual heating beginning, perhaps in 1850, but the earth’s climate is chaotic with a fractal temperature rise that has been going on for the last 15,000 years (see figure).

Recent temperature variation as measured from the Greenland Ice. A previous post had the temperature variation over the past 600,000 years.

Recent temperature variation as measured from the Greenland Ice. Like the stock market, it shows aspects of chaos.

Over a larger time scale, the earth’s temperature looks, chaotic and cyclical (see the graph of global temperature in this post) with ice ages every 120,000 years, and chaotic, fractal variation at times spans of 100 -1000 years. The earth’s temperature is self-similar too; that is, its variation looks the same if one scales time and temperature. This is something that is seen whenever a system possess feedback and complexity. It’s seen also in the economy (below), a system with complexity and feedback.

Manufacturing Profit is typically chaotic -- something that makes it exciting.

Manufacturing Profit is typically chaotic — and seems to have cold spells very similar to the ice ages seen above.

The economy of any city is complex, and the world economy even more so. No one part changes independent of the others, and as a result we can expect to see chaotic, self-similar stock and commodity prices for the foreseeable future. As with global temperature, the economic data over a 10 year scale looks like economic data over a 100 year scale. Surprisingly,  the economic data looks similar to the earth temperature data over a 100 year or 1000 year scale. It takes a strange person to guess either consistently as both are chaotic and fractal.

gomez3

It takes a rather chaotic person to really enjoy stock trading (Seen here, Gomez Addams of the Addams Family TV show).

Clouds and ice play roles in the earth’s feedback mechanisms. Clouds tend to increase when more of the sun’s light heats the oceans, but the more clouds, the less heat gets through to the oceans. Thus clouds tend to stabilize our temperature. The effect of ice is to destabilize: the more heat that gets to the ice, the more melts and the less of the suns heat is reflected to space. There is time-delay too, caused by the melting flow of ice and ocean currents as driven by temperature differences among the ocean layers, and (it seems) by salinity. The net result, instability and chaos.

The sun has chaotic weather too. The rate of the solar reactions that heat the earth increases with temperature and density in the sun’s interior: when a volume of the sun gets hotter, the reaction rates pick up making the volume yet-hotter. The temperature keeps rising, and the heat radiated to the earth keeps increasing, until a density current develops in the sun. The hot area is then cooled by moving to the surface and the rate of solar output decreases. It is quite likely that some part of our global temperature rise derives from this chaotic variation in solar output. The ice caps of Mars are receding.

The change in martian ice could be from the sun, or it might be from Martian dust in the air. If so, it suggests yet another feedback system for the earth. When economic times age good we have more money to spend on agriculture and air pollution control. For all we know, the main feedback loops involve dust and smog in the air. Perhaps, the earth is getting warmer because we’ve got no reflective cloud of dust as in the dust-bowl days, and our cities are no longer covered by a layer of thick, black (reflective) smog. If so, we should be happy to have the extra warmth.

The Gift of Chaos

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Many, if not most important engineering systems are chaotic to some extent, but as most college programs don’t deal with this behavior, or with this type of math, I thought I might write something on it. It was a big deal among my PhD colleagues some 30 years back as it revolutionized the way we looked at classic problems; it’s fundamental, but it’s now hardly mentioned.

Two of the first freshman engineering homework problems I had turn out to have been chaotic, though I didn’t know it at the time. One of these concerned the cooling of a cup of coffee. As presented, the coffee was in a cup at a uniform temperature of 70°C; the room was at 20°C, and some fanciful data was presented to suggest that the coffee cooled at a rate that was proportional the difference between the (changing) coffee temperature and the fixed room temperature. Based on these assumptions, we predicted exponential cooling with time, something that was (more or less) observed, but not quite in real life. The chaotic part in a real cup of coffee, is that the cup develops currents that move faster and slower. These currents accelerate heat loss, but since they are driven by the temperature differences within the cup they tend to speed up and slow down erratically. They accelerate when the cup is not well stirred, causing new stir, and slow down when it is stirred, and the temperature at any point is seen to rise and fall in an almost rhythmic fashion; that is, chaotically.

While it is impossible to predict what will happen over a short time scale, there are some general patterns. Perhaps the most remarkable of these is self-similarity: if observed over a short time scale (10 seconds or less), the behavior over 10 seconds will look like the behavior over 1 second, and this will look like the behavior over 0.1 second. The only difference being that, the smaller the time-scale, the smaller the up-down variation. You can see the same thing with stock movements, wind speed, cell-phone noise, etc. and the same self-similarity can occur in space so that the shape of clouds tends to be similar at all reasonably small length scales. The maximum average deviation is smaller over smaller time scales, of course, and larger over large time-scales, but not in any obvious way. There is no simple proportionality, but rather a fractional power dependence that results in these chaotic phenomena having fractal dependence on measure scale. Some of this is seen in the global temperature graph below.

Global temperatures measured from the antarctic ice showing stable, cyclic chaos and self-similarity.

Global temperatures measured from the antarctic ice showing stable, cyclic chaos and self-similarity.

Chaos can be stable or unstable, by the way; the cooling of a cup of coffee was stable because the temperature could not exceed 70°C or go below 20°C. Stable chaotic phenomena tend to have fixed period cycles in space or time. The world temperature seems to follow this pattern though there is no obvious reason it should. That is, there is no obvious maximum and minimum temperature for the earth, nor any obvious reason there should be cycles or that they should be 120,000 years long. I’ll probably write more about chaos in later posts, but I should mention that unstable chaos can be quite destructive, and quite hard to prevent. Some form of chaotic local heating seems to have caused battery fires aboard the Dreamliner; similarly, most riots, famines, and financial panics seem to be chaotic. Generally speaking, tight control does not prevent this sort of chaos, by the way; it just changes the period and makes the eruptions that much more violent. As two examples, consider what would happen if we tried to cap a volcano, or provided  clamp-downs on riots in Syria, Egypt or Ancient Rome.

From math, we know some alternate ways to prevent unstable chaos from getting out of hand; one is to lay off, another is to control chaotically (hard to believe, but true).

 

The martian sky: why is it yellow?

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In a previous post, I detailed my calculations concerning the color of the sky and sun. Basically the sun gives off light mostly in the yellow to green range, with fairly little red or purple. A lot of the blue and green wavelengths scatter leaving the sun  looking yellow because yellow looks yellow and the red plus blue also looks yellow because of additive color.

If you look at the sky through a spectroscope, it’s pretty blue with some green. Sky blue involves a bit of an eye trick of additive color so that we see the scattered blue + green as sky blue and not aqua. At sundown, the sun becomes reddish and the majority of the sky becomes greenish-grey as more green and yellow light gets scattered. The sky near the sun is orange as the atmosphere is thick enough to scatter orange, while the blue and green scatters out.

Now, to talk about the color of the sky on Mars, both at noon and at sunset. Except for the effect of the red color of the dust on Mars I would expect the sky to be blue on Mars, just like on earth but a lighter shade of blue as the atmosphere is thinner. When you add some red from the dust, one would expect the sky to be grey. That is, I would expect to find a simple combination of a base of sky blue (blue plus green), plus some extra red-orange light scattered from the Martian dust. In additive colors, the combination of blue-green and red-orange is grey, so that’s the color I’d expect the Martian sky to be normally. Some photos of the Martian sky match this expectation; see below. My guess is this is on a day when there was not much dust in the air, though NASA provides no details here.

martian sky; looks grey

On some days (high dust days, I assume), the Martian sky is turns a shade of yellow-green. I’d guess that’s because the red-dust absorbs the blue and some of the green spectrum, but does not actually add red. We are thus involved with subtractive color and, in subtractive color orange plus blue-green = butterscotch, not grey or pink.

Martian sky color

I now present a photo of the Martian sky at sunset. This is something really peculiar that I would not have expected ahead of time, but think I can explain now that I see it. The sky looks yellow in general, like in the photo above, but blue around the sun. I could explain this picture by saying that the blue and green of the Martian sky is being scattered by the Martian air (CO2, mostly), just like our atmosphere scatters these colors on earth; the sky near the sun looks blue, not red-orange because the Martian atmosphere is thinner (at noon there is less air to scatter light, but at sun-down the atmosphere is the same thickness as ours, more or less). The red of the dust does not show up in the sky color near the sun since the red-color is back scattered near the sun, and not front scattered. The Martian sky is yellow elsewhere where there is some front scatter of the reddish light reflecting off of the dust. This sounds plausible to me; tell me what you think.

Martian sky at sunset

Martian sky at sunset

As an aside, while I have long understood there was an experimental difference between subtractive and additive color, I have never quite understood why this should be so. Why is it that subtractive color combinations are different, and uniformly different from additive color combinations. I’d have thought you’d get more-or-less the same color if you remove red from one part of a piece of paper and remove blue from another as if you add red, purple, and yellow. A mental model I have (perhaps wrong) is that subtractive color looks like it does because of the details of the spectral absorption of the particular pigment chemicals that are typically used. Based on this model, I expect to find someday some new red and green pigments where the combination looks yellow when mixed on a page. I’ve not found it yet, but that’s my expectation — perhaps you know of a really good explanation for why additive color is so different from subtractive color.

For parents of a young scientist: math

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It is not uncommon for parents to ask my advice or help with their child; someone they consider to be a young scientist, or at least a potential young scientist. My main advice is math.

Most often the tyke is 5 to 8 years old and has an interest in weather, chemistry, or how things work. That’s a good age, about the age that the science bug struck me, and it’s a good age to begin to introduce the power of math. Math isn’t the total answer, by the way; if your child is interested in weather, for example, you’ll need to get books on weather, and you’ll want to buy a weather-science kit at your local smart-toy store (look for one with a small wet-bulb and dry bulb thermometer setup so that you’ll be able to discuss humidity  in some modest way: wet bulb temperatures are lower than dry bulb with a difference that is higher the lower the humidity; it’s zero at 100%). But math makes the key difference between the interest blooming into science or having it wilt or worse. Math is the language of science, and without it there is no way that your child will understand the better books, no way that he or she will be able to talk to others who are interested, and the interest can bloom into a phobia (that’s what happens when your child has something to express, but can’t speak about it in any real way).

Math takes science out of the range of religion and mythology, too. If you’re stuck to the use of words, you think that the explanations in science books resemble the stories of the Greek gods. You either accept them or you don’t. With math you see that they are testable, and that the  versions in the book are generally simplified approximations to some more complex description. You also get to see that there the descriptions are testable, and that are many, different looking descriptions that will fit the same phenomena. Some will be mathematically identical, and others will be quite different, but all are testable as the Greek myths are not.

What math to teach depends on your child’s level and interests. If the child is young, have him or her count in twos or fives, or tens, etc. Have him or her learn to spot patterns, like that the every other number that is divisible by 5 ends in zero, or that the sum of digits for every number that’s divisible by three is itself divisible by three. If the child is a little older, show him or her geometry, or prime numbers, or squares and cubes. Ask your child to figure out the sum of all the numbers from 1 to 100, or to estimate the square-root of some numbers. Ask why the area of a circle is πr2 while the circumference is 2πr: why do both contain the same, odd factor, π = 3.1415926535… All these games and ideas will give your child a language to use discussing science.

If your child is old enough to read, I’d definitely suggest you buy a few books with nice pictures and practical examples. I’d grown up with the Giant Golden book of Mathematics by Irving Adler, but I’ve seen and been impressed with several other nice books, and with the entire Golden Book series. Make regular trips to the library, and point your child to an appropriate section, but don’t force the child to take science books. Forcing your child will kill any natural interest he or she has. Besides, having other interests is a sign of normality; even the biggest scientist will sometimes want to read something else (sports, music, art, etc.) Many scientists drew (da Vinci, Feynman) or played the violin (Einstein). Let your child grow at his or her own pace and direction. (I liked the theater, including opera, and liked philosophy).

Now, back to the science kits and toys. Get a few basic ones, and let your child play: these are toys, not work. I liked chemistry, and a chemistry set was perhaps the best toy I ever got. Another set I liked was an Erector set (Gilbert). Get good sets that they pick out, but don’t be disappointed if they don’t do all the experiments, or any of them. They may not be interested in this group; just move on. I was not interested in microscopy, fish, or animals, for example. And don’t be bothered if interests change. It’s common to start out interested in dinosaurs and then to change to an interest in other things. Don’t push an old interest, or even an active new interest: enough parental pushing will kill any interest, and that’s sad. As Solomon the wise said, the fire is more often extinguished by too much fuel than by too little. But you do need to help with math, though; without that, no real progress will be possible.

Oh, one more thing, don’t be disappointed if your child isn’t interested in science; most kids aren’t interested in science as such, but rather in something science-like, like the internet, or economics, or games, or how things work. These areas are all great too, and there is a lot more room for your child to find a good job or a scholarship based on their expertise in theses areas. Any math he or she learns is certain to help with all of these pursuits, and with whatever other science-like direction he or she takes.   — Good luck. Robert Buxbaum (Economics isn’t science, not because of the lack of math, but because it’s not reproducible: you can’t re-run the great depression without FDR’s stimulus, or without WWII)

Why isn’t the sky green?

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Yesterday I blogged with a simple version of why the sky was blue and not green. Now I’d like to add mathematics to the treatment. The simple version said that the sky was blue because the sun color was a spectrum centered on yellow. I said that molecules of air scattered mostly the short wavelength, high frequency light colors, indigo and blue. This made the sky blue. I said that, the rest of the sunlight was not scattered, so that the sun looked yellow. I then said that the only way for the sky to be green would be if the sun were cooler, orange say, then the sky would be green. The answer is sort-of true, but only in a hand-waving way; so here’s the better treatment.

Light scatters off of dispersed small particles in proportion to wavelength to the inverse 4th power of the wavelength. That is to say, we expect air molecules will scatter more short wavelength, cool colors (purple and indigo) than warm colors (red and orange) but a real analysis must use the actual spectrum of sunlight, the light power (mW/m2.nm) at each wavelength.

intensity of sunlight as a function of wavelength (frequency)

intensity of sunlight as a function of wavelength

The first thing you’ll notice is that the light from our sun isn’t quite yellow, but is mostly green. Clearly plants understand this, otherwise chlorophyl would be yellow. There are fairly large components of blue and red too, but my first correction to the previous treatment is that the yellow color we see as the sun is a trick of the eye called additive color. Our eyes combine the green and red of the sun’s light, and sees it as yellow. There are some nice classroom experiment you can do to show this, the simplest being to make a Maxwell top with green and red sections, spin the top, and notice that you see the color as yellow.

In order to add some math to the analysis of sky color, I show a table below where I divided the solar spectrum into the 7 representative colors with their effective power. There is some subjectivity to this, but I took red as the wavelengths from 620 to 750nm so I claim on the table was 680 nm. The average power of the red was 500 mW/m2nm, so I calculate the power as .5 W/m2nm x 130 nm = 65W/m2. Similarly, I took orange to be the 30W/m2 centered on 640nm, etc. This division is presented in the first 3 columns of the following table. The first line of the table is an approximate of the Rayleigh-scatter factor for our atmosphere, with scatter presented as the percent of the incident light. That is % scattered = 9E11/wavelength^4.skyblue scatter

To use the Rayleigh factor, I calculate the 1/wavelength of each color to the 4th power; this is shown in the 4th column. The scatter % is now calculated and I apply this percent to the light intensities to calculate the amount of each color that I’d expect in the scattered and un-scattered light (the last two columns). Based on this, I find that the predominant wavelength in the color of the sky should be blue-cyan with significant components of green, indigo, and violet. When viewed through a spectroscope, I find that these are the colors I see (I have a pocket spectroscope and used it an hour ago to check). Viewed through the same spectroscope (with eye protection), I expect the sun should look like a combination of green and red, something our eyes see as yellow (I have not done this personally). At any rate, it appears that the sky looks blue because our eyes see the green+ cyan+ indigo + purple in the scattered light as sky blue.220px-RGB_illumination

At sunrise and sunset when the sun is on the horizon the scatter percents will be higher, so that all of the sun’s colors will be scattered except red and orange. The sun looks orange then, as expected, but the sky should look blue-green, as that’s the combination of all the other colors of sunlight when orange and red are removed. I’ve not checked this last yet. I’ll have to take my spectroscope to a fine sunset and see what I see when I look at the sky.

Why isn’t the sky green and the sun orange?

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Part of the reason the sky isn’t green has to do with the color of the sun. The sun’s color, and to a lesser extent, the sky color both are determined by the sun’s surface color, yellow. This surface color results from black body radiation: if you heat up a black object it will first glow red, then orange, yellow, green etc. Red is a relatively cool color because it’s a low frequency (long wavelength) and low frequencies are the lowest energy photons, and thus are the easiest for a black body to produce. As one increases the temperature of a black object, the total number of photons increases for all wavelengths, but the short wavelength (high frequency) colors increase faster than the of long wavelength colors. As a result, the object is seen to change color to orange, then yellow, or to any other color representative of objects at that particular temperature.

Our star is called a yellow sun because the center color of its radiation is yellow. The sun provides radiation in all colors and wavelengths, even colors invisible to the eye, infra red and ultra violet, but because of its temperature, most of the radiated energy appears as yellow. This being said, you may wonder why the sky isn’t yellow (the sky of Mars mostly is).

The reason the sky is blue, is that some small fraction of the light of the sun (about 10%) scatters off of the molecules of the air. This is called Rayleigh scatter — the scatter of large wavelegth waves off of small objects.  The math for this will be discussed in another post, but the most relevant aspect here is that the fraction that is scattered is proportional to the 4th power of the frequency. This is to say, that the high frequencies (blue, indigo, and violet) scatter a lot, about 20%, while the red hardly scatters at all. As a result the sky has a higher frequency color than the sun does. In our case, the sky looks blue, while the sun looks slightly redder from earth than it does from space — at least that’s the case for most of the day.

The sun looks orange-red at sundown because the sunlight has to go through more air. Because of this, a lot more of the yellow, green, and blue scatter away before we see it. Much more of the scatter goes off into space, with the result that the sky to looks dark, and somewhat more greenish at sundown. If the molecules were somewhat bigger, we’d still see a blue sky, maybe somewhat greener, with a lot more intensity. That’s the effect that carbon dioxide has — it causes more sunlight to scatter, making the sky brighter. If the sun were cooler (orange say), the sky would appear green. That’s because there would be less violet and blue in the sunlight, and the sky color would be shifted to the longer wavelengths. On planets where the sun is cooler than ours, the sky is likely green, but could be yellow or red.

Rayleigh scatter requires objects that are much smaller than the light wavelength. A typical molecule of air is about 1 nm in size (1E-9 of a meter), while the wavelength of yellow light is 580 nm. That’s much larger than the size of air molecules. Snow appears white because the size of the crystals are the size of the sun wavelengths, tor bigger, 500-2000 nm. Thus, the snow looks like all the colors of the sun together, and that’s white. White = the sum of all the colors: red + orange + blue + green + yellow + violet + indigo.

Robert Buxbaum  Jan. 27, 2013 (revised)

Newfie joke (Newfie’s are Canadians from Newfinland)

Here’s a Newfie joke; it was originally another joke, but I tweaked it because I was in Frankenmuth today watching folks snow sculpt and ice-fish. As for Newfinland, it’s basically the Appalachia of Canada. The folks there aren’t considered to be particularly bright. Anyway.

So this Newfie decides it’s time he took up ice fishing. It’s winter and all his other Newfie friends ice fish. So he gets a setup with an ice drill, a few short fishing poles, a plastic pail to sit on, and a little tent, and he goes to set it up on the ice early one morning.

He finds a nice, empty spot, but as soon as he’s set up the pail and tent but he hears a voice from somewhere around him, “Go home, there are no fish under the ice.” He looks all around but he can’t see anyone. Who said that? Was that comment for him? He goes back to start drilling, and he hears the voice again. “Go home; there are no fish under the ice.” Well, he still can’t see anyone, but figures that maybe he should go to a different spot. He picks up his stuff, moves about 50 feet away and starts to set up again, when he hears the voice again, just as loud. “Go home, there are no fish under the ice.” “Who are you?” yells the Newfie to no-one in particular, “God?” “I’m the rink manager. There are no fish under the ice.”