# Why is it hot at the equator, cold at the poles?

Here’s a somewhat mathematical look at why it is hotter at the equator that at the poles. This is high school or basic college level science, using trigonometry (pre calculus), a slight step beyond the basic statement that the sun hits down more directly at the equator than at the poles. That’s the kid’s explanation, but we can understand better if we add a little math.

Solar radiation hits Detroit or any other non-equator point at an angle, As a result, less radiation power hits per square meter of land.

Lets use the diagram at right and trigonometry to compare the amount of sun-energy that falls on a square meter of land at the equator (0 latitude) and in a city at 42.5 N latitude (Detroit, Boston, and Rome are at this latitude). In each case, let’s consider high-noon on March 21 or September 20. These are the two equinox days, the only days each year when the day and night are equal length, and the only times when it is easy to calculate the angle of the sun as it deviates from the vertical by exactly the latitude on the days and times.

More specifically the equator is zero latitude, so on the equator at high-noon on the equinox, the sun will shine from directly overhead, or 0° from the vertical. Since the sun’s power in space is 1050 W/m2, every square meter of equator can expect to receive 1050 W of sun-energy, less the amount reflected off clouds and dust, or scattered off or air molecules (air scattering is what makes the sky blue). Further north, Detroit, Boston, and Rome sit at 42.5 latitude. At noon on March 31 the sun will strike earth at 42.5° from the vertical as shown in the lower figure above. From trigonometry, you can see that each square meter of these cities will receive cos 42.5 as much power as a square meter at the equator, except for any difference in clouds, dust, etc. Without clouds etc. that would be 1050 cos 42.5 = 774 W. Less sun power hits per square meter because each square meter is tilted. Earlier and later in the day, each spot will get less sunlight than at noon, but the proportion is the same, at least on one of the equinox days.

To calculate the likely temperature in Detroit, Boston, or Rome, I will use a simple energy balance. Ignoring heat storage in the earth for now, we will say that the heat in equals the heat out. We now ignore heat transfer by way of winds and rain, and approximate to say that the heat out leaves by black-body radiation alone, radiating into the extreme cold of space. This is not a very bad approximation since Black body radiation is the main temperature removal mechanism in most situations where large distances are involved. I’ve discussed black body radiation previously; the amount of energy radiated is proportional to luminosity, and to T4, where T is the temperature as measured in an absolute temperature scale, Kelvin or Rankin. Based on this, and assuming that the luminosity of the earth is the same in Detroit as at the equator,

T Detroit / Tequator  = √√ cos 42.5 = .927

I’ll now calculate the actual temperatures. For American convenience, I’ll choose to calculation in the Rankin Temperature scale, the absolute Fahrenheit scale. In this scale, 100°F = 560°R, 0°F = 460°R, and the temperature of space is 0°R as a good approximation. If the average temperature of the equator = 100°F = 38°C = 560°R, we calculate that the average temperature of Detroit, Boston, or Rome will be about .927 x 560 = 519°R = 59°F (15°C). This is not a bad prediction, given the assumptions. We can expect the temperature will be somewhat lower at night as there is no light, but the temperature will not fall to zero as there is retained heat from the day. The same reason, retained heat, explains why it is warmer will be warmer in these cities on September 20 than on March 31.

In the summer, these cities will be warmer because they are in the northern hemisphere, and the north pole is tilted 23°. At the height of summer (June 21) at high noon, the sun will shine on Detroit at an angle of 42.5 – 23° = 19.5° from the vertical. The difference in angle is why these cities are warmer on that day than on March 21. The equator will be cooler on that day (June 21) than on March 21 since the sun’s rays will strike the equator at 23° from the vertical on that day. These  temperature differences are behind the formation of tornadoes and hurricanes, with a tornado season in the US centering on May to July.

When looking at the poles, we find a curious problem in guessing what the average temperature will be. At noon on the equinox, the sun comes in horizontally, or at 90°from the vertical. We thus expect there is no warming power at all this day, and none for the six months of winter either. At first glance, you’d think the temperature at the poles would be zero, at least for six months of the year. It isn’t zero because there is retained heat from the summer, but still it makes for a more-difficult calculation.

To figure an average temperature of the poles, lets remember that during the 6 month summer the sun shines for 24 hours per day, and that the angle of the sun will be as high as 23° from the horizon, or 67° from the vertical for all 24 hours. Let’s assume that the retained heat from the summer is what keeps the temperature from falling too low in the winter and calculate the temperature at an .

Let’s assume that the sun comes in at the equivalent of 25° for the sun during the 6 month “day” of the polar summer. I don’t look at equinox, but rather the solar day, and note that the heating angle stays fixed through each 24 hour day during the summer, and does not decrease in the morning or as the afternoon wears on. Based on this angle, we expect that

TPole / Tequator  = √√ cos 65° = .806

TPole = .806 x 560°R = 452°R = -8°F (-22°C).

This, as it happens is 4° colder than the average temperature at the north pole, but not bad, given the assumptions. Maybe winds and water currents account for the difference. Of course there is a large temperature difference at the pole between the fall equinox and the spring equinox, but that’s to be expected. The average is, -4°F, about the temperature at night in Detroit in the winter.

One last thing, one that might be unexpected, is that temperature at the south pole is lower than at the north pole, on average -44°F. The main reason for this is that the snow on south pole is quite deep — more than 1 1/2 miles deep, with some rock underneath. As I showed elsewhere, we expect that, temperatures are lower at high altitude. Data collected from cores through the 1 1/2 mile deep snow suggest (to me) chaotic temperature change, with long ice ages, and brief (6000 year) periods of warm. The ice ages seem far worse than global warming.

Dr. Robert Buxbaum, December 30, 2017

# 18 year pause in global warming

Here is an updated version of the climate change graph. It’s now 18+ years, and as was true with last year’s version, 17+ years of no climate change, I see no significant climate change. Similar to this: Global Warming takes a 15 year rest.

18+ years of Global Temperature anomaly to July 2015. The climate seems to have stopped changing.

Though the average planetary temperature has remained constant, there is local variation. It’s been warm in California for the past 2+ years, but cold in Michigan. Before that, it was warm in Michigan and California was cold. The Antarctic ice is at record high levels while the arctic ice has shrunk enough that we should make a Northwest passage.

Climate vs weather, from the blog of Steven Goddard. It’s funny because…

Theory suggests we should see global warming because of increased CO2 trapping of atmospheric heat 2 miles up or so. The problem with the theory is that it doesn’t include clouds. A few extra clouds, e.g. from Chinese industry, could have more cooling power than a lot of CO2 has heating power. It seems that the effects cancel, and temperature 2-3 miles up is about what you’d expect from entropy.

My biggest climate fear, BTW, is global cooling: a new ice age. They come every 110,000 years or so and we seem overdue.

Global temperatures from the antarctic ice show ice ages every 110,000 years. cyclic chaos and self-similarity.

Robert Buxbaum, July 22, 2015. You may not have noticed, but there have been relatively few hurricanes — something that could change at any minute. Here’s a link to 1/2 hour lecture by a Nobel physicist, Ivar Giaever on the subject. Like me, he notices no change, and thinks warmer is better.

# The 2013 hurricane drought

News about the bad weather that didn’t happen: there were no major hurricanes in 2013. That is, there was not one storm in the Atlantic Ocean, the Caribbean Sea, or the Gulf of Mexico with a maximum wind speed over 110 mph. None. As I write this, we are near the end of the hurricane season (it officially ends Nov. 30), and we have seen nothing like what we saw in 2012; compare the top and bottom charts below. Barring a very late, very major storm, this looks like it will go down as the most uneventful season in at least 2 decades. Our monitoring equipment has improved over the years, but even with improved detection, we’ve seen nothing major. The last time we saw this lack was 1994 — and before that 1986, 1972, and 1968.

Hurricanes 2012 -2013. This year there were only two hurricanes, and both were category 1 The last time we had this few was 1994. By comparison, in 2012 we saw 5 category 1 hurricanes, 3 Category 2s, and 2 Category 3s including Sandy, the most destructive hurricane to hit New York City since 1938.

In the pacific, major storms are called typhoons, and this year has been fairly typical: 13 typhoons, 5 of them super, the same as in 2012.  Weather tends to be chaotic, but it’s nice to have a year without major hurricane damage or death.

In the news, a lack of major storms lead to the lack of destruction of the boats, beaches, and stately homes of the North Carolina shore.

The reason you have not heard of this before is that it’s hard to write a story about events that didn’t happen. Good news is as important as bad, and 2013 had been predicted to be one of the worst seasons on record, but then it didn’t happen and there was nothing to write about. Global warming is supposed to increase hurricane activity, but global warming has taken a 16 year rest. You didn’t hear about the lack of global warming for the same reason you didn’t hear about the lack of storms.

Here’s why hurricanes form in fall and spin so fast, plus how they pick up stuff (an explanation from Einstein). In other good weather news, the ozone hole is smaller, and arctic ice is growing (I suggest we build a northwest passage). It’s hard to write about the lack of bad news, still Good science requires an open mind to the data, as it is, or as it isn’t. Here is a simple way to do abnormal statistics, plus why 100 year storms come more often than once every 100 years.

Robert E. Buxbaum. November 23, 2013.

# Arctic and Antarctic Ice Increases; Antarctic at record levels

Good news if you like ice. I’m happy to report that there has been a continued increase in the extent of both Antarctic and Arctic Ice sheets, in particular the Antarctic sheet. Shown below is a plot of Antarctic ice size (1981-2010) along with the average (the black line), the size for 2012 (dotted line), and the size for 2013 so far. This year (2013) it’s broken new records. Hooray for the ice.

Antarctic ice at record size in 2013, after a good year in 2012

The arctic ice has grown too, and though it’s not at record levels, the Arctic ice growth  is more visually dramatic, see photo below. It’s also more welcome — to polar bears at least. It’s not so welcome if you are a yachter, or a shipping magnate trying to use the Northwest passage to get your products to market cheaply.

Arctic Ice August 2012-2013

The recent (October 2013) global warming report from NASA repeats the Arctic melt warnings from previous reports, but supports that assertion with an older satellite picture — the one from 2006. That was a year when the Arctic had even less ice than in 2012, but the date should be a warning. From the picture, you’d think it’s an easy sail through the Northwest passage; some 50 yachts tried it this summer, and none got through, though some got half way. It’s a good bet you can buy those ships cheap.

I should mention that only the Antarctic data is relevant to Al Gore’s 1996 prediction of a 20 foot rise in the sea level by 2100. Floating ice, as in the arctic, displaces the same amount of mass as water. Ice floats but has the same effect on sea level as if it were melted; it’s only land-based ice that affects sea level. While there is some growth seen in land-ice in the arctic photos above — compare Greenland and Canada on the 2 photos, there is also a lot of glacier ice loss in Norway (upper left corners). The ocean levels are rising, but I don’t think this is the cause, and it’s not rising anywhere near as fast as Al Gore said: more like 1.7mm/year, or 6.7 inches per century. I don’t know what the cause is, BTW. Perhaps I’ll post speculate on this when I have a good speculation.

Other good news: For the past 15 years global warming appears to have taken a break. And the ozone hole shrunk in 2012 to near record smallness. Yeah ozone. The most likely model for all this, in my opinion, is to view weather as chaotic and fractal; that is self-similar. Calculus works on this, just not the calculus that’s typically taught in school. Whatever the cause, its good news, and welcome.

Robert E. Buxbaum, October 21, 2013. Here are some thoughts about how to do calculus right, and how to do science right; that is, look at the data first; don’t come in with a hypothesis.

# Ozone hole shrinks to near minimum recorded size

The hole in the ozone layer, prominently displayed in Al Gore’s 2006 movie, an inconvenient truth has been oscillating in size and generally shrinking since 1996. It’s currently reached its second lowest size on record.

South pole ozone hole (blue circle in photo), shrinks to its 2nd smallest size on record. Note outline of antarctica plus end of south america and africa. Photo Credit: BIRA/IASB

The reason for the oscillation is unknown. The ozone hole is small this year, was large for the last few years, and was slightly smaller in 2002. My guess is that it will be big again in 2013. Ozone is an alternate form of oxygen containing three oxygen atoms instead of the usual two. It is an unstable compound formed by ions in the upper atmosphere acting on regular oxygen. Though the ozone concentration in the atmosphere is low, ozone is important because it helps shield people from UV radiation — radiation that could otherwise cause cancer (it also has some positive effects on bones, etc.).

An atmospheric model of ozone chemistry implicated chlorofluorocarbons (freons) as a cause of observed ozone depletion. In the 1980s, this led to countries restricting the use of freon refrigerants. Perhaps these laws are related to the shrinkage of the ozone hole, perhaps not. There has been no net decrease in the amount of chlorofluorocarbons in the atmosphere, and the models that led to banning them did not predicted the ozone oscillations we now see are common — a fault also found with models of global warming and of stock market behavior. Our best computer models do not do well with oscillatory behaviors. As Alan Greenspan quipped, our best models successfully predicted eight of the last five recessions. Whatever the cause, the good news is that the ozone hole has closed, at least temporarily. Here’s why the sky is blue, and some thoughts on sunlight, radiation and health.

by Dr. Robert E. Buxbaum, dedicated to bringing good news to the perpetually glum.

# Global warming takes a 15 year rest

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 COconcentrations 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.

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. CObehaves 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 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).

# Chaos, Stocks, and Global Warming

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. 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 — 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.

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.

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.

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?

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.

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.

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

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

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)