Solar Impulse

Have you heard of the Solar Impulse? It’s a Swiss aircraft that’s powered entirely by solar energy. The ambitious goal of this project is to fly around the world using only solar power. On May 1, they’ll begin a trip from San Francisco to New York City, with multiple stops along the way. They’ve already pulled off a 26 hour flight, as well as an inter-continental journey from Spain to Morocco, powered only by sunshine. (They use battery packs to store the spare energy and power the plane at night.)

When I first heard about this, I was kind of astonished that this is even possible. Are solar panels really sufficient to power an aircraft? And when can I expect to fly in one?

To find out how they managed to pull off this feat, let’s crunch some numbers.

How much power can you get from the sun?

First, let’s work out how much power the plane captures from sunlight. The Solar Impulse has about the same wingspan as a 747 airplane, and its wings are covered in nearly 12,000 solar cells. That’s about 200 square meters of solar cells.

Solar Impulse

Now, the amount of power delivered by sunshine is a well known number. If you ignore clouds, and average over day and night, it comes to about 250 Watts delivered to every square meter of land. This number, 250 Watts/square meter is how much power we, sitting here on earth, can extract directly from the sun.

Put the two numbers together, and we get 250 Watts/square meter × 200 square meters = 50,000 Watts. This is the maximum amount of power that this airplane can theoretically capture from the sun, given its wingspan.

But we don’t have the technology to tap into all of this power. The best commercially available solar cells are about 20% efficient at capturing solar power, and then there are further losses in the batteries and the electric motors, all of which waste some power. Overall, the kind folks at Solar Impulse tell us that 12% of the incoming solar power is pumped out by the electric motors. That’s 12% of 50,000 Watts, leaving us with 6,000 Watts of useful power. Remember that number, we’ll come back to it.

How much power do you need to fly a plane?

So far so good. On average, we’ve got 6,000 Watts being pumped out to fly this plane. But is that enough? To answer this, we need to figure out how much power it takes to fly a plane. There are really two components to answering this question.

1. Weight.

Heavier planes need more power to fly them. That’s because planes fly by throwing air downwards. They need to throw enough air downwards to counteract their own weight, so heavier planes need to ‘work harder’ on throwing air down. To stay afloat, a heavier plane needs to fly faster than a lighter plane, so it can ram into more air each second, and hurl the air downwards, counteracting its own tendency to drop out of the sky. (If you’d like to read more on how this works, see my post entitled Can we build a more efficient airplane? Not really, says physics.)

So it takes more power to fly a heavier plane. That’s pretty intuitive. (By the way, since we’re on the subject, this is also why it’s wrong to say something like “The plane was flying anyway, so my flying on it was carbon-neutral.” No – it takes extra energy to carry your extra weight! Not to mention that airlines would fly fewer planes if there were fewer people flying.)

But weight isn’t the whole story.

Aerodynamic World. Print by Milatree.

2. Aerodynamics, or the ability to glide.

Some planes are just better at staying up than others.  If I throw a paper airplane, it’ll glide across the room. If I take that same piece of paper, crush it into a ball, and throw it with same force, it won’t go nearly as far. The difference is that the paper airplane is more aerodynamic – it’s better able to throw air downwards and keep itself afloat. This is also why, if a 747 were to run out of fuel, it wouldn’t just fall out of the sky like a rock, but would glide about as effectively as a paraglider.

This ability to glide is captured by a number called the glide ratio. Here’s how it works. Imagine that you switch off the engines in a Boeing 747 mid-flight (don’t try this at home). It will end up falling 1 foot for every 12 feet that it moves forward. This means that it has a glide ratio of 12/1 = 12. An albatross glides 20 feet forward for every foot it falls (glide ratio of 20), while a sparrow glides 4 feet forward for every foot of descent (glide ratio of 4). Here’s a table of some more examples.

The larger the glide ratio, the more energy-efficient the airplane, because it means that you have more lift and less drag. Think of the albatross versus a sparrow. The most straightforward way of increasing your glide ratio is by increasing your wingspan, because then you can throw a lot more air down.

Let’s put these two concepts together. A heavier plane (more weight) consumes more energy to fly. A more aerodynamic plane (higher glide ratio) consumes less energy to fly. Divide those two numbers – the weight and the glide ratio – and voilà!  You’ve just found out how much energy it takes to fly a plane a certain distance.

This equation captures a simple idea. The total energy that it takes to fly depends on a plane’s weight, and is inversely related to its ability to glide. If you wanted to build an energy-efficient airplane, you’d want to make it really light, and have a large glide ratio. This is exactly what the Solar Impulse pulls off with a very light body and a huge wingspan. It’s quite a technical feat to balance these opposing demands. You could call it the albatross’s dilemma.

Now, the equation above tells you how much energy it takes to fly a certain distance. To convert this into a power rating, you need to multiply it by the speed of the plane.

The power consumed by a plane depends essentially on three numbers – its speed, its weight, and its glide ratio.

We’re finally ready to plug in the numbers for the Solar Impulse. The mass of the plane is 1600 kg, which corresponds to a weight of 15,680 Newtons (to go from kilograms to weight on Earth, you have to multiply by the conversion factor of 9.8 Newtons/kg). The glide ratio of the Solar Impuse is 40. How about the speed? Their website tells us that it has an average flying speed of 43 miles per hour, which is about 19 meters/second.

Plugging the numbers into our equation, we get that the total power needed to fly this plane is about 7,500 Watts.

Uh-oh.

Remember the 6,000 Watts of power that we got from the solar panels? The plane seems to consume more power than it produces. Our model predicts that plane doesn’t have enough power to stay up in the air. But this contradicts experiment – we know that this plane is able to fly. What’s going wrong?

Here’s what I think happened. I suspect the average speed that they offer on the website is a tad on the high side. Perhaps they measured this during a daytime flight, when the solar power is twice as high as the day-and-night average? I don’t know. Instead, let’s get the average speed from real flight data. In particular, the New York Times reports that for the Solar Impulse’s 26 hour voyage (day and night), it flew at an average speed of 26 miles/hour (nearly 12 meters/second).

Plug that number in, and you get a power consumption of 4,700 Watts. That’s safely within the 6,000 Watts that the solar panels can produce. Phew.

In reality, this number could be a little higher because the plane might have met some headwind along the way. And there are some assumptions in our calculation that put our estimate on the lower side of things (I assumed that the plane is flying at its most energy-efficient speed, and didn’t take into account the extra energy for takeoff and landing). But even with wiggle room of 25%, it’s less than 6000 Watts. The plane isn’t going to run out of juice, unless it meets some clouds or some serious tailwind.

Toy models keep the essence and leave out the nitty-gritty details. Avioncitos by José Manuel Ríos Valiente

In summary, we reasoned our way through a quantitative estimate of the power production and consumption of a solar-powered plane. Happily, the two numbers match. This sort of back-of-the-envelope calculation is what physicists call a toy model – you leave out the nitty-gritty details, and strip a problem down to its bare essentials. If it works, you can get a lot of insight with not a whole lot of work. For example, we didn’t need to delve into any hairy details of fluid dynamics, or get bogged down in drag coefficients of an airplane, the physics of solar cells, and so on. Yet we were able to make a reasonable, testable prediction.

So, when do I get my solar plane?

Before we get ahead of ourselves and envision a world where we can zip around in our carbon neutral planes, let’s think of a few issues, and see what physics has to say about them.

1. Scalability. As we’ve seen above, the reason this plane can run on solar power is because it’s light and it has a large wingspan. Could we ever get this to work on a more practical scale – could we build a solar-powered equivalent of a 747? No – because carrying more people would mean that you’d have to increase the weight of the plane, and so you’d need more power to fly. But the amount of power you can provide is limited by the solar panels on the wings, so you just won’t be able to meet the demand. Replacing solar panels with more efficient panels won’t help much either – that can buy you energy gains of up to a factor of 2 or so, and that isn’t nearly sufficient to cope with the added weight.

2. Speed. This is one of the biggest limiters – the Solar Impulse will fly across the US at something like 30 or 40 miles per hour.

Planes are typically optimized to fly at the speed that minimizes fuel consumption. It turns out that the lighter the plane and the longer the wingspan, the slower this optimum speed. I’ve argued that solar planes have to stay light and have a large wingspan, so the physics of flight demands that they must fly slowly as well. If you try and speed up a plane past its optimum speed, you’ll have to spend a lot more energy on pushing air out of the way (drag forces).

This isn’t good news for our solar powered plane. If we’re stuck at highway speeds, we might as well just take a train, or a bus. (Here’s an interesting fact – it would cost about the same amount of energy per mile if you took all the people in a 747 and put them in cars, with two people in each car.)

3. Range. The range of a plane is the maximum distance it can go without re-fuelling. You might imagine that bigger planes always have a larger range because they have more fuel, but this isn’t correct, because they’re also heavier, and so they need to use their fuel faster. It turns out that there’s a maximum range that a plane (or a bird) can attain, and it depends on its glide ratio and on the energy density of the fuel.

Energy density is just a number that tells you how much energy you can get out of a kilogram of fuel. Gasoline has an energy density of 40 million Joules/kg = 40 MJ/kg. Plugging in the numbers for a gasoline powered plane gives a maximum range of about 13,000 km. This is about the distance from the USA to India, and there are direct flights that take you that distance.

If you could always fly in the sunshine, solar planes would have an unlimited range. But solar planes have to rely on batteries to fly at night, and this is what limits their range.

The most energy dense batteries around today are Zinc-air batteries, and they have an energy density of about 1.6 million Joules/kg = 1.6 MJ/kg. The Solar Impulse uses Lithium-ion batteries that are about half as energy dense. Plugging in the numbers for Zinc-air batteries gives a range of about 1,000 km, or about a fifth of the distance from San Francisco to New York - this is the furthest that a solar plane can fly in a night! Incidentally, this explains why the Solar Impulse needs to make so many stops to fly from San Francisco to New York. Of course, things may change if we develop phenomenally more energy dense batteries, but we’re still a long, long way from the energy density of gasoline.

Betrand Piccard, co-founder and co-pilot of Solar Impulse, was asked at a press conference whether solar energy would every power mainstream aircrafts. His response:

“It would be crazy to answer yes and stupid to answer no. Today we couldn’t have a solar-powered plane with 200 passengers. Maybe one day.”

Sadly, we still have a long way to go in building a viable, greener alternative to conventional flight. In another blog post, I’ve argued that it isn’t even possible – commercial airplanes are about as energy efficient as they’re ever going to get. The Solar Impulse is certainly an impressive technical feat, and it gets us to think more clearly about what really matters when it comes to building a better airplane.

References:

If you’d like to dig deeper into the math behind these arguments, check out this technical chapter in David Mackay’s book.

His immensely readable book Sustainable Energy – without the hot air is the best resource I’ve seen for thinking clearly and quantitatively about renewable energy.

I’ve previously written on the topic of airplane efficiency. Can we build a more efficient airplane? Not really, says physics.

As I watched this miniature world self-assemble on my windshield like an alien landscape, I wondered about the physics behind these patterns. I learned later that these patterns of ice are related to a rich and very active current area of research in math and physics known as universality. The key mathematical principles that belie these intricate patterns lead us to some unexpected places, such as coffee rings, growth patterns in bacterial colonies, and the wake of a flame as it burns through cigarette paper.

Let’s start with a simple example. Imagine a game similar to Tetris, but where you only have one kind of block – a 1 x 1 square. These identical blocks fall at random, like raindrops. Here’s a question for you. What pattern of blocks would you expect to see building up at the bottom of the screen?

You might guess that since the blocks are falling randomly, you should end up with a smooth, uniform pile of blocks, like the piles of sand that collect on a beach. But this isn’t what happens. Instead, in our make-believe Tetris world, you end up with a rough, jagged skyline, where tall towers sit next to deep gaps. A tall stack of blocks is just as likely to sit next to a short stack as it is to sit next to another tall stack.

This doesn’t look much like what I saw on my windshield. For one thing, there aren’t any gaps or holes. But we’ll get to that later.

This Tetris world is an example of what’s known as a Poisson process, and I’ve written about these processes before. The main point is that randomness doesn’t mean uniformity. Instead, randomness is typically clumpy, just like the jagged skyline of Tetris blocks that you see above, or like the clusters of buzzbombs dropped over London in World War II.

This Tetris example might seem a bit abstract, so let me introduce you to a guy who takes abstract ideas and connects them to real-world examples. His name is Peter Yunker, and he’s a physicist at Harvard who’s also really into his coffee.

What’s the science behind these stains? Coffee stain typeface by Mark Mustaine

Yunker was curious about what causes these ring shaped coffee stains. In 1997, a group of physicists worked out the reason that coffee forms this ring. As a drop of coffee evaporates, liquid from the center rushes outwards to the edge of the drop, carrying coffee particles with it. The drop starts to flatten. Eventually, all you’re left with is a thin ring, as the coffee particles have all rushed to the edge of the drop. Here’s a (wonderfully trippy) video of work by Yunker’s team, showing what this process looks like.

What Yunker demonstrated is really pretty neat. He discovered that the reason that coffee makes a ring has to do with the shape of the coffee particles. Look at a drop of coffee under a microscope, and you’ll find tiny, round coffee particles suspended in water. If you zoom into the edge of an evaporating coffee drop, you’ll see coffee particles sliding past each other, just like the blocks in our Tetris world. In fact, Yunker demonstrated mathematically that the pattern of growth of these coffee particles exactly mirrors that of our randomly falling Tetris blocks!

And here’s the crazy thing. Yunker and his colleagues also discovered that if you replaced all the spherical coffee particles with new particles that are more elongated, sort of like ovals, then you get an entirely different pattern. Instead of a ring, you get a solid blotch. You can see this happening in the video above.

If the coffee particles are round (spheres), you get a coffee ring, but if they’re oval (ellipsoids) you get a coffee blotch instead. Image Credit: Yunker et al (2011)

In one case you get a coffee ring, and in the other case you get a solid blotch. So why does tweaking the shape of the particle change the overall pattern of growth? To understand why the oval particles behave differently from the spherical ones, we first need to tweak our Tetris game. Let’s call the new version Sticky Tetris.

In sticky Tetris, a block keeps falling until it touches another block. As soon as the falling block touches another block, even if only from the side, it immediately sticks into place.

It’s a small modification to the rules, but it has a pretty big consequence. In regular Tetris, it takes very many blocks to fill a deep gap, in sticky Tetris, you can fill a gap with a single block. Very quickly, the height differences between towers start to even out. Instead of the jagged, rough skyline of our regular Tetris world, the skyline in the sticky Tetris world is more smooth.

That looks a lot more like the pattern on my windshield!

And here’s the point. While the spherical coffee particles behave like regular Tetris pieces, the oval shaped particles behave just like these sticky Tetris pieces. The moment an oval coffee particle touches another one, it sticks in place. Instead of the jagged skyline from before, you get this Swiss cheese like pattern, an intricate structures of sprawling filaments separated by holes and gaps.

Oval shaped coffee particles form blotches, mirroring the intricate patterns formed by sticky Tetris blocks. Image credit: Felice Macera

So here we have essentially two distinct kinds of growth processes. On the one hand we have things that accumulate like Tetris blocks, or like particle of coffee in a coffee ring. Here’s an animation of real data from Yunker’s lab showing what this looks like.

On the other hand, we have things that accumulate like Sticky Tetris blocks or like oval shaped coffee particles. The growth of these particles looks like this (again, this is real data).

It’s clear that these are two qualitatively different kinds of patterns.

But it’s also a quantitative difference. Remember that in the Tetris world, you end up with a jagged skyline, while in the sticky Tetris world, the skyline is more smooth. By studying how the topmost layer of particles (the skyline) widens over time, physicists can classify growth processes into different categories. In the jargon of the field, processes that grow at different rates really belong into different Universality Classes.

If the skyline of a growth process widens according to the blue curve, it falls into the same universality class as Tetris. If it widens according to the purple curve, it falls into the same universality class as Sticky Tetris.

You can think of universality classes like a sort of mathematical filing cabinet. Say that you’re studying how ice particles clunk together on your windshield. If the rate at which the skyline widens matches the blue curve above, ice clunking is in the same universality class as Tetris. If it matches the purple curve, then ice clunking is in the same universality class as Sticky Tetris. Now, there are other universality classes out there, and not all growth processes can be neatly filed into a universality class. But the key point is that many seemingly different physical systems, when analyzed mathematically, show identical patterns of growth. This slightly mysterious tendency for very different things to behave in very similar ways is the essence of universality.

What’s more, there is a rich mathematical theory behind this sticky Tetris universality class, described by an equation known as the Kardar–Parisi–Zhang (KPZ) equation. To give you a sense of how current this research is, it was as late as 2010 that mathematicians managed to prove that this KPZ equation is in the same universality class as sticky Tetris.

These deep connections between coffee rings and the KPZ equation took Peter Yunker by surprise. In Yunker’s words, “Alexei Borodin, a mathematician from MIT, contacted us after we published a paper on how particle shape affects particle deposition regarding the coffee-ring effect. He saw our experimental videos online and was reminded of simulations that he has performed. I think this is a great example of the value of reaching out across disciplines – we never would have studied this topic without Alexei bringing it to our attention.”

And this sticky Tetris universality class has turned up in all sorts of odd places. One example involves burning paper. A physics experiment in 1997 took sheets of copier paper, carefully lit them on fire from one end, and recorded the flame front as it burnt through the paper. Here’s a sketch of what they saw. You’re looking at multiple snapshots of the flame, as it burns through the paper.

Snapshots of a flame as it burns through copier paper. J. Maunuksela et al., Phys. Rev. Lett. 79, 1515 (1997).

As the flame burns through the paper, it develops a smooth, wavy pattern. And when the physicists studied the growth of this flame front in detail, they found that it exactly matches the predictions of the KPZ equation. They repeated their experiment using cigarette paper as well as copier paper, and saw the same results. In their words, “The second set of experiments on the cigarette paper gave results consistent with those for the copier paper despite the fact that the cigarette paper is strongly anisotropic and may contain nontrivial correlations.” (Always gotta watch out for those nontrivial correlations in cigarette paper.)

And another example that’s pretty neat and unexpected – bacterial colonies. A team of Japanese physicists showed in 1997 that in certain nutrient conditions, the edge of a bacterial colony grows outwards in exactly the manner predicted by the KPZ (sticky Tetris) universality class. Here’s an animated gif of this in action, adapted from their paper. What you’re looking at is a zoomed in photograph of the edge of a bacterial colony, as it grows in a petri dish.

Now, if you think about it, there’s something deeply puzzling here. Bacterial colonies, travelling flames, and coffee particles are all totally different systems, and there’s no reason to expect that they should obey the same mathematical laws of growth. So what’s behind this mysterious universality? Why do such different beasts play by the same rules?

You might have noticed that all these examples look a little, well, fractal-esque. It turns out that the phenomenon of universality is intricately connected to the fact that these systems are each self-similar, like fractals. As I zoomed my camera into the ice particles on my windshield, the overall pattern looked basically the same. The same is true for the front of the flame, the edge of the bacterial colony, or the skyline of sticky Tetris. Here’s an example of a curve that’s self-similar (or scale-invariant, as physicists like to call it).

Fractals of the world, Unite! Self-similarity is at the heart of universality.

Surprisingly, this self-similarity implies that many of the nitty-gritty physics details of bacteria, flames, or coffee turn out to be irrelevant. According to Peter, “the fractal nature of these growth processes is essential to their universality. In order to be universal, a system cannot depend on its microscopic details, like particle size or typical interaction lengthscale. Thus, a universal system should be scale-invariant.”

Which brings me back to the ice particles on my windshield. They clumped together in these wonderfully fractal-esque patterns that, to my eye, looked a lot like sticky Tetris. I wanted to know if there’s a connection between these ice particles and the KPZ universality class. I put the question to Peter Yunker.

He responded, “These videos are fantastic. I agree with you that the underlying process occurring here appears quite similar to a KPZ process. However, this may be a great example of why it is difficult to identify KPZ processes in real experiments. The rearrangements of these structures have a strong effect on how the interface is developing. Thus, it is very unlikely that this system exhibits the same growth exponents as a KPZ process.”

It seems that the very piece of physics that makes these ice patterns short-lived is also what makes them so hard to study. And so, let me end with a very short video, a tiny meditation on the theme of growth and longevity.

References

Coffee Stains Test Universal Equation. Physics 6, 7 (2013) - an excellent readable account on the research of Yunker, Yodh, Borodin and colleagues

In Mysterious Pattern, Math and Nature Converge. Natalie Wolchover does a really great job of covering Universality from a totally different angle. If you’re not reading her stuff, you ought to!

Ace mathematician Terrence Tao has written a good explainer on Universality. It’s a long read that’s packed with insights.

Animated gifs of Tetris simulations and coffee deposition data were made with permission from data by Yunker et al. (2013)

Academic References

Effects of Particle Shape on Growth Dynamics at Edges of Evaporating Drops of Colloidal Suspensions. Yunker, Lohr, Still, Borodin, Durian and Yodh, Phys. Rev. Lett. 110, 035501 (2013)

Suppression of the coffee-ring effect by shape-dependent capillary interactions. Yunker, Still, Lohr and Yodh, Nature 476, 308–311 (2011)

The Kardar-Parisi-Zhang equation and universality class by Ivan Corwin - Although very mathematical, this an excellent and clearly written review of the KPZ equation and its connection to Universality, written by one of the experts in the field.

Self-Affinity for the Growing Interface of Bacterial Colonies. Wakita, Itoh, Matsuyama and Matsushita, J. Phys. Soc. Jpn. 66 (1997)

Kinetic Roughening in Slow Combustion of Paper. Maunuksela, Myllys, Kähkönen, Timonen, Provatas, Alava and Ala-Nissila, Phys. Rev. Lett. 79, 1515–1518 (1997)

Wow. I’m really excited that Henry Reich, who’s behind the absolutely brilliant series of animated physics explainers Minute Physics, included me in his video list of “the most consistently awesome and creative science storytellers, explainers and teachers”. I got a chance to catch up with Henry at Science Online (more on that later), and it was really great to get his perspective on science communication, on physics explainers, and on the rapidly growing following that his work is amassing. Minute Physics recently crossed a *million* followers – it just blows my mind that a video series on physics can have that reach, and it speaks to Henry’s tremendous gifts as a smart, talented and funny science communicator. The traffic from Henry’s referral actually knocked my blog off the internet, and I had to frantically scramble to get things going again (too much love is a good kind of problem, in my book .

Do check out the video. It includes many of my favorite places on the internet, including Radiolab‘s amazingly engrossing science storytelling and Sean Carroll‘s deliciously idea-dense blog.

In other shamelessly self-promoting news, I’m really floored to be listed in Byliner’s Best of Journalism list of 2012. It’s very cool for me to see this under-two-year-old blog included up there with so many mainstream journalistic organizations. I write this blog in my ever-dwindling free time, and do it for the love of writing and explaining science. It’s been a wild ride, and I’m excited to keep playing. Looking ahead, over the next few months I’m collaborating on a really fun blog-related experiment, so watch this space!

What can these three teach us about temperature?

There’s been a lot of buzz lately in the science blogosphere about a recent experiment where physicists created a gas of quantum particles with a negative temperature – negative as in, below absolute zero. This is pretty strange, because absolute zero is supposed to be that temperature at which all atomic motion ceases, where atoms that normally jiggle about freeze in their places, and come to a complete standstill. Presumably, this is as cold as cold can be. Can anything possibly be colder than this?

Here’s the short answer. It is possible to create negative temperatures. It was actually first done in 1951. But it’s not what it sounds like – these temperatures aren’t colder than absolute zero. For instance, you can’t keep cooling something down to make its temperature drop below absolute zero. In fact, as I’ll try to explain, objects at a negative temperature actually behave as if they’re HOTTER than objects that are at any positive temperature.

To understand this, we first need to know what physicists mean by temperature. You may remember from high school physics or chemistry that temperature measures the average kinetic energy of motion of particles. When you heat a substance, you’re speeding up its molecules, and when you cool it down, you’re slowing the molecules down.

This definition really made sense to me when I could see it for myself, so here is a simulation where you can play around with gas molecules. Go switch on the heater, and then turn up or down the heat, and see what happens.

So far, so good. But physicists realized that this definition of temperature doesn’t always work, because there are more types of energy than kinetic energy of motion. There are even situations where an object has an energy, but there isn’t really anything moving around in the conventional sense, like the magnetic spins in a magnet, or the ones and zeros on your hard disk. These are essentially quantum systems, where it doesn’t really make sense to talk about stuff moving, but you can still write down how much energy it has. It became clear that physics needed a more fundamental definition of temperature, that would make room for these possibilities.

Here’s the new definition that they came up with. Temperature measures the willingness of an object to give up energy. Actually, I lied. This isn’t how they really define temperature, because physicists speak math, not english. They define it as  which says, in words, that the temperature is inversely proportional to the slope of the entropy vs. energy curve.

Now, if you don’t speak math, I’m going to let you in on a little secret. You don’t need to know any math or physics to understand how temperature works. You can use a surprisingly accurate analogy. I first heard this analogy as an undergraduate, in an excellent thermal physics textbook by Daniel Schroeder.

Picture a world where people are constantly exchanging money to attain happiness. This is probably not that hard for you to imagine. But there’s a small twist.

The people in this society have agreed that they will work to maximize happiness – not just their own happiness, but the total happiness in the society. This has surprising consequences. For example, there might be some people who get very happy when they earn a little money. We could call them greedy. Other people don’t really care much about money – they become a little happier when they earn some money, and a little sadder when they lose it. These people are generous - if they’re playing by the rules of the game, they ought to give money to greedy people, to raise the overall happiness of society.

So why am I inventing this socialist utopia with rampant income redistribution? It’s because this is closely analogous to the physics of heat (as Steven Colbert put it, reality has a well know liberal bias).

Here’s the analogy. The socialist commune is what physicists call an isolated system. The people are the objects in this system. The money that they exchange is really energy – a quantity whose total amount is conserved, but that is constantly being exchanged. Happiness is entropy – just as the society wants to maximize happiness, physical systems are driven to maximize their total entropy. And finally, generosity is temperature, the willingness of people (i.e. objects) to give up money (i.e. energy).

This is a lot to swallow, so here’s a handy dictionary that lets you translate from our analogy to real physics:

By looking up this dictionary, everything that we say about our commune is translated into a statement about physics.

Now, imagine that our society consists of people like Warren Buffett. Initially, when they’re poor, getting money makes them very happy. But as they get wealthier, the same amount of money doesn’t make them nearly as happy. If you plot the happiness of these Buffett-like people versus their wealth, it would look something like this.

For the Buffetts, happiness per dollar (greediness) falls as you earn more money

In this world, every dollar earns you less happiness than the last one. So to raise the overall happiness, a rich Buffett should give money to a poor Buffett. This is a world where people become more generous as they acquire money. Or a system whose temperature rises as it gains energy.

The Buffett curve describes normal particles that we know and love, whose temperatures rise as you heat them. These are the jiggling atoms in solids, liquids, or gases.

Now, instead, consider a world of people who are misers, like Uncle Scrooge. Every dollar they earn makes them more happy than the previous dollar did.

For Scrooges, happiness per dollar (greediness) rises with more money

Unlike the Buffetts, if a rich Scrooge gives a dollar to a poor Scrooge, this would lower the overall happiness of Scrooges. In other words,  the Scrooges generosity decreases as they acquire more money. Using our dictionary, this is a system whose temperature drops as it gains energy.

Chew on that last thought for a moment. Could you really have an object that gets colder as you give it energy?

This really happens, when you have a bunch of particles that attract each other. Stars are held together by gravity, and they behave in just this way. As a star loses energy, its temperature rises. Give a star energy, and you’re actually cooling it down. Black holes also behave in this odd way – the more energy you feed them, the bigger they get, and yet, the colder they get.

And if that wasn’t counter-intuitive enough for you, here’s another scenario. Picture a world of people who have attained enlightenment – they actually become happier when they lose money.

In this example, every dollar that the Dalai Lama receives actually makes him sadder. The natural tendency, then, is to give away all his money to whoever is willing to take it. This odd, inverted curve, is exactly the situation that results in negative temperatures – just relabel happiness to entropy and money to energy (mathematically, the curve has negative slope, so it must have negative temperature).

What happens when a negative temperature object meets a positive temperature object? To find out, imagine that the Dalai Lama meets Warren Buffett.

Paradoxically, the Dalai Lama will give his money away to the billionaire, because losing money will make the Dalai Lama happier, and gaining money will make Waren Buffett just a teeny bit happier. In this strange exchange, the net happiness goes up. Using our dictionary, energy flows from a negative temperature object to an object that is at any positive temperature!

This might sound like something dreamt up by an over-zealous theorist. But there are real materials where the entropy versus energy curve looks like the Dalai Lama’s happiness versus money curve, i.e. where the temperature is negative.

To get here, you first need to engineer a system that has an upper limit to its energy. This is a very rare thing – normal, everyday stuff that we interact with has kinetic energy of motion, and there is no upper bound to how much kinetic energy it can have.

Systems with an upper bound in energy don’t want to be in that highest energy state. Just as the Dalai Lama is not happy with a lot of money, these systems have low entropy in (i.e. low probability of being in) their high energy state. You have to experimentally ‘trick’ the system into getting here.

This was first done in an ingenious experiment by Purcell and Pound in 1951, where they managed to trick the spins of nuclei in a crystal of Lithium Fluoride into entering just such an unlikely high energy state. In that experiment, they maintained a negative temperature for a few minutes. Since then, negative temperatures have been realized in many experiments, and most recently established in a completely different realm, of ultracold atoms of a quantum gas trapped in a laser.

From black holes to quantum gases, this analogy shows us that temperature is a lot more subtle than what we measure on a thermometer.

References

Here’s a very charming blog post that explains temperature using Leprechauns and Laser Beams.

The money and happiness analogy is not my own, but borrowed from the marvelous physics textbook Thermal Physics by Daniel Schroeder.

Statistical Mechanics by Pathria and Beale has a nice discussion on how magnetic systems can realize negative temperatures (via Tim Prisk).

John Baez’s blog on negative temperature and the entropy of stars.

Personally, I found the last trick the hardest to believe (3:42 onwards). I wasn’t alone in my skepticism. Here’s what the New York Times had to say about it:

The most eye-popping trick is saved for last. Rugland punts one ball high into the air and then quickly kicks a second ball off a tee. The balls collide in midair.

“That last kick, it took about eight tries,” Rugland said. “The basketball kick, I wanted it to go straight in, but it kept hitting the rim. That actually took a while. That could have been like 40 tries.”

Rugland is so accurate on so many difficult kicks that his video almost seems too good to be true. It brings to mind doctored videos featuring other athletes, like one of the Los Angeles Lakers star Kobe Bryant leaping over a speeding Aston Martin (Bryant never would have risked his knees). But Rugland insisted his video was real. He said that NRK, Norway’s public broadcasting network, reviewed the raw videos and concluded they were legitimate.

So, inspired by Rhett Allain’s blog posts, I decided to try my hand at analyzing this video with physics.

I downloaded a clip of the last trick, and opened it up in Tracker, an open source physics toolkit for video analysis.

The first problem is that there is a pretty massive perspective distortion in the video. The video camera is pretty close to Rugland, and it’s inconveniently positioned at an angle. Fortunately, tracker has a handy tool that lets you morph the video to correct for this perspective distortion. (Here’s Rhett explaining how to use it).

Here’s the video before correcting for perspective:

And here it is afterwards:

Before correction, the ‘parallel lines’ of the treetops, fence and the turf aren’t really parallel – they converge to a point. After the correction, they seem more or less parallel.

The next step is to track the two footballs. I made a video of what the trick shot looks like when you do this. The first ball is in red, the second in light blue, and the green dots show you the center of mass of the two balls (The center of mass is the midpoint of the line that connect the two balls).

So far, so good. Now, on to the physics. If these trickshots are legitimate, they should come close to obeying the laws of projectile motion. In particular, if you plot the height of each projectile over time, you should get a parabola described by the equation

Here is time, is the vertical launch speed of the ball at time zero, and is the one number that everyone remembers from a physics course  - the acceleration due to gravity, which is .

If you haven’t seen this equation before, all you need to know is that it represents a parabola, and that you can test whether an object is really in free fall by fitting this equation to the data. What’s more, you can try and extract the known acceleration due to gravity.

To do this, take the coefficient of the term in that equation, and multiply it by two. You should recover the acceleration due to gravity .

Does this work for the trick shot? The first thing I need to do is set the scale in the video, so we can convert on-screen distances to real life distances. To do this, I assumed that Rugland is about 6 feet tall (1.8 meters), and am guessing this is accurate to about 20% or so. So I don’t expect any result I get to be more accurate than this.

Update: Rugland told me on twitter that he is 1.9 meters tall, so this guess is well within 10 percent.

Now, to the plots! First up is the plot of the height of the first football (vertical axis), plotted versus time (horizontal axis).

Tracker fits this curve to a parabola, and you can see that the trajectory of the ball (red line) is quite close to the parabola (pink line). I only used data from BEFORE the collision (in yellow) to fit the curve. After the collision, you wouldn’t expect it to stay on the same parabolic path. The curve fit is surprisingly good, considering that there’s definitely some wind resistance, lens distortion, and remaining issues with perspective.

Do we recover the value of gravitational acceleration ()  from this curve? If I take the parameter A from the curve fit and double it, I get . That’s just 5 percent away from the actual value, which is far more accurate than we have any reason to expect.

How about the second ball? Here it is the curve for its height vs. time:

Same trick as before. I used Tracker to fit the second ball’s curve to a parabola (considering only data up till the collision). Then, I just multiply the parameter A times two to get the acceleration due to gravity. This time I get  , which is about 17 percent away from the known value. Again, not too shabby. (The pink line is what you would expect if you extrapolated the balls trajectory to after the collision. In reality, of course, it smacked into the other ball and made a significant course adjustment).

Before we take the next step, I need to introduce a new concept. Imagine that you have a firework in your hand, and you light it and throw it into the air. It begins to trace out a nice, neat parabola. What happens after it explodes? Suddenly, instead of one particle you have dozens, and everything looks like a mess. There is a way out of this mess, and it involves the concept of center of mass.

What physics tells us is that after the firecracker explodes, if we considered the average position of all the little exploded chunks of firecracker, then that average position (the center of mass) will still trace out a parabola. It doesn’t matter if it’s a tiny firecracker or a spectacular fireworks display, all the internal forces of the explosion will cancel out, and the center of mass will trace out a boring, old parabola.

What does this have to do with the two footballs? Well, you can think of a collision as an explosion in reverse. (Update: Added in that link, via Ed Yong on Twitter.) The same idea holds – the center of mass of the two footballs isn’t bothered by the collision. Now, of course, the forces in the collision will dramatically alter the trajectory of each football – they’re bumping into each other, after all. BUT, if you consider the two footballs as one extended system, then these bumps are internal forces, and they cancel each other out (Heck yeah, Newton’s 3rd Law). The upshot is that if we plot the center of mass of the two footballs, we should see a parabola that isn’t really affected by the collision.

Here’s a plot of both balls (red and blue), and the center of mass of the two balls (in green).

Fireworks in reverse?

After the collision, the two footballs converge to their center of mass. (This is what physicists call a highly inelastic collision, because the two particles basically stick to each other. It means that the energy of motion, kinetic energy, isn’t being conserved, probably because the balls start to spin wildly and are therefore bleeding energy in to the rotational motion).

Now, I’m going to take the curve traced by the center of mass (in green) and fit the data points before the collision to a parabola. If this collision is really obeying the laws of physics, then the center of mass shouldn’t care about the collision, and the green curve after the collision should stay on the same path.

Here’s what I get:

The pink curve is the predicted trajectory, based on extrapolating the center of mass motion from before the collision. The green curve (sandwiched between the red and blue) is the real data. It’s not dead on, but it’s no too far either.

One possible reason for the discrepancy is that after the collision, the footballs might move sideways to some extent (i.e. perpendicular to the plane of the camera). This would make the center of mass calculation inaccurate after the collision. Also, at this point the balls are at their furthest from the camera, so the perspective correction might not be so great at this distance.

I’m going to go ahead and say that this video is for real. No one would fake a video while also bothering to preserve the center of mass trajectory!

Kudos to you Håvard Rugland, and I hope you kick some ass in that NFL tryout!

Nerdy footnote:

When you have a hammer, it’s fun to hammer things. For no particular reason, here are a few more numbers that we can infer from the data. Rugland kicked Ball 1 at an angle of about 64 degrees at a speed of about 32 mph. About 1.5 seconds later, and 1.5 meters ahead, he kicked Ball 2 at an angle of 40 degrees and at a speed of about 38 mph. It’s a pretty cool testament to Rugland’s abilities that he’s basically able to solve a physics problem in his head that would give most undergrads a severe headache!

For more gratuitous (and hopefully fun) physics, check out my post on the physics of leaping lemurs, where I solve for the launch speed and launch angle of a sifaka lemur.

In the wake of the tragic massacre at Sandy Hook Elementary School, there’s been a lot of discussion about whether mass shootings in the United States are on the rise. Some sources argue that mass shootings are on the rise, while others argue that the rate has stayed more-or-less constant.

Steven Pinker, author of The Better Angels of Our Nature: Why Violence Has Declined was recently interviewed by CNN. When asked whether incidents such as the Sandy Hook massacre represent a real rise in mass shootings, he responded:

It’s not clear whether we’re seeing a real uptick, or just a cluster of events that are more or less distributed at random. You’ve got to remember – random events will occur in clusters just by sheer chance. So we don’t really know whether the fact that there are many of them in the year 2012 represents a trend or just a very unlucky year.

In this article, I’d like to use data available online to address this question.

I recently wrote a post about randomness and rare events. The main lesson from that article is that randomness isn’t the same thing as uniformity. For example, if on average, sharks attack swimmers 3 times a year, then just by chance, you will expect to see years in which no swimmers are attacked, and years in which 7 swimmers are attacked. To our eyes, streaks like this don’t seem random. But, as I argue in my previous post, we are typically not good judges of randomness. In particular, we vastly underestimate the likelihood of such streaks. And so the question is, how can you test whether a set of events is random?

Here’s how. There is a formula that tells you how many times you expect to see streaks arise from a random process. It’s called the Poisson distribution, and it assumes that your events are rare, have a fixed average rate, and are independent (i.e. that events are just as likely to occur at any time). You can then compare the number of predicted streaks to the real number of streaks in your data, and mathematically test whether a set of events is random or not.

To summarize: if the incidences of mass shootings in the US match a Poisson distribution, then this argues that the streaks (years with unusually high number of shootings) are expected due to chance. If the data doesn’t fit a Poisson distribution, then this suggests that it violates one of the assumptions – either mass shootings are not independent events, or the rate is falling, or it’s on the rise.

The data. I downloaded data for mass shootings in the United States occurring from 1982 to 2012, from this comprehensive Mother Jones article on mass shootings. I used their numbers because they compiled information from multiple credible sources, and they clearly outlined the criteria they used to classify a crime as a mass shooting. (Update: this link has the data in easily accessible formats)

Their data shows a total of 62 mass shootings in 31 years – an average of 2 mass shootings per year. However, 2012 was the most violent year on record, clocking in 7 mass shootings. Is this an outlier, or would you expect to see streaks this large, simply due to chance?

To get at this question, I counted years in which there were 0 mass shootings, 1 mass shooting, 2 mass shootings, and so on..

Out of 31 years of data, we find one year with 7 mass shootings, and four three years with no mass shootings. Are these values consistent with an average of 2 mass shootings a year?

To find out, we can compare these counts to a Poisson distribution with an average value of 2.

In the graph above, the blue bars represent the observed instances of 0,1,2,3.. mass shootings in a year. For example, the long blue bar tells us that there were 10 years with one mass shooting per year. The red dotted curve is the Poisson distribution – these are the outcomes that one expects from a random process with an average value of 2 per year. To my eye, the red curve sort of fits the data, but not quite.

But instead of trusting my eye, we can use statistics to compare these two curves. I used a chi-squared test to test whether the two distribution were significantly different, and found a p-value of 0.18 0.09. What does this mean? It suggests that there is no isn’t strong evidence of clustering beyond what you would expect from a random process. In other words, the occurrences of mass shootings from 1982-2012 are consistent not inconsistent with the assumption that shootings are independent events, occurring at an average rate of 2 per year. However, a p-value of 0.18 0.09 is not particularly high, and if we see a few more years another year as extreme as 2012, it’s likely that this will rule out the hypothesis that mass shootings are random events.

What do I conclude from this? If mass shootings are really occurring at random, then this suggests that they are extreme, hard-to-predict events, and are perhaps not the most relevant measure of the overall harm caused by gun violence.  (Update: That last claim is my deduction and not a conclusion of the above analysis – In response to some of the comments at hackernews, I wanted to clarify this point.) I agree with Steven Pinker’s take, and with this analysis by Chris Uggen, who says:

a narrow focus on stopping mass shootings is less likely to produce beneficial changes than a broader-based effort to reduce homicide and other violence. We can and should take steps to prevent mass shootings, of course, but these rare and terrible crimes are like rare and terrible diseases — and a strategy to address them is best considered within the context of more common and deadlier threats to population health.

We are compelled to pay attention to extreme events. In the words of Steven Pinker, “we estimate risk with vivid examples that we recall“. But as much as we should try to prevent these horrific extreme events from taking place, we should not use them as the sole basis for making inferences that determine policy. The outliers are a tragic part of the overall story, but we also need to pay attention to the rest of the distribution.

On 13 June 1944, a week after the allied invasion of Normandy, a loud buzzing sound rattled through the skies of battle-worn London. The source of the sound was a newly developed German instrument of war, the V-1 flying bomb. A precursor to the cruise missile, the V-1 was a self-propelled flying bomb, guided using gyroscopes, and powered by a simple pulse jet engine that gulped air and ignited fuel 50 times a second. This high frequency pulsing gave the bomb its characteristic sound, earning them the nickname buzzbombs.

From June to October 1944, the Germans launched 9,521 buzzbombs from the coasts of France and the Netherlands, of which 2,419 reached their targets in London. The British worried about the accuracy of these aerial drones. Were they falling haphazardly over the city, or were they hitting their intended targets? Had the Germans really worked out how to make an accurately targeting self-guided bomb?

Fortunately, they were scrupulous in maintaining a bomb census, that tracked the place and time of nearly every bomb that was dropped on London during World War II. With this data, they could statistically ask whether the bombs were falling randomly over London, or whether they were targeted. This was a math question with very real consequences.

Imagine, for a moment, that you are working for the British intelligence, and you’re tasked with solving this problem. Someone hands you a piece of paper with a cloud of points on it, and your job is to figure out if the pattern is random.

Let’s make this more concrete. Here are two patterns, from Steven Pinker’s book, The Better Angels of our Nature. One of the patterns is randomly generated. The other imitates a pattern from nature. Can you tell which is which?

Thought about it?

Here is Pinker’s explanation.

The one on the left, with the clumps, strands, voids, and filaments (and perhaps, depending on your obsessions, animals, nudes, or Virgin Marys) is the array that was plotted at random, like stars. The one on the right, which seems to be haphazard, is the array whose positions were nudged apart, like glowworms

That’s right, glowworms. The points on the right records the positions of glowworms on the ceiling of the Waitomo cave in New Zealand. These glowworms aren’t sitting around at random, they’re competing for food, and nudging themselves away from each other. They have a vested interest against clumping together.

Update: Try this out for yourself. After reading this article, praptak and roryokane over at hacker news wrote a script that will generate random and uniform distributions in your browser, nicely illustrating the point.

Try to uniformly sprinkle sand on a surface, and it might look like the pattern on the right. You’re instinctively avoiding places where you’ve already dropped sand. Random processes have no such prejudices, the grains of sand simply fall where they may, clumps and all. It’s more like sprinkling sand with your eyes closed. They key difference is that randomness is not the same thing as uniformity. True randomness can have clusters, like the constellations that we draw into the night sky.

Here’s another example. Imagine a professor asks her students to flip a coin 100 times. One student diligently did the work, and wrote down their results. The other student is a bit of a slacker, and decided to make up fake coin tosses instead of doing the experiment. Can you identify which student is the slacker?

Student 1:

THHHTHTTTTHTTHTTTHHTHTTHT

HHHTHTHHTHTTHHTTTTHTTTHTH

TTHHTTTTTTTTHTHHHHHTHTHTH

THTHTHHHHHTHHTTTTTHTTHHTH

Student 2:

HTTHTTHTHHTTHTHTHTTHHTHTT

HTTHHHTTHTTHTHTHTHHTTHTTH

THTHTHTHHHTTHTHTHTHHTHTTT

HTHHTHTHTHTHHTTHTHTHTTHHT

Take a moment to reason this through.

The first student’s data has clusters – long runs of up to eight tails in a row. This might look surprising, but it’s actually what you’d expect from random coin tosses (I should know – I did a hundred coin tosses to get that data!) The second student’s data in suspiciously lacking in clusters. In fact, in a hundred coin tosses, they didn’t get a single run of four or more heads or tails in a row. This has about a 0.1% chance of ever happening, suggesting that the student fudged the data (and indeed I did).

Image by Scott Adams.

Trying to work out whether a pattern of numbers is random may seem like an arcane mathematical game, but this couldn’t be further from the truth. The study of random fluctuations has its roots in nineteenth century French criminal statistics. As France was rapidly urbanizing, population densities in cities began to shoot up, and crime and poverty became pressing social problems.

In 1825, France began to collect statistics on criminal trials. What followed was perhaps the first instance of statistical analysis used to study a social problem. Adolphe Quetelet was a Belgian mathematician, and one of the early pioneers of the social sciences. His controversial goal was to apply probability ideas used in astronomy to understand the laws that govern human beings.

In the words of Michael Maltz,

In finding the same regularity in crime statistics that was found in astronomical observations, he argued that, just as there was a true location of a star (despite the variance in the location measurements), there was a true level of criminality: he posited the construct of l’homme moyen (the “average man”) and, moreover, l’homme moyen moral. Quetelet asserted that the average man had a statistically constant “penchant for crime,” one that would permit the “social physicist” to calculate a trajectory over time that “would reveal simple laws of motion and permit prediction of the future” (Gigerenzer et al, 1989).

Quetelet noticed that the conviction rate of criminals was slowly falling over time, and deduced that there must be a downward trend in  the “penchant for crime” in French citizens. There were some problems with the data he used, but the essential flaw in his method was uncovered by the brilliant French polymath and scientist Siméon-Denis Poisson.

Poisson’s idea was both ingenious and remarkably modern. In today’s language, he argued that Quetelet was missing a model of his data. He didn’t account for how jurors actually came to their decisions. According to Poisson, jurors were fallible. The data that we observe is the rate of convictions, but what we want to know is the probability that a defendant is guilty. These two quantities aren’t the same, but they can be related. The upshot is that when you take this process into account, there is a certain amount of variation inherent in conviction rates, and this is what one sees in the French crime data.

In 1837, Poisson published this result in “Research on the Probability of Judgments in Criminal and Civil Matters“. In that work, he introduced a formula that we now call the Poisson distribution. It tells you the odds that a large number of infrequent events result in a specific outcome (such as the majority of French jurors coming to the wrong decision). For example, let’s say that on average, 45 people are struck by lightning in a year. Feed this in to Poisson’s formula, along with the population size, and it will spit out the odds that, say, 10 people will be struck by lightning in a year, or 50, or a 100. The assumption is that lightning strikes are independent, rare events that are just as likely to occur at any time. In other words, Poisson’s formula can tell you the odds of seeing unusual events, simply due to chance.

By Randall Munroe

One of the first applications of Mr. Poisson’s formula came from an unlikely place. Leap sixty years ahead, over the Franco-Prussian war, and land in 1898 Prussia. Ladislaus Bortkiewicz, a Russian statistician of Polish descent, was trying to understand why, in some years, an unusually large number of soldiers in the Prussian army were dying due to horse-kicks. In a single army corp, there were sometimes 4 such deaths in a single year. Was this just coincidence?

A single incidence of death by horse kick is rare (and assumedly independent, unless the horses have a hidden agenda). Bortkiewicz realized that he could use Poisson’s formula to work out how many deaths you expect to see. Here is the prediction, next to the real data.

See how well they line up? The sporadic clusters of horse-related deaths are just what you would expect if horse-kicking was a purely random process. Randomness comes with clusters.

By Ryan North

I decided to try this out for myself. I looked for publicly available datasets for deaths due to rare events, and came across the International Shark Attack File, that tabulates worldwide incidents of sharks attacking people. Here’s the data of shark attacks in South Africa.

The numbers are fairly low, with an average of 3.75. But compare 2008 and 2009. One year has zero shark attacks, and the next has 6. And then in 2010, there are 7. You can already imagine the headlines crying out, “Attack of the sharks!“. But is there really a shark rebellion, or would you expect to see these clusters of shark attacks due to chance? To find out, I compared the data to Mr. Poisson’s prediction.

“Anyone else see the shark fin?” Nice catch, by @Gareth_Elms

In blue are the observed counts of years with a 0,1,2,3.. shark attacks. For example, the long blue bar represents the 3 years in which there were 4 shark attacks (2000, 2005 and 2006). The red dotted line is the Poisson distribution, and it represents the outcomes that you would expect if the shark attacks were a purely random process. It fits the data well – I found no evidence of clustering beyond what is expected by a Poisson process (p=0.87). I’m afraid this rules out the great South African shark uprising of 2010. The lesson, again, is that randomness isn’t uniform.

Which brings us back to the buzzbombs. Here’s a visualization of the number of bombs dropped over different parts, reconstructed by Charles Franklin using the original maps in the British Archives in Kew.

Note: A clarification. The plot above shows the distribution of bombs that were dropped over London. The question I’m asking is, if you zoom in to the part of the city most heavily under attack (essentially the mountain that you see in the figure above), are the bombs being guided more precisely, to hit specific targets?

It’s far from a uniform distribution, but does it show evidence of precise targeting? At this point, you can probably guess how to answer this question. In a report titled An Application of the Poisson Distribution, a British statistician named R. D. Clarke wrote,

During the flying-bomb attack on London, frequent assertions were made that the points of impact of the bombs tended to be grouped in clusters. It was accordingly decided to apply a statistical test to discover whether any support could be found for this allegation.

Clarke took a 12 km x 12 km heavily bombed region of South London, and sliced it up in to a grid. In all, he divided it into 576 squares, each about the size of 25 city blocks. Next, he counted the number of squares with 0 bombs dropped, 1 bomb dropped, 2 bombs dropped, and so on.

In all, 537 bombs fell over these 576 squares. That’s a little under one bomb falling per square, on average. He plugged this number into Poisson’s formula, to work out how much clustering you would expect to see by chance. Here’s the relevant table from his paper:

Compare the two columns, and you can see how incredibly close the prediction comes to reality. There are 7 squares that were hit by 4 bombs each - but this is what you would expect by chance. Within a large area of London, the bombs weren’t being targeted. They rained down at random in a devastating, city-wide game of Russian roulette.

The Poisson distribution has a habit of creeping up in all sorts of places, some inconsequential, and others life-altering. The number of mutations in your DNA as your cells age. The number of cars ahead of you at a traffic light, or patients in line before you at the emergency room. The number of typos in each of my blog posts. The number of patients with leukemia in a given town. The numbers of births and deaths, marriages and divorces, or suicides and homicides in a given year. The number of fleas on your dog.

From mundane moments to matters of life and death, these Victorian scientists have taught us that randomness plays a larger role in our lives than we care to admit. Sadly, this fact offers little consolation when the cards in life fall against your favor.

“So much of life, it seems to me, is determined by pure randomness.” - Sidney Poitier

References

Shark attacks and the Poisson approximation. A nice introduction to using Poisson’s formula, with applications including the birthday paradox, one of my favorite examples of how randomness is counter-intuitive.

From Poisson to the Present: Applying Operations Research to Problems of Crime and Justice. A good read about the birth of operations research as applied to crime.

Applications of the Poisson probability distribution. Includes a list of many applications of the Poisson distribution.

Steven Pinker’s book The Better Angels of our Nature has many great examples of how our intuition about randomness is generally wrong.

Want to know more about the accuracy of the flying bombs? The story is surprisingly rich, involving counterintelligence and espionage. Here’s a teaser.

Satellite images of Hurricanes Camille (left) and Katrina (right). Source: NOAA

As Hurricane Katrina surged towards New Orleans, people faced the unthinkable prospect of abandoning their homes and finding shelter. Worst affected were some of the city’s most vulnerable citizens, the poor and the elderly, parents with young children, people without cars, and people living in flood-prone areas. Among those who stayed back, many were old enough to remember Hurricane Camille, a category 5 storm that devastated the region in 1969. Many homes were spared from flooding then, so it stood to reason that they should hold up to Katrina, also a category 5 storm that was demoted to a category 3 by the time it hit land. Sadly, they were mistaken, as the category rating of the hurricane was not the best measure of the raw destructive power of the storm.

The Saffir-Simpson rating system

In the western hemisphere, hurricanes are all rated on the Saffir-Simpson scale, an empirical measure of storm intensity devised in 1971 by civil-engineer Herbert Saffir and meteorologist Bob Simpson. To compute a storm’s category rating, you have to measure the highest speed sustained by a gust of wind for an entire minute. The wind’s speed is measured at a height of 10 meters  because wind speeds increase as you climb higher, and it is here that they do the most damage. Based on how large this maximum speed is, a storm is assigned to one of five different categories.

Source: Wikipedia

The problem with this number is that it only captures one aspect of a storm’s intensity – the highest speed that it can sustain. Not only is it tricky to measure this peak speed, but different organizations may come to different conclusions about it, depending on their coverage of the wind data. This number doesn’t tell you anything about the size of the storm, nor about how the wind-speeds are distributed overall.

Consider a tale of two storms – the first is fierce but more contained, whereas the second is larger, and though it has lower peak wind speed, these wind speeds are spread over a larger area. The SS scale would give the first storm a higher score, even though the latter may be more destructive. Based on the rating, people might have expected Katrina to be about as destructive as Camille.

A tale of two storms. On the left is Hurricane Camille, a category 5 storm that struck the Gulf Coast in 1969. On the right is Katrina, a category 3 as it hit the same coast in 2005. Warmer colors correspond to higher wind speeds. Although peak wind speed was higher in Camille, high winds spread over a larger region in Katrina, leading to more widespread destruction. Source: NOAA

A rip in the wind tapestry

So how can one take the true measure of a storm? Storms are dangerous because of the energy carried in the moving air. Unless you live in a windy city, or drive your car at high speed on an interstate, you probably don’t think of air as something that carries much energy. But in a storm, strong winds ram into stationary objects, like trees, buildings, or the surface of the ocean, and impart some of their energy of motion. Some structures can safely absorb this energy, while others will give way.

The Wind Map for the morning after Hurricane Sandy made landfall in the US

As Hurricane Sandy made its way through the US, many turned to this incredible real-time wind map to get a larger picture of the storm. I watched Sandy as it made landfall, and was mesmerized by the unexpected beauty that underlies this destructive force. On most days, if you look at the wind map, you’ll find a seamless tapestry made up of delicate threads and broader, sweeping strokes. The wind weaves its way through the central mountains, and brushes through the plains in wide swathes, leaving trails like a comb pulled through the hair of an unruly child. It’s a flow that is sculpted by geography and powered by the ebb and flow of weather systems. Visualizing this flow is like watching a globe-sized zen garden rearrange itself, tended not by any individuals, but by the blind, mathematical laws of fluid dynamics.

On this day, as Hurricane Sandy pummeled through the north-eastern US states, the winds started to pick up outside my window in New Jersey, and the trees swayed violently as the gusts grew stronger. On the wind map, there seemed to be a giant bald spot, a rip in the wind tapestry where from where threads had started to fray.

In essence, a hurricane creates a vortex, like the kind that forms when you drain the water from your bath tub.  Vortices are strange, unwieldy creatures, a consequences of the non-linear equations that govern the flow of fluids. In normal situations, these vortices die out, as their energy drains out to the fluid around them. But hurricanes are self-sustaining, fed by evaporating columns of air rising from warm ocean water.

Source: Geophysical Fluid Dynamics Laboratory

Scientists struggle to adequately model the dynamics of hurricanes. It’s a mathematical balancing act. One a larger scale, you have to worry about the flow of the atmosphere that’s responsible for steering the hurricane. On a finer-scale, you have to grapple with interactions near the core that give the storm its strength. You have to include just enough essential mathematics to reproduce the behavior of a real storm, while leaving out the details that can bog you down into a mire of calculations.

To get a small sense of the complexity of the task, this visualization shows a computer simulation of Hurricane Katrina forming (cue to unnecessarily dramatic blockbuster music)

And here’s a more down-to-earth video of an impressive computer simulation that was able to reproduce the known seasonal cycle of tropical hurricanes (or, as they call them, hurricane-like-vortices):

The measure of a storm

Predicting a storm is one half of the scientific story. The other part is figuring out how destructive it’s going to be. We can get a sense of a storms’ strength with a little bit of high school physics. You might remember that every object in motion carries a certain amount of energy, known as its kinetic energy. The kinetic energy of an object depends on the square of its speed, and is directly proportional to the mass of the object.

Imagine a gust of wind blows by your house, travelling at 50 miles per hour. How much energy does this wind gust carry, and what does that mean in terms of damage it can inflict?

Well, think of a storm as being built out of moving parcels of air. Each of these parcels has a certain amount of kinetic energy. To work this out, we first need the mass of each chunk (in kilograms). We can get this using a little trick. Let’s rewrite a kilogram in an odd way:

What we just did is write a kilogram as the product of kilogram per cubic meter — a density — times a cubic meter — a volume. So, the mass of an object is just its density multiplied by the volume. That means,

Kinetic Energy stored in a chunk of storm wind

Or,

Energy per cubic meter of storm 

This equation tells you how much energy is sitting in each cubic meter of air that whizzes by your window. So let’s try plugging in some approximate numbers. Moisture laden air has a density of about 1 kilogram per cubic meter, and 50 miles per hour is about 22 meters per second.

Plugging in numbers, this tells us that a standard 2-liter coke bottle worth of storm wind at 50 mph carries about half a Joule of kinetic energy. That may not sound like a lot, and it isn’t (it’s about the energy you’d expend in poking someone). But consider what happens when you make the sizes bigger.

Imagine the trunk of a small car filled with storm wind, and you get up to 100 Joules of wind energy. That’s about the kinetic energy carried by a child accidentally stumbling into a glass door. Now take a Volkswagen Beetle, fill it with storm wind, and you get nearly 700 Joules of Wind Energy. That’s the kinetic energy contained in a shot put thrown by the world record holder. Finally, picture many times that energy ramming into a tree or a building, and you get a sense of the kind of damage a storm can deliver.

In other words, by adding up the contributions of these ‘bottles of wind energy’ over the real size of a storm, you start to get a sense of its true destructive potential. 

A wind map for Sandy as it struck the US coast line on 29 October 2012. The kinetic energy carried by the storm is measured in tens or hundreds of Terajoules (TJ). One Terajoule is a trillion (thousand billion) Joules. To put that number in context, the atomic bomb over Hiroshima released about 63 Terajoules of energy. Source: NOAA

What makes this method very different from the SS scale is that different ‘wind bottles’ can have different wind speeds (think of bottling up all the different colors in the wind map above). By adding it all up, you learn much more than just the peak intensity of a storm. This number takes into account how the wind speeds are distributed throughout the bulk of a storm.

What I’ve described here is essentially a measure called the Integrated Kinetic Energy of a storm, and it is used by the National Oceanic and Atmospheric Administration to measure the strength of a storm. (See here for a good review. The main difference is that they only add up the energy for bottles of wind moving faster that a certain minimum speed, and the ‘bottles’ are much bigger than Volkswagens.)

This leads us to a question that was bothering me a few days ago. You may have heard people predicting that Hurricane Sandy may be the biggest storm to hit the US, yet it’s only a category 1 storm. How can this be? Well, this plot helps make things clear.

It shows the integrated kinetic energy for hurricanes hitting the United States, measured when they strike land. I came across it on twitter via Brian McNoldy (@BMcNoldy), an atmospheric researcher at University of Miami.

#Sandy‘s integrated kinetic energy at landfall was second only to Isabel (2003), but was higher than Katrina (2005). twitter.com/BMcNoldy/statu…

— Brian McNoldy (@BMcNoldy) October 31, 2012

You can see that by this measure, Sandy exceeds Katrina in energy content, exceeded in the US only by Hurricane Isabel.

What’s more, the integrated kinetic energy of a storm is a good predictor of its potential to cause storm surges, where sea levels rise dangerously as the wind pushes against the ocean’s surface. Category ratings based on the SS scale, on the other hand, do not predict storm surges well.

As a storm’s kinetic energy rises, so does its potential to cause flooding through storm surges. Source: DOI:10.1175/BAMS-88-4-513

By now, I hope I’ve convinced you that when you hear about a tropical storm or a hurricane, you should look for numbers beyond the SS category rating.

But there’s still another side to the story. Hurricanes bring on an onslaught of energy, but nature manages to fight back. Trees don’t topple like dominoes. Far from being passive agents, it turns out that trees have tricks up their sleeves (or rather, up their trunks) that help them stay rooted. To learn more, stay tuned for the next post on the science of hurricanes.

References

Powell, M., & Reinhold, T. (2007). Tropical Cyclone Destructive Potential by Integrated Kinetic Energy Bulletin of the American Meteorological Society, 88 (4), 513-526 DOI: 10.1175/BAMS-88-4-513

Sandy Wind Analyses at the Hurricane Research Division, NOAA

Operational Hurricane Track and Intensity Forecasting at the Geophysical Fluid Dynamics Laboratory, NOAA

Boeing recently launched a new line of aircraft, the 787 Dreamliner, that they claim uses 20% less fuel than existing, similarly sized planes.

How did they pull off this sizeable bump in fuel efficiency? And can you always build a more fuel-efficient aircraft? Imagine a hypothetical news story, where a rival company came up with a new type of airplane that used half the fuel of its current day counterparts. Should you believe their claim?

More generally, do the laws of physics impose any limits on the efficiency of flight? The answer, it turns out, is yes.

Jet Man, by Ben Heine

There’s something about flying that doesn’t sit well with us. If we never saw a bird fly, it may never have occurred to us to build flying machines of our own.

Here’s where I think this sense of unease comes from. It takes stuff to support stuff. Everyday objects fall unless other things get in their way. Take the floor away, and you’ll plummet to your doom – the air below your feet isn’t going to do much for you. We move through air so effortlessly, that we barely notice it’s there. So what keeps a plane up? There doesn’t seem to be enough ‘stuff’ there to hold up a bird, let alone a Boeing aircraft weighing up to 500,000 pounds. To put that last number in context, its more than the weight of an adult blue whale!

Why is it that planes fly and whales typically don’t? The answer is easy to state, but its consequences are rather surprising. Planes fly by throwing air down. That’s basically it. It’s an important point, so I’ll say it again. Planes fly by throwing air down.

As a plane hurtles through the air, it carves out a tube of air, much of which is deflected downwards by the wings. Throw down enough air fast enough, and you can stay afloat, just as the downwards thrust of a rocket pushes it up. The key is that you have to throw down a lot of air (like a glider or an albatross), or throw it down really fast (like a helicopter or a hummingbird).

A physicist’s two-step guide to flight (it’s simple, really!)

Let’s make this idea more quantitative. Following David MacKay’s wonderful book on Sustainable Energy, I’m going to build a toy model of flight. A good model should give you a lot of bang for the buck. The means being able to predict relevant quantities about the real world while making a minimum of assumptions.

Toy models gone wrong. By Randall Munroe at XKCD.

Step 1: Sweep out a tube of air

As a plane moves, it carves out a tube of air. This air was stationary, minding its own business, until the airplane rammed into it. This costs energy, for the same reason your car’s fuel efficiency drops when you speed up on the highway. Your car has to shove air out of its way.

Exactly how much energy does this cost? You might remember from high school physics that it takes an amount of energy equal to to bring stuff with mass up to a speed .

In our case, we have

There’s still this mysterious factor of the mass of the air tube. To work this out, we can use a favorite trick in the toolbox of a physicist – unit cancellation. We can re-write the humble kilogram as a seemingly complicated product of terms.

What we’ve done here is to express an unknown mass of air in terms of other quantities that we do know. Each of these terms makes sense. Air that’s more dense will weigh more. A fatter plane (larger cross-sectional area) sweeps out more air, as does a faster plane. We’ve arrived at a meaningful result, just by playing around with units. In the words of Randall Munroe, unit cancellation is weird.

Put these two ideas together and here’s what you find:

Here’s a graph of what that looks like.

If you’re with me so far, we just found that for a plane to plow through air, it has to expend an amount of energy proportional to the speed of the plane to third power. (The extra factor of v comes from the fact that faster planes sweep out a larger mass of air.) If you want to go twice as fast, you need to work 8 times as hard to shove air out of your way.

We’ve arrived at a general rule about the physics of drag. This holds true for a car on the highway, or for a swimmer or cyclist in a race. It’s why drag racing cars get only about 0.05 miles to a gallon! If we want to reduce overall energy consumption by cars, one option is to lower the speed limits on highways.

What does this mean for our toy plane? It would seem that the slower the plane, the higher its efficiency. So are airplane speed limits also in order? Absolutely not! To see why, read on to the second half the story..

Step 2: Throw the air down

In order to fly, a plane must throw air downwards. This generates the lift that a plane needs to stay up. It turns out that slower planes have to throw air harder to stay afloat. That’s why slow moving hummingbirds and pigeons have to flap their wings frenetically. It’s also why planes extend flaps while landing – they’re not throwing the air fast enough, so they compensate by throwing more of it.

More precisely, for a plane to stay afloat, the speed of the air jettisoned downwards must be inversely proportional to the speed of the plane. (You can take my word for this, although if you want to see where it comes from, take a look at David MacKay’s book.)

So we can now work out the second part of the puzzle. How much energy does it take to throw air down? As before, this is given by

Just as we did in the first step, let’s express things in terms of the speed of the plane.

In words, the energy spent in generating lift is inversely proportional to the speed of the plane. Here’s what this looks like on a graph.

You can see from the plot that, as far as lift is concerned, slower flight is less efficient than faster flight, because you have to work harder in throwing air downwards.

There’s a lot to chew on here. To summarize, we’ve discovered that in making a machine fly, you have to spend energy (really fuel) in two ways.

1. Drag: You need to spend fuel to push air away. This keeps you from slowing down.
2. Lift: You need to spend fuel to throw air down. This is what keeps the plane afloat.

The total fuel consumption is the sum of these two parts.

If you fly too fast, you’ll spend too much fuel on drag (think of a drag racer or an F-16). Fly too slow, and you’ll have to spend too much fuel on generating lift, like a hummingbird furiously flapping its wings, powered by high calorie nectar. However, at the bottom of this curve there is a happy minimum, an ideal speed that resolves this tradeoff. This is the speed at which a plane is most efficient with its fuel. Be it through the ingenuity of aircraft engineers, or the ruthless efficiency of natural selection,  airplanes and birds are often fine-tuned to be as energy efficient as possible.

Here’s a plot of experimental data of the power consumption of different birds, as their flight speed varies.

You can see that it matches the qualitative predictions of the toy model.

But we can do more than this, and actually extract quantitative predictions from the model. An undergraduate schooled in calculus should be able to work out that special optimal speed at which energy consumption is a minimum. David MacKay plugs in the numbers in  his book, and finds that the optimal speed of an albatross is about 32 mph, and for a Boeing 747 is about 540 mph. Both these numbers are remarkably close to the real values. Albatrosses fly at about 30-55 mph, and the cruise speed of a Boeing 747 is about 567 mph. 

That’s a lot of mileage from a toy model!

And with this physicsy interlude into the world of albatrosses, hummingbirds, and jet planes, we come back to the question of the fuel efficiency of Boeing’s new aircraft.

You can actually use the model to work out the fuel efficiency of a plane. What you find is that it really just depends on a few factors: the shape and surface of the plane, and the efficiency of its engine. And of these factors, the engine efficiency plays the biggest role. So we would predict that engine efficiency, followed by improvements in body design might drive Boeing’s fuel savings.

This agrees with Boeing’s own assessment.

New engines from General Electric and Rolls-Royce are used on the 787. Advances in engine technology are the biggest contributor to overall fuel efficiency improvements.

New technologies and processes have been developed to help Boeing and its supplier partners achieve the efficiency gains. For example, manufacturing a one-piece fuselage section has eliminated 1,500 aluminum sheets and 40,000 – 50,000 fasteners.

Try as we like, we can’t squeeze a lot of improvement out of airplanes. Engines are already remarkably efficient, and you certainly can’t shrink the size of a plane by much, as economy class passengers can well attest. New manufacturing techniques could cut the amount of drag on the plane’s surface, but these improvements would only raise fuel efficiency by about 10%.

To quote David Mackay,

The only way to make a plane consume fuel more efficiently is to put it on the ground and stop it. Planes have been fantastically optimized, and there is no prospect of significant improvements in plane efficiency.

A 10% improvement? Yes, possible. A doubling of efficiency? I’d eat my complimentary socks.

References

I based this blog post on material I learnt from David MacKay’s fantastically clear book, Sustainable Energy without the Hot Air. It’s available online for free, and is highly recommended for anybody looking to use numbers to understand energy.

David MacKay (2009). Sustainable Energy – Without the Hot Air UIT Cambridge Ltd

I used this tip to make those XKCD style plots.

The Maasai and their Diet

Maasai tribe member drinking blood. Image credit: Rita Willaert

The Maasai are a pastoralist tribe living in Kenya and Northern Tanzania. Their traditional diet consists almost entirely of milk, meat, and blood. Two thirds of their calories come from fat, and they consume 600 – 2000 mg of cholesterol  a day. To put that number in perspective, the American Heart Association recommends consuming under 300 mg of cholesterol a day. In spite of a high fat, high cholesterol diet, the Maasai have low rates of diseases typically associated with such diets. They tend to have low blood pressure, their overall cholesterol levels are low, they have low incidences of cholesterol gallstones, as well as low rates of coronary artery diseases such as atherosclerosis.

Even more remarkable are the results of a 1971 study by Taylor and Ho. Two groups of Maasai were fed a controlled diet for 8 weeks. One group – the control group – was given food rich in calories. The other group had the same diet, but with an additional 2 grams of cholesterol per day. Both diets contained small amounts of a radioactive tracer (carbon 14). (You’d never get approval for a study like this today, and for good reason.) By monitoring blood and fecal samples, the scientists discovered that the two groups had basically identical levels of total cholesterol in their blood. In spite of consuming a large dose of cholesterol, these individuals had the same cholesterol levels as the control group.

Here is how the authors concluded their study:

This led us to believe, but without direct proof, that the Masai have some basically different genetic traits that result in their having superior biologic mechanisms for protection from hypercholesteremia

Motivated by these results, we set out to identify genes under selection in the Maasai as a result of these unusual dietary pressures. We scanned the genome looking for genetic signatures of natural selection at work.

The Data

Our data comes from the International HapMap Project, a collaborative experimental effort to study the genetic diversity in humans. The HapMap Project has collected DNA from groups of people from genetically diverse human populations with ancestry in Africa, Asia and Europe. Their anonymized data is publicly available for free. One of the HapMap populations is a group of Maasai from Kinyawa, Kenya  (n=156), and this is the population that we focus on.

DNA sequences on a part of Chromosome 7 from two random individuals, with the differences shown in red.

HapMap does not sequence full genomes, as this would have been prohibitively expensive at the time of data collection. Instead, they employ a shortcut. If you take my DNA sequence and line it up against yours, the two sequences will be about 99.9% similar. But every once in a thousand letters, or so, there will be a difference. You may have an A where I have a C. The HapMap group measures the DNA sequence at these very locations, where humans are known to vary from each other. In essence, they’re sampling the genome, looking only at sites where we expect to see variation. In the jargon of the field, this method is called looking for Single Nucleotide Polymorphisms, or SNPs (pronounced snips).

Hunting for signatures of selection in genetic data

Once you have the data, what can you do with it? We wanted to detect signs of natural selection. The basic idea behind detecting selection in genomic data is quite simple, and it has to do with sex. Every sperm or egg cell that you produce contains a single genome, which is formed by shuffling together the two sets of genomes that you inherited from your parents. Viewed this way, the role of sex is to shuffle together the genomes in a population into new combinations. If you compare the DNA sequences of a group of people, you will see signs of this shuffling.

The effect of sex is to shuffle genomes, in a process known as genetic recombination.

Now lets add natural selection to the mix. What happens if an individual is born with a new mutation that benefits their survival? Over time, you’d expect to see this mutation rise in frequency. Descendants of this individual will be over-represented in the population, as the fraction of people with this beneficial mutation goes up. In essence, the fingerprint of such selection is a reduction of genomic diversity. (I’m describing a particular model of selection here, known as positive natural selection. Some other types of selection can increase diversity, such as the selection on viruses to evade recognition by their host’s immune system.)

A new beneficial mutation arises in an individual (shown in red). It will rise in frequency in the population, leading to a characteristic reduction in diversity. Over time, genetic recombination and new mutations will build back the diversity, and the signal is lost.

Eventually, new mutations will creep in again, and generations of sexual reproduction would build back the diversity. However, if the loss of diversity was sudden enough (strong selection) and not too long ago, you can still detect it today. There are statistical tests (Fst, iHS, XP-EHH) that can formally detect if the reduction in diversity at a given region is sufficient to infer selection. Sabeti et al have a nice review paper that discusses the different methods available to detect selection using genomic data.

Our Results

We used three different methods to detect selection, and our top candidate regions under selection are considered significant by at least two of the methods.

The strongest signal of selection, detected by all 3 methods, was a region on Chromosome 2 containing the Lactase gene (LCT), responsible for breaking down the lactose present in milk. Mutations in a neighboring gene in the cluster, MCM6, are associated with the ability to digest lactose in adulthood.

The strongest signal of selection was a region on Chromosome 2 that contained the LCT gene producing lactase, the enzyme that breaks down the lactose in milk. Interestingly, the default state in all adult mammals is to stop producing lactase in adulthood – our ancestors were all ‘lactose intolerant’. This makes sense from an evolutionary point of view, it forces children to wean from milk, and frees up the mothers resources. It turns out that different sets of mutations arose that gave European and African pastoralists the ability to digest milk. Those of us whose ancestors weren’t pastoralists still have trouble digesting milk.

This is a classic example of a selective sweep – a mutation confers an advantage (the ability to digest milk), and then sweeps through a population like wildfire. This result has been previously described in European populations, and also in African populations (including the Maasai) by Sarah Tishkoff and collaborators. Given that the Maasai consume large amounts of milk, it is not surprising that we see a very strong signal at this locus. We sequenced DNA in this region to confirm this result and, sure enough, we found that one of the lactase persistence conferring mutations identified by Tishkoff was present in the HapMap Maasai samples.

Two of the tests for selection that we used require that you make comparisons with another population. We chose the Luhya of Kenya as a our reference population. Among all the protein-altering mutations present in the data, the one that showed the largest population difference between the Maasai and Luhya (as measured by Fst) sits in the gene for a fatty acid binding protein FABP1. This protein is expressed in the liver, and the variant that occurs at higher frequency in the Maasai is associated with a lowering of cholesterol levels in Northern German women (n = 826) and in French Canadian men consuming a high fat diet (n = 623). Furthermore, studies in mice fed a high fat, high cholesterol diet showed that deactivating the FABP1 protein leads to protection against obesity, and lower levels of triglycerides in the liver, when compared to normal mice on an identical diet. These results suggest that this protein plays a role in regulating lipid homeostasis, and its selection in the Maasai may be diet-related.

On Chromosome 7, two of the methods we used to detect selection identified a cluster of genes that fall in the Cytochrome P450 Subfamily 3A (CYP3A). This family of genes is involved in drug metabolism, in oxidizing fatty acids, and in synthesizing steroids from cholesterol.

What’s next?

Computational methods can only take you so far. We have identified genes in candidate regions undergoing positive natural selection in the Maasai, possibly arising due to their unusual diet. But the case for selection can only be definitively made with an experimental study targeted to address the role of these genes in maintaining cholesterol homeostasis. We’re hoping to collaborate with experimental biologists to take these hypotheses forward and investigate their role in the evolutionary history of the Maasai.

So head over to PLOS, check out the paper, and let us know what you think.

Update: Here’s another blog post that discusses the paper, focusing more on the mixed genetic makeup of the Maasai.

References:

Kshitij Wagh, Aatish Bhatia, Gabriela Alexe, Anupama Reddy, Vijay Ravikumar, Michael Seiler, Michael Boemo, Ming Yao, Lee Cronk, Asad Naqvi, Shridar Ganesan, Arnold J. Levine, Gyan Bhanot (2012). Lactase Persistence and Lipid Pathway Selection in the Maasai PLOS ONE, 7 (9) : 10.1371/journal.pone.0044751

If you’d like to read more about selective sweeps, you may enjoy my post Why moths lost their spots, and cats don’t like milk. Tales of evolution in our time.