The Earth’s Gravity, and How the Media Got It Wrong

Note: an edited and expanded version of this post is now online at ScientificAmerican.com.

This video is no April’s fool joke: the Earth really is shaped like a potato. Only that the shape you see here is, hum, slightly exaggerated to highlight its irregularities. Another caveat: what is depicted here is not the shape of the planet, but rather the shape of an idealized sea-level surface extending around the entire globe — a surface that Earth scientists call the geoid.

The video is the most accurate reconstruction of the geoid to date, and was released by the European Space Agency at a scientific meeting in Munich on March 31 (see video of press conference here). It is based on data collected by ESA’s Gravity Field and Steady-State Ocean Circulation Explorer (GOCE). The five-meter long, arrowhead-shaped probe has been in a low orbit for almost exactly two years, taking painstakingly accurate measurements of the gravitational field of Earth.

Alas, many of the web sites and blogs that reported on GOCE’s results yesterday got it wrong. Contrary to what was stated in news stories and in blogs, the geoid is not a surface on which the strength of gravity is the same everywhere.

The geoid is, in the words of an oceanographer on the GOCE team, a surface such that if you placed a marble anywhere on it, it would stay there rather than rolling in any direction. Another way of saying it is, imagine you were an engineer traveling around the world with a spirit level. Then wherever you go, the level would be exactly parallel to the geoid at that place. Yet another equivalent definition: it is a surface that’s everywhere perpendicular to the direction of a plumb-bob, or in other words, to the gravitational field.

Gravity need not have the same strength everywhere on the geoid. In other words, if you could walk on the geoid you would see gravity always pointing exactly downwards, but your weight could slightly change from one region to another.

The misunderstanding may have stemmed from the confusion of two concepts from calculus: a function and its derivative. In this case, it’s actually two concepts from multivariable calculus: a vector field and its potential. The potential is the gravitational potential; the vector field is the gravity field. At any point in space, the gravity field represents the direction in which the gravitational potential rises fastest. Its magnitude, or length, is the rate of change of the potential. The field is called the gradient vector field of the potential and it is the 3-D version of a derivative. (To be pedantic: for historical reasons, the gravity field is defined to be the opposite vector to the gradient of the potential.)

If you were to follow a field line of the field – a curve which is at every point tangent to the vector field at that point – you would follow a curve of steepest ascent on the gravitational potential. That’s why it is called the gradient. (Note that because of inertia, the field lines of the gravitational field are not necessarily the trajectories of a body in free fall. The field won’t tell you in which direction you’ll be moving — only in which direction you will accelerate.)

Gradient vector fields are difficult to grasp in 3-D, because picturing the potential (to visualize the ascent or descent) would require a fourth dimension. But a 2-D analogy might help. Think of how water trickles down a hilly surface. Neglecting inertia, a droplet of water would follow a curve of steepest descent, which is a vector field made of vectors along the surface. In this analogy, the elevation of the droplet is the analogue of the potential at that point. And the analogue of the geoid is an elevation contour on a topographical map: the potential – the height — is constant along the contour. Obviously, some parts of this contour can be on steep terrain, whereas others can be on mild gradients. (On a topographical map, the slope is steeper where many different contour lines are crowded together.) The magnitude of the gradient vector field – which represents the steepness of a slope – is not the same at all points of the contour.

Thus, gravity does change in strength along the geoid, and so the geoid is not “a shape where the gravity is the same no matter where you stood on it,” as one blogger put it. Nor is it “how the Earth would look like if its shape were distorted to make gravity the same everywhere on its surface,” as reported by a major news site, which went on to add that “areas of strongest gravity are in yellow and weakest in blue,” apparently not noticing the contradiction with the previous sentence.

There are many reasons why the geoid isn’t a sphere. First and foremost, the Earth itself isn’t a sphere. It is closer to an ellipsoid, being flattened at the poles by the centrifugal force of its own rotation. But the planet isn’t an ellipsoid either, because of topography. Mountains and valleys are asymmetrical distributions of mass. That mass distribution affects the gravitational field and makes the geoid asymmetrical too.

Then there is the Earth’s inner and outer structure. The oceanic crust and the landmasses of the continental shelves are made of different materials that have different densities; and farther down, the mantle is not uniform either, with regions that have slightly different compositions and temperatures, and thus slightly different densities. “The variations,” Reiner Rummel, a senior GOCE scientist, told me in an email, “reflect processes in the deep earth mantle such as descending tectonic plates and hot mantle plumes.”

All of which affects the strength of the gravitational field, and the GOCE’s orbit. The probe measures these effects thus it measures the gravitational field.

The geoid is not to be confused with anomalies due to the topography of the land – although it is affected in part by it. Also – and here is where things get really tricky – it is not to be confused with the actual sea level.

Wait a minute, you say, but isn’t the ocean supposed to be at sea level by definition? Except for relatively small perturbations such as waves or tides, you say, shouldn’t the surface be exactly horizontal (that is, perpendicular to the gravitational field) everywhere? Isn’t that, after all, what “sea level” means? Yes, to some approximation. However, the oceans are not homogenous. Differences of salinity and temperature make them more or less dense. Moreover, the Earth’s rotation produces forces that keep the oceans in constant motion.

Just like the water in a river is not all at the same level – after all, it flows to the sea for a reason – the water in the ocean also has differences in height. And by height, it should be clear by now, I mean elevation relative to the geoid, which can be positive or negative. Oceanographers thus speak of the “topography” of the ocean, meaning that it really has hills and valleys. In fact, here is where the “ocean circulation” part of GOCE’s name comes into play: one of the main goals of the mission is to measure the oceans’ topography and from that deduce the structure of the oceanic currents.

The deviations of the geoid from the simplified, ellipsoidal model of the Earth are substantial: they range from 100 meters below (dark blue in the video) to 80 meters above (yellow), Rummel says.

The surface in the video is the geoid amplified by a factor 7,000, says GOCE mission manager Rune Floberghagen.

John Milnor and His Spherical Exotica

John Milnor
MFO via Creative Commons License
John Milnor’s discovery of exotic smooth spheres unleashed the exploration of new mathematical riches. He was just awarded the 2011 Abel prize.

On March 23, the Norwegian Academy of Science and Letters announced that it was awarding the 2011 Abel Prize — Norway’s mathematical answer to Sweden’s Nobels — to John Milnor. The Stony Brook mathematician, 80, had already won a Fields Medal and nearly every other major math prize there is.

As most math grad students have done over the past half-century — at least those students specializing in topology and geometry — when I was in grad school I learned much of what I know from Milnor’s canonized textbooks. (The most standard ones are probably “Characteristic Classes,” which he co-authored with James Stasheff, and “Morse Theory,” but my favorite one was “Lectures on the h-Cobordism Theorem,” a typewritten gem that has long been out of print but can be found online if you google it. Some day I’ll blog about why I liked it so much.)

And perhaps as many other topology students have done over the years, I have often tried to imagine what it must have felt to be a topologist in 1956, when Milnor announced his discovery of a 7-dimensional sphere that was homeomorphic, but not diffeomorphic, to the standard 7-dimensional sphere.

What does that mean? I have tried my best to explain the basic idea in my Scientific American article on the solution of the Kervaire invariant problem and then again in the article I co-wrote with my colleague John Matson on the announcement of Milnor’s Abel prize for ScientificAmerican.com.

Maxwell’s Demon Lives! And He’s Devilishly Cool

Maxwell's Demon
Click on this image to see a slightly more modern take.

In the 19th-century, James Clerk Maxwell came up with a thought experiment that still befuddles physics students (as well as the rest of us) to this day. The great Scottish physicist theorized the existence of a “demon” that seemed able to concentrate the atoms of a gas into a smaller volume without raising their temperature, thus reducing their entropy. That feat seemed to violate the second law of thermodynamics, according to which entropy can never decrease.

What’s the catch?

To find out, get your hands on the March issue of Scientific American, which is now hitting the newsstands. In the article “Demons, Entropy and the Quest for Absolute Zero,” physicist Mark G. Raizen describes how he built a working version of the demon: a one-way gate that only lets atoms go through in one direction only, forcing them accumulate on one side of a container.

Raizen’s is not the first physical realization of a Maxwell’s demon. The molecular machinery of living cells often appears to rely on the related concept of a Brownian ratchet, which has inspired chemists to build molecules that harvest the energy of Brownian motion, in seeming violation of the second law of thermodynamics (see the article on the nanomachines of life and their manmade imitations, which I wrote for Science News three years ago). But the gate Raizen built is particularly elegant in that it is pretty much as efficient as it gets: it sorts atoms using a single photon for each atom.

Raizen used his device to cool a rarefied gas down to temperatures of just millionths of a degree above absolute zero. He started out with a gas that had already been cooled to one one-hundredth of a kelvin (using a device called an atomic coilgun, also described in the text) and placed it in a magnetic trap. He then used his one-way gate to bring the temperature closer to absolute zero by many orders of magnitude.

To see how that works, check out the interactive single-photon cooling animation on SciAm’s web site. It shows how a one-way gate can help cool a gas in two steps: First let the gate concentrate atoms into a smaller volume (but without raising their temperature), then allow them to expand to the original volume (which brings their temperature down).

Century-Old Enigma on Mathematical Partitions Is Solved

Ono and Kent
Carol Clark/Emory U.
Ken Ono and Zach Kent were part of a team that cracked an enigma formulated by the Indian mathematician Ramanujan

For someone who died at the age of 32, the largely self-taught Indian mathematician Srinivasa Ramanujan left behind an impressive legacy of insights into the theory of numbers—including many claims that he did not support with proof. One of his more enigmatic statements, made nearly a century ago, about counting the number of ways in which a number can be expressed as a sum, has now helped researchers find unexpected fractal structures in the landscape of counting.

Ramanujan’s statement concerned the deceptively simple concept of partitions—the different ways in which a whole number can be subdivided into smaller numbers. Ken Ono of Emory University and his collaborators have now figured out new ways of counting all possible partitions, and found that the results form fractals—namely, structures in which patterns or shapes repeat identically at multiple different scales. “The fractal theory we’ve discovered completely answers Ramanujan’s enigmatic statement,” Ono says. The problems his team cracked were seen as holy grails of number theory, and its solutions may have repercussions throughout mathematics.

One way to think of partitions is to consider how a set of any (indistinguishable) objects can be divided into subsets. For example, if you need to store five boxes in your basement, you can pile them all into a single stack; lay them individually on the floor as five subsets containing one box apiece; put them in one pile, or subset, of three plus one pile of two; and so on—you have a total of 7 options:

5, 1+1+1+1+1, 1+1+1+2, 1+1+3, 1+4, 1+2+2 or 2+3.

Mathematicians express this by saying p(5) = 7, where p is short for partition. For the number 6 there are 11 options: p(6) = 11. As the number n increases, p(n) soon starts to grow very fast, so that for example p(100) = 190,569,292 and p(1,000) is a 32-figure number. (The WolframAlpha knowledge engine calculates partitions for numbers as large as one million.)

Partitions
R. A. Nonenmacher/WikiMedia
The partitions of 1 through 8.

The concept is so basic and fundamental that it is central to number theory and pops up in most other fields of math as well. In the figure to the right you can see all partitions of the numbers 1 through 8.

Mathematicians have long known that the sequence of numbers made by the p(n)’s for all values of n is far from being random. Ramanujan and others after him found formulas to predict the value of any p(n) with good approximation, for example. And general “recursive” formulas have long existed to calculate p(n), but they don’t speed up calculations very much because to find p(n) you first need to know p(n – 1), p(n – 2) and so on. “That’s impractical even with the help of a computer today,” Ono says.

A direct formula for calculating the exact value of p(n) could in principle be faster. Another advantage of a direct formula would be the ability to compare values of p(n) for arbitrarily large n’s and thus to prove the existence of patterns, such as properties that repeat along an entire infinite sequence.

According to Ono, for example, empirically it appears that exactly half of all partition numbers are even and that one-third are divisible by 3. These empirical patterns however have yet to be proven to be true throughout the entire set of natural numbers.

Ramanujan’s original statement, in fact, stemmed from the observation of patterns, such as the fact that p(9) = 30, p(9 + 5) = 135, p(9 + 10) = 490, p(9 + 15) = 1,575 and so on are all divisible by 5. Note that here the n’s come at intervals of five units.

Ramanujan posited that this pattern should go on forever, and that similar patterns exist when 5 is replaced by 7 or 11—there are infinite sequences of p(n) that are all divisible by 7 or 11, or, as mathematicians say, in which the “moduli” are 7 or 11.

Then, in nearly oracular tone Ramanujan went on: “There appear to be corresponding properties,” he wrote in his 1919 paper, “in which the moduli are powers of 5, 7 or 11…and no simple properties for any moduli involving primes other than these three.” (Primes are whole numbers that are only divisible by themselves or by 1.) Thus, for instance, there should be formulas for an infinity of n’s separated by 5^3 = 125 units, saying that the corresponding p(n)’s should all be divisible by 125.

In the years since, mathematicians were able to prove the simple cases based on Ramanujan’s statement. Thus, every fifth partition number is divisible by five. (It actually seems that about 32 percent are divisible by 5, which would be more than the 20 percent minimum that’s guaranteed by Ramanujan’s formula: the formula only says that one every 5 is divisible by 5, but doesn’t say anything about the numbers in between.)

As to what “no simple properties” could mean, that was anybody’s guess—until now.

Working with Jan Hendrik Bruinier of Darmstadt Technical University in Germany, Ono has developed the first exact formula for calculating p(n) for any n. And in a separate paper with Zachary A. Kent, also at Emory, and Amanda Folsom of Yale University, he has identified patterns that probably even Ramanujan could not have dreamed of.

The patterns link certain sequences of p(n) where the n’s are separated by powers of any prime number beyond 11. For example, take the next prime up, 13, and the sequence p(6), p(6 + 13), p(6 + 13 + 13) and so on. Ono’s formulas link these values with those of p(1,007), p(1,007 + 13^2), p(1007 + 13^2 + 13^2) and so on. The same formulas link the latter sequence with one where the n’s come at intervals of 13^3—and so on for larger and larger exponents. (The formulas are slightly more subtle than just saying that the p(n) are multiples of a prime.) Such recurrence is typical of fractal structures such as a Mandelbrot set, and is the number theory equivalent of zooming into a fractal, Ono explains.

Ono unveiled the discoveries January 21 at a specially convened symposium at Emory. By that afternoon the news had made waves across the math world, and his in-box had filled with 1,500 e-mails from mathematicians, reporters and “cranks,” he says. (Ono, Folsom and Kent posted their proof on the Web site of the American Institute of Mathematics and also submitted it to a journal. The full proof of Ono and Bruinier’s new formula is still being written up, Ono says.)

“Ken is a phenomenon,” comments George E. Andrews, a partitions expert at The Pennsylvania State University. The new fractal view of partitions, Andrews adds, “provides a superstructure that no one anticipated just a few years ago.”

Do Ono et al.’s discoveries have any practical use? Hard to predict, Andrews says. “Often deep understanding of underlying pure mathematics takes awhile to filter into applications.” In the past methods developed to understand partitions have later been applied to physics problems such as the theory of the strong nuclear force or the entropy of black holes.

Meanwhile, mathematicians are left to contemplate Ramanujan’s mind. Many of his claims, Ono points out, have turned out to be incorrect, but his work still illuminates so much of what number theorists study today. “All of this stuff that we’re studying right now for some crazy reason was anticipated by Ramanujan,” he says.

“He was a magical genius,” Andrews adds, “and the rest of us wish we knew how he was able to see so deeply.”

(Here you can see a video of Ono explaning the discoveries.)

A slightly shorter version of this story was posted today at ScientificAmerican.com.

Citizen Satellites

CubeSatCalifornia Polytechnic State University
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Developing, testing, launching and operating a space science mission typically costs hundreds of millions to billions of dollars, but with this new breed of satellites lowers you can have your own space mission for just $100,000 or so.

These cubic satellites, called CubeSats, take up just one liter of space, weigh one kilogram, are designed to be launched in batches, and can piggyback on other space missions, factors that together combine to dramatically reduce launch costs. CubeSats originated in a set of technical specifications proposed as a standard in 2000 by aerospace engineer Bob Twiggs, formerly of Stanford University’s Space and Systems Development Laboratory, and Jordi Puig-Suari of California Polytechnic State University, San Luis Obispo.

The two engineers wanted to make space science affordable for most science laboratories and even for college and high school projects. CubeSats have been embraced as an educational tool, because a team of students can design and build one in just two years, and students can get a holistic feeling of what space science is about.

The concept has spread—even NASA and the National Science Foundation have joined the club—and dozens of teams have started CubeSat projects. At least two dozen have already completed their missions successfully and many more are at various planning stages.

Members of the CubeSat community build on each other’s experience, sharing design tips. As Twiggs and Alex Soojung-Kim Pang wrote in “Citizen Satellites,” their February 2011 Scientific American article, “When developers find something that works—one model of ham radio that works in space longer than another, for example—they share their findings with other CubeSat designers.”

Meanwhile, several companies have started selling off-the-shelf CubeSat components, such as flight electronics, transceivers, solar panels and structural elements. Teams of scientists and engineers can now focus on designing science instruments that are particular to their own CubeSats, rather than having to design entire spacecraft from scratch.

You can see how CubeSats are made in this slide show, including some by undergraduate students and even one by an eight-grade student, Bryan Fewell from Hawaii.

Controlling a Mind, Neuron by Neuron

C. elegans neuron controlSamuel Lab/Harvard University
Laser light activated neurons in this genetically engineered worm and forced it to stop swimming. The worm resumed when the laser was turned off.

Scientists have come a step closer to gaining complete control over a mind, even if that mind belongs to a creature the size of a grain of sand. A team at Harvard University has built a computerized system to manipulate worms—making them start and stop, giving them the sensation of being touched, and even prompting them to lay eggs, as seen in the videos here—by stimulating their neurons individually with laser light, all while the worms swim freely in a petri dish. The technology may help neuroscientists for the first time gain a complete understanding of the workings of an animal’s nervous system.

Andrew Leifer, a graduate student in biophysics at Harvard who conducted the experiment, hopes the technique could some day help scientists create complete simulations of the organism’s behavior. “Hopefully, we’ll be able to make a computational model of the entire nervous system,” he says.

In a way, that would be like “uploading a mind,” even if a rudimentary one.

Read the rest of my story (and see videos!)

Titan Volcanoes, Light Pollution, Lightning X-rays, and Wonders of the Ocean

volcanoes on titanNASA/JPL-Caltech/University of Arizona
Cassini’s radar mapped three potential volcanoes in a region called Sotra Facula on Titan, Saturn’s largest moon

The Cassini mission to Saturn has mapped a region of the moon Titan and found the best evidence yet for icy water volcanoes in the solar system. Night-time city lights may indirectly raise daytime smog. The first X-ray movie of a lightning. And dozens of potential new species seen in the Sulawesi Sea, off Borneo. These are the stories I filed this week from the 2010 Fall Meeting of the American Geophysical Union in San Francisco.

Also from the meeting, Harvey Leifert has an update on Voyager 1, which is now 115 times more distant from Earth than Earth is from the Sun.

The Great Masters of Microscopy

Spike Walker in BioScapesSpike Walker
Can you guess what this is?

The December issue of Scientific American is about to come out. The issue will feature our pick of images from the 2010 Olympus BioScapes Digital Imaging Competition, but you can already see those online as they were posted today, which is also the day of the announcement of the winners.

I helped choose the images that went into our feature (which don’t coincide with Olympus’s own top ten picks) iand I wrote the captions for them, after interviewing each of their creators. In the process I discovered that there is a whole subculture of serious microscopy hobbyists, such as Spike Walker, who took the beautifully abstract shot here, which is actually … well, you’ll have to click to find out.

To see all nine images that will appear in print, plus an online bonus of ten more, visit Life Unseen.

Come to Cairo and Let’s Get Physical

WCSJ 2011

The World Federation of Science Journalist is convening the World Conference of Science Journalists in Cairo next summer.  I was invited to join a panel on covering physics (title: “Let’s Get Physical”), together with Alex Witze, Neil Tourok and Sean Carroll.

The idea will be to try to demistify physics for journalists who feel intimidated by it. Physics writers feel intimidated by it too. We’ve all gone through moments of panic when interviewing physicists. Like when you ask someone to please dumb down their model of superconductivity for you and they say “let’s make an analogy with something simpler, like string theory …” That’s when you want to kill yourself.

Fortunately, you’re not alone. There are people who work at national labs like SLAC and at professional organizations like the American Physical Society or the American Institute of Physics who will be happy to help you wade through “simple” concepts like string theory.  I know this because I have worked (or interned) at each of the organizations I’ve mentioned, and people there were always more than happy to get queries from journalists. Think of it as a suicide hotline for science writers.

Hawking vs. God

Castelvecchi on Hawking

The November 2010 issue of Scientific American is now out, with my brief article on the controversy surrounding Stephen Hawking’s new book with Leonard Mlodinow.

The book stated that it’s possible to answer questions such as why the laws of physics are what they are and how the universe arose from nothingness “purely within the realm of science, and without invoking any divine beings.”

Those statements, which were not at all central to the book (which contained statements that we at SciAm expected to be much more controversial) were seen as denying the existence of God, and created a storm of controversy. The whole story was especially curious to us because we published an adaptation of the book in our October issue (The Elusive Theory of Everything).

In the article I quote an interview Hawking gave to Larry King on CNN and Marcelo Gleiser’s NPR blog, as well as my own interviews with Mlodinow, Leonard Susskind, and the theologist Robert E. Barron.

For this story I had also interviewed Deepak Chopra (who reviewed Hawking and Mlodinow’s book in the Huffington Post and debated Mlodinow on Larry King), but sadly that paragraph got cut out because of space limitations.

Chopra told me that he shared Hawking’s disbelief in the type of anthropomorphic God that dispenses miracles–a God that he says has been “killed” multiple times in the history of science, not by Nietsche but by the likes of Laplace, Charles Darwin and Richard Dawkins. (“How many times can you kill God?” he said.) But according to Chopra the Hawking and Mlodinow’s materialism and excessive reliance on determinism leaves no space for phenomena such as free will, the existence of which they dismiss too “cavalierly.”

Incidentally, I loved the book (despite some substantial flaws such as a complete lack of bibliographical references or of mentions of the work of many important physicists). I found it to have some of the clearest explanations of fundamental physics I’ve seen anywhere, especially of the quest for a grand unified theory of the fundamental forces.

What I found shocking in the book is that the authors essentially say that, after decades of search for a theory of everything, it is time for physicists to give up (although they disputed that characterization).

What happened is that in their quest to unify all the laws of Nature, physicists have discovered many different possible theories–among them, several different versions of string theory–that each apply in a limited range of situations.

That could be as good as it gets, according to Hawking. Physicists hope to find one all-encompassing theory, called M-theory, that would subsume all those string theories and constitute the final theory of everything. But Hawking seems to believe that there cannot be a way to write down such a single theory, and that we have to content ourselves with a “network” of string theories.

The book also is one of the places where one can read about Hawking’s own fascinating but arcane vision of “top-down cosmology,” something I wrote about years ago in Physics News Update.