Vladimir Arnold, he of the A in KAM Theory, wrote a classic graduate text entitled Mathematical Methods of Celestial Mechanics. This, as one might imagine, is a book that is not exactly a storehouse of easy homework assignments. There are, however, a scattering of problems that offer insights while, at the same time, not actually requiring the tough-guy methods that are the text’s primary focus.

During his walk in outer space [as part of the Voskhod 2 Mission on 18 March 1965], the cosmonaut Alexey Arkhipovich Leonov threw the lens cap of his movie camera toward the Earth. Describe the motion of the lens cap with respect to the spacecraft, taking the velocity of the throw as 10 m/s. Neglect the asphericity of the Earth.

(Ria Novosti/Science Photo Library)

Leonov’s space walk tipped off a hair-rising adventure which began with his being nearly unable to re-enter the spacecraft, and ended with a frigid way-off-course landing in the Siberian Tiaga, all of which is covered in a recent BBC documentary.

One can hand-crank the problem by noting that the radially directed, \({\bf v}_{i}=10\,{\rm m\,s^{-1}}\), launch of the lens cap exerts no torque, so that \({\bf r}\times{\bf v}_{i}=0\), whereas the total specific energy of the lens cap’s initial orbit, \(-GM_{\oplus}/2a\) is augmented by \(\frac{1}{2}v_{i}^{2}\). Given the new semi-major axis, \(a_{\rm new}\) and the before-and-after conservation of \(h=(GM_{\oplus}a_{\rm new}(1-e^2))^{1/2}\), one can solve for \(e\), and then proceed to all four orbital elements by noting that \(r=a(1-e\cos E)\) and \(M=E-e\sin E\), and then working out the longitude of pericenter, \(\varpi\), relative to the reference direction defined by the radius vector from the Earth’s center to the point where the lens cap was thrown. Clumsy.

The guiding center approximation revives the old idea of epicycles to describe the motion of a particle (in this case, the lens cap) on a low eccentricity orbit. For modest \(e\), the true Keplerian motion is approximated as a compound of the circular motion of a “guiding center” and the counter-directed motion about the guiding center on a 2:1 ellipse, where the semi-minor axis is oriented radially, and has length \(ae\). Both motions complete once per orbital period. Here’s the basic idea, drawn for an orbit with \(e=0.3\), which is actually quite a substantial eccentricity:

For the lens cap problem, the guiding center materializes at a distance \(2ae\) ahead of the launch point. The motion associated with the guiding center is the superposition of two simple harmonic motions. For the radial oscillation, \(\frac{1}{2}v_{i}^{2}=\frac{1}{2}n^{2}x^{2}\rightarrow v_{i}=nae\), which works out to \(e=0.0013\). To first (and very good) approximation, the lens cap arrives back 88 minutes later in the close vicinity of the spacecraft, after inward and outward radial excursions of 8.4 km, and after leading the spacecraft by as much as \(4ae=34\) km. The small total gain in orbital energy lengthens the lens cap’s orbital period slightly, which means that the cap fails to catch up by a few tens of meters at the end of a full one-orbit epicyclic oscillation.

In Michael Rowan-Robinson’s Cosmology (3rd ed.) Oxford University Press. pp. 62–63, one finds, “It is evident that in the post-Copernican era of human history, no well-informed and rational person can imagine that Earth occupies a unique position in the universe.”

If one insists on a strictly inertial frame, I guess that’s true, but non-inertial frames often have more value. Thousands of extrasolar planets have been found, and not one of them is remotely habitable. Many lines of evidence are beginning to point toward an Earth that is unique, probably in the galaxy, and perhaps, even, in the accessible universe. In the post-post Copernican era, epicycles (2:1, rather than 1:1) have a certain appeal. And indeed, there’s nothing inherently wrong about the Tychonic model of the solar system, it simply subscribes to an Earth-centered point of view. Don’t we all?

On the topic of epicycles, I have to say I wasn’t a fan of the article on “retrograde beliefs” that appeared a few weeks ago in the New York Times Magazine. Obviously, astrology is a bunch of bunk. It’s known to be wrong. One can make the argument that taking it down in the genteel and informed confines of the NYT magazine amounts to shooting fish in a barrel. Satire is probably the best approach. Aside from this stylistic quibble, the writing in the NYT piece seems incoherent, somehow second-hand and artless. An intricate 1756 diagram by James Ferguson (who is no longer around to defend his work) is given a rather underhanded misrepresentation:

The full title of Ferguson’s book is Astronomy Explained Upon Sir Isaac Newton’s Principles, And Made Easy to Those who Have Not Studied Mathematics. The diagram reproduced in the NYT is not the product of some recalcitrant Ptolemaic view as implied in the caption, but rather appears in a chapter entitled “The Phenomena of the Heavens as seen from Different Parts of the Solar System”. It shows the intricate motions of the inferior planets in a co-rotating Earth-centered frame, and Ferguson gives a fascinating description of the analog-computational method he used to create the diagram:

One tends to roll one’s eyes when the topic turns to Georges-Louis Leclerc, Comte de Buffon, the French encyclopedist and pre-revolutionary intellectual luminary.

Buffon sounds regrettably similar to Buffoon, especially considering that The Comte is best-known for some memorable blunders. For example, Georges-Louis came out on the losing side of a tussle with Thomas Jefferson regarding the general valor of the New World fauna. From the wikipedia article:

At one point, Buffon propounded a theory that nature in the New World was inferior to that of Eurasia. He argued that the Americas were lacking in large and powerful creatures, and that even the people were less virile than their European counterparts. He ascribed this inferiority to the marsh odors and dense forests of the American continent. These remarks so incensed Thomas Jefferson that he dispatched twenty soldiers to the New Hampshire woods to find a bull moose for Buffon as proof of the “stature and majesty of American quadrupeds”

Buffon also speculated, in 1778, that the solar system’s planets were the result of a collision between a comet and the Sun, a hypothesis that is completely incorrect. Even in the 1750s, perturbation analyses (such as those carried out by Alexis Claude Clairaut in connection with the successful predictions of the return ephemeris for Halley’s Comet) had made it clearly evident that cometary masses are far smaller than planetary masses.

Buffon, however, was definitively not a buffoon. He came remarkably close to having a full command of all the scientific disciplines, and some of his efforts still sparkle. He calculated that the chance of the Sun rising tomorrow is \(1-(1/2)^x\), where \(x\) is the number of consecutive days that it has risen to date. In his treatment of probability theory, he also stated that one chance in 10,000 is the lowest practical probability — an enormously useful bon mot, on par, say, with Andy Warhol’s remark that “when you think about it, Department Stores are kind of like Museums.”

Buffon’s Histoire Naturelle, which aimed to exhaustively cover all of the natural sciences, ran to 44 quarto volumes, eight of which were written and which appeared after he died, and all of which were out of date the moment they were printed. Even in the 1700s, scientific knowledge was accumulating so rapidly that it was impossible to keep up.

It has been recently hammered home to me that the same situation also now holds true for extrasolar planets. Jack Lissauer and I just finished a review article on exoplanets for the forthcoming second edition of Elsevier’s Treatise on Geophysics. A pre-print is up on today’s arXiv listing. In writing the article, it was painfully clear just how large the literature is, and how fast it is growing…

When I lived in Japan, I visited Hokkaido University in Sapporo to give an astronomy colloquium. While there, I immediately noticed that an odd motto, “Boys, Be Ambitious!” is attached (in English) with great frequency to the various affairs, both large and small, of the University. One of the astronomy graduate students had the phrase written on a post-it note attached to the screen of his computer. In another building, there was a large mural showing a stern, stiffly dressed 19th-century gentleman exhorting a group of reverent students with a longer version of the phrase:

“Boys, be ambitious! Be ambitious not for money or for selfish aggrandizement, not for that evanescent thing which men call fame. Be ambitious for that attainment of all that a man ought to be.”

Which, upon reflection, seems to be reasonable advice…

The gentleman in the mural, it turns out, is William Clark Smith, the founder and first president of the University of Amherst, Massachusetts. In the mid 1870s, he was enlisted by the Japanese Meiji Restoration government as an Oyatoi Gaikokujin, or “hired foreigner”, to establish an agricultural college in Sapporo (now Hokkaido University) and he made an impression that has lasted well over a century. The Wikipedia article is extensive and quite interesting. On the origination of the motto:

“On the day of Clark’s departure, April 16, 1877, students and faculty of SAC rode with him as far as the village of Shimamatsu, then 13 miles (21 km) outside of Sapporo. As recalled by one of the students, Masatake Oshima, after saying his farewells, Clark shouted, “Boys, be ambitious!”

Upon returning to the United States, and flush with the organizational successes and appreciation that he had garnered in Japan, Clark left his academic career, cultivated an interest in gold and silver mining, and embarked on an abrupt, ambitious, and ultimately disastrous foray into the business world. In 1880, he teamed up with a junior partner, John R. Bothwell, to found what might best be described as a 19th-century incarnation of a metals hedge fund. From offices on the corner of Nassau and Wall Streets in Manhattan, the firm of Clark & Bothwell acquired interests in a slew of silver and gold mines across North America, for which they assumed management and issued stock. Clark, as president, got his contacts and colleagues to invest in the venture, and for a period during 1881, the stocks issued by Clark and Bothwell ran up into multi-million dollar valuations. A classic example of a bubble.

Clark travelled around the country, promoting the company, acquiring new mines, and seeing to their management, while Bothwell appears to have been responsible for back-office operations. Clark, who had no experience in finance, and little real knowlege of mining geology seems to have spun his wheels, while Bothwell, who had a shady history, actively mismanaged the companies. The operation got into debt, with the outcome being all too typically familiar along the lines of When Genius Failed. By the Spring of 1882, they were facing insolvency, investor lawsuits, fraud allegations, and various other problems. Bothwell disappeared on a train trip to San Francisco, never to be seen again, leaving Clark holding the bag. The story played out to the delight of the Massachusetts and national press.

From the Springfield Republican, May 29, 1882:

… it appears form the beginning that he, as manager of the mines has allowed Bothwell, as treasurer, absolute control of the books and finances of the several companies. It doesn’t appear that he ever examined the books, nor had anybody do so for him, or inquired into the financial condition of each mine, or what was being done with their profits; neither has he required from Bothwell such bonds as the latter’s position should require for the safe handling of moneys entrusted to him..

The scandal made the New York Times, which wrote several articles about the affair, including this one, from May 29th, 1882, which I dug out of the archive:

The scandals eventually ruined Clark’s health, and he died four years later, in 1886, at age 60. A cautionary tale for academics everywhere with ambitions to leave the Ivory Tower in search of glittering lucre…

I’ve been putting the finishing touches on a review article covering extrasolar planets that will be posted to arXiv in a few days. The list of to-do’s involves updating the figures, including the one shown just below, which charts \(M\sin i\)‘s of the RV-sourced planets in dark gray and simple radius-derived mass estimates of the transit-sourced planets in red. The steady Moore’s Law-like progression toward ever-lower masses has definitively reached Earth-mass (not to be confused with Earth-like) planets. The process took up only two decades, and was among the more impressive scientific advances of the recent past.

Here’s an elaboration of the above figure that doesn’t make it into the article, but is interesting nonetheless. On the y-axis is \(K/rms\), which is reasonably well correlated with the signal strength of Doppler velocity discoveries. One can certainly detect planets with confidence at low \(K/rms\), but it requires a large number of independent Doppler velocity measurements. The color corresponds to “astrobiological interest” — surely naive, and probably misplaced, but nonetheless quantifiable by my planet valuation formula.

Credit: NASA/JPL

It feels increasingly awkward and embarrassing to read LaTeXed, peer-reviewed articles that quantify and delineate the habitable zone — the special region surrounding a star that is invariably (and rather fittingly) linked to a particular fairy tale from the Brothers Grimm.

Evolutionary psychologists have speculated that the concept of the afterlife might be inextricably entwined to the evolution of the mind’s ability to reason about the minds of others. A rational world view, however, frustrates ingrained atavistic yearnings and a belief in the supernatural. Habitable planets provide a respectable stopgap to assuage the discomfort of these incompatible poles. Could it be a mere coincidence that the ancient Greek and classical depictions of Elýsion pedíon, the Elysian Fields, are part and parcel the very image of the habitable zone?

Credit: NASA/SETI/JPL

And they live untouched by sorrow in the islands of the blessed along the shore of deep-swirling Ocean, happy heroes for whom the grain-giving earth bears honey-sweet fruit flourishing thrice a year, far from the deathless gods…

— Hesiod, Works and Days (170)

The submerged summit of the Detroit Seamount ranks among the planet’s gloomiest spots. East of Kamchatka, a mile beneath the waves at 51 51′ N, 167 45′ E, it is second-to-last in the long line of Emperors. Inch by inch, it creeps toward destruction in the Aleutian Trench.

Detroit’s glory days were the late Cretaceous. Back then, it was an active Hawaiian volcano.

Live it fast, you’re gonna get there soon. Kauai is five million years old, but underground, the lights have gone out. Over half of the original height and the original land area have disappeared. Rivers gush sediment into the sea. Waimea Canyon juxtaposes verdure and an erosive wasteland. Four wheel drive claws and rends the red dirt.

Beyond Kauai, the next islands in the chain are Nihoa,

Necker,

and the La Perouse Pinnacle,

whose resemblance to a sinking ship is not just metaphoric.

Before humans arrived, the Hawaiian islands had strange flightless birds. Indeed, each island in the chain developed its own odd avian inhabitants, sculpted by natural selection, and then driven conveyor-like to extinction. Not once, in forty, fifty, sixty tries, did the birds respond by evolving intelligence and doing something about their situation. Probably, there was never enough time.

Or perhaps, that’s something that rarely, if ever, happens.

It’s worth a scramble to get a window seat on a Hawaiian inter-island flight. The views are full of craggy green cliffs, porcelain ocean, and wispy masses of fog and cloud. Sometimes, several islands are visible at once, and it’s not hard to imagine that the archipelago might extend over the entire globe.

That would be a very different planet, and, in fact, a world covered by hotspot volcanoes might have a surface elevation profile somewhat reminiscent of the WMAP image of the temperature fluctuations in the cosmic microwave background. The WMAP image brings to mind a planet covered in Hawaiian islands.

Any distribution, \(f(\theta,\phi)\), on the surface of a sphere, be it of temperature, or elevation, or the density of IP addresses, can be expressed as a weighted sum of spherical harmonics

$$f(\theta,\phi)=\sum_{l,m} a_{l,m} Y(\theta,\phi)_{l}^{m}\, ,$$
where the coefficients corresponding to the individual weights, \(a_{l,m}\) are given by
$$a_{l,m}=\int_{\Omega}f(\theta,\phi)Y(\theta,\phi)_{l}^{m \star}d\Omega\, ,$$
and the power, \(C_{l}\) at angular scale \(l\) is
$$C_{l}=\frac{1}{2l+1}\sum_{m=-l}^{l}a_{l,m} {a_{l,m}}^{\star}\, .$$

The power spectrum of the CMB anisotropies peaks at \(l\sim 200\), which corresponds to an angular scale on the sky of \(\Delta \theta \sim 1^{\circ}\), which is very close to the solid angle subtended by the Big Island of Hawaii on the surface of the spherical Earth.

Here’s a recent version of the CMB temperature anisotropy spectrum from the Planck Mission website

The peaks in the spectrum of CMB temperature anisotropies stem from acoustic oscillations and diffusion damping in the early universe, and they encode all sorts of information about the fundamental cosmological parameters. This, of course, is very well-known stuff: a search on all literature in the ADS database published since 2000, and ranked by citations, lists Spergel et al. 2003, First-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Determination of Cosmological Parameters at #1, with 7,914 citations and (rapidly) counting.

Given the similarity between the angular scales of the Hawaiian islands and the main CMB peak, it’s interesting to compute the angular power spectrum of Earth’s bedrock elevation profile. A global relief dataset with one arc-minute resolution is available from NOAA as a 4GB (uncompressed) file. Downsampling by a factor of 100, and applying the “terrain” color map yields a familiar scene

Computing the power in the first 108 angular modes of the relief distribution in the above data set gives a spectrum that is weighted toward continents and ocean basins rather than archipelagos. There is a pronounced peak at \(l=5\) that reflects the typical angular scale of continents and ocean basins.

Here is the global relief distribution obtained by summing just the \(l=5\) contributions. It’s right for more or less the same reason that Crates of Mallus was right:

Using all 108 angular mode families to reconstruct the image gives a fairly credible-looking world map. It’s as if the watercolors ran slightly before they dried. Most critically, the \(l=108\) reconstruction fails to capture the highest peaks and the lowest ocean trenches, and hence more of the dynamic range of the color map is distributed across the globe.

Degree-wide islands like Hawaii are the exception rather than the rule on Earth’s surface. I believe that this was the concept that former US Vice President Dan Qualye was struggling to express in one of his much-ridiculed pronouncements:

Hawaii has always been a very pivotal role in the Pacific. It is IN the Pacific. It is a part of the United States that is an island that is right here.

(See also his comments on Mars.)