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Over the past two days, I got the opportunity to fly to Hawaii to give two talks for the Keck Observatory’s Evening With Astronomers series. The talks focused on extrasolar planets (here’s a link to the slides in Quicktime format, ~40MB , along with the audio files of (1) a planetary system in a 2:1 resonance, (2) an unstable planetary system, and (3) another unstable system). Both talks were on Kona coast of the Big Island, where, behind the palm trees, Mauna Kea looms up 13,796 feet in the hazy volcanic distance.

The landscape here resembles nothing so much as a habitable, terraformed Mars. Hardened ropes of lava run down to the water’s edge:

In the pre-dawn light this morning, the air was totally silent, and it was easy to imagine that I was actually on Mars, before the water was gone, when a Northern hemispheric ocean lapped up against the lava of the lowermost slopes of Elysium Mons:

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In the last few years the Martian landscape has become much more familiar, as the Spirit and Opportunity rovers crawl across the surface and radio home their photographs:

At Kona, looking out toward the lava fields, the view is positively Martian, with the most immediate difference being a sky that is a hazy blue-white rather than a hazy salmon-white. Here, the Ala Loa trail recedes into the jagged distance of what could easily be Mars:

On Mars, however, one generally has a fairly reasonable sense of what the 360-degree panorama will look like even if only part of the horizon is in view. On Earth, the situation can be quite different. Here’s the view that one gets simply by turning and looking in the opposite direction down the Ala Loa trail:

(On a marginally related note, our Alpha Centauri ApJ paper is starting to pick up some news coverage. Here’s a link to a story by National Geographic News.)

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The last mile of the San Lorenzo river in Santa Cruz is strongly affected by the twice-daily ebb and flow of the tides.

It’s always startling to see the tidal bore, a solitary breaking wave that runs upstream at a ~8 minute per mile pace when the tide is coming in. The San Lorenzo bore is small, usually six to nine inches high, but dramatic nonetheless. In its wake, there’s a turbulent froth of whitewater, whose eddies eventually cascade into viscous dissipation, turning the kinetic energy of organized flow into a slight heating of the water. As the Moon recedes, the Earth spins down, and the bore expends itself in a swirl of eddies.

The energy that powers the bore was all imparted during the Moon-forming impact, in which a Mars-sized object collided with Earth, leaving the planet violently shaken and stirred and spinning crazily through days that were originally just a few hours long. Now, 4.5 billion years later, the bore running up the river is a distant echo of the impact that was large enough to cause Earth to glow with the temperature of a red dwarf star.

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There’s a nice discussion of tidal bores in the 1899 popular-level book The Tides and Kindred Phenomena in the Solar System, by Sir G. H. Darwin (son of the naturalist). The book in its entirety can be downloaded from The Internet Archive.

The Moon-forming impact, which occurred somewhere between 10 and 100 million years after the collapse of the pre-solar molecular cloud core, essentially marked the end of terrestrial planet formation in our own solar system. From a dynamical standpoint, a system undergoes a lot of evolution during a time scale of 100 million orbits. By contrast, the Milky Way galaxy is only about 40 orbits old, and is still in an effectively pristine, dynamically unrelaxed configuration.

At Darwin’s time, the first photographs of spiral galaxies were appearing, and there’s a remarkably good photo of the Andromedae galaxy on page 339 of the book:

Darwin writes:

There is good reason for believing that the Nebular Hypothesis presents a true statement in outline of the origin of the solar system, and of the planetary subsystems, because photographs of nebulae have been taken recently in which we can almost see the process in action. Figure 40 is a reproduction of a remarkable photograph by Dr. Isaac Roberts of the great nebula in the constellation of Andromeda. In it we may see the lenticular nebula with its central condensation, the annulation of the outer portions, and even the condensations in the rings which will doubtless at some time form planets. This system is built on a colossal scale, compared with which our solar system is utterly insignificant. Other nebulae show the same thing, and although they are less striking we derive from them good grounds for accepting this theory of evolution as substantially true.

In 1899, the extragalactic distance scale hadn’t been established, and so Darwin thought that M31 was a lot closer than it actually is. In dynamical terms, he would have guessed that it’s many thousands of orbits old rather than only a few dozen. Nevertheless, it’s interesting to think about what will happen to an isolated spiral galaxy by the time it’s 10^18 years old…

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The Gliese 876 system is remarkable for a number of reasons. It makes a mockery of the notion that the minimum-mass solar nebula has a universal validity. It harbors one of the lowest-mass extrasolar planets known (discovered by our own Eugenio Rivera). And of course, the outer two planets are famously caught in a 2:1 mean motion resonance, in which the inner 0.8 Jupiter-mass planet makes (on average) exactly two trips around the red dwarf for every one trip made by the outer 2.5 Jupiter-mass planet.

As users of the console know, the planet-planet interactions between the Gliese 876 planets are strong enough so that one needs a self-consistent dynamical fit to the system. Even on the timescale of a single outer planet orbit, the failure of the Keplerian model can be seen on a 450-pixel wide .gif image:

The following three frames are from a time-lapse .mpg animation of the Gliese 876 system over a period of roughly one hundred years:

Each frame strobes the orbital motion of the planets at 50 equally spaced intervals which subdivide the P~60 day period of the outer planet. Upon watching the movie, it’s clear that the apsidal lines of the outer two planets are swinging back and forth like a pendulum. This oscillation has an amplitude (or libration width) of 29 degrees, and acts like a fingerprint identifier of the Gliese 876 system.

The derangement of the orbits is reflected in their continual inability to maintain an exact 2:1 orbital commensurability. The first figure up above shows that when planet c has finished exactly two orbits, it has already managed to lap planet b, which was still dawdling down Boardwalk prior to passing GO.

Planet b, however, doesn’t always run slow. The gravitational perturbations between the two planets provide a second pendulum-like restoring action which prevents the bodies from straying from the average period ratio of 2:1, which, over the long term, is maintained exactly. The degree to which the orbits themselves librate, combined with the planets’ abilities to run either ahead or behind exact commensurability is captured by the resonant arguments of the configuration. These can be defined as,

where the lambdas are mean longitudes and the curly pi’s are the longitudes of periastron. The two resonant arguments capture the simultaneous libration of the mean motions and the apsidal lines. The smaller the arguments, the more tightly the system is in resonance.

In the Gliese 876 system, the resonant arguments are both librating with amplitudes of less than 30 degrees. This is evidence that a dissipative mechanism was at work during the formation of the system. Interestingly, however, when one looks at the other extrasolar planetary systems that are thought to be in 2:1 resonance, one finds that the libration amplitudes in every case are much larger. In fact, in the HD 73526 system and in the HD 128311 system, only one of the arguments is librating, while the other is circulating. In this state of affairs, the apsidal lines act like a pendulum that is swinging over the top. In addition, the orbital eccentricites are higher, and the sum of planet-planet activity is strikingly greater (see this animation of the evolution of the HD 128311 system).

A gas disk seems to be the most likely mechanism for pushing a planetary system into mean-motion resonance. Protoplanetary disks are likely, however to experience turbulent density fluctuations. These density fluctuations lead to stochastic gravitational torques, which provide a steady source of orbital perturbations to any planets that are embedded in a disk. For a reasonable spectrum of turbulent fluctuations, it turns out that it’s rather difficult to wind up with a planetary system that is as deeply in resonance as Gliese 876. The conclusion, then, is that Gliese 876-like configurations should be quite rare. Indeed, 2:1 resonances of every stripe should constitute only a minor fraction of planetary systems, and the majority that do exist should either large libration widths or only a single argument in resonance.

If you’re interested in more detail, we’ve submitted a paper that goes into much more detail (Adams, Laughlin & Bloch, ApJ, 2008 Submitted).

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Messenger flew by Mercury last week, and photographed vast swaths of terrain that, until now, had never been seen. The new landscapes, as expected, are cratered, barren, and utterly moonlike. The galaxy could contain a hundred billion planets that would be hard, at first glance, to distinguish from Mercury, and within our cosmic horizon, there are probably of order as many Mercury-like worlds as there are sucrose molecules in a cube of sugar.

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Nevertheless, we do gain something extraordinary whenever a new vista onto a terrestrial world is opened up. Galileo was the first to achieve this, when he turned his telescope to the Moon and saw its three-dimensional relief for the first time. Mariner 4 and Mariner 9 accomplished a similar feat for Mars. The Magellan spacecraft revealed the Venusian topography. And once Messenger has photographed the full surface of Mercury, there will be a profoundly significant interval before we get our next up-close view of an unmapped terrestrial planet. My guess is that it’ll be Alpha Centauri B b.

The Messenger website is well worth a visit. I was particularly struck by the movie that the spacecraft made of the Earth during the close fly by of March 2005. During the course of 24 hours, the spinning Earth recedes into the black velvet distance and space travel seems like the real thing.

Mercury’s orbit, with its 88 day period and its eccentricity of 0.2 could slip unnoticed into the distribution of known exoplanets. It’s vaguely comparable, for example, with the orbit of HD 37605 b. This Msini=2.3 Mjup gas giant has an apoastron distance similar to Mercury’s, but dives much closer to its star during periastron.

We’ve been interested in HD 37605 b lately because its orbit dips in and out of the insolation zone where water clouds are expected to exist. At the far point of the 55 day orbit, it should be possible for white clouds to form out of a clear steamy atmosphere. At close approach, the clouds are turning to steam.

Jonathan Langton’s models for this planet show persistent polar vortices, which sequester cooler air, and which may remain cloudy even during the hot days surrounding periastron. The vortices are tenaciously long-lived, and tracer particles seeded into the vortices leak out only slowly. It would be interesting to know what sort of chemistry is brewing in the steamy hothouse environment of trapped and noxious air.

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On the UCSC Science Library shelves, we have an 1828 edition of Pierre Simon de Laplace’s Oeuvres that includes the five-volume Mecanique Celeste. At moments like this, it’s great to have a camera on one’s cellphone:

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Laplace’s identification of the 5:2 near-resonance between Jupiter and Saturn allowed him to augment the exisiting second-order Laplace-Lagrange secular analysis to produce a theory of planetary motion that was in extraordinary agreement with the observations of the late eighteenth century. His success in explaining the so-called Great Inequality was likely a contributing factor in the development the concept of Laplacian determinism, of a clockwork universe.

In 1802, during William Herschel’s visit to Paris, Herschel and Laplace had a meeting with Napoleon, who, like Thomas Jefferson, appears to have been not much taken with a system of the world created and dictated by natural law:

The first Consul then asked a few questions relating to Astronomy and the construction of the heavens to which I made such answers as seemed to give him great satisfaction. He also addressed himself to Mr. Laplace on the same subject, and held a considerable argument with him in which he differed from that eminent mathematician. The difference was occasioned by an exclamation of the first Consul, who asked in a tone of exclamation or admiration (when we were speaking of the extent of the sidereal heavens): “And who is the author of all this!” Monsieur De la Place wished to shew that a chain of natural causes would account for the construction and preservation of the wonderful system. This the first Consul rather opposed.

[Source: Herschel’s diary of his visit to Paris in 1802, as found in C. Lubbock’s _The Herschel Chronicle_, p. 310, see here for a nice background.]

I like the extrasolar planet game because it’s simultaneously up-to-the-minute and steeped in tradition. With systems like Gliese 876, we’re approaching roughly the same effective degree of refinement in our detection of planet-planet orbital perturbations that was possible in the late eighteenth century for Jupiter and Saturn. As a result, someone like Laplace, were he to materialize (see today’s NYT) in the Interdisciplinary Sciences Building here at UCSC, could roll up his french cuffs and immediately begin contributing publishable work. The same would certainly not be true if one of his equally luminous scientific contemporaries, say Antoine Lavoisier, were to suddenly walk in to a modern-day chemistry lab.

Will be making an effort to post more frequently. Thanks for your continued readership and participation as oklo.org heads into its third year.