systemic

Gough Island. Image Source.

Urbana, Illinois, the quintessential Midwestern University town, was a fine place to grow up, but it is sited in a landscape that is neither remote nor exotic.

Lifting up from Willard Airport just south of town, the near-absolute flatness of the landscape, planed by the last glacial advance, extends in a patchwork of corn and soybean fields to every horizon.

Something about the first-glance monotony of the Illinois landscape gradually instills a heightened sensitivity to the subtle detail inherent in a sense of place. Ray Bradbury, in Something Wicked This Way Comes, captures the essence of this perfectly. I think that living in Illinois also instilled a fascination with maps of the distant and rugged corners of the world.

I spent a lot of time poring over the maps that come with National Geographic. I’ve always been particularly drawn to the region corresponding roughly to the South Atlantic Anomaly, the vast expanse of the Southern Ocean that spans the temperate through subarctic latitudes. In the region roughly equidistant from South America, Africa and Antarctica, the maps show only a few specks of land: St. Helena, Tristan da Cunha, Gough, Bouvet. These islands, on the basis of their latitudes alone, seemed like they might be “habitable”, but the colossal scale imposed by millions of square miles of deep water, left them completely unresolved.

Within a few years, we’ll also know about extrasolar planets that just might be habitable. That is, we’ll have specific, concrete knowledge of worlds with radii and masses similar to Earth, on orbits within their parent star’s so-called habitable zones. But in all likelihood, for quite a while after that, a few spare, unadorned facts will constitute the bulk of our information about those planets — it’ll be left to extrapolation, to flights of conjecture and guesswork, to fill in the details.

The situation seems oddly parallel to the maps of the Southern Ocean. I can remember ranging over the names and coordinates of the the cryptic dots in the expanse of blue, and wondering, what are they like? There was nothing about Inaccessible I. in the public library. There was hardly a mention, of St. Helena I. (U.K.), other than a few maddeningly sketchy fragments in the Encyclopedia Britannica. Napoleon, after Waterloo, had been famously dispatched there, precisely because of its remoteness and isolation. Almanacs are invariably fond of listing the fact that Bouvet is the most isolated spot of land on Earth.

Like a current-day version of the TPF mission, the advent of Google and the Internet have brought the worlds of the Southern Ocean into focus.

Tristan da Cunha. Image Source.

Tristan da Cunha is dominated by a steep-sided 2000-meter volcano that last erupted in 1961. Two hundred and sixty people live on the island, making it the most isolated permanently inhabited spot on Earth. With Google, it’s possible to explore in great detail, although actually going there is not easy. There’s no airstrip. The only way in is by boat.

To get a better sense of scale, I superimposed the island on Urbana, Illinois, for a personalized juxtaposition of the exotic and the familiar.

Even more remote, is Gough Island. Until last year, it was hard to find good pictures of Gough. The views all seemed the same — a craggy heap of lava in the misty distance from the decks of ships. Recently, though, Google pointed me to an absolutely fantastic set of annotated photos, taken by Chantal Steyn, who spent an entire year during 2008-2009 on the island as part of an 8-person team that staffed a South African weather station on the Island. Suddenly, Gough comes spectacularly to life, the very picture of a habitable, yet alien world.

Mount Zeus on Gough Island. Image Source.

Further south, and far more formidable, is Bouvet. Nobody seems to be there, but oddly, the island has a top-level internet domain code (.bv) for which there are six registered hosts…

Image Source.

Controversy generates revenue for exoplanet weblogs and supermarket tabloids alike, so I’m always happy when planet-related press releases roll out dramatic, far-reaching claims. Last week’s ESO press release — “Turning Planetary Theory Upside Down” — was quite satisfactory in this regard…

Upon digging into the back story, one finds that the observations underlying the press release are fully uncontroversial — it’s the big-picture interpretation that’s turning heads. Using Doppler velocity measurements taken during transit, Triaud et al. (preprint here) have measured the sky-projected misalignment angles, λ, for six of the transiting planets discovered by the SuperWASP consortium.

After an initial run of nine transiting planets were found to have sky-projected misalignment angles close to zero, the current count now has 8 out of 26 planets sporting significant misalignment. In the standard paradigm where hot Jupiters form beyond the ice line and migrate inward to reach weekend-length orbits, one would expect that essentially all transiting planets should be more or less aligned with the equators of their parent stars.

The standard migration paradigm, however, leaves at least two questions rather vaguely answered. First, why do the hot Jupiters tend to halt their inward migration just at the brink of disaster? The distribution of orbital periods — slew of selection biases aside — shows a durable peak near ~3 days. Second, why are transiting planets with well-characterized companions so scarce? In general, if one finds a giant planet with a period of ~10 days or more, the odds are excellent that there are further planets to be found in the system. For the known aggregate of transiting planets, and for hot Jupiters in general, additional planets with periods of a few hundred days or less are only infrequently found.

HD 80606b provides a clue that processes other than disk migration might be generating the observed population of hot Jupiters. The planet HD 80606b, its parent star HD 80606, and the binary companion HD 80607 form a “hierarchical triple” system, in which the two large stars provide an unchanging Keplerian orbit that drives the orbital and spin evolution of HD 80606b. If we imagine that HD 80606b and HD 80606 are both subject to small amounts of tidal dissipation, then to plausible approximation, this paper by Eggleton & Kiseleva-Eggleton argues that (i) the orbital evolution of “b”, (ii) the spin vector of “b”, and (iii) the spin vector of HD 80606 itself  can be described by a set of coupled first-order ordinary differential equations:

where e and h are vectors describing the planetary orbit, and where Ω_1 and Ω_2 are the spin vectors for HD 80606 and HD 80606b. The equations are somewhat more complicated than they appear at first glance, with expressions such as:

making up the various terms on the right hand sides.

Numerical integrations of the ODEs indicate that solutions exist in which the e and h vectors for `606b are bouncing like a ’64 Impala. Check out, for example, this solution vector animation by Dan Fabrycky (using initial conditions published by Wu and Murray 2003) which shows the leading scenario for how HD 80606b came to occupy its present state.

HD 80606b is imagined to have originally formed in a relatively circular orbit that was roughly 5 AU from its parent star, and which happened to be at nearly a right angle to the plane of the HD 80606-HD80607 binary orbit:

The large mutual inclination led to Kozai oscillations in which ’606b was cyclically driven to very high eccentricity. During the high-eccentricity phases, tidal dissipation within the planet gradually drained energy from the orbit and decreased the semi-major axis:

Eventually, the orbital period became short enough so that general relativistic precession was fast enough to destroy the Kozai oscillations, and the planet was marooned on a high-eccentricity, gradually circularizing orbit that is severely misaligned with the stellar equator — exactly what is observed:

With HD 80606b, the case for Kozai-migration is pretty clear cut. The guilty party — the perturbing binary companion — is sitting right there in the field of view, and the scenario provides an easy explanation for anomalously high orbital orbital eccentricity. The only “just-so” provision is the requirement that the planet-forming protoplanetary disk of HD 80606 started out essentially perpendicular to the orbital plane of its wide binary companion.

The Triaud et al paper and the press release draw the much more dramatic conclusion that Kozai cycles with tidal friction could be the dominant channel for producing of the known hot Jupiters. From the abstract of their paper:

Conclusions. Most hot Jupiters are misaligned, with a large variety of spin-orbit angles. We observe that the histogram of projected obliquities matches closely the theoretical distributions of using Kozai cycles and tidal friction. If these observational facts are confirmed in the future, we may then conclude that most hot Jupiters are formed by this very mechanism without the need to use type I or II migration. At present, type I or II migration alone cannot explain the observations.

Can this really be the case? Might it be time to start reigning in the funding for studies of Type II migration in protostellar disks?

A key point to keep in mind is that Rossiter-McLaughlin measurements yield the sky-projected misalignment angle, λ, between the stellar spin and planetary orbital angular momentum vectors, and not the true misalignment angle, ψ, in three-dimensional space. That is, with transit spectroscopy alone, you can’t discern the difference between the following configurations:

In a paper published in 2007, Dan Fabrycky carried out integrations of the Eggleton-Kiseleva-Eggleton equations for an ensemble of a thousand star-planet-star systems that experience HD80606-style Kozai migration coupled with tidal friction. From the results of the integrations, he constructed a histogram showing the distribution of final misalignment angles, ψ:

The first nine Rossiter-McLaughlin observations of transiting planets all produced values for λ that were close to zero, in seeming conflict with Fabrycky’s distribution for ψ. The jump-the-gun conclusion, then, was that Kozai-migration is not an important formation channel for hot Jupiters.

With the spin-orbit determinations that appear in the Triaud et al. paper, there are now a total of 26 λ determinations. A fair fraction of the recent results indicate severely misaligned systems, and Triaud et al. show a histogram over λ (or in their notation, β):

In order to compare the observed distribution of λ measurements with Fabrycky’s predicted distribtion of Kozai-migration misalignments, ψ, Triaud et al. assume that the distribution of spin axes for the transit-bearing stars is isotropic. With this assumption, one can statistically deproject the λ distribution and recast it as a ψ distribution, giving a startlingly good match between Fabrycky’s theory (blue dashed line) and observation:

When I first saw the above plot, I had a hard time believing it. The assumption that the spin axes of transit-bearing stars are isotropically distributed seems somewhat akin to baking a result into the data. Nevertheless, it is true that if Kozai migration produces the hot Jupiters, then the current ψ distribution is right in line with expectations.

In early 2009, Fabrycky and Winn did a very careful analysis of the 11 Rossiter measurements that were known at that time. Among those first 11 measurements, only XO-3 displayed a significant sky-projected spin-orbit misalignment. From the sparse data set, Fabrycky and Winn concluded that there were likely 2 separate populations of transit-bearing stars. One population, in which the spins and orbits are all aligned, constitutes (1-f)>64% of systems, whereas a second population, sporting random alignments, is responsible for f<36% of systems (to 95% confidence).

Bottom line conclusion? More Rossiter-McLaughlin measurements are needed, but I think its safe to say that Kozai-migration plays a larger role in sculpting the planet distribution than previously believed. If I had to put down money, I’d bet f=50%.

Competition keeps everyone on their toes, and the exoplanet Doppler detection game is no exception.

The California Planet Search has recently done a major overhaul of their exoplanets.org website, and the results are impressive. The redesigned site is now fully interactive, and it must be seeing a lot of traffic. Certainly, I can count myself as a frequent visitor!

Perhaps the most exciting feature of the site is a plotting applet that seamlessly connects to an up-to-date and curated database of the known extrasolar planets. In the “advanced” mode, one can get very finely tuned plots that can tell interesting stories. As an example, here is a plot of RV half-amplitudes of the known planets plotted against the RMS of the residuals to the fits. The color of the points corresponds to discovery year (cool = back in the day) and the size of each point corresponds to the number of published RV data points for the planet (those five big points correspond to 55 Cancri b-f which has a very extensive data set).

The plot shows that progress comes in part from competition. As the competing Doppler surveys push to lower Ks, there has a been a trend toward decreased signal-to-noise for the detections. It looks like oklo.org posts a few years from now will likely be discussing systems with K~60 cm/sec. At that amplitude, one is plausibly talking habitable worlds.

Another interesting plot comes from plotting parent star metallicity against planet mass. As with most of the interesting diagrams, a logarithmic scaling is required. The parent star masses are keyed to the sizes of the individual points, and color is assigned to eccentricity. The software has the nice feature that a cursor placed on a dot informs you of the planet name. This plot shows the benefit of looking at lower mass stars, and it shows how the metallicity correlation is diminished as one pushes below roughly a Saturn mass (evidence, of course, for core accretion):

The exoplanets.org site also contains a very useful planet table, which is giving the competition (in this case, exoplanet.eu) a run for its money.

The question of how the world’s top Doppler teams match up in league play is something that I imagine comes up quite a bit in exoplanet-related water-cooler discussions. A suitable scoring system is therefore in order, and the tables on exoplanets.org make this a very doable proposition.

After some thought, I’ve decided to adopt the system used for cross-country running, with the K‘s of the team’s planets replacing the times of the team’s runners. (The image for this post is from a 1983 dual meet between two high school teams from Central Illinois. If you look carefully, you can see that the coach is hurling an acorn at yours truly, presumably because of the much wider-than-expected gap between runners #2 and #3.) In the exoplanet context, the cross-country scoring system encourages fluid changes of lead — one or two high-grade multiple super-Earth systems can catapult a team to the top of the board. From the wikipedia article:

When two or more teams of cross country runners compete, a score may be compiled to determine which team is the better. Points are awarded to the individual runners of eligible teams, equal to the position in which they cross the finish line (first place gets 1 point, second place gets 2 points, etc). Teams are considered ineligible to score if they have fewer than the meet’s required number of scorers, which is typically five. Only the first five runners in for a team are counted towards that team’s score; the points for these runners are summed, and the teams are ranked based on the total, with lowest being best. In the event of a tie, the rules vary depending on the competition; often the team that closes scoring first wins, though in the US NCAA ties are possible. In high school competition, if two teams tie, then the victor is decided by whose sixth runner, the first one whose score does not count, finished first.

The lowest possible score in a five-to-score match is 15 (1+2+3+4+5), achieved by a team’s runners finishing in each of the top five positions. If there is a single opposing team then they would have a score of 40 (6+7+8+9+10), which can be considered a “sweep” for the winning team. In some competitions a team’s sixth and seventh runner are scored in the overall field and are known as “pushers” or “displacers” as their place can count ahead of other runners. In the above match, if there are two non-scoring runners and they came 6th and 7th overall, the opponent’s score would be 50 (8+9+10+11+12). Accordingly, the official score of a forfeited dual meet is 15-50.

According to the above rules, there are currently three RV teams in the running. The Geneva Extrasolar Planet Search (whose planets I’ve listed with SWISS on the table), the California Planet Search (planets listed with CPS), and the Earthbound Planet Search (who I’ve marked as EPS):

The score as of this morning? SWISS 25, EPS 47, CPS 62…