systemic

trying to keep up to date5
greg posted in Uncategorized on November 1st, 2007

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This has not been the best month to get swamped with work and as a result essentially ignore my extrasolar planet weblog. The discoveries have been coming thick and fast, and many of them have some very interesting ramifications. So I’m going to make an effort to get back to a regular schedule of posts.

Staying up-to-the-minute on extrasolar planets can involve quite a bit of work. Fortunately, for the past year, Mike Valdez has been combing astro-ph each day as soon as the new mailings are released. He applies a strict standard of applicability to select the papers relevant to extrasolar planets, and reports the most interesting ones here on the systemic backend. I recommend this service to everyone.

HAT P-5b, WASP-3, OGLE-TR182b, WASP-4, HAT P-6b, WASP-5. Man. It seems like the transit detection rate is ramping up significantly. In all probability, the bottleneck is now the pace of RV confirmation with 8-meter class telescopes, rather than any shortage of transits themselves. It’ll be very interesting to see what the correlation diagrams and the planet catalog looks like one year from now.

Earlier this month, Ruth Murray-Clay visited UCSC, and gave an interesting talk about work that she’s been doing with Eugene Chiang on a model for the winds that flow off of hot Jupiters. Back in 2003, the Hubble Space Telescope was used to observe the HD 209458 b transit in the ultraviolet region of the spectrum surrounding the Lyman-alpha line. It turns out that the HD 209458 b transit has a depth of order 15% in Lyman alpha, indicating that a comet-like wind of hydrogen is flowing off the planet. Here’s a cartoon view:

The Murray-Clay and Chiang model assumes a steady-state flow, which allows them to adopt a time-independent treatment of the equations of hydrodynamics. It would be interesting to relax the time-independence and extend the analysis to the recently detected transits of HD 17156b. Because HD 17156b has such an eccentric orbit, any comet-like wind that it produces should be time-variable in nature. It should thus be possible to make some interesting predictions that can be tested when the community eventually regains the capability of observing transits in the ultraviolet.

Gigantic0
greg posted in Uncategorized on August 7th, 2007

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The TrES survey announced the discovery of a new transiting planet today, raising the number of known transits to twenty (including Mercury and Venus). The new planet, “TrES-4″, has a mass of order 84% that of Jupiter, and with a radius of 1.67 Rjup, it’s pumped to nearly five times Jupiter’s volume:

The false color image of Jupiter was produced from near-infrared data obtained with the Gemini telescope. The even more luridly false-color representation of TrES-4 is based on a vorticity map from one of Jonathan Langton’s recent simulations.

In order for TrES-4 to be swollen to its current size, it needs to be experiencing heating of order 6×10^27 ergs per second. One way to do this is to have a significant perturbing companion which drives large time-averaged variations in TrES-4’s orbital eccentricity. So far, there are only four published radial velocities for TrES-4, so the orbit could easily be non-circular. More provocatively, if strong orbital forcing is indeed occurring, then there’s a reasonable chance that the perturber might also be observable in transit. I recommend that Transitsearch.org observers keep this bad boy under constant supervision.

planeticity vs. metallicity14
greg posted in Uncategorized on July 6th, 2007

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Mike Valdez pointed me to an interesting paper by Pasquini et al. that was posted to astro-ph today. The authors examined the frequency with which Jovian-mass planets are detected around giant stars and dwarf (that is, ordinary main sequence) stars as a function of the metallicity of the host star. Their main result is summed up in this redrawn figure:

The red histogram shows the well-known result that detectable Jovian-mass planets are preferentially found around metal-rich stars. The blue histogram shows a result that seems surprising at first glance. It indicates that for giant stars, the metallicity effect essentially goes away. The distribution in the blue histogram is not much different from the overall distribution of stellar metallicities in our local galactic neighborhood.

Pasquini et al. give several possible explanations for their result. Their favored interpretation is that the planet-metallicity correlation is due not to high intrinsic metallicity, but rather to stellar pollution. The idea is that after a planet-bearing star forms, its thin convective envelope is enriched by the accretion of heavy elements. The planet-bearing stars that have metal-rich spectra are in actuality ordinary stars sheathed in enriched envelopes. As polluted stars evolve off the main sequence, their convective envelopes grow deeper, and the apparent metallicity enhancements largely disappear.

As an inveterate adherent of the core-accretion hypothesis for the bulk of giant planet formation, my knee-jerk reaction is to be unhappy with a pollution interpretation. Disks and (by extension) stars that are metal-rich are more capable of building planetary cores while there’s still gas remaining in the protoplanetary disk. The planet-metallicity connection is thus a natural consequence of the core accretion hypothesis.

Pasquini et al. point out that the giant stars in their sample are systematically more massive than the main-sequence stars for which the planet-metallicity connection has been established. This leads them to speculate:

Since the fraction of planet-hosting giants is basically independent of metallicity, it is feasible that intermediate mass stars favor a planet formation mechanism, such as gravitational instability, which is independent of metallicity. One could speculate that such a mechanism is more efficient in more massive stars, which (likely) have more massive disks.

I don’t completely agree with this interpretation either, but I do think that the correct explanation is tied into a systematic difference in stellar mass between the giant sample and the dwarf sample. While it’s somewhat difficult to get accurate masses for giants, its reasonable to assume that the average mass of the giants in the above histogram is ~2 solar masses. If we assume that protostellar disks scale in mass with the mass of the parent star, then the average disk around a 2 solar mass star had roughly twice the surface density of solids than the average disk around a solar mass star. This is equivalent to a 0.3 dex increase in metallicity in a disk around a solar mass star, neatly explaining the magnitude of the offset between the red and the blue histograms.

The paucity of planets around high-metallicity giants probably stems in part from small number statistics and from the fact that there are very few super-metal-rich giants in the survey. Note that the histograms plot the distributions in metallicity for planet-bearing stars, and not the fraction of planet-bearing stars in a complete sample as a function of metallicity Although a detailed Monte-Carlo experiment is definitely in order, I think that Pasquini et al.’s result will end up being fully in line with the expectations of the core-accretion theory.

This argument would have had a lot more weight if I’d done a detailed Monte-Carlo analysis in advance, rather than monday-morning-armchair-quarterbacking (that is, blogging) with a smug postdiction. I think, however, that the core-accretion theory indicates that these general trends will all continue to hold true:

N equals L10
greg posted in Uncategorized on July 2nd, 2007

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Last weekend, I participated in the “Future of Intelligence in the Cosmos” workshop at NASA Ames. In an age of ultra-specialized conferences, the focus for this one bucked the trend by pulling back for the really big picture:

The Future of Intelligence in the Cosmos” is an interdisciplinary two-day workshop that seeks to elucidate potential scenarios for the evolution of intelligent civilizations in our galaxy and thus, perhaps, to find a resolution for this seeming paradox. The probability that intelligent civilizations exist has been succinctly stated by the Drake Equation. While the first few terms in the equation, such as the number of stars in the Milky Way Galaxy, the fraction of stars that have planets, and the number of planets in the habitable zone, are becoming better known, the last three terms that depict the fraction of planets that evolve intelligent life, the fraction that communicate, and the fraction of the lifetime of the Milky Way Galaxy over which they communicate, are not well known. It is these last three terms in the Drake Equation that are the focus of the workshop.

In most venues, extrasolar planets veer toward the esoteric. At this workshop, however, the galactic planetary census was perhaps the most nuts-and-bolts topic on the agenda. We know that planet formation is common in the galaxy, and its increasingly clear that the “great silence” isn’t stemming from a lack of Earth-mass worlds.

Here’s a link to a .pdf document containing the slides from my talk.

In an upcoming post, I’ll try to pull together a synopsis of what emerged from the conference. Perhaps the most startling moment for me came in Paul Davies‘ talk, when he described the extent to which the simulation argument has been developed.

When I was in graduate school, Frank Drake was a faculty member in our Department. I noticed right away that the license plate on his car read “neqlsl”. I always read this as “n equals one”, until I finally asked him which term was responsible for thwarting all the alien civilizations.

“It’s not N equals one,” he said, “it’s N equals L”.

time series3
greg posted in Uncategorized on May 1st, 2007

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It’s remarkable how Keplerian fitting functions can be pushed to model a wide variety of time series. Anyone recognize this particular data stream?

It shows complicated behavior on timescales ranging from days to years, superimposed on an autoregressive tendency. The downloadable systemic console’s periodogram points to significant power at low frequencies, reflecting the gradual overall decline during the duration of the time series. There are also a number of distinct peaks at higher frequencies.

A crazy (read eccentric) six-planet Keplerian system does a credible job of fitting the data.

largely because the periastron passages of eccentric planets are capable of producing peaks that ramp up and then decay. To fit a particular peak, the five keplerian parameters can be varied to produce an enormous variety of waveforms.

The Keplerian model can be evaluated at any forward time to make a prediction, albeit in this case, one with presumably zero physical justification…

backlog1
greg posted in Uncategorized on April 5th, 2007

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It was the end of the Winter quarter here at UCSC last week, and then I went on a trip, and then bam! More than a week with no posts… In the interim, there have been a number of interesting developments related to extrasolar planets. Here’s a brief run-down of some topics that I want to look at in more depth in the very near future:

Thanks to continuing efforts from the back-end user base, we’re accumulating a highly useful database of stable, low chi-square fits to the synthetic radial velocity data sets that comprise the Systemic Jr. catalog. Stefano has run a preliminary analysis and interpretation of the data. There are interesting implications for the overall eccentricity distribution of extrasolar planets, and there also appears to be a robust criterion for determining with confidence when you’ve extracted a real, previously unannounced planet from a given data set. We’re putting together a full report, which will appear quite soon. In the meantime, please keep submitting fits for systems that haven’t yet been adequately characterized.

The detection of another Neptune-mass planet orbiting a nearby red-dwarf was announced today. Yet more evidence for the core-accretion theory of planet formation! The discovery paper stops short of tabulating the radial velocities, but as I write this, Eugenio is busy dextering them onto the systemic back-end and onto the downloadable systemic console.

The theoretical case for the existence of Alpha Centauri B b is getting stronger by the day.

This year’s first ‘606 day is coming up next week, with a transit opportunity following on April 17th. I didn’t do enough to get the word out last December, but I’m hoping for good photometric coverage of the star during the upcoming window. A central transit for HD 80606b would last roughly 18 hours, so participation from observers worldwide will be required to definitively rule out transits.

Impact!5
greg posted in Uncategorized, non-technical posts on March 7th, 2007

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The news out of the Planetary Defense Conference is that NASA has the ability to locate all the potentially dangerous asteroids in our solar system by the year 2020, but that the cash to carry out the project is not currently forthcoming.

CNN and many other news outlets carried the story, along with a dramatic artist’s rendition of an asteroid striking the Earth.

That’s a rather large asteroid.

Measuring the circumscribed circles, it appears that the impacting body is 1/10th the Earth’s diameter, or approximately 600 km in radius. A bolide of this magnitude would currently rank as the ~25th largest object in the solar system, larger than Uranus’s moon Umbriel (584 km radius), smaller than Saturn’s moon Iapetus (736 km in radius), and nearly exactly the same size as Pluto’s moon Charon (606 km in radius). Ceres, the largest object in the main asteroid belt, by contrast, has a radius of ~475 km.

Based on the location of the late-afternoon catastrophe relative to the day-night terminator, the impactor seems to have had an orbit that was highly inclined relative to the solar system’s angular momentum plane. Perhaps it was undergoing Jupiter-driven Kozai oscillations prior to striking the Earth.

The last impact of the magnitude shown in the illustration was probably the Moon-forming impact ~4.4 Billion years ago, in which a Mars-sized body struck the Earth. Kevin Zahnle of the NASA Ames Research Center has estimated the distribution of giant impacts after the Moon-forming event. It’s likely that the largest strikes were by bodies with roughly half the radius of the object shown in the above picture.

The consequences of even a 300 km object hitting the Earth are severe. Such an impact is energetic enough to entirely vaporize the Earth’s oceans and create a temporary rock-vapor atmosphere with a surface pressure of ~100 bars. For a period of several months to a few years, the Earth would radiate with a temperature of ~2000 Kelvin — hot enough to glow brightly in the visible region of the spectrum. And depressingly, our planet would be fully sterilized by the event.

spectra1
greg posted in Uncategorized on February 22nd, 2007

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So, uhh, yeah, the oklo blog went through a dry spell with no posts last week. This was primarily a consequence of the fact that Spitzer GO-4 proposals were due last Friday. I teamed up with Drake Deming of GSFC and UCSC physics grad student Jonathan Langton to propose a 30-hour observation of HD 80606b during the ‘606 day that’ll occur next November 20th. In an upcoming article, I’ll be pushing the reasons why we’re really excited about the possibility of observing HD 80606 b during its big periastron swing.

The Spitzer Space Telescope has turned out to be a regular wellspring of exo-planet results. It’s providing very interesting and often surprising constraints on the weather conditions at the surfaces of the hot Jupiters, and another big new result was announced today. Three different teams released the first-ever observations of emergent infrared spectra from two observational runs on HD 189733 and HD 209458.

The transiting planet HD 189733 b was discovered by the Swiss team in 2005. Of the fourteen known transiting planets, HD 189733 b is the best-suited for detailed follow-up observations. The parent star lies only 19 parsecs away, the orbital period is a skimpy 2.1 days, and the 1.15 Jupiter-mass planet has a radius fully 15% the size of the primary star’s radius. Like the other transiting systems, the planet, the orbit, and the star can all be drawn completely to scale on a “saved for web” diagram that’s only 420 pixels across:

Grillmair et al.’s Spitzer spectrum of HD 189733 was obtained with 12 hours of observation, in which the brightness of the star at infrared wavelengths between 7 and 14 microns is compared in and out of the secondary transit:

The observed flux distribution from the planet is nearly completely flat as a function of wavelength! Models of the atmospheres of hot Jupiters had all predicted that the presence of water vapor in the planetary atmosphere would lead to a prominent absorption feature at ~8 microns. No hint of the predicted dip was seen. The overall amount of infrared light coming from the planet during the secondary transit indicates that heat is probably being efficiently redistributed between the day and night sides of the planet.

A Nature paper by Jeremy Richardson and collaborators reported a very similar result for HD 209458 b. Their spectrum runs between 7.5 and 13.2 microns, is similarly devoid of absorption features, and also suggests a modest day-night temperature difference.

So how to interpret these results? One possibility is that the lack of absorption lines is caused by a high, uniformly emitting cloud layer, perhaps made of silicate grains. A problem with this interpretation, however, is that the cloud decks would have to be extremely dark and unreflective in the optical. Hot Jupiters absorb nearly 95% of the radiation that they receive from their parent stars. Another possibility, put forward by Jonathan Fortney, is that the atmospheres of these planets are isothermal down to large optical depths. Because we can’t actually see to a hotter underlying layer, there’s no mechanism for deep absorption lines to form.

And finally, another, rather startling, interpretation of the results was offered several hours ago by CBS News:

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300B0
greg posted in Uncategorized on January 24th, 2007

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The 200-odd extrasolar planets that have been discovered with the radial velocity method are orbiting stars that lie within a few hundred light years of the Sun. The light we now see coming from GJ 876 left that red dwarf back in early August 1991. When you’re in the bars drinking to celebrate the periastron passages of HD 80606 b, it’s easy to forget that last December’s periastron passage actually occurred in September 1817.

By galactic standards, however, a distance of 300 light years is still right next door. For every star within 300 light years of the sun, the Milky Way contains roughly 300,000 additional stars that are farther away. All told, adopting the latest rules on what constitutes a planet, our galaxy likely contains about 300 billion planets, of which perhaps 500 million are hot Jupiters.

Right now, 51 Peg, HD 209458, Upsilon Andromedae, et al. count among the Sun’s local galactic neighbors, but this hasn’t always been the case. The velocity dispersion of stars in the solar neighborhood is ~20 kilometers per second. A kilometer per second is a parsec per million years, which means that in a mere 15 million years, the roster of nearby planets will contain very few familiar names. HD 209458b is transiting now, but in a few hundred thousand years, it’s likely that the line of sight to the system will no longer allow Earthbound observers to watch that dip every 3.5247542 days.

So get out there while there’s still time! Due to a computer glitch, the transitsearch candidates table failed to get its nightly update for the past several nights. I’ve fixed the problem, and the ephemerides are all up to date.

Roll ‘em out…12
greg posted in Uncategorized, exoplanet detection on December 31st, 2006

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The discovery of new planets is rarely clear cut. No sooner does a new world (Vesta, Neptune, Pluto) emerge, than the wrangling for the credit or the naming rights starts. And it’s usually possible to find a reason why the prediction (or even the planet itself) wasn’t really valid in the first place.

The trans-Uranian planet predicted by Urbain J. J. Le Verrier and John Couch Adams happened to coincide quite closely with Neptune’s actual sky position in September 1846, but the orbital periods of their models were too long by more than 50 years. Le Verrier’s predicted planetary mass, furthermore, was too large by nearly a factor of three, and Adams’ mass prediction was off by close to a factor of two.

In England, following the announcement of Neptune’s discovery, and with the glory flowing to Le Verrier in particular and France in general, the Rev. James Challis and the Astronomer Royal George Airy were denounced for not doing enough to follow up Adams’ predictions, “Oh! curse their narcotic Souls!” wrote Adam Sedgwick, professor of geology at Trinity College.

Nowadays, with the planet count up over 200, the prediction and discovery of a new world doesn’t quite carry the same freight as it did in 1846. No editorial cartoons, no Orders of Empire, and no extravagant public praise to the discoverer, such as that heaped by Camille Flammarion on Le Verrrier, who wrote, “This scientist, this genius, has discovered a star with the tip of his pen, without other instrument than the strength of his calculations alone!”

Nevertheless, I don’t want to be shoehorned into the ranks of the “narcotic souls” as a result of not properly encouraging the bringing to light of any potential planetary discoveries in the systemic catalog of real stellar radial velocity data sets. As of Dec. 30th, 2006, over 3,680 orbital fits have been uploaded to the systemic backend. It’s definitely time to start sifting carefully through the results that the 518 registered systemic users have produced. Over the next few weeks we’ll be introducing a variety of analysis and cataloging tools that will make this job easier, but there are some interesting questions that can be answered right away. Foremost among these is: what are the most credible (previously unannounced) planets in the database?

The backend uses the so-called reduced chi-square statistic as a convenient metric for rank-ordering fits:

In the above expression, N is the number of radial velocity data points, and M is the number of activated fitting parameters. As a rule of thumb, a reduced chi-square value near unity is indicative of a “good” fit to the data, but this rule is not exact, and should hence be applied with caution. The observational errors likely depart from a normal distribution, and more importantly, the tabulated errors don’t incorporate the astrophysical radial velocity noise produced by activity on the parent star. Furthermore, it’s almost always possible to lower the reduced chi-square statistic by introducing an extra low-mass planet.

Eugenio recently implemented the downloadable console’s F-test, which can provide help in evaluating whether an additional planet is warranted. The F-test is applied to two saved fits and returns a probability that the two fits are statistically identical. As an example, pull up the HD 69830 data set and obtain the best two planet fit that includes the 8.666-planets and 31-day planets. Save this fit to disk. Next, add the 200-day outer planet and save the resulting 3-planet fit to disk (using a separate name). Clicking on the console’s F-test button allows the F-test to be computed using the two saved fits:

In the case of HD 69830, there’s a 1.7% probability that the 2-planet fit and the 3-planet fit are statistically identical. This low probability indicates that the third planet is providing a significant improvement to the characterization of the data. It’s likely really out there orbiting the star.

So here’s the plan: Let’s comb through the systemic “Real Star” catalog, and find the systems that (1) contain an unannounced planet(s) in addition to the previously announced members of the system (see the exoplanet.eu catalog for the up-to-date list). (2) have a F-test probability of less than 2% of being statistically identical, and (3) are dynamically stable for at least 10,000 years. If you find a system that meets these requirements, post your findings to the comments section of this post.

Disclaimer: this exercise is for the satisfaction of obtaining a better understanding of the planetary census, and also for fun. When the planets do turn up, I’m going to sit back with a bottle full of bub and enjoy any scrambles for priority from a safe distance.

Happy New Year, y’all!

Gift idea for ‘606 day0
greg posted in Uncategorized on December 8th, 2006

Tis the season! If you’re like me, you’re probably looking for ways to minimize your exposure to malls, crowds, and overloaded sleighs. If so, we here at oklo have devised a one-stop solution for all of your holiday gifts. On Dec. 11th, Taylor and Francis publishers is releasing Numerical Methods in Astrophysics, by Peter Bodenheimer, Michal Rozyczka, Hal Yorke, and myself.

From the publishers description:

This guide develops many numerical techniques for solving major astrophysics problems. After an introduction to the basic equations and derivations, the book focuses on practical applications of the numerical methods. It explores hydrodynamic problems in one dimension, N-body particle dynamics, smoothed particle hydrodynamics, and stellar structure and evolution. The authors also examine advanced techniques in grid-based hydrodynamics, evaluate the methods for calculating the gravitational forces in an astrophysical system, and discuss specific problems in grid-based methods for radiation transfer. The book incorporates user manuals and a CD-ROM of the numerical codes.

It should start shipping Dec. 11th, order yours today!

Watch the Skies3
greg posted in Uncategorized on December 7th, 2006

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Sorry about this long lapse in new posts. The end of the academic quarter has left me awash in deadlines and scrambling to get everything done.

Nevertheless, we’ve been making progress behind the scenes. The new oklo server has been delivered, configured, and slotted into a rackspace in a dedicated server room. To use the vernacular, it’s hecka fast. Over the next several days, we’ll be transferring the site over to the new machine, and then it’ll be bye-bye bluehost.

HD 80606 is looking more interesting all the time. I’m working on a writeup of what we’ve been learning. It really has the potential to give us an unambiguous value for the radiative time constant appropriate to the atmospheres of hot Jupiters. The next ‘606 day is December 26th, and I’ll be sending out a circular to the transitsearch.org observers to get a definitive confirmation that it doesn’t transit. Here’s the promotional poster (inspired by the SAO Moonwatch program, while simultaneously attempting to achieve a retro cold-war-flying-saucers feel):

Finally, keep fitting the last batch of Systemic Jr. systems. We need to get a full range of good fits for all of the data sets in order to carry out some very interesting analyses…

zoom1
greg posted in Uncategorized on October 28th, 2006

Good news for the systemic user base! Eugenio has posted a new version of the downloadable systemic console. This most recent update fixes several bugs, and offers a better graphical interface for those working at limited display resolutions. Progress overall has been rapid during the past several days, and next week we’re planning to roll out both a fully threaded version of the console as well as the Systemic Jr. catalog of synthetic radial velocity data sets. Systemic Jr. will be a testbed for the full Systemic simulation, and will allow us to answer a number of interesting questions regarding the fidelity of planetary models as a function of orbital parameters and observational sampling. Put oklo.org on your bookmark list and tell your friends to drop by. We’ve manufactured plenty of consoles to hand out.

By tomorrow I’ll be back on the extrasolar planets beat, but I thought it would be interesting to show a few more results connected to the strange orbits detailed in the previous post.

It’s clear from the sample of eight orbits that were charted that the m=1 singular isothermal disk potential supports an extensive variety of orbital families: tube orbits, box orbits, chaotic orbits, resonant orbits, and Enron orbits just to name a few. Is it possible to design a map that shows the regions of parameter space that are delineated by the different kinds of orbits?

The best method that I’ve been able to devise consists of what I’ll call an “excursion map”. We can clasify orbits by the total angle that they accumulate over time. For example, a loop orbit (such as the first trajectory shown in the previous post) experiences a steady accumulating of total angle — 360 degrees worth per orbital period. A box-type orbit on the other hand (like the seventh and eighth trajectories shown in the previous post) oscillates back and forth across the x-axis and never accumulates more than 90 degrees or so of total angular excursion. Chaotic orbits (such as the sixth example trajectory) execute a random walk in angular excursion, and on average accumulate a total absolute angular excursion (either positive or negative) which is proportional to the square root of the time.

We can thus pack information about the orbital structure of the potential function into a single diagram. We choose intial starting conditions parameterized by position on the x-axis and the e-parameter of the potential function. A given starting condition corresponds to a point on a two-dimensional diagram, and also defines an orbit. The orbit can be integrated for a characteristic time (t=1000, say) and the total angular excursion or the orbit can be logged. A color code can then be assigned: white for orbits that accumulate positive angle in direct proportion to the time, gray for orbits that accumulate angle in proportion to the square root of the time, and dark gray for orbits that never get beyond plus or minus 90 degrees. With this coding, the excursion map looks like this:

The numbers label the locations of the 8 different orbits shown in the previous post.

Take orbit 3, for example. It corresponds to a loop-type orbit within an island of similar loop orbits surrounded by a sea of chaotic orbits. If we zoom in on the island with a magnification factor of ten, we see structure emerge. Tiny changes in the initial conditions determine whether an orbit is stable (white) or chaotic (gray). Two jagged fingers of box orbits jut up into the map.

Zooming in by another factor of ten shows that the map has a fractal structure, with detail emerging on every level of magnification:

It’s strange to realize how so much bizarre structure is inherent within such a simple potential function. Somehow, encapsulated into one simple formula, the dynamics are all folded up like an inifinite series of orgami cranes, waiting patiently to be observed…

weird orbits in a weird potential3
greg posted in Uncategorized on October 26th, 2006

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The academic quarter is pulling up to the half-way point at UCSC, and it’s getting tougher to keep up with everything that I’m supposed to do. I’ve been spending a lot of time getting the lectures together, so in the interest of sticking to a schedule of posts, I thought I’d veer from the all-planets-all-the-time approach and show some scanned transparencies from Friday’s class.

The orbit of an idealized planet around an idealized star is a keplerian ellipse. A planet on an elliptical trajectory conserves its eccentricity, orbital period, longitude of periastron, inclination, and line of nodes. The only orbital element that changes over time is the mean anomaly. We can thus say that the Keplerian orbit contains five integrals (constants) of the motion.

If the potential arises from a mass distribution that’s not a perfect point mass, then in general we won’t have five integrals of motion. It’s interesting to look at a subsection of the weird variety of planar orbits that occur in a two-dimensional potential distribution that looks like this:

The ln(r) term causes this potential goes to negative infinity at the origin while remaining unbounded at large radii. The second term, in which e is specified to take on values between 0 and 1, lends a modulation that makes the force law non-axisymmetric with respect to the origin. One can roughly think of orbits in this potential as the motion of a marble rolling in a funnel-shaped, lopsided bowl.

The potential function does not change with time, and so the energy of an orbiting particle is conserved. Further, because of the self-similarity of the potential, the structure of orbits at one energy will be an exact copy of the orbit structures at all other energies. Thus, there’s no loss in generality by sampling orbits having only a single total energy (kinetic + potential). In the following sampler of pictures, I integrate the trajectories of single particles launched from the long (+x) axis of the potential with initial velocities always perpendicular to the long axis. The magnitude of the velocities are determined by the total energy choice: particles starting closer to the origin must have a higher initial kinetic energy to offset their more negative gravitational potential energy. I also vary the parameter e.

For e=0.2 and an initial position x=0.78, the orbit is reasonably circular, and steady precession smears the excursion of the particle over many orbits into a thin annular region centered on the origin.

For e=0.42, x=0.55, the particle starts fairly close to its zero velocity curve. It thus falls inward almost to the origin before making a second loop, and then a second approach to the origin which sends it rocketing back up close to its initial position.

Taking e=0.2, x=0.78 leads to a single loop orbit that dives in very close to the origin.

Here’s the result of taking e=0.42 and x=0.55. It’s a good thing the Earth isn’t orbiting in this potential with these particular starting conditions.

This one, which arises from e=0.81 and x=0.85 is pretty cool. Most of the time it runs counterclockwise as viewed from above, but before close approach to the origin it switches to clockwise. One’s tempted to classify it as an Enron orbit. From all appearances it appears to be clocking a steady increase in angle, whereas in reality, when the books are finally audited, it’s accumulating -2pi radians every period.

Choosing e-0.30, x=0.03 launches the particle on a highly chaotic trajectory. This orbit is uniformly sampling the entire area allowed to it by its total energy constraint.

For larger values of the e parameter the orbits often show a fundamentally different behavior. Choosing e=0.72, x=0.22 leads to a motion which is restricted to oscillations of a narrow angular range centered on the long-axis of the potential. The particle is basically rolling back and forth in the narrow valley provided by the high-e potential.

e=0.90, x=0.20 gives an orbit with a similar quality:

easy money2
greg posted in Uncategorized on August 3rd, 2006

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Unless lightning strikes, the lower layers of the Earth’s atmosphere contain very small fraction of charged particles. The air is electrically neutral, and indeed is a fairly good insulator. This state of affairs is something to be thankful for.

Imagine what would happen if the air started to carry a tiny ionization fraction. That is, imagine if one out of every million air molecules were stripped of an electron. The ionized air molecules and the electrons would experience an immediate desire to spiral around the Earth’s magnetic field lines. In doing so, they would bash into the surrounding sea of neutral particles and drag them along with their motion.

Bulk motion of charged particles drags magnetic field lines along and vice-versa. Magnetic field lines, however, don’t like being compressed or twisted, and have a tendancy - verging on insistence - to spring back into shape. If the Earth’s atmosphere had a small magnetic field, the jet stream would rapidly wind up the Earth’s magnetic field, which would angrily resist the winding and pull backward on the jetstream. Our normal weather patterns would be thrown into complete and utter disarray.

In the inner regions of a protostellar disk, the temperature is high enough for trace elements such as sodium to lose their outer electrons. This raises the ionization fraction of the disk gas to the point where the ambient magnetic field begins to play an important role. This, in turn, leads to an interesting situation.

Imagine two parcels of disk gas on a circular orbit. Imagine also, that the two parcels are connected by a weak magnetic field line. Next, perturb the leading parcel by pulling backward on it slightly. Such a pull drains orbital energy from the parcel and causes it to drop down to a lower orbit. A lower orbit, however, has a faster rotational velocity. The faster rotational velocity causes the parcel to run forward. This pulls on the magnetic field line, which pulls back, forcing the particle even further down into the gravitational well. Clearly, we have the condition for a runaway situation.

This process, known as the magnetorotational instability was discussed by Chandrasekhar in the late 1950’s, and appears in his monograph on Hydrodynamic and Hydromagnetic Stability, and was brilliantly revived in the context of disks in the early 1990’s by Steve Balbus and John Hawley. The nonlinear outcome of the magnetorotational instability is turbulence in the disk. This turbulence may play an important role in allowing mass to slip down and accrete onto the star.

The magnetorotational instability is a simple consequence of the remarkable fact that self-gravitating systems have a negative heat capacity. Balbus and Hawley completely cleaned up by recognizing the importance of the instability within the context of accretion disk physics. Their 1991 paper has now garnered 960 citations. I’m of the opinion that there may be some similarly useful gems ready to be mined out of several of Chandrasekhar’s more opaque books. In fact, I’m going to put on my mining helmet and stake some claims inside of Ellipsoidal Figures of Equilibrium.

Lone Star0
greg posted in Uncategorized on April 14th, 2006

Frequent visitors to oklo.org will have noticed that the new posts have dried up over the past several days. I was out of town to attend the 2nd annual Mitchell Institute Symposium at Texas A&M. This is a conference that brings together speakers from a broad range of sub-disciplines in Astronomy and Physics. Ten gallon hats off to Texas! I had a great time. Warm weather, informative talks, and the Aggies all called me “Sir”. My plan for next week is to get the UCSC Banana Slugs to start up with that tradition.

As part of the conference, I was asked to give a public talk on Extrasolar Planets. It was an all-day scramble on the laptop to get all my slides together into a coherent whole, but the talk ended up being a lot of fun. The audience was highly informed and engaged. The TAMU Physics Department definitely got the word out. I was completely stunned this morning to find that I was on the the front page of the Bryan-College Station Eagle, and I was even recognized at the College Station Airport cafe while I was waiting for my flight out. Unbelievable.

Here’s a link to a quick-time movie, as well as a .pdf file with the slides that I showed during the talk. I’ve also put the sound files (you had to be there to know what I’m talking about) here, here, and here in .wav format. A future oklo post will go into much more detail about what’s being heard in these files, and how they are generated.

If you’re new to the site, here’s a bit of information. Oklo.org is the home base for the systemic collaboration, which is a public participation research project aimed at obtaining a better characterization and understanding of extrasolar planets. Everyone is invited to participate, and details and updates are given regularly in our systemic faq posts.

We have been developing both the oklo.org site, as well as the systemic console using a Mac OS-X platform. We have been testing both the site and the console using Internet Explorer, and we have gotten generally good results, but it is clear that some users are experiencing problems. We are working hard to clear these issues up. We’re astronomers by trade, and, and sadly, at the moment, it’s strictly amateur hour when it comes to website development. As an example, you should see a menu of links directly to your right. I recently saw the oklo.org site on a Windows-IE combination in which the links had been mysteriously pushed all the way down to the bottom of the page. I had to scroll all the way down to even see them.

Also, if you are a Macintosh user, run the console in Safari. There is a still a Java issue with the Firefox on OS X. Firefox should, however, work fine on both Linux and Windows machines if your Java libraries are up to date…

systemic 0020
greg posted in Uncategorized, systemic faq on April 5th, 2006

There’s a new data set on the Systemic Console. To access it, launch the console, and select systemic002 from the system menu (it’s the second from the bottom of the list).

Let’s just say I’ve often wondered whether these particular data can be modeled by a stable planetary system.

stop-action stop-gap0
greg posted in Uncategorized on March 20th, 2006

We’ve noticed that fresh content encourages regular return visits to oklo.org.

With that sentiment in mind, here’s a stop-action .mpeg4 animation of the newly discovered 2:1 resonant planetary system orbiting HD73526. The planets are represented by red and green peppercorns, and a kumquat stands in for the central star:

hd73526.mov

If the version above won’t load in your browser, try this one. Rest assured that the systemic team is hard at work on more substantive posts (including some very interesting new exoplanet-related results), so check back frequently!

Planets at the AAS Meeting0
greg posted in Uncategorized on January 16th, 2006

Frequent visitors to oklo.org will have noticed a definite fall-off in the number of recent posts. This was a direct result of the start of the winter quarter here at UCSC, but now things are rolling, and the systemic team is working hard to prepare the next phase of the collaboration.

Last week was also the 207th meeting of the American Astronomical Association. I took a one-day trip to Washington in order to give a talk at Tuesday’s extrasolar planets session entitled, “From Hot Jupiters to Hot Earths“. I teach class on both Monday and Wednesday mornings, so the trip was more of a lightning raid.

I arrived at Dulles Airport at 6 am, after an overnight flight. My talk wasn’t finished, so I sat in an empty departure lounge for several hours and worked on the slides. By mid-morning, I realized that I had better head to the venue. I took a cab to the conference hotel, tapping on the laptop for most of the way.

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I Saw the Light….1
paul posted in Uncategorized on December 19th, 2005

Hi All,

Happy Holidays, as we close in on the Winter Solstice-

As Greg noted, the growing pandemic of light trespass is certainly an issue for every astronomer, whether exoplanet searching is one’s vocation or avocation…. IDA leads the way in light pollution defense, and on their website, you can find a number of tools to put in your toolkit in the “Global War on Light Trespass”. You can also find out what your area is supposed to look like pollution-wise, by going to another site… In fact, the site is primarily a weather site – that other nemesis of getting good visual observations; try it out.

Now, to look at its excellent weather information – and light pollution maps for any US location, you find your state (or, D.C. for me), and pick your site (again, USNO for me) at this link. You’ll get something like so:

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Hello Exo-world…1
paul posted in Uncategorized on November 17th, 2005

Good to see this up!

Paul