oklo

delineating connections

New worlds to conquer

November 25, 2006

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The California Carnegie planet search team posted a data-rich paper on astro-ph this week. The new article is scheduled to appear in the February 2007 issue of the Astrophysical Journal. Eugenio, exercising his usual diligence, has added the new velocity tables to both the downloadable systemic console and the systemic back-end stellar catalog.

The highlight of the paper is a new two-planet system orbiting HIP 14810, a metal-rich solar-mass star lying 53 parsecs away. The inner planet in the system has a period of 6.66 days, and tips the scales with least 3.84 Jupiter Masses. The outer planet is less massive (Msin(i)=0.76 Mjup), and goes around the star every 95.3 days.

The secular interaction between the two planets compels them to trade angular momentum back and forth. As a result, the inner planet cycles between an eccentricity of 0.02 and 0.15 on a relatively short 5000-year timescale. It’s currently in the high-eccentricity phase of its orbit. The large radial velocity signal-to-noise for the planet means that its eccentricity can be measured quite precisely (have a look at it with the console). The fact that the orbit is clearly non-circular would be strong evidence for the presence of planet c, even if there weren’t enough data to detect c directly. If planet b was the only significant planet in the system, its orbit would have circularized via tidal dissipation on a timescale that is less than the age of the star.

Short-period planets with masses greater than three Jupiter masses are intrinsically rare. Tau Boo b (with a mass of at least 3.9 Jupiter masses and an orbital period of 3.3 days) is the only other object with roughly similar properties. By contrast, 32 planets with periods of less than a week and minimum masses less than Jupiter’s mass are currently known.

In my opinion, the two most robust statistical correlations that have emerged from the first decade of extrasolar planet detection are (1) the planet-metallicity connection and the (2) dearth of high-mass short-period planets. The planet-metallicity correlation makes perfect sense. It’s the natural, expected outcome of the core-accretion process and the fact that Jovian-mass (as opposed to Neptunian-mass) planet formation is a threshold phenomenon. The paucity of high-mass short-period planets, on the other hand, is both weird and completely unexplained. It’s telling us something about the process of planetary formation and migration. We just don’t know what it is.

Noise Floor

November 20, 2006

concrete sky

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Giant planets are interesting. Terrestrial planets are more interesting. Habitable terrestrial planets are the most interesting of all, and it’s nearly guaranteed that we’re living in the age when the first genuinely Earthlike worlds beyond our solar system will be discovered. The only question is which technique will wind up doing it. The big money is on space-based transit photometry, but I think that ground-based RV might take the prize.

The Systemic Challenge 004 system was designed to be a futuristic idealization of what the Sun’s reflex velocity would like if it were observed with high precision from a neighboring star for more than two decades.

The individual radial velocity uncertainties for the 1172 velocities the Challenge 004 datset are each of order 10 centimeters per second. Errors this small are still safely smaller than the sub-meter per second precision that is currently being obtained by the Swiss team (with HARPS) and the California Carnegie team (at Keck). Given the rapid improvement in the radial velocity technique over the past decade, however, it’s not at all unreasonable to expect instrumental precisions of 10 cm/s fairly soon. Many console users were able to extract the four largest-amplitude solar-system planets — Jupiter, Saturn, Earth, and Venus — out of the challenge004 dataset, suggesting that it’s only a matter of time before instrumental precisions and observational baselines arrive at the threshold where truly habitable, Earth-mass planets can be detected from the ground using the radial velocity technique.

A potential show-stopper for this rosy predictive picture is the astrophysical radial velocity noise produced by the stars themselves. If you want to detect a planet with the mass and period of Earth (which induces a radial velocity half-amplitude of only 9 cm/sec) then you need to be assured that the star is quiet enough for the low-amplitude terrestrial planet signal to be detectable. It’s therefore natural to ask the question: what does the Sun’s reflex velocity look like?

The GOLF experiment on the SOHO satellite provides one set of measurements. A massive time-series of radial velocity observations (from 1996 through 2004) has been published, and is now publicly available. The data set contains over seven million radial velocities taken at a 20-second cadence. The main goal in obtaining this data was to study the Sun’s spectrum of p and g-type modes, which show strongest oscillations at periods of a few minutes.

Three alternate calibrations of the GOLF dataset are posted on the project website. Two of these have clearly been processed to filter out low-frequency, long-period radial velocity variations. It’s interesting, however, to look at what the one unfiltered dataset suggests is happening over timescales of a year or more. I sampled the unfiltered data at a cadence of one velocity measurement per several days, and then loaded the resulting time-series into freshly downloaded version of the systemic console:

radial velocities from the GOLF experiment

According to the above time series, the Sun is a pretty noisy star. I scoured the papers on the GOLF site, and could not find any discussion regarding how much of the variation shown above is believed to come from instrumental effects and how much is belieived to be actually intrinsic to the Sun. The fact that both the scatter and the amplitude of the variations seem to be increasing during the run of the data tend to indicate that intrumental effects relating to the aging of the detector play an important role. If anyone has more specific information on this issue (or if anyone is aware of a preferred calibration) please post to the comments section of the post.

What happens to the detectability of planets that are placed in the GOLF time series? To date, the most precise RV detection of an extrasolar planetary system is the Swiss Team’s discovery of the three Neptune-mass planets orbiting HD 69830. As a control experiment, we relabeled the published HD 69830 dataset at systemic003, and placed it on the backend for Systemic users to evaluate. As expected, nearly all of the twelve submitted fits recovered the published configuration, with chi-square reaching down to about 1.20.

For the systemic004 system, we took the published HD 69830 3-planet orbital model and integrated it forward in time to make a synthetic radial velocity curve. We then perturbed this curve with noise values drawn from the unfiltered GOLF dataset (We averaged the velocities into 15-minute blocks to simulate rapid-fire multiple observations that average over high-frequency p-modes). As of Sunday night, there have been 21 fits uploaded for systemic004 [thanks, y’all, -ed.]. None of them manage a chi-square below 2.5, and aside from the innermost planet, none of them make a convincing case for the presence of the planets that were placed in the dataset. Log in to the backend, call up systemic004 from the “real stars” catalog, and you’ll see what I mean.

The conclusion, then, is that if the GOLF data-set gives a realistic determination of the intrinsic radial velocity variation of the Sun, then the Sun is a far noisier star than HD 69830 (and other similarly old, early K-dwarfs). Indeed, you would even be hard-pressed to believe the presence of Jupiter in the GOLF time-series, unless you’ve got the luxury of waiting for at least several Jovian orbital periods.

dialing 411

November 16, 2006

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Update: The original version of this post tagged systemic003 as the target system and systemic004 as the control system. It’s actually the reverse. In any case, we’re interested in getting multiple fits to both systems. -GL

No post today, just a request:

We’ve been doing an analysis of the detectability of low-mass planets around certain types of stars. In the course of this work, we’ve generated a radial velocity data set, systemic004, which may (or may not) harbor a planetary system. I’d like to ask everyone to (1) download the latest version of the console, and (2) submit your fits to the systemic004 system to the backend. We’ve also included a control system, systemic003, which may look familiar. It would be very useful to have your fits to that system as well.

Thanks in advance! Once we get a batch of fits, I’ll write a post that explains the motivation underlying this request…

year 2.0

November 15, 2006

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As of tomorrow, oklo.org will have been on the air for one year. We’re pleased with the response that we’ve gotten thus far, and we’re looking forward to rapid progress during year two. A big thank-you is definitely in order to everyone who’s either worked on the site or made regular visits or participated in the ongoing collaborative research!

In recent weeks, the user base on the systemic back-end has grown substantially, and we’ve been pushing the limits of what our ISP is geared to provide. Bluehost provides a very cost-effective package for hosting weblogs and running small-scale sites, but it’s become abundantly clear that one can’t expect to run a web 2.0 startup for $6.95 per month. At that level of expenditure, we’ve been limited to the use of 20% of one processor with a maximum job length of 60 seconds. Stefano has stretched our ration with clever use of cron command, but nevertheless,

has become a refrain tiresomely familiar to backend users, and our attempts over the past week to shift the backend to alternate stop-gap servers have been thwarted by various software incompatabilities.

I’m thus very happy to report that an order has been placed for a dedicated server that will obliterate the current problems. It will be located in downtown Santa Cruz on a high-speed T3 line. We should have everything up and running on it within 2 weeks. It’s spec’d to run the full systemic simulation, the new connection is ready to handle a hoped-for shout-out from boing-boing or slashdot, and the joint package should deliver a much more satisfying end-user experience.

In the meantime, however, keep sending in those fits. Neither sleet, nor snow, nor server overloads shall… We’re very eager to build up a solid distribution of fits for Systemic Junior.

Viewed from afar (Challenge 004)

November 13, 2006

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The fourth systemic challenge turned out to be somewhat less challenging than the first three. Quite a few entrants figured out that the data-set corresponds to our own solar system. Among a large number of excellent models, Mark Kilner turned in the fit with the lowest chi-square: 1.0401. In addition to Jupiter, Saturn, Earth, and Venus, he topped off his system with a spurious Mercury-mass planet in a 5.62 day orbit, which allowed him to take the prize. Nice one, Mark!

Eugenio created the challenge 004 synthetic data set after a conversation in which we decided that it’ll soon be feasible to push the precision of the radial velocity method down to an instrumental error of 0.1 m/s. Even more optimistically, we assumed that the Sun, viewed from afar, exhibits negligible radial velocity noise (more on that soon).

Our Solar System, expressed in the Jacobi orbital elements used by the console, is given by:

The true three-dimensional model that Eugenio actually integrated to produce the synthetic data set also includes the correct values for the planetary inclinations and nodes. Because of the sin(i) degeneracy for Keplerian orbits, the current version of the downloadable systemic console does not include the inclinations and nodes as fitting parameters.

The synthetic data set was created with the KeckTAC program, which mimics realistic observing strategies. In an all-out effort on a particular star, one would combine repeated individual observations to get a composite observation that averages over the effect of short-period oscillations (p-modes) of the star itself. This is the strategy that is being currently used by the Swiss team in their campaigns on stars such as HD 69830 and HD 160691. In the challenge004 dataset, there are 1171 radial velocity measurements spread out over 24 years.

Eugenio describes the procedure he used to fit the data:

The periodogram (and the data) shows Jupiter clearly. Saturn appears as a trend, but the periodogram of the residuals after fitting Jupiter gives a good guess for Saturn’s period. After removing Saturn, Earth pops out in the residuals periodogram. I did not find it easy to fit Jupiter, Saturn, and Earth, but after succeeding, Venus very clearly appears in the residuals. I kept on fooling around with the 4-planet fit to see if there was any chance of finding Mars even though the RMS was telling me that 4 planets was the best that I would likely do. I was hoping that N would be large enough to let me get Mars, but I was not able to see a (significant) signal in the residuals periodogram. If anything, Mercury seemed to be more easily detectable. However, after fooling around with the eccentricities of Saturn, Earth, and Venus, the (weak) signal for Mercury disappeared.

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With the contests wrapped up, we’re now in the business of getting the fits completed for the Systemic Jr. data set. Eugenio recently incorporated an F-test module into the console, which can be used to determine whether the addition of a planet is warranted. We’ll have a post up shortly that explains in detail how this works. In the meantime, see the discussion on the backend, or download a new console and give its new modules a whirl.

Inky Black Dot

November 9, 2006

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It was a brilliantly clear November afternoon today, and the fact that the Sun’s rays were diminished by 0.0026% mattered not one jot. Mercury was in transit.

I was fortunate to get a glimpse of the event through a telescope. It was about an hour before third contact, and Mercury was clearly visible as a tiny, perfectly round, perfectly black dot set against the pale yellow immensity of the solar disk. There was something about the simplicity of the situation that was quite striking.

Transits, with their odd cadences, link the timescales of human activity to the flow of astronomical time. The June 2004 transit of Venus, for example, took place on the day before the final exam for my introductory Astronomy class. In the weeks leading up to the exam, I would point to Venus shining in the early evening sky, and urge the students to study, “See how it’s getting noticeably lower in the Sky every night at sunset? You can use the angular distance between Venus and the Sun as a countdown clock to the final!” (At which point they’d roll their eyes.)

The next Venus transit is in 2012. The one prior to 2004 took place in 1882, when William Harkness wrote,

We are now on the eve of the second transit of a pair, after which there will be no other till the twenty-first century of our era has dawned upon the Earth, and the June flowers are blooming in 2004. When the last transit season occurred the intellectual world was awakening from the slumber of ages, and that wondrous scientific activity which has led to our present advanced knowledge was just beginning. What will be the state of science when the next transit season arrives, God only knows. Not even our children’s children will live to take part in the astronomy of that day. As for ourselves, we have to make do with the present.

Accurate weather predictions are good for no more than a few days, but transit predictions can be made a long time in advance. For example (according to the Wikipedia) simultaneuous transits of Mercury and Venus will occur in the years 69163 and 224508.

Motions in the inner solar system are nevertheless chaotic, though, with a Lyapunov timescale of order several million years. Our lack of absolutely precise knowledge regarding the positions of the planets at the present moment gradually exponentiates into much larger uncertainties. As a result, we can predict transits millions of years into the future, but we have no ability to predict when the transits of hundreds of millions of years from now will occur.

In fact, there’s even a (thankfully small) chance that the solar system will become dynamically unstable before the Sun swells into a red giant. This afternoon, Mercury seemed utterly insignificant and completely remote when pitched against the solar disk. In the final hours before a collision with the Earth, however, it would present an altogether different sort of impression.

Apsidal

November 7, 2006

In 1999, Upsilon Andromedae burst onto the international scene with the first known multiple-planet system orbiting a sunlike star. Eight years later, we know of twenty-odd additional multiple-planet systems, but Ups And remains a marquee draw. No other system evokes quite its exotic panache. No other extrasolar planets have garnered names that have stuck.

High in the cold and toxic atmosphere of Fourpiter, Upsilon Andromedae shines with a brilliance more dazzling than the Sun. Twopiter is occasionally visible as a small disk which, near conjunction, subtends about one-tenth the size of the full Moon in Earth’s sky. Dinky, which lies about four times closer to the star than Mercury’s distance to our Sun is lost in the glare.

To date, Upsilon Andromedae has accumulated a total of 432 published radial velocities from four different telescopes. The full aggregate of data is available on the downloadable systemic console as upsand_4datasets_B06L. The velocities span nearly two decades, during which the inner planet, “Dinky”, has executed well over 1000 orbits.

In earlier versions of the console, use of the zoom slider on an extensive data set would reveal a badly undersampled radial velocity curve at high magnification. Eugenio’s latest console release has addressed this problem, however, and the radial velocity model curve now plots smoothly even with the zoom slider pulled all the way to the right.

It’s interesting to look at the best radial velocity fit to all four data sets. The planets are very well separated in frequency space, and so it’s a straightforward exercise to converge on the standard 3-planet fit. Upsilon Andromedae itself is a little too hot (6200K) to be an ideal radial velocity target star, and so the chi-square for the best fit to the system is above three, with a likely stellar jitter of a bit more than 14 meters per second. If Ups And were a slightly cooler, slightly older star, we’d potentially be able to get a much more precise snapshot of the planet-planet interactions. (In that Department, however, there’s always 55 Cancri.)

The best fit shows that the apsidal lines of the two outer planets are currently separated by 30 degrees, and are executing very wide librations about alignment. This configuration continues to support the formation theory advanced two years ago by Eric Ford and his collaborators. They hypothesize that Ups And originally had four giant planets instead of the three that we detect now. The outer two (Fourpiter and, uh, “Outtathere”) suffered a close encounter followed by an ejection of Outtathere. Fourpiter, being the heavier body, was left with an eccentric orbit. Now, 2.5 billion years later, the memory of this disaster is retained as the system returns every ~8,000 years to the eccentricity configuration that existed just after the disaster.

Systemic Jr.

November 5, 2006

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The activity this week has all been under the hood, and as a result, the systemic front-end has languished without news. Apologies for that! A slew of updates are on the spike.

Stefano is now officially on the roster at Oklo HQ, and we’re very happy to have him here. The atmosphere is caffeine-fueled startup. He’s already implemented numerous updates and improvements to the systemic backend which, when coupled with Eugenio’s progress on the console, put our web 2.0 story into high gear.

There were a number of times last week when the oklo.org site was temporarily unavailable. Our ISP restricts us to no more than 20% of a full processor load, and exceeding this causes the site to shut down for 5 minutes. We’re now in the process of temporarily mirroring the backend on a machine at Lick Observatory, and quite soon we’ll have a dedicated server up and running.

The systemic Junior datasets have now been added to the downloadable systemic console. Eugenio writes (see the backend discussion forum for the full description):

Systemic Jr. is now included in systemic.zip. You will see two drop down boxes in the upper right region of the main console. One is used to choose a real star system, while the other one is used to pick a Systemic Jr. system. Note that while both boxes are enabled, only one data set is actually selected. In the systemic directory, you will see two new items: “sysjrSystems.txt” and the directory “sysjrdatafiles.” These hold the information needed for Systemic Jr.

As soon as the Lick Observatory server is online, the backend will be able to accept fits to the Systemic Jr. data sets. In the meantime, please save your fits on your local machine. Some of the Systemic Jr. systems may seem familiar. It’s best however, if all of the datasets are approached without a pre-conceived notion of what might be generating them. Once the Systemic Jr. data sets have been fitted, we’ll be able to do a very interesting analysis which will give us some much-wanted information about the nature of the galactic planetary census.

Threaded console available!

October 30, 2006

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This weekend, Eugenio posted an updated version of the downloadable systemic console. The Java code in this version is fully multithreaded, which means that we’re finally able to provide the much-needed and much-requested “stop” button.

For previous console releases, clicking multi-parameter minimization — “polish” — with integration enabled would often cause the console to effectively freeze as the computer worked it’s way through an exceedingly long bout of computation. With the new version, progress is indicated both by a graphical redrawing of the fit, and by a running tally of the number of Levenberg-Marquardt iterations that have been completed. If things appear to be progressing too slowly, it’s now possible to abort to the latest model state by pressing the stop button.

A “back” button will be activated shortly, which will allow you to step backward through your work to revisit earlier model configurations in the session. These features should significantly improve the overall usability of the console.

Another area where progress has been rapid is in the stability checker. Eugenio has put a lot of detailed information on this new functionality on the general discussion section of the backend. In short, the stability checker can now be used as a full fledged integrator which can write time series data to user-specified files. In a post that will go up shortly, we’ll look at how the stability checker can be used to answer some interesting dynamical questions.

Systemic Jr. is also just about ready to go. Assuming that there’s no unforseen snags, we’re looking to launch it on Nov. 1 (next week). In the meantime, download a fresh console, and give the new features a spin.

zoom

October 28, 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 potential

October 26, 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:

challenge 4

October 25, 2006

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Eugenio has put the fourth (and final) systemic challenge system on the downloadable systemic console. This dataset is somewhat easier to decipher than the first and second challenges, which were rather esoteric in their planetary configurations. We hope that you’ll find that this one’s a little more down to Earth. I’d like to have your entries in by Oct 31, 23:59 UT. As with our previous three contests, Sky and Telescope is awarding a Star Atlas to the person who achieves the best model of the system.

For this system, it’s likely possible to drive the chi-square arbitrarily close to unity by successively adding spurious, very low-mass planets that act to soak up random noise in the data. We’re currently working on incorporating some standard statistical test utilities into the console which will make it easier to determine whether adding an extra planet is truly necessary. (This will be the topic of an upcoming post, and see the comment thread on Sunday’s post.) For this contest, however, if there are multiple submissions with reduced chi-square near unity, then the prize will be awarded to the fit that also gets the total number of planets in the underlying model correct.

If you haven’t downloaded the console recently, we’re encouraging you to grab a fresh copy. A number of improvements have been added, and there are also a number of additional radial velocity data sets that have been added in recent weeks. Eugenio has been posting a running commentary on the backend describing the console improvements. We’re also putting the final touches on the Systemic Jr. datasets, which we’re hoping to release at the end of next week.

As a result of some articles in the press and on the Internet, we’ve been continuing to see a large increase in the oklo user base. If you’re visiting the site for the first time, you’ll find information about the project and about our goals on the links to the right. Welcome aboard!

1:2:4

October 22, 2006

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The third Systemic Challenge closed to entries on Friday, and I’ve gone through and evaluated the submitted fits. The results were very encouraging. Eight out of twenty-five submissions corresponded to both the correct orbital configuration and the correct number of planets in the underlying dynamical model.

For challenge 003, we looked to our own solar system for inspiration, and tapped the four Gallilean satellites of Jupiter. Eugenio writes:

The system is a scaled-up version of Jupiter and the four Galilean satellites. To generate the model, I first set the central mass to 1 solar mass. The (astrocentric) period of Callisto was set to 365.25 days, and I required that the mass and (astrocentric) period ratios in the system would remain the same. Here’s the resulting model (using Jacobi elements, with i~88 deg):

The Challenge 003 System
Parameter “Io” “Europa” “Ganymede” “Callisto”
Period (days) 38.77079 77.77920 156.65300 365.42094
Mass (Jupiters) 0.04926 0.02646 0.08175 0.05936
Mean Anomaly (deg) 99.453 50.772 285.591 47.538
eccentricity 0.003989 0.009792 0.001935 0.007547
omega (deg) 31.229 205.427 303.460 359.879

Among the eight entries that got both the total number of planets and their periods correct, there was a fair amount of variation among fits that had nearly equivalent values for the chi-square statistic. Chuck Smith (among others) turned in a configuration that bears a very strong resemblance to the actual input system. The four planets in his fit all have nearly circular orbits:

and the resulting radial velocity curve does a very good job of running through the data, with a chi-square value for the integrated fit equal to 1.1005:

A number of other users turned in very similar configurations.

Because of random measurement errors in the data, the true underlying planetary configuration will not necessarily provide the best fit to a given set of radial velocity observations. Often, a better fit can be found for a configuration that is different from the system that generated the data. Steve Undy, for example, achieved a slightly lower chi-square value for his fit by giving a very significant eccentricity to his “Europa”:

The winner of the contest, however, was Eric Diaz, who submitted a 6-planet fit that achieves an integrated chi-square value of 1.04. In addition to the four planets that are actually present in the model, Eric added small planets with periods of 1.06 days and 18.11 days. These objects soaked up some of the residual noise in the fit, allowing for a lower chi-square value, and a copy of the Sky and Telescope star atlas. Nice job Eric!

The contest raises some interesting issues. First, at what point should one stop adding planets to a fit? The chi-square statistic penalizes the inclusion of additional free parameters in a fit, but it’s clear that chi-square can nearly always be lowered by adding additional small bodies to the fit. Second, its very encouraging to see that subtle, but substantially non-interacting systems can be pulled out of radial velocity data sets. In this system, the masses of the planets are small enough so that their dynamical interactions with eachother are not significant over the time-frame that the system is observed. This is in stark contrast to systems such as GJ 876 and 55 Cancri where it is vital to take interactions into account (by fitting with the integrate button clicked on). Finally, I think that we’ll soon see examples of the 1:2:4 Laplace resonance as competitive fits within the existing catalog of radial velocity data sets on the systemic backend.

Gamma Cephei

October 19, 2006

Image Source.

Guillermo Torres of the CfA recently posted an interesting article on astro-ph in which he takes a detailed look at the planet-bearing binary star system Gamma Cephei.

Gamma Cephei has a long history in the planet-hunting community. In 1988, Campbell, Walker and Yang published radial velocity measurements which show that Gamma Cephei harbors a dim stellar-mass companion with a period of decades. More provocatively, they also noted that the star’s radial velocity curve shows a periodicity consistent with the presence of a Jupiter-mass object in a ~2.5 year orbit around the primary star. In a 1992 paper, however, they adopted a cautious interpretation of their dataset, and argued that the observed variations were likely due to line-profile distortions caused by spots on the stellar surface. From their abstract:

In 1988 Gamma Cep was reported as a single-line, long-period spectroscopic binary with short-term periodic (P = 2.7 yr) residuals which might be caused by a Jupiter-mass companion. Eleven years of data now give a 2.52 yr (K = 27 m/s) period and an indeterminate spectroscopic binary period of not less than 30 yr. While binary motion induced by a Jupiter-mass companion could still explain the periodic residuals, Gamma Cep is almost certainly a velocity variable yellow giant because both the spetrum and (R – I) color indices are typical of luminosity class III. T eff and the trigonometric parallax give 5.8 solar radii independently.

In October 1995, 51 Peg b was announced, and exoplanet research was off to the races. The Walker team, with their futuristic RV surveys had seemingly come close to success, but had not managed to snag the cigar.

In the Fall of 2002, however, the planetary interpretation for the Gamma Cephei radial velocity variations was revived by Hatzes et al., who used McDonald Observatory to extend the data set. They showed that the 2.5 year signal has stayed coherent over two decades, thus effectively ruling out starspots or other stellar activity as the culprit. The planet clearly exists.

Aside from providing a pyrrhic victory for the Walker team, the Gamma Cephei planet is a remarkable discovery in its own right. Its presence showed that gas giants can form in relatively long-period orbits around binary stars of moderate period. In their discovery paper, Hatzes et al. assumed that the binary companion orbits with a period of 57 years, but other estimates varied widely. Walker et al. (1992), for example, adopted 29.9 years, whereas Griffin (2002) use 66 years. The mystery is strengthened by the fact that to date, the companion star has never been seen directly.

The details of the orbit of the binary star are of considerable interest. For configurations where the periastron approach is relatively close, simulations show that the star-planet-star configuration can easily be dynamically unstable.

In his new article, Torres methodically collects all of the available information on the star, and shows that the binary companion to Gamma Cephei has a 66.8 +/- 1.4 year period, an eccentricity of e=0.4085 +/- 0.0065, and a mass of 0.362 +/- 0.022 solar masses. The orbital separation thus lies at the high end of the previous estimates, and renders the stability situation for the system considerably less problematic.

We’re stoked about the Torres paper because it provides references to some truly ancient radial velocities, dating all the way back to a compendium published by Frost and Adams in 1903:

who report 3 measurements made at the University of Chicago’s Yerkes Observatory:

Eugenio has tracked down the various references in the Torres paper, and has recently added all of the available old-school RV’s for Gamma Cephei to the downloadable console. You can access the full dataset by clicking on “GammaCephei_old”:

It’s straightforward to manually adjust the offset sliders to put the radial velocities on a rough baseline. You can then build a rough binary star fit with the sliders, followed by repeated clicking on the Levenberg-Marquardt polish button, with the five orbital elements and the five velocity offsets as free parameters. This gives an Msin(i)=386 Jupiter masses, a period of 24,420 days, and an eccentricity, e=0.4112. Try it! The values that you’ll derive are in excellent agreement with the Torres solution:

With the binary fitted out, try zooming in on the more recent data from the past 10-20 years. You’ll see that the modulation of the radial velocity curve arising from the planet is faintly visible even to the eye. It’s interesting to go in and find the best-fit planetary model…

Follow Ups And other items…

October 17, 2006

Image Source.

It’s very gratifying to see an increasing number of people logging in to the Systemic Backend, and downloading the console. We’ve also been getting a lot of good feedback from users, which we’ll be incorporating into updated versions of the software.

Several people have noted that the backend is currently assigning chi-square values of zero to uploaded fits! We’re highly aware of this problem, and it likely stems from the fact that we may be exceeding our CPU allocation at our ISP. The back-end code integrates all submitted fits to verify the chi-square statistic for purposes of ranking. For submitted systems with long time baselines and short-period planets, these calculations can wind up being fairly expensive. We’ll let you know as soon as this issue gets resolved. In the meantime, it’s fine to submit fits, but if you get a good one, please save a copy in your own fits directory for the time being.

We’ve been getting a lot of entries for the Challenge 003 system. At the end of this week, I’ll tally up the results, so if you’ve got a fit to submit, go ahead and send ‘er in (using the e-mail address listed on the web-page given in the print version of the October Sky and Telescope). It’s fine to submit multiple fits — I’ll use your best one to determine the final ranking. The challenge 003 system represents an interesting dynamical configuration of a type not yet observed for planets in the wild, and so it’ll be very interesting to see what people pull out. Look for Challenge 004 to appear this weekend on the downloadable console, and shortly thereafter, warm up those processors for the advent of the 100 star Systemic Jr. release.

Yesterday’s post is generating an interesting and vigorous discussion thread. Jonathan Langton and I were hopeful yesterday that his benchmark Cassini-State 1 simulation might show an appropriately asymmetric light curve when viewed from lines of sight inclined to the planetary equator (as is the case for the Ups And observations). Frustratingly, however, when the model light curves are actually computed, they wind up drearily sinusoidal, and the phase offset is independant of viewing inclination:

We’re holding out hope, though, for Cassini-State 2. In that case, there are two angles to vary (the orientation of the pole in the orbital plane, and the viewing inclination) and so it may well be possible to dredge up a good fit to the data. After-the-fact parameter tweaking, however, is highly unsatisfactory! I’m looking very much forward to seeing more data sets like Ups And’s. In particular, HD 189733, should give a very nice full-phase curve, and further down the line HD 80606 should be even more interesting.