dialing 411

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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

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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)

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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

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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

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.

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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!

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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

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

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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

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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!