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If you’re a new visitor to the site, welcome aboard! Yesterday’s post talks about the systemic collaboration, and gives an overview of how you can participate.

The interpretation of radial velocity data sets is confounded by the existence of many different model planetary systems that all do a good job of fitting the data from a given star. If you really want to know whether a particular fit is the correct interpretation of the system, then you need to wait for (or make) more observations to see if your fit’s predicted radial velocity curve is confirmed.

For a real planetary system orbiting a real star, it can take years for enough confirming observations to be made, and so it’s useful to have as many criteria as possible for evaluating whether a particular fit is a contender. Orbital stability provides one such criterion.

On the backend, there are many orbital models that have been submitted that give excellent fits to the given data sets. For example, the four configurations shown in the picture just below are all acceptable models for the 14 Her system.

One immediately notices that these orbital configurations look “crowded”. The orbits make close approaches and sometimes even cross. If we let these model systems run forward in time, then we find that the mutual gravitational pulls between the planets lead to catastrophe within a few decades or less. Instead of behaving in an orderly fashion, the orbits execute motions like this:

which lead inevitably to collisions and ejections. While it’s theoretically possible that we happen to be observing a particular system just before it experiences disaster, Occams razor strongly suggests that wildly unstable fits are likely spurious. We can safely exclude any configuration that lasts for only a tiny fraction of the stellar ages (which are generally in the 2-10 billion year range).

Participants in the systemic collaboration can evaluate the stability of their models by using the “check long-term stability” function on the console. Stefano has also recently implemented a robot that crawls through the systems residing in the backend database and integrates all of the submitted fits. So far, it has sorted out which systems are unstable on timescales of less than a century, and as time goes on, it’s pushing the integration times to longer horizons. It turns out that a 100-year integration can catch a majority of the systems that eventually go unstable. After that, we expect roughly equal numbers of systems to be lost in each factor-of-ten increase of integration time.

Although we don’t expect to see orbital instabilities play out on our watch, it’s nevertheless likely that planet-planet interactions and their associated instabilities have played an important past role in sculpting the systems that we now observe. For example, Eric Ford and his collaborators have published a highly plausible theory for the formation of the Upsilon Andromedae planetary system that involves a dramatic instability. In their scenario, the system starts out with four planets, and eventually ejects one of them. The outer two survivors are left stunned and reeling, and the dynamical imprint of the disaster survives to the present day. They’ve made an engaging animation (available here) that shows the action blow-by-blow.

This brings up a relevant question. If orbital instability exists among the extrasolar planets, might our own solar system eventually go unstable? Is it possible that Earth will find itself getting dramatically tossed around the solar system in the manner that was experienced by the unfortunate Upsilon Andromedae E?

The question isn’t new, and the stability of the solar system has been at the forefront of interest for the last 350 years. It was first tackled by Newton, who wanted to understand how the orbits of the Jupiter and Saturn would behave over long periods if their mutual interactions were taken into account. Newton put a lot of effort into the problem, and eventually decided that:

To consider simultaneously all these causes of motion, and to define these motions by exact laws admitting of easy calculation exceeds, if I am not mistaken, the force of any human mind.

Newton’s fame, and the fact that he’d written off the problem as too difficult, was a big motivation for succeeding generations of mathematicians. Pierre Simon de Laplace eventually solved the problem of the motions of Jupiter and Saturn, and fully explained their orbits to the accuracy that could be observed in the late 1700s. In Laplace’s model, the solar system is completely stable, and the inherent predictability of his planetary motions contributed to the concept of a rational determinism, and the idea of a clockwork universe.

During its first three hundred years, the problem of the stability of the solar system was attacked using pen and paper. In the past few decades, however, the advent of computers has provided a powerful new tool. We can now make accurate simulations of the trajectories of the planets through space, and look in detail at the solar system’s possible futures. By the 1980s, when hardware and algorithms had progressed to the point were it was possible to integrate the planets millions of years forward into the future, it was found that the solar system is chaotic in a sense originally envisioned by Poincaré. If the position of a planet, the Earth say, is given a tiny change in the computer, then as millions of years elapse, this slight perturbation grows erratically larger. If Earth is displaced in its orbit by a centimeter, then, after several million years, Earth will likely be located somewhere within 2 centimeters of where it would have been had it been given no push at all. After several million years, the degree of uncertainty doubles again, this time to 4 centimeters.

Worrying about such tiny buildups of uncertainty in the position of Earth on its orbit sounds utterly absurd. Nevertheless, like interest compounding in a forgotten account, the accumulation of uncertainty is guaranteed to eventually become significant. After a hundred million years, which is much less than the 4.5 billion year age of the solar system, the position of Earth in its orbit becomes completely impossible to predict. For times 100 million years in the future, we have no firm knowledge of Earth’s trajectory. We have no idea whether January 1, 100,000,000 AD will occur in the winter or in the summer, or even whether Earth will be orbiting the Sun at all.

Poincaré’s great insight was that the realistic physical description of non-trivial systems can involve what we now call chaotic behavior. The weather is an excellent example. Overnight weather forecasts are generally quite accurate. Three-day forecasts are certainly of some utility. Two-week forecasts, on the other hand, are essentially worthless. Although we have a very clear understanding of the laws of physics that govern the behavior of Earth’s atmosphere, we can’t sample global weather conditions with enough precision to make forecasts accurate beyond a few days. If you let out a deep sigh at the complexity of it all, then the air current that you exhale will spur subtle deviations in the flow of air and moisture of the Earth’s surface that become increasingly magnified over time. The aggravated swirl of air from a slap at a mosquito can career into divergences that visit a hurricane on Miami rather than spinning it out into oblivion over the North Atlantic. Although we can’t accurately predict how the pattern of weather fronts and daily high temperatures will look on the 10:00 p.m. News two weeks from today, we do have some idea of what the weather will be. If it is in the middle of the summer, Texas will be hot. Duluth, in January, will be cold. The pattern of erratic day-to-day weather is superimposed over solidly predictable seasonal and regional climates.

We can thus ask the question: Are the movements of the planets predictably chaotic in the same sense as the weather? That is, over billions of years, will the planets wander only within circumscribed bounds, or is a more wild chaos, with orbit crossing, ejections, collisions and the like – a real possibility?

The answer will be a statistical statement. To high probability, the planets will remain more or less on their present courses until the Sun becomes a red giant. Exactly how high a probability is not fully clear. Stay tuned…

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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!

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When J. Edgar Hoover was getting on in years, his aides would often tell scheduled visitors to his office that he was unable to meet with them because he was “in conference”. In reality, this meant that Hoover was napping at his desk.

It might seem that the refrain of, “we’re busy working on the systemic back-end” is an equally convenient euphemism for long lapses between posts on the front end. Nevertheless, we have been busy getting the new oklo xserve quad xeon up and running. The whole site has now been replicated and tested, the server is live and on air, and very shortly, we’ll be flipping the switch. Can’t wait, man!

With the vast increase in processing power afforded by the xserve, we’ll be able to provide a much more extensive suite of research tools to oklo visitors. In particular, it’ll be possible to dynamically generate the kinds of correlation diagrams that are currently only available from our estimable continental competition: exoplanet.eu.

It’s always interesting to look through the latest versions of the correlation diagrams to see whether the various trends and hints of trends are holding up. The a-e plot is worth examining, as is the plot that charts the number of planetary discoveries per year over the past decade. As of today, exoplanet.eu lists 192 planets that have been detected with the radial velocity method. Plotting the masses of these planets against their periods on a log-log plot (and running the resulting screenshot through Illustrator) yields the following:

For Keplerian orbits, the relationship between the radial velocity half-amplitude of the parent star and the orbital period of the planet is given by:

If we assume that the mass of the planet is negligible in comparison to the mass of the star and if we further assume edge-on, circular orbits around solar mass stars, then we get the dashed lines in the figure that show detection thresholds for K=3 m/s and K=1 m/s. The three planets orbiting HD 69830 stand out in this diagram as the most striking discoveries of 2006.

To the eye, there are two curious clusters of planets in the diagram. At short periods (P~3d) we have the hot Jupiters. Most of these have masses (times the sine of the unknown inclination) somewhat less than Jupiter. At longer periods (P>100d) we have a second prominent clump of planets. These are the Eccentric Giants, and their masses average out at a significantly higher value (between 2 and 3 times the mass of Jupiter). Part of the difference in mass is due to selection bias, but nevertheless there is a real effect. Like the planet-metallicity connection, this effect is telling us something about either planet formation or planet migration (probably the latter).

Anyone got an idea regarding what’s going on? Let’s get a discussion going in the comment section. Over the past week, I’ve been flooded by depressingly clumsy attempts at comment spam from single-minded robots with mechanical enthusiasms for satellite TV service and online poker, e.g. “Great blog, keep it comming.” It’d be nice to see some signal in the noise…

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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.

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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:

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.

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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…