systemic

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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.

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…

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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:

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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!

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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):

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.

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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…

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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.

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Last week, Joseph Harrington and his collaborators published a paper in Science that announced the results of a very interesting set of observations of Upsilon Andromedae with the Spitzer Space Telescope.

As console users know, Upsilon Andromedae is accompanied by three Jovian planets. The innermost body (officially known as “Dinky“) has at least 70% of Jupiter’s mass and orbits with a period of 4.6 days. Observers have checked to see whether Dinky passes directly in front of the parent star. They found that transits don’t occur, and so the orbital geometry likely looks something like this (as seen from Earth, with the planet grossly not to scale):

Harrington and collaborators made careful measurements of the infrared brightness of the star in the 24-micron band at five known phases during the planetary orbit. These phases are marked with small yellow circles in the above plot.

When the data were reduced, it was found that the brightness of the star was varying in phase with the orbital period of the planet. The brightness is lower when Dinky is in front of the star (near “inferior conjunction”) and higher when more of the planet’s illuminated surface is in view.

The difference in brightness during the course of the orbit is consistent with a temperature difference of order 1000 K between the illuminated dayside and the dark night side. The planet should be spin-synchronized, so that one side always faces the star and the other face is always pointed away. Harrington et al. showed that the data could be understood if it is assumed that the planet transfers very little heat to the night-side, thus allowing the large temperature difference to be maintained. In fact, they were able to get a good model of the brightness variations by assuming that the night-side was not radiating at all. Such a model curve looks like this:

Intuitively, this result seems to make perfect sense. You’d expect a spin-synchronized planet to be hottest at the subsolar point, and coldest at the antistellar point, and this picture is fully consistent with the five observed fluxes. The results are surprising, however, when we take into account the fact that there should be hellacious winds on the planetary surface which should disgorge heat onto the night side.

UCSC graduate student Jonathan Langton has been studying the surface flows on hot Jupiters using a hydrodynamic technique known as the shallow water approximation. A often-seen feature of his models is that the hottest point on the surface of a synchronously rotating planet is well eastward from the substellar point. (A similar state of affairs is predicted by Cooper and Showman, who use a full 3D GCM-type model.)

Similarly, the coldest spot on the night side, is also displaced eastward from the anti-stellar point:

These models predict a smaller day-night temperature difference than the no-redistribution model that Harrington et al. fitted to the data. A smaller day-night temperature difference can indeed be accomodated by the observations, but the predicted phase shift seems highly inconsistent at first glance. Eastward-displaced hot and cold spots give a (edge-on inclined) lightcurve that is clearly out of phase:

Taken at face value, the observations thus seem to suggest that the flows on the planet are very effective at radiating heat. That is, the upper layers that we can actually observe seem to have a short radiative time constant. In a set of upcoming posts, we’ll have a closer look at the interpretation of this very interesting new result.

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SETI and the idea of alien life are the stuff of endlessly fascinating speculation. I remember wild late-night conversations with my freshman dorm mates when we should have been writing lab reports and studying for chemistry exams. To date, however, the SETI hasn’t turned up anything, and the Fermi Paradox seems as perplexing as ever. Proponents of the conventional SETI approach argue that this is because we’ve barely scratched the surface in terms of the number of stars that we’ve observed. Build a bigger telescope, they argue, scan more stars, and success will come.

If I look at my own behavior, the trend has been toward increasingly frequent correspondence with more and more people. The cell phone rings many times a day. I send a lot of e-mails via wireless internet. I look on flickr to see if my photos have accumulated views or comments. My life revolves around connectivity. I rarely send letters through regular post, and I have little interest in conversations with a response time of 8.78 years. I’m not inclined to beam coded messages to the sky, and I don’t shine high-power collimated lasers at nearby stars. My behavior is similar in aggregate to many, many others here on Earth.

It seems reasonable, then, that the most promising strategy for a succesful SETI is to look for behaviors that resemble our own. I think it’s much more likely to detect another civilization through their signal “leakage” rather than through reception of a directed message. If I knew that it was going to take at least 8.78, and in all likelihood millions of years for my photos to accumulate views, I’d soon start neglecting to post them.

When I was at the CfA last week, I had an interesting conversation with Avi Loeb, who pointed out that at present, the largest sources of artificial terrestrial radio emission are military radars, FM radio broadcasts, and television broadcasts, all of which emit their power in the frequency range between about 40 and 800 Mhz. SETI searches, on the other hand, have focused in the frequency range above 1 Ghz.

Loeb is involved in the Mileura Wide Field Array (MWA), which is a low-frequency radio telescope designed to study highly redshifted 21 centimeter emission from hydrogen. By mapping the spatial distribution and redshift distribution of 21 centimeter emission, the Mileura project will be able to make a 3-dimensional map of the distribution of atomic hydrogen in the early universe.

The MWA will provide an enormous increase in sensitivity at exactly the frequencies that we here on Earth broadcast. Loeb recently received a grant from the FQXi foundation to run a SETI-search on data obtained during the course of MWA survey observations. The cosmic signals received will be combed for telltale artificial emissions from nearby stars. The array will be sensitive enough to detect Earth-like leakage from more than 1000 of the nearest stars, a list that includes oklo.org Southern Hemisphere favorites such as Alpha Centauri B, Beta Hyi, GJ 780, and Tau Ceti.

Loeb informs me that he’s posted an overview paper on astro-ph. Look for it on Sunday night, 5PM PST.

While we’re on the topic, I recently participated in a panel discussion on SETI that closed up the AIAA Space 2006 meeting in San Jose. I argued that the resolution of the Fermi Paradox lies in the fact that we’re inward bound. My understanding is that the video of the discussion will go up on the web at some point, but for the moment, here’s a .pdf (4MB) file with the transparencies that I showed in my 10 minute summary.

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UCSC just put out a press release on the Systemic Project, so if you’re a first-time visitor to oklo.org, welcome aboard. You’ll find information about the project in the list of page links just to the right.

Last week, during my visit to Harvard CfA, I talked to Eric Ford, who has been exploring the idea of searching for trojan companions to extrasolar planets. He pointed out that the discovery of a body in a trojan configuration with a known extrasolar planet would provide an important test of theories of hot Jupiter formation. Here’s a link to his paper.

One way to make a hot Jupiter is to form the planet through the standard core-accretion method at a large radius in the protostellar disk. In this scenario, a newborn gas giant planet starts as a core of rock and ice, which grows to a size of 5-10 Earth masses and begins to rapidly accrete gas from the surrounding nebula. As the planet increases in size, it begins to clear a annular gap in the parent disk. Hydrodynamical simulations (such as the ones reported here, and reproduced in the illustration below) show that L4 and L5, the so-called trojan points located 60 degrees ahead and 60 degrees behind the forming planet, are the last regions of the gap to be cleared out.

It’s possible that co-orbital planets can form from the slow-to-clear material at L4 and L5. When the gas is gone, these objects will remain in stable trojan orbits.

If a pair of planets is caught in a trojan configuration, then they will migrate inward together through the disk, and the migration process will not cause them to become dynamically unstable. Eric points out that the observed presence of a trojan companion to a hot Jupiter would thus be evidence that the hot Jupiter arrived at its short-period orbit via migration. Other possibilities for forming hot Jupiters, such as dynamical instability followed by orbital circularization, do not allow for trojan companions.

A trojan pair of planets presents an interesting conundrum for planet hunters. Normally, a single planet on a circular orbit goes through its radial velocity zero point at the moment when the planet lies on the plane containing the line of sight from the Earth to the parent star. If we have a trojan, however, a planetary transit will be offset from the radial velocity zero point, which is associated with the orbit of a “ghost body” that combines the gravitational effect of the primary planet and its trojan companion. Using the console, try obtaining a two-planet perfect trojan (60 degree separation) fit to a well-known hot Jupiter data set such as that for HD 187123. You’ll find that it’s perfectly possible. The resulting configurations have utterly indistinguishable radial velocity signatures.

Trojans can be detected, however, if the primary planet happens to transit. The presence of the trojan companion can be inferred by measuring the lag between the center of the transit and the zero crossing of the radial velocity curve. For planets with an equal mass ratio, this would amount to a full 1/12 of an orbital period (6 hours for a 3-day orbit).

Which brings up an interesting project for transitsearch.org observers. Most of the known hot Jupiters have been checked photometrically for transits. These transit searches, however, are performed in the time window surrounding the radial velocity zero point. In the (admittedly unlikely) case that some of these objects are trojan pairs with near-equal mass ratios, the transits would have been missed using this approach. To fully rule out transits, one should cover the full 1/6th of an orbital period surrounding the nominal predicted transit time…