systemic

Uranus and Neptune have returned to nearly the configuration that they were in at the time of Neptune’s discovery in 1846. Using Solar System Live, it’s easy to see where the planets were located when Galle and d’ Arrest turned the Berlin Observatory’s 9-inch Fraunhofer refractor to the star fields of the ecliptic near right ascension 22 hours:

In 2011, Neptune, with its 165-year period period, will have made one full orbit since its discovery. Uranus, with an 84-year period, will have gone around the Sun almost two times.

Because the planets are fairly close to conjunction, Neptune has recently gone through the phase of its orbit where it exerts its largest perturbation on the motion of Uranus. This was similarly true in the years running up to 1846, and was responsible for LeVerrier’s sky predictions bearing such a stunning proximity to the spot where Neptune was actually discovered by Galle.

LeVerrier (and Adams) were quite fortunate. Without a computer, multi-parameter minimization is hard, and both astronomers cut down on their computational burden by assuming an incorrect distance for Neptune (based on Bode’s “law”). Their solutions were able to compensate for this incorrect assumption by invoking masses for Neptune that were much too large. They carried out remarkable calculations, but nevertheless, luck (in form of the fact that Uranus and Neptune had recently been near conjunction) played a considerable role.

Predictably, as soon as the real orbit of Neptune was determined, the playa haters tried to rush the stage. Benjamin Peirce of Harvard, in the Proceedings of the American Academy of Arts and Sciences 1, 65 (1847) described LeVerrier’s accomplishment as a mere “happy accident”:

I personally think that’s going a bit far. In any case, it’s interesting to compare the two independent predictions with the actual orbit of Neptune. I pulled the LeVerrier and Adams data in the following table from Baum and Sheehan’s book “In Search of Planet Vulcan” :

There’s been no shortage of hard work, and there’s been no shortage of predictions and false alarms, but nevertheless, nobody has managed to discover another solar system planet via analysis of gravitational perturbations. With the extrasolar planets, however, the prospects look a lot better. In particular, the Systemic Backend collaboration can team up with amateur observers to do the trick.

On the Systemic Backend, there are many candidate planets that have had their orbits characterized. As is usually the case with planet predictions, most of the candidates will wind up being spurious, but it’s definitely true that real planets orbiting real stars have been detected by the Backend user base. For example, Gliese 581 c was accurately characterized by the Systemic users several months before it’s announcement by the Swiss (see this post) and the same holds true for 55 Cancri f (see this post).

In the happy circumstance that a candidate planet is part of a system with a known transiting planet, then there’s an increased probability that if the candidate planet exists then it can also be observed in transit. This provides a channel for detection that completely circumvents the need for professional astronomers to carry out confirming radial velocity observations. Amateur observers are currently pushing the envelope down to milli-mag precision. Here’s an out-of-transit observation of the parent star of XO-1b by Bruce Gary:

This photometry is potentially good enough to confirm a Neptune-sized planet in transit across a Solar-type star, which is absolutely amazing.

An initial proof-of-concept observation has recently been carried out. On the systemic backend, the users have been investigating the HD 17156 system, which contains a known transiting planet. User “japf ” (José Fernandes) found that a lower chi-square fit to the published radial velocity data can be obtained if there’s a 6.2 Earth-mass companion on a 1.23 day orbit.

The best-fit eccentricity of the planet would bring it to a hair-raising 2 stellar radii of HD 17156, and if the planet is made of rock or water, it’ll be too small to detect, but nevertheless, it’s at least worth having a look. Jose sent the ephemeris to Bruce Gary, who observed on the opportunity falling on April 20, 04.5 UT.

No transit detected. This in itself was not at all surprising, given the long-shot nature of this particular candidate planet. What’s exciting, though, is that the full pipeline is now in place. There will definitely be strong candidates emerging over the coming months, and I think it’s quite probable that we’ll see a prediction-confirmation that is at least as good a match as was obtained for Neptune in 1846…

A few nights ago, we were looking at the skies through a 10-inch telescope set up in our backyard. The neighbor’s security light made a mockery of any pretense of dark-sky observering, but nevertheless, there’s something remarkable about stepping outside and having your retina absorb light that’s been on the wing for 10 million years.

Using averted vision, I could just make out M81 and M82. They look like this:

On the Astronomy Picture of the Day, one sees a lot more detail:

With the aid of lurid false color, the sense of galactic catastrophe is unmistakable. M82, in particular, emanating distended neon-red lightning bolts, looks positively unwell. The two galaxies, of course, are in the process of merging, and over the next billion years, will convert their delicate dynamical structures into the frenzied agglomeration of orbits that constitutes an elliptical galaxy.

But I like the fact that through the telescope, it’s just two faint misty patches. Static. Unhurried. Completely calm. A billion years is an incredibly long time. The view gives a good illustration of Eisenhower’s remark that “the urgent is seldom important and the important is seldom urgent.”

Saturn, too, was high in the sky, and looked like this.

After seeing M81 in the Miocene, it’s slightly jarring to note that the light from Saturn had left the planet after dinner while I was doing the dishes.

With the low-power telescope view, it’s easy to see why Galileo was puzzled when he first saw Saturn under magnification. Huygens’ accomplishment in figuring out the true geometry of an inclined planet with rings suddenly seems much more impressive. And now, there’s spacecraft all the way out there, sending photo after incredible photo back to the Deep Space Network. I was very happy to hear that Cassini’s first mission extension was approved.

Image Source.

M81 and the rings of Saturn are separated by an enormous expanse of scale and time, but they are both excellent examples of disks whose detailed structures are created by a combination of external forces and self-gravity. The protostellar disk that gave rise to the solar system falls in this same category of object.

An important issue in the study of protostellar disks is the identification of when a disk is massive enough to experience the development of spiral instabilities. Stefano (in addition to all the work he’s been doing on the systemic project) has been doing a detailed study of this problem. He’s found that the presence of a gap in a self-gravitating disk makes the disk far more prone to spiral instabilities than it would otherwise be. Gaps are unavoidable if a massive planet is forming in the disk. The spiral instabilities generate mass and angular momentum transport that efficiently attempt to fill in the gap. This new phenomenon has potentially very important ramifications for our understanding of giant planet formation and protostellar disk evolution.

Stefano’s paper has been accepted for publication in the Astrophysical Journal Letters, and will be appearing on astro-ph very shortly. In the meantime, here’s an advance copy in .pdf format.

Also, be sure to check out the website that Stefano has set up to explain this research. He has some very cool animations of protostellar disks succumbing to catastrophic instabilities, and he provides a link to the slides for his recent FLASH seminar on his work. My personal favorite is the graphical rendering of the solution to the thorny integro-differential equation that has to be solved to determine the growth rates, the pattern speeds and the overall appearances of the unstable spiral modes:

In a nutshell, here’s the question: “What are the odds that the planets will experience a dramatic orbital instability before the Sun turns into a red giant and destroys the Earth?”

In a nutshell, here’s the answer: “About 1%.”

I’m very happy that it’s now possible to write a full follow-up report on last summer’s post about UCSC physics undergraduate Konstantin Batygin’s work on the long-term stability of the solar system.

Recapping last summer’s post:

The long-term stability of the planetary orbits has been a marquee-level question in astronomy for more than three centuries. Newton saw the ordered structure of the solar system as proof positive of a benign deity. In the late 1700s, the apparent clockwork regularity of interaction between Jupiter and Saturn helped to establish the long-standing concept of Laplacian determinism. In the late Nineteenth Century, Poincaré’s work on orbital dynamics provided the first major results in the study of chaotic systems and nonlinear dynamics, and began the tilt of the scientific worldview away from determinism and toward a probabalistic interpretation.

In recent years, it’s become fairly clear that the Solar System is dynamically unstable in the sense that if one waits long enough (and ignores drastic overall changes such as those wrought by the Sun’s evolution or by brushes with passing stars) the planets will eventually find themselves on crossing orbits, leading to close encounters, ejections and collisions.

Desktop PCs are now fast enough to integrate the eight planets into the future for time scales that exceed the Sun’s hydrogen burning lifetime. This makes it possible to explore future dynamical trajectories for the solar system. Over the long term, of course, the planetary orbits are chaotic, and so for durations longer than ~50 million years into the future, it becomes impossible to make a deterministic prediction for exactly where the planets will be. The butterfly effect implies that we can have no idea whether January 1, 100,000,000 AD will occur in the winter or in the summer. We can’t even say with complete certainty that Earth will be orbiting the Sun at all on that date.

We can, however, carry out numerical integrations of the planetary motions. If the integration is done to sufficient numerical accuracy, and starts with the current orbital configuration of the planets, then we have a possible future trajectory for the solar system. An ensemble of integrations, in which each instance is carried out with an unobservably tiny perturbation to the initial conditions, can give a statistical indication of the distribution of possible long-term outcomes.

Here’s a time series showing the variation in Earth’s eccentricity during a 20 billion year integration that Konstantin carried out. In this simulation, the Earth experiences a seemingly endless series of secular variations between e=0 and e=0.07 (with a very slight change in behavior at a time about 10 billion years from now). The boring, mildly chaotic variations in Earth’s orbit are mostly dictated by interactions with Venus:

Mercury, on the other hand, is quite a bit more high-strung:

These two plots suggest that the Solar System is “good to go” for the foreseeable future. Indeed, an analysis (published in Science in 1999) by Norm Murray and Matt Holman suggests that the four outer planets have a dynamical lifetime of order one hundred quadrillion years (ignoring, of course, effects of passing stars and the Sun’s evolution).

Work by Jacques Laskar, on the other hand, who is Laplace’s dynamical heir at the Bureau des Longitudes in Paris, suggests that the inner solar system might be on far less stable footing.

Laskar performed the following experiment (described in this 1996 paper, which is well worth reading). Using an extremely fast (but approximate) numerical code which incorporates more than 50,000 secular perturbation terms involving the eight planets, Laskar integrated the current configuration of the Solar System 2 billion years into negative time. He then made four “realizations” of the solar system in which Earth’s position was shifted by a mere 150 meters in different directions. These four nearly identical variations of the Solar System were each integrated backward in time for a further 500 million years. Due to the highly chaotic nature of the system, each of Laskar’s four simulations spent most of the computational time exploring entirely different dynamical paths within the Solar System’s allowed phase space.

When the four integrations were complete, Laskar examined the individual orbital histories and selected the trajectory in which Mercury’s eccentricity achieved its largest value. The Solar system configuration at the time of this greatest eccentricity excursion was then used as a starting condition for a second set of four individual 500-million year integrations. At the end of this second round of calculations a new set of starting conditions was determined by again selecting the configuration at which Mercury’s excursion was the largest.

Here’s a diagram that flowcharts (using positive time) the basic idea underlying Laskar’s bifurcation method:

After 18 rounds, which when pieced together yielded a 6 billion year integration, Laskar observed that Mercury’s eccentricity had increased to e>0.5. Mercury, and indeed the entire inner solar system, had gotten itself into extremely serious trouble. A secular integration scheme can’t handle close encounters, though, and so the final gory details were left to the imagination. Nevertheless, it was clear that by the end of Laskar’s simulation, Mercury was in line to suffer a close encounter with Venus, or a collision with the Sun, or an ejection from the Solar System. The 1996 Laskar integration was the first explicit demonstration of the Solar System’s long-term dynamical instability. In essence, it brought a 300-year quest to a dramatic head.

I read Laskar’s paper in 1999, shortly after the discovery of the Upsilon Andromedae planetary system spurred me into a crash-course study of orbital dynamics. His calculations seemed to raise some really interesting questions. What is the dynamical mechanism that destabilized the inner Solar System? Was the elevation of Mercury’s eccentricity a consequence of the secular perturbation approach that he applied? Would his bifurcation strategy find a similar result when used with direct numerical integration of the equations of motion?

Two years ago, I told Konstantin about Laskar’s experiment, and we decided to see if we could answer the questions that it raised. As a first step, Konstantin set about replicating Laskar’s simulation strategy with full numerical integrations. All told, this required over a year of computing, including a lot of effort to make sure that the buildup of numerical error was kept under control.

Our version of Laskar’s method works as follows (and is shown in the flow chart above). First, a direct integration spanning 500 million years, ~100 Earth Lyapunov times, is made using the current Solar System configuration as a starting point. Picking up at the integration’s endpoint, five solutions for 500 million years are computed. Four of these use initial conditions in which Earth’s position is shifted, while one uses the unaltered solution. Because initial uncertainties diverge exponentially with time, a shift of 150 meters in Earth’s position 500 million years from now corresponds to an initial error today of order 10^-42 meters — ten orders of magnitude smaller than the Planck scale. After the five bifurcated trajectories are computed, the solution in which Mercury attains the its highest eccentricity is preserved to the nearest whole million years, and five new trajectories are started.

Much to our amazement, the bifurcation strategy is capable of showing Mercury the door in a hurry. In our first complete experiment, only three Laskar steps were required in order to coax Mercury into a collision with Venus at a time 861.455 million years from now:

And it wasn’t only Mercury that ran into problems. At t=822 million years, shortly after Mercury’s entrance into a zone of severe chaos, Mars — rovers and all — was summarily ejected from the Solar System:

This is some pretty heavy stuff. We have a direct numerical solution of Newton’s equations in which the solar system goes unstable well before life on Earth is expected to perish. (Can GR save the day? Read the paper.)

So what’s the mechanism that causes the instability?

At first, we thought that the dynamics were stemming from an overlap of mean motion resonances, but we were able to show that isn’t the case. In the end, Konstantin used the technique of synthetic secular perturbation theory to demonstrate that the culprit is a linear secular resonance with Jupiter. In short, Mercury winds up in a situation where the resonant argument (omega_1 – omega_5) librates between +19.8 and -43.56 degrees for three million years. The result is a steady increase in Mercury’s eccentricity to a dangerously high value:

The evolution of Mercury’s orbit is driven both directly by Jupiter, and to a greater extent by Jupiter’s influence transmitted through Venus. It’s an amazing, scary possibility, and the full details are in the paper.

Needless to say, we were thrilled when the full picture came together. We wrote up our work and submitted it to the Astrophysical Journal in mid-January. I got in touch with the UCSC public affairs office with an eye toward issuing a press release once our paper cleared the refereeing process.

Then, to our total astonishment and dismay, we were scooped! It turns out that Jacques Laskar himself has also been working on the problem. On February 22nd, he posted an astro-ph preprint of a paper that will be appearing in Icarus. He beat us to the punch with a basic result that’s fully in line with what we found. Here’s his astro-ph abstract:

A statistical analysis is performed over more than 1001 different integrations of the secular equations of the Solar system over 5 Gyr. With this secular system, the probability of the eccentricity of Mercury to reach 0.6 in 5 Gyr is about 1 to 2 %. In order to compare with (Ito and Tanikawa, 2002), we have performed the same analysis without general relativity, and obtained even more orbits of large eccentricity for Mercury. We have performed as well a direct integration of the planetary orbits, without averaging, for a dynamical model that do not include the Moon or general relativity with 10 very close initial conditions over 3 Gyr. The statistics obtained with this reduced set are comparable to the statistics of the secular equations, and in particular we obtain two trajectories for which the eccentricity of Mercury increases beyond 0.8 in less than 1.3 Gyr and 2.8 Gyr respectively. These strong instabilities in the orbital motion of Mecury results from secular resonance beween the perihelion of Jupiter and Mercury that are facilitated by the absence of general relativity. The statistical analysis of the 1001 orbits of the secular equations also provides probability density functions (PDF) for the eccentricity and inclination of the terrestrial planets.

Rather ironically, Laskar did not use his bifurcation method to solve the problem. By sticking with his secular code, he’s able to get a big speedup over direct numerical integration, which allowed him to perform a suite of 1001 straight-line integrations of the secular equations. The resulting statistics of these allow him to place a 1-2% probability of Mercury going haywire within 5 billion years. (With general relativity included, this number is probably closer to 1%, although his integrations in the GR case haven’t finished yet.)

So sadly, no UCSC press release will be forthcoming. Priority of discovery goes to the Bureau of Longitudes, and our paper, which will be appearing in the Astrophysical Journal, will be providing dramatic confirmation of the mechanism by which the Solar System can come undone.

Our paper (Batygin, K. & Laughlin, G. 2008, Astrophysical Journal, In Press.) is available on astro-ph.

Back in 2002, Keith Horne gave a talk at the Frontiers in Research on Extrasolar Planets meeting at the Carnegie Institute in Washington and showed an interesting table:

At that time, there were more than two dozen active searches for transiting extrasolar planets, but only a single transiting planet — HD 209458 b — had been detected. Transits were generating a lot of excitement, but paradoxically, the community was well into its third straight year with no transit detections. The photometric surveys seemed to be just on the verge of really opening the floodgates, with a total theoretical capacity to discover ~200 planets per month.

It’s been six years, and the total transiting planet count is nowhere near 14,000. Most of the surveys on the table have had a tougher-than-expected time with detections because of the large number of false positives, and because of the need to obtain high-precision radial velocities on large telescopes to confirm candidate transiting planets. Indeed, the surveys that were sensitive to dimmer stars have largely faded out. It’s just too expensive to get high-precision velocities for V>15 stars. With the exception of the OGLE survey (which had been set up to look for microlensing during the 1990s, and which had established a robust pipeline early on) none of the surveys that employed telescopes with apertures larger than 12 cm have been successful. The currently productive photometric projects: TrES, XO, HATnet, and SuperWASP all rely on telescopes of 10 to 11 cm aperture to monitor tens of thousands to hundreds of thousands of stars, and all are sensitive to planets transiting stars in the V~10 to V~12 magnitude range. This magnitude range is the sweet spot: there are plenty of stars (and hence plenty of transits) and the stars are bright enough for reasonably efficient radial velocity confirmation.

Yesterday, SuperWASP rolled out 10 new transits at once, dramatic evidence of the trend toward planetary commoditization and of the fact that it’s getting tougher to make a living out on the discovery side. The detection of new planets is growing routine enough that in order to generate a news splash, you need multiple planets, and the more the better. This inflationary situation for new transit news is highly reminiscent of where the Doppler surveys were at seven years ago. For example, on April 4, 2001, the Geneva team put out a press release announcing the discovery of eleven new planets (including current oklo fave HD 80606b).

I’d like to register some annoyance with this latest SuperWASP announcement. There are no coordinates for the new planets, making it impossible to confirm the transits. There is no refereed paper. The data on the website are inconsistent, making it hard to know what’s actually getting announced. I was astonished, for example, that WASP-6 is reported on the website to have a radius 50% that of Jupiter, and a mass of 1.3 Jovian masses:

That’s nuts! If the planet is so small, why is the transit so deep? And a 2200 K surface temperature for a 3.36d planet orbiting a G8 dwarf? Strange. Perhaps the radius and mass have been reversed? In addition, there are weird inconsistencies between the numbers quoted in the media diagram and in the tables. For example, the diagram pegs WASP-7 at 0.67 Jovian masses, whereas the table lists it at 0.86 Jovian masses. WASP-10 has a period of 5.44 days in the table and 3.093 days in the summary diagram. Putting out a press release without the support a refereed paper is never a very good idea, even when there’s a danger that another team will steal your thunder with an even larger batch of planets.

Despite the difficulty in getting accurate quotes from the exchange, it’s interesting to see how the ten new planets stack up in the transit pricing formula. Using the data from the new WASP diagram (except for the 0.66 day period listed for WASP-9) and retaining the assumption that USD 25M has been spent in aggregate on ground-based transit searches, the 46 reported transits come out with the following valuations:

The ten new WASP planets (assuming that the correct parameters have been used) contribute about 1/10th of the total catalog value. There will likely be interesting follow-up opportunities on these worlds from ground and from space, but its unlikely that they’ll rewrite the book on our overall understanding of the field.

It’s interesting to plot the detection rate via transits in comparison to the overall detection rate of extrasolar planets. (The data for the next plot was obtained using the histogram generators at the Extrasolar Planets Encyclopaedia, which are very useful and are always up-to-date.)

It’s a reasonable guess that 2008 will be the first year in which the majority of discoveries arrive via the transit channel, especially if CoRoT comes through with a big crop. Radial velocity, however holds an edge in that it’s surveying the brightest stars, and (so far) has been responsible for progress toward the terrestrial-mass regime. I think that we might be seeing planets of only a few Earth masses coming out of the RV surveys during the coming year. Certainly, everything else being equal, a planet orbiting an 8th magnitude star is far more useful for follow-up characterization than a planet orbiting a 13th magnitude star.