oklo

In preparing my talk for the Torun meeting, it seemed appropriate to take a careful look at the book that got the whole planetary systems business going — De revolutionibus orbium coelestium (On the Revolutions of Heavenly Spheres) by Copernicus.

Being not in possession of a classical education, that meant settling for an English translation, but it’s interesting to look at the original Latin editions (which are dramatically out of copyright, and hence available from the ether in the departure lounge at O’Hare if one is willing to fork out for a wi-fi connection). Here’s the frontispiece of Harvard’s edition:

The text translates to:

Diligent reader, in this work, which has just been created and published, you have the motions of the fixed stars and planets, as these motions have been reconstituted on the basis of ancient as well as recent observations, and have moreover been embellished by new and marvelous hypotheses. You also have most convenient tables from which you will be able to compute those motions with the utmost care for any time whatever. Therefore, buy, read and enjoy.

To a modern sensibility, the exhortation to buy the book seems to run at cross purposes with the warning just below (written in Greek for heightened effect):

Let no one untrained in geometry enter here.

Certainly, in trying to make sense of the text, it’s clear that the warning is no empty threat. The book, with its arduous descriptions of ephemerides is tough going. Section 17 of Book V presents a typical example:

Now it was made clear above that in the last of Ptolemy’s three observations Mars, by its mean movement as at 244.5 deg, and its anomaly of parallax was at 171 deg, 26′. Accordingly during the year between there was a movement of 5 deg 38′ besides the complete revolutions. Now for the 2nd year of Antoninus on the 12th day of Epiphi the 11 month by the Egyptian calendar 9 hours after mid-day, i.e. 3 equatorial hours before the following midnight, with respect to the Cracow meridian, to the year of Our Lord 1523 on the 8th day before the Kalends of March 7 hours before noon, there were 1384 Egyptian years 251 days 19 minutes [of a day]. During that time there were by the above calculation 5 deg 38′ and 648 complete revolutions of anomaly of parallax. Now the regular movement of the sun was held to be 257 1/2 deg. The subtraction from 257 1/2 deg of the 5 deg 38′ of the movement of parallax leaves 251 deg 52′ as the mean movement of Mars in longitude. And all that agrees approximately with what was set down just now.

By connecting observations from the Ptolemaic era with his own (and other contemporary) observations, Copernicus was able to achieve a great improvement in timing accuracy. Remarkably, his combination of timing data and positional measurements for solar system planets such as Mars give a signal-to-noise quite similar to the modern data that we currently have for transiting hot Jupiters such as HD 149026b. These extrasolar planets have been observed over hundreds of orbits with both ground-based photometry (for timing) and with radial velocities (for elucidating the orbital figure).

Given that the distances to the planet-bearing stars are millions of times larger than the distances to the solar system planets, this is a testament both to how far we’ve come in 500 years, and simultaneously, to the durability of the Copernican accomplishment.

Sometimes, when I give a talk, I’m asked why the extrasolar planets don’t have evocative names.

Names and labels carry a heavy freight and they get people worked up. The agonized IAU deliberations vis-à-vis Pluto’s status as a plutoid or a planet or a dwarf planet constituted by far the biggest planet news of 2006, dwarfing, for example, the discovery of the triple Neptune system orbiting HD 69830. It’s unlikely that New Horizons would have gotten its congressional travel papers in order had Pluto been a plutoid right from the start.

When new comets and asteroids are discovered, their names generally follow on fairly quickly. Comets are bestowed with the name of the discoverer(s), and as a result, Dr. Hale and Mr. Bopp are entwined together in immortality. With asteroids, the discoverer gets the naming rights (subject to certain IAU rules), resulting in both some cool choices, (99942) Apophis, (3040) Kozai, as well as a Kilroy-was-here sloop of John B’s: (6830) Johnbackus, (20307) Johnbarnes, (4525) Johnbauer, (15461) Johnbird, (12140) Johnbolton, (16901) Johnbrooks, (11652) Johnbrownlee, (26891) Johnbutler, etc. etc.

Galileo, in sighting the moons of Jupiter, made the first telescopic discovery of solar system objects. Ever on the eye for an angle, he tried to increase his odds of patronage by naming his new moons “The Medicean Stars” in reference to Cosimo II de’ Medici, fourth Grand Duke of Tuscany. It’s now generally agreed that Mr. Medici, whatever his merits, was rather dramatically undeserving of the following accolades:

Serenissimo Grand Duke, “scarcely have the immortal graces of your soul begun to shine forth on earth than bright stars offer themselves in the heavens, which, like tongues [longer lived than poets] will speak of and celebrate your most excellent virtues for all time.”

Later in the seventeenth century, when Giovanni Cassini discovered Saturn VIII, V, III, and IV, he tried the same tactic. Three hundred and twenty two years later, his prose reads like a purple toad:

In the Conclusion, the Discoverer considers that the Antient Astronomers, having translated the Names of their Heroes among the Starrs, those Names have continued down to us unchanged, notwithstanding the endeavour of following Ages to alter them; and that Galileo, after their Example, had honoured the House of the Medici with the discovery of the Satellites of Jupiter, made by him under the Protection of Cosmus II; which Starrs will be always known by the Name of Sidera Medicea. Wherefore he concludes that the Satellites of Saturn, being much more exalted and more difficult to discover, are not unworthy to bear the Name of Louis le Grand, under whose Reign and in whose Observatory the same have been detected, which therefore he calls Sidera Lodoicea, not doubting but to have perpetuated the Name of that King, by a Monument much more lasting than those of Brass and Marble, which shall be erected to his Memory. [1]

In order to forestall just these sorts of embarrassments, the current IAU naming convention specifies that, the names of individuals or events principally known for political or military activities are unsuitable until 100 years after the death of the individual or the occurrence of the event.

The Medicean Stars are neither medicean nor stars, and so it’s not surprising that the name failed to stick. In 1847, the names of the Sidera Lodoicea were finally standardized to Iapetus, Rhea, Tethys, and Dione, all of which just sound right. It’s remarkable that nearly two hundred years elapsed before the final names were assigned.

At present, there’s no IAU sanction for naming extrasolar planets. Sometimes astronomers give it a go anyway, as seen here in the abstract for astro-ph/0312382:

Three transits of the planet orbiting the solar type star HD209458 were observed in the far UV at the wavelength of the HI Ly-alpha line. The planet size at this wavelength is equal to 4.3 R_Jup, i.e. larger than the planet Roche radius (3.6 R_Jup). Absorbing hydrogen atoms were found to be blueshifted by up to -130 km/s, exceeding the planet escape velocity. This implies that hydrogen atoms are escaping this “hot Jupiter” planet. An escape flux of >~ 10^10g/s is needed to explain the observations. Taking into account the tidal forces and the temperature rise expected in the upper atmosphere, theoretical evaluations are in good agreement with the observed rate. Lifetime of planets closer to their star could be shorter than stellar lifetimes suggesting that this evaporating phenomenon may explain the lack of planets with very short orbital distance.

This evaporating planet could be represented by the Egyptian God “Osiris” cut into pieces and having lost one of them. This would give us a much easier way to name that planet and replace the unpleasant “HD209458b” name used so far.

The name Osiris doesn’t seem to have caught on, perhaps because (5×10^9)(3.17×10^7)(1×10^10) is a good deal less than (1.4×10^30). Also, I’d tend to disagree that HD 209458b is “unpleasant”. A sequence of letters and numbers carries no preconception, underscoring the fact that these worlds are distant, alien, and almost wholly unknown — K2 is colder and more inaccessible than Mt. McKinley, Vinson Massif or Everest.

Ray Bradbury, in several of his stories, tapped into the profound significance of names. In the 2035-2036 section of The Martian Chronicles, he wrote:

The old Martian names were names of water and air and hills. They were the names of snows that emptied south in the stone canals to fill the empty seas. And the names of sealed and buried sorcerers and towers and obelisks. And the rockets struck at the names like hammers, breaking away the marble into shale, shattering the crockery milestones that named the old towns, in the rubble of which great pylons were plunged with new names: Iron Town, Steel Town, Aluminum City, Electric Village, Corn Town, Grain Villa, Detroit II, all the mechanical names and the metal names from Earth.

I think we’ll eventually reach the extrasolar planets, and in so doing, we’ll find out what their true names are.

Yikes! It was brought to my attention this morning that the transitsearch.org domain name expired last week. The robots at Network Solutions were apparently posting their anxious renewal demands off into the great unknown. Visitors to transitsearch.org are now presented with a blandly science and astronomy themed page with links to topics such as “Save the planet” and “NASA Jobs”. Vaguely curious, I clicked on “NASA Jobs” and discovered that astronauts can earn online degrees in as fast as one year.

Renewal of the Transitsearch domain is now in progress using a sepulchural fax-based procedure. An inevitable credit card payment and a few days lag time, and everything should be back in working order. In the meantime, you can always access the candidates page at the oklo server, where the transit tables continue to be brewed anew every ten minutes:

http://207.111.201.70/transitsearch/dynamiccontent/candidates.html

And oklo.org? A chronic lack of posts, yes, but no, we’re not on vacation. The referee’s report on our HD 80606 results. A better transitsearch algorithm and table design. The rejuvenation of the console and the systemic backend. Doppler Survey schedule optimization. Etc. Etc., all soaking up much more time than expected.

I’m very hopeful, though, that the solution to the anagram can be revealed sooner rather than later…

Visitors to oklo.org over the past several weeks have frequently been greeted with page loading from the 2400 baud era, or worse yet, with the dreaded grey and yellow “highload.html” page.

I’ve been fully distracted with other projects, and so I didn’t really give it much attention. In what’s best described as a case of wishful thinking, I chalked the slowdown to the traffic spike that came in when the anagram post got written up at the NY Times site.

Over the weekend, things suddenly got much worse. It was clear that something was wrong. A little digging revealed that the wordpress installation was out of date, and was being exploited to the hilt by link spammers. The normally lightfooted footer.php script in the themes directory was staggering under a 427 kB load of grungy links. Oklo dot org was under full scale assault by dreary robots with single minded enthusiasms for cia1is and satellite TV.

A complete reinstall seems to have fixed the problem. It’d be tragic if those evil robots win.

On a related note, it might be worthwhile to sift through all those CoRoT lightcurves for photometric banner ads. As Luc Arnold writes in the abstract for “Transit Lightcurve Signatures of Artifical Objects” (astro-ph/0503580):

The forthcoming space missions, able to detect Earth-like planets by the transit method, will a fortiori also be able to detect the transit of artificial planet-size objects. Multiple artificial objects would produce lightcurves easily distinguishable from natural transits. If only one artificial object transits, detecting its artificial nature becomes more difficult. We discuss the case of three different objects (triangle, 2-screen, louver-like 6-screen) and show that they have a transit lightcurve distinguishable from the transit of natural planets, either spherical or oblate, although an ambiguity with the transit of a ringed planet exists in some cases. We show that transits, especially in the case of multiple artificial objects, could be used for the emission of attention-getting signals, with a sky coverage comparable to the laser pulse method. The large number of expected planets (several hundreds) to be discovered by the transit method by next space missions will allow to test these ideas.

Download 1017 x 761 px version here.

This dry range is near Gabbs, Nevada.

I remember stopping at a bar in Gabbs on a Saturday night in October 1993. We were low on gas, having foolishly skipped a possibility to fill up at Walker Lake. We’d been driving all day. In the deserted gravel lot, the sky was freezing black and spangled with stars.

I drank a beer and talked to the only other patron — a grizzled Vietnam veteran who worked at the molybdenum mine. The word molybdenum sounded strange, exotic. In 1993, the price of molybdenum was in free fall, and in 1994, it would reach a low of $3,510 per metric ton ($1.59 per pound). The mine was laying off workers and was in danger of closing.

The gas station in Gabbs was closed. The bartender called the nearest possibility, the old Pony Express station Middlegate, 50 miles north. “You’re in luck, they’ve got gas.”

The current spot price for Molybdenum oxide is 33.50 dollars per pound, a less-noticed example from the many changes that make 1993 seem increasingly a part of a bygone millennium. Hundreds of extrasolar planets, e-mail inboxes that routinely receive hundreds of messages (mostly spam) per day, and this uneasily growing realization that the raw materials may be the deciding factor after all.

I wonder whether the extrasolar planets will ever have a flatly practical economic value. The scramble to detect new planets often feels like a land rush, but is there a real possibility that we’ll eventually pack up and go to these systems that are showing up in the correlation diagrams? Do the economics of interstellar travel ever work out?

In this context, it’s slightly disconcerting to remember that the molybdenum has already made the interstellar journey (see e.g. here). The most abundant Mo isotope is molybdenum-98, which constitutes 24.14% of Earth’s molybdenum. These atoms were produced both via the s-process, which takes place in red giant stars, and where a chain of slow neutron captures is interspersed with beta decays, and by the r-process, which occurs in supernovae.

The fact that the resources made the trip for free makes it seem a little more likely that we may well be able to get more, but only if we pay…

Man! Like everyone else over the past 24 hours, I’ve been thinking about that new crop of Superearths.

The conventional wisdom (over which I was waxing enthusiastic a mere 36 hours ago) holds that Mayor’s new population of planets are essentially failed giant planet cores which began forming at considerably larger radii in the protostellar disk and then experienced significant inward migration as they built themselves up. In this scenario, the Superearths arise from more or less the same sort of process (but with a different outcome) that formed the giant planets in our own solar system.

What’s struck me, however, is the odd resemblance between a multiple-planet system like HD 40307 and the regular satellite systems of the Jovian planets. In both cases, the characteristic orbital period is of order a week, and the system mass ratio (satellites-to-central-body) is of order 2 parts in 10,000.

In the Ward-Canup theory, the regular satellites of the Jovian planets are thought to have formed more or less in situ in gas-starved disks (see here for more discussion). If the new population of planets is somehow the result of an analogous formation process, then they really will be superEarths, as opposed to subNeptunes, and as a consequence, their transit depths will be small.

“Getting scooped” is an ongoing occupational hazard for astronomers. An interesting idea pops into your head, or a significant peak starts to emerge in a periodogram, and you drop everything to do an analysis and write up your idea or discovery for submission. If your idea seems to work, and as your story takes shape on paper, it occurs to you that there are plenty of other colleagues who could easily have latched on to what you’ve just done. After all, there are only so many nearby red dwarfs in the sky!

The invention of the telescope at the beginning of the seventeenth century led to very rapid progress in astronomy, and because telescopes are relatively straightforward to make once the principle is understood, astronomers suddenly faced heightened competition, and with it, the ever-unnerving possibility of getting scooped.

Anagrams were brought into use as a method of protecting one’s priority of discovery while simultaneously keeping a discovery under wraps in order to obtain further verification. Galileo was an early adopter of anagrams. After observing Saturn, he circulated the following jumble of letters:

s m a i s m r m i l m e p o e t a l e u m i b u n e n u g t t a u i r a s

When he was ready to announce that Saturn has a very unusual shape when seen through his small telescope, he revealed that the letters in the anagram can be rearranged to read, Altissimum planetam tergeminum observavi, or “I have observed the highest planet tri-form.”

Galileo’s telescope wasn’t powerful enough to allow him to decode what he was actually seeing when he observed Saturn. The true configuration as a ringed planet was first understood by Christiaan Huygens, who, in 1656, with the publication of the discovery of Titan in De Saturni luna observatio nova, also circulated an anagram to protect his claim to discovery:

a a a a a a a c c c c c d e e e e e h i i i i i i i l l l l m m n n n n n n n n n o o o o p p q r r s t t t t t u u u u u.

In 1659, Huygens revealed that the anagram can be decoded to read, Annulo cingitur, tenui, plano, nusquam cohaerente, ad eclipticam inclinato, or “It is surrounded by a thin flat ring, nowhere touching, and inclined to the ecliptic.”

The most appealing anagrams rearrange the true sentence into a satisfyingly oblique haiku-like clue. In connection with his discovery of the phases of Venus, Galileo issued an anagram that read, Haec immatura a me iam frustra leguntur, or “These immature ones have already been read in vain by me.” When properly reconstructed, the letters reveal that, Cynthiae figuras aemulatur Mater Amorum, or “The Mother of Loves [i.e. Venus] imitates the figures of Cynthia [i.e. the moon]”.

So, in service to this venerable tradition, but without adhering to the hoary custom of couching everything in Latin, let me just say that,

Huge Applet, Unsearchable Terrestrials!

Note that according to the wikipedia,

The disadvantage of computer anagram solvers, especially when applied to multi-word anagrams, is that they usually have no understanding of the meaning of the words they are manipulating. They are therefore usually poor at filtering out meaningful or appropriate anagrams from large numbers of nonsensical word combinations.

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.