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“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.

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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.

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Over the past two days, I got the opportunity to fly to Hawaii to give two talks for the Keck Observatory’s Evening With Astronomers series. The talks focused on extrasolar planets (here’s a link to the slides in Quicktime format, ~40MB , along with the audio files of (1) a planetary system in a 2:1 resonance, (2) an unstable planetary system, and (3) another unstable system). Both talks were on Kona coast of the Big Island, where, behind the palm trees, Mauna Kea looms up 13,796 feet in the hazy volcanic distance.

The landscape here resembles nothing so much as a habitable, terraformed Mars. Hardened ropes of lava run down to the water’s edge:

In the pre-dawn light this morning, the air was totally silent, and it was easy to imagine that I was actually on Mars, before the water was gone, when a Northern hemispheric ocean lapped up against the lava of the lowermost slopes of Elysium Mons:

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In the last few years the Martian landscape has become much more familiar, as the Spirit and Opportunity rovers crawl across the surface and radio home their photographs:

At Kona, looking out toward the lava fields, the view is positively Martian, with the most immediate difference being a sky that is a hazy blue-white rather than a hazy salmon-white. Here, the Ala Loa trail recedes into the jagged distance of what could easily be Mars:

On Mars, however, one generally has a fairly reasonable sense of what the 360-degree panorama will look like even if only part of the horizon is in view. On Earth, the situation can be quite different. Here’s the view that one gets simply by turning and looking in the opposite direction down the Ala Loa trail:

(On a marginally related note, our Alpha Centauri ApJ paper is starting to pick up some news coverage. Here’s a link to a story by National Geographic News.)

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The last mile of the San Lorenzo river in Santa Cruz is strongly affected by the twice-daily ebb and flow of the tides.

It’s always startling to see the tidal bore, a solitary breaking wave that runs upstream at a ~8 minute per mile pace when the tide is coming in. The San Lorenzo bore is small, usually six to nine inches high, but dramatic nonetheless. In its wake, there’s a turbulent froth of whitewater, whose eddies eventually cascade into viscous dissipation, turning the kinetic energy of organized flow into a slight heating of the water. As the Moon recedes, the Earth spins down, and the bore expends itself in a swirl of eddies.

The energy that powers the bore was all imparted during the Moon-forming impact, in which a Mars-sized object collided with Earth, leaving the planet violently shaken and stirred and spinning crazily through days that were originally just a few hours long. Now, 4.5 billion years later, the bore running up the river is a distant echo of the impact that was large enough to cause Earth to glow with the temperature of a red dwarf star.

Adapted from: Source.

There’s a nice discussion of tidal bores in the 1899 popular-level book The Tides and Kindred Phenomena in the Solar System, by Sir G. H. Darwin (son of the naturalist). The book in its entirety can be downloaded from The Internet Archive.

The Moon-forming impact, which occurred somewhere between 10 and 100 million years after the collapse of the pre-solar molecular cloud core, essentially marked the end of terrestrial planet formation in our own solar system. From a dynamical standpoint, a system undergoes a lot of evolution during a time scale of 100 million orbits. By contrast, the Milky Way galaxy is only about 40 orbits old, and is still in an effectively pristine, dynamically unrelaxed configuration.

At Darwin’s time, the first photographs of spiral galaxies were appearing, and there’s a remarkably good photo of the Andromedae galaxy on page 339 of the book:

Darwin writes:

There is good reason for believing that the Nebular Hypothesis presents a true statement in outline of the origin of the solar system, and of the planetary subsystems, because photographs of nebulae have been taken recently in which we can almost see the process in action. Figure 40 is a reproduction of a remarkable photograph by Dr. Isaac Roberts of the great nebula in the constellation of Andromeda. In it we may see the lenticular nebula with its central condensation, the annulation of the outer portions, and even the condensations in the rings which will doubtless at some time form planets. This system is built on a colossal scale, compared with which our solar system is utterly insignificant. Other nebulae show the same thing, and although they are less striking we derive from them good grounds for accepting this theory of evolution as substantially true.

In 1899, the extragalactic distance scale hadn’t been established, and so Darwin thought that M31 was a lot closer than it actually is. In dynamical terms, he would have guessed that it’s many thousands of orbits old rather than only a few dozen. Nevertheless, it’s interesting to think about what will happen to an isolated spiral galaxy by the time it’s 10^18 years old…

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The Gliese 876 system is remarkable for a number of reasons. It makes a mockery of the notion that the minimum-mass solar nebula has a universal validity. It harbors one of the lowest-mass extrasolar planets known (discovered by our own Eugenio Rivera). And of course, the outer two planets are famously caught in a 2:1 mean motion resonance, in which the inner 0.8 Jupiter-mass planet makes (on average) exactly two trips around the red dwarf for every one trip made by the outer 2.5 Jupiter-mass planet.

As users of the console know, the planet-planet interactions between the Gliese 876 planets are strong enough so that one needs a self-consistent dynamical fit to the system. Even on the timescale of a single outer planet orbit, the failure of the Keplerian model can be seen on a 450-pixel wide .gif image:

The following three frames are from a time-lapse .mpg animation of the Gliese 876 system over a period of roughly one hundred years:

Each frame strobes the orbital motion of the planets at 50 equally spaced intervals which subdivide the P~60 day period of the outer planet. Upon watching the movie, it’s clear that the apsidal lines of the outer two planets are swinging back and forth like a pendulum. This oscillation has an amplitude (or libration width) of 29 degrees, and acts like a fingerprint identifier of the Gliese 876 system.

The derangement of the orbits is reflected in their continual inability to maintain an exact 2:1 orbital commensurability. The first figure up above shows that when planet c has finished exactly two orbits, it has already managed to lap planet b, which was still dawdling down Boardwalk prior to passing GO.

Planet b, however, doesn’t always run slow. The gravitational perturbations between the two planets provide a second pendulum-like restoring action which prevents the bodies from straying from the average period ratio of 2:1, which, over the long term, is maintained exactly. The degree to which the orbits themselves librate, combined with the planets’ abilities to run either ahead or behind exact commensurability is captured by the resonant arguments of the configuration. These can be defined as,

where the lambdas are mean longitudes and the curly pi’s are the longitudes of periastron. The two resonant arguments capture the simultaneous libration of the mean motions and the apsidal lines. The smaller the arguments, the more tightly the system is in resonance.

In the Gliese 876 system, the resonant arguments are both librating with amplitudes of less than 30 degrees. This is evidence that a dissipative mechanism was at work during the formation of the system. Interestingly, however, when one looks at the other extrasolar planetary systems that are thought to be in 2:1 resonance, one finds that the libration amplitudes in every case are much larger. In fact, in the HD 73526 system and in the HD 128311 system, only one of the arguments is librating, while the other is circulating. In this state of affairs, the apsidal lines act like a pendulum that is swinging over the top. In addition, the orbital eccentricites are higher, and the sum of planet-planet activity is strikingly greater (see this animation of the evolution of the HD 128311 system).

A gas disk seems to be the most likely mechanism for pushing a planetary system into mean-motion resonance. Protoplanetary disks are likely, however to experience turbulent density fluctuations. These density fluctuations lead to stochastic gravitational torques, which provide a steady source of orbital perturbations to any planets that are embedded in a disk. For a reasonable spectrum of turbulent fluctuations, it turns out that it’s rather difficult to wind up with a planetary system that is as deeply in resonance as Gliese 876. The conclusion, then, is that Gliese 876-like configurations should be quite rare. Indeed, 2:1 resonances of every stripe should constitute only a minor fraction of planetary systems, and the majority that do exist should either large libration widths or only a single argument in resonance.

If you’re interested in more detail, we’ve submitted a paper that goes into much more detail (Adams, Laughlin & Bloch, ApJ, 2008 Submitted).

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Messenger flew by Mercury last week, and photographed vast swaths of terrain that, until now, had never been seen. The new landscapes, as expected, are cratered, barren, and utterly moonlike. The galaxy could contain a hundred billion planets that would be hard, at first glance, to distinguish from Mercury, and within our cosmic horizon, there are probably of order as many Mercury-like worlds as there are sucrose molecules in a cube of sugar.

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Nevertheless, we do gain something extraordinary whenever a new vista onto a terrestrial world is opened up. Galileo was the first to achieve this, when he turned his telescope to the Moon and saw its three-dimensional relief for the first time. Mariner 4 and Mariner 9 accomplished a similar feat for Mars. The Magellan spacecraft revealed the Venusian topography. And once Messenger has photographed the full surface of Mercury, there will be a profoundly significant interval before we get our next up-close view of an unmapped terrestrial planet. My guess is that it’ll be Alpha Centauri B b.

The Messenger website is well worth a visit. I was particularly struck by the movie that the spacecraft made of the Earth during the close fly by of March 2005. During the course of 24 hours, the spinning Earth recedes into the black velvet distance and space travel seems like the real thing.

Mercury’s orbit, with its 88 day period and its eccentricity of 0.2 could slip unnoticed into the distribution of known exoplanets. It’s vaguely comparable, for example, with the orbit of HD 37605 b. This Msini=2.3 Mjup gas giant has an apoastron distance similar to Mercury’s, but dives much closer to its star during periastron.

We’ve been interested in HD 37605 b lately because its orbit dips in and out of the insolation zone where water clouds are expected to exist. At the far point of the 55 day orbit, it should be possible for white clouds to form out of a clear steamy atmosphere. At close approach, the clouds are turning to steam.

Jonathan Langton’s models for this planet show persistent polar vortices, which sequester cooler air, and which may remain cloudy even during the hot days surrounding periastron. The vortices are tenaciously long-lived, and tracer particles seeded into the vortices leak out only slowly. It would be interesting to know what sort of chemistry is brewing in the steamy hothouse environment of trapped and noxious air.