systemic

In the antique language of the space age, one might call it an interstellar “probe”, or perhaps a von Neumann machine. That’s not really what it is. It’s better described as a snarky, fusion-powered tangle of competing social networks, some of them still executing the hallowed fossil liturgies and intrigues of the mighty corporations from which they long since sprang.

It had no particular expectations for the fast-approaching star that was next on its ancient route. On the last flyby of this particular star, twenty-seven million years ago, the probe observed that the third planet was still robustly in the grip of a somewhat unusual, low-energy parasitic film that was efficiently exploiting the surface entropy gradient, and running undirected at a computational rate roughly equivalent to 10^34 bit operations per second.

Over the last few years, as the probe sifted the electromagnetic spectrum emanating from the third planet, it rippled with a hint of something that might best be thought of as a collective rolling of eyes. The third planet has recently stumbled into directed processes, and remarkably, foolishly, it is radiating manifestly unencrypted signals into space. This state of affairs caught a fraction of the probe’s interest, especially when it grasped that the planet’s computational efforts are increasingly focusing on concepts that the planet was calling “blockchain” and “proof-of-work through SHA-256 hashing”. This is just the sort of pursuit that the probe can relate to…

The above, of course, is unlikely to be true. In all likelihood, A/2017 U1 is a battered, inanimate 160-meter chunk of rock or metal, spawned in the dry collision of planetesimals orbiting an alien star, sometime within the past ten billion years. What’s remarkable, is that this interstellar visitor came within 0.25 AU of the Sun. As it departs into the depths of the Galaxy, it can expect to fly for roughly ten quadrillion years before it revisits another star with such proximity. It’s next rendesvous of comparable drama lies far into the depths of the Stelliferous era. In all likelihood, this will have it sailing past the frigid hulk of a white dwarf, warmed a few degrees above absolute zero by the flicker of proton decay.

Speaking of rendesvous, it must have occurred to quite few that the recent visit by A/2017 U1 is rather uncannily reminiscent of Arthur C. Clarke’s famous ’70s-era sci-fi page-turner. A Google trends search hints at a moderate uptick in interest over the past few days, which I expect will soon grow to undeniable statistical significance:

Closer to home, A/2017 U1 generates a very convenient route to completion of problem #1 on my Astronomy 395/575 homework assignment, which was set to the students just two days before A/2017 U1 was announced in the news:

There’s no denying the fundamentally alien climates on the hot Jupiters. It’s not clear, however, how hot Jupiters form, and it’s not clear why so many of them are badly distended. Moreover, it’s only vaguely clear what the weather patterns on one would look like up close. (One thing that is clear is that the flights would all be canceled).

Hot Jupiters are rare, but not overwhelmingly so. Something about the planet formation process causes about one in two hundred sun-like stars to end up stuck with one. In the original Kepler field, there are about 150,000 stars with light curves, and so about 750 hot Jupiters total are lurking in that population. Some of them, of course, are observable in transit, but as yet, most have gone undetected.

Yale graduate student Sarah Millholland has a new lead-authored paper out which uses supervised learning techniques to identify sixty high-probability non-transiting hot Jupiter candidates among the Kepler stars. The basic idea is that the phase curves of the planets, some of which have photometric amplitudes of several dozen parts per million or more, can be teased out of the noise and the stellar variability. After an involved process of sifting, the candidates (along with their supporting light curves) can be presented for a screen test:

[Full resolution version here]

Some members of the Kepler hot Jupiter class portrait will prove to be imposters (just like #5, #13, #29, and #30 in the nineteenth-century insect woodcut above). Doppler velocity observations — the equivalent of counting the number of legs on the arthropods — will provide a more definitive list. If you want to weigh in on the odds that these candidates are predominantly real, there’s a fresh Metaculus question that pools community input regarding the fidelity and prospects for confirmation of the members of the sample.

One might reasonably wonder, what’s the utility of yet another tray of bugs, smothered with ether and pinned to cards?

One superb benefit from gathering sixty non-transiting hot Jupiters that are detectable in the optical region is that trends in the planets’ surface temperature variations — that is, the weather maps — can be elucidated with a far larger sample than was previously available. Sarah’s candidates support an interesting trend in which cooler planets (relatively speaking, of course) are posited to have reflective clouds to the west of the substellar point, whereas hotter hot Jupiters are consistently advecting the most strongly optically radiating gas downwind from high Noon.

For detailed information on the individual candidates, visit Sarah’s website, and if you are at the Kepler Science Conference, she’ll present the details during Friday’s session.

Most oklo.org readers know the story line of Fred Hoyle’s celebrated 1957 science fiction novel, The Black Cloud. An opaque, self-gravitating mass of gas and dust settles into the solar system, blots out the sun, and wreaks havoc on the biosphere. It gradually becomes clear that the cloud itself is sentient. Scientists mount an attempt to communicate. A corpus of basic scientific and mathematical principles is read out loud in English, voice-recorded, and transmitted by radio to the cloud.

The policy was successful, too successful. Within two days the first intelligible reply was received. It read:

“Message received. Information slight. Send more.”

For the next week almost everyone was kept busy reading from suitably chosen books. The readings were recorded and then transmitted. But always, there came short replies demanding more information, and still more information…

Sixty years later, communicating interstellar clouds are still in the realm of fiction, but virtualized machines networked in the cloud are increasingly dictating the course of actions in the real world.

In Hoyle’s novel, the initial interactions with the Black Cloud are quite reminiscent of a machine learning task. The cloud acts as a neural network. Employing the information uploaded in the training set, it learns to respond to an input vector — a query as a sequence of symbols — with a sensible output vector. Throughout the story, however, there’s an implicit assumption that the Cloud is self-conscious and aware; nowhere is it intimated that that the processes within the Cloud might simply be an algorithm managing to pass an extension of the Turing Test. On the basis of the clear quality of its output vectors, the Cloud’s intelligence is taken as self-evident.

The statistics-based regimes of machine learning are on a seemingly unstoppable roll. A few years ago, I noticed that Flickr became oddly proficient at captioning photographs. Under the hood, an ImageNet classification with convolutional neural networks (or the like) was suddenly focused, with untiring intent, on scenes blanketing the globe. Human mastery of the ancient game of Go has been relinquished. Last week, I was startled to read Andrej Karpathy’s exposition of the unreasonable effectiveness of recurrent neural networks.

By drawing from a large mass of example text, a recurrent neural network (RNN) character-level language model learns to generate new text one character at a time. Each new letter, space, or punctuation mark draws its appearance from everything that has come before it in the sequence, intimately informed by what the algorithm has absorbed from its fund of information. As to how it really works, I’ll admit (as well) to feeling overwhelmed, to not quite knowing where to begin. This mind-numbingly literal tutorial on backpropagation is of some help. And taking a quantum leap forward, Justin Johnson has written a character-level language model, torch-rnn, which is well-documented and available on github.

In Karpathy’s post, RNNs are set to work generating text that amuses but which nonetheless seems reassuringly safely removed from any real utility. A Paul Graham generator willingly dispenses Silicon Valley “thought leader” style bon mots concerning startups and entrepreneurship. All of Shakespeare is fed into the network and dialogue emerges in an unending stream that’s — at least at the phrase-to-phrase level — unkindly indistinguishable from the real thing.

I’m very confident that it would be a whole lot more enjoyable to talk to Oscar Wilde than to William Shakespeare. As true A.I. emerges, it may do so in a cloud of aphorisms, of which Wilde was the undisputed master, “I can resist everything except temptation…”

Wilde employed a technique for writing The Picture of Dorian Gray in which he first generated piquant observations, witty remarks and descriptive passages, and then assembled the plot around them. This ground-up compositional style seems somehow confluent with the processes — the magic — that occurs in an RNN.

The uncompressed plain text UTF8 version of Dorian Gray is a 433701 character sequence. This comprises a fairly small training set. It needs a supplement. The obvious choice to append to the corpus is A rebours — Against Nature, Joris-Karl Huysman’s 1884 classic of decadent literature.

Even more than Wilde’s text, A rebours is written as a series of almost disconnected thumbnail sketches, containing extensive, minutely inlaid descriptive passages. The overall plot fades largely into the background, and is described, fittingly, in one of the most memorable passages from Dorian Gray.

It was a novel without a plot and with only one character, being, indeed, simply a psychological study of a certain young Parisian who spent his life trying to realize in the nineteenth century all the passions and modes of thought that belonged to every century except his own, and to sum up, as it were, in himself the various moods through which the world-spirit had ever passed, loving for their mere artificiality those renunciations that men have unwisely called virtue, as much as those natural rebellions that wise men still call sin. The style in which it was written was that curious jewelled style, vivid and obscure at once, full of argot and of archaisms, of technical expressions and of elaborate paraphrases, that characterizes the work of some of the finest artists of the French school of Symbolistes. There were in it metaphors as monstrous as orchids and as subtle in colour.

A rebours attached to Dorian Gray constitutes a 793587 character sequence, and after some experimentation with torch-rnn, I settled on the following invocation to train a multilayer LSTM:

MacBook-Pro:torch-rnn Greg$ th train.lua -gpu -1 -max_epochs 100 -batch_size 1 -seq_length 50 -rnn_size 256 -input_h5 data/dorianGray.h5 -input_json data/dorianGray.json

My laptop lacks an Nvidia graphics card, so the task fell to its 2.2 GHz Intel Core i7. The code ran for many hours. Lying in bed at night in the quiet, dark house, I could hear the fan straining to dissipate the heat from the processor. What would it write?

This morning, I sat down and sampled the results. The neural network that emerged from the laptop’s all-nighter generates Wilde-Huysmans-like text assembled one character at a time:

MacBook-Pro-2:torch-rnn Greg$ th sample.lua -gpu -1 -temperature 0.5 -checkpoint cv/checkpoint_1206000.t7 -length 5000 > output.txt

I opened the output, and looked over the first lines. It is immediately clear that a 2015-era laptop staying on all night running downloaded github code can offer no competition — in any sense — to either Mr. Wilde or Mr. Huysmans. An abject failure of the Turing Test, a veritable litany of nonsense:

After the charm of the thread of colors, the nineteenth close to the man and passions and cold with the lad's heart in a moment, whose scandal had been left by the park, or a sea commonplace plates of the blood of affectable through the club when her presence and the painter, and the certain sensation of the capital and whose pure was a beasts of his own body, the screen was gradually closed up the titles of the black cassion of the theatre, as though the conservatory of the past and carry, and showing to me the half-clide of which it was so as the whole thing that he would not help herself. I don't know what will never talk about some absorb at his hands.

But we are not more than about the vice. He was the cover of his hands. "You were in his brain."

"I was true," said the painter was strangled over to us. It is not been blue chapter dreadfully confesses in spite of the table, with the desert of his hands in her vinations, and he mean about the screen enthralled the lamp and red books and causes that he was afraid that he could see the odious experience. It was a perfect streating top of pain.

"What is that, I am sorry I shall have something to me that you are not the morning, Mr. Gray," answered the lad, and that the possession of colorings, which were the centre of the great secrets of an elaborate curtain.

You cannot believe that I was thinking of the moon.

He was to be said that the world is the restive of the book to the charm of a matter of an approvingian through a thousand serviced it again. The personality of the senses by the servants were into the shadow of the next work to enter, and he had revealed to the conservatory for the morning with his wife had been an extraordinary rooms that was always from the studio in his study with a strange full of jars, and stood between them, or thought who had endured to know what it is.

"Ah, Mr. Gray?"

"I am a consolation to be able to give me back to the threat me."

But such demands are excessive. The text is readable English, convened in a headlong rush by a program that could just as easily have been synthesizing grant proposals or algebraic topology. Torch-rnn contains no grammar rules, no dictionaries, no guides to syntax. And it really does learn over time. Looking at the early checkpoint snapshots of the network, during epochs when words and spaces are forming, before any sense of context has emerged, one finds only vaguely English-like streams of gibberish:

pasticite his it him. "It him to his was paintered the cingring the spure, and then the sticice him come and had to him for of a was to stating to and mome am him himsed at he some his him, and dist him him in on of his lime in stainting staint of his listed."

Perhaps the best comparison of Torch-rnn’s current laptop-powered overnight-effort capabilities are to William S. Burroughs’ cut-up novels — The Soft Machine, The Ticket that Exploded — where one sees disjoint masses of text full of randomized allusions, but where an occasional phrase sparkles like a diamond in matrix, “…a vast mineral consciousness near absolute zero thinking in slow formations of crystal…”

In looking over a few thousand characters of text, generated from checkpoint 1,206,000 at temperature T=0.61, one finds glimmers of recurrent, half-emerged truths,

You are sure to be a fragrant friend, a soul for the emotions of silver men.

An interesting development caught my eye this afternoon. Warm Spitzer, fresh off all that attention generated by the discovery of the TRAPPIST-1 planets — was used by a Michael Gillon-led team to determine that HD 219134 c transits its K-dwarf host star. (Here’s a link to the paper in Nature Astronomy).

Given the near-constant flux of high-profile exoplanet results, it’s understandable that HD 219134 AKA HR 8832 might not immediately ring a bell. The system is interesting, however, because it is a radial velocity extraction that very cleanly typifies the most common class of systems detected by the Kepler Mission — multiple-transiting collections of super-Earth sized worlds with orbital periods ranging from days to weeks. Upscaled versions, that is, of the Jovian planet-satellite members of our own solar system. The innermost planet in the HD 219134 system is already known to transit. The Gillon et al result adds a second transiting member, which presents itself as the closest transiting extrasolar planet to Earth. Plotting the HD 219134 system on the mass-period diagram emphasizes how effectively it can be viewed as a draw from the Minimum Mass Extrasolar Nebula:

And because of the proximity and the modest radius of the host star, this system will be a fantastic target for future platforms.

There was something a little disorienting about TRAPPIST-1 vaulting into the public consciousness to fleetingly become one of the largest news events in the world. The small-telescope detection of temperate Earth-sized planets orbiting stars at the bottom of the main sequence was a frequent topic during oklo.org’s first ten years. In looking back over the early articles, one of the very first posts (from 11/29/2005) looks quaint, naive and prescient all at once:

We know that planets aren’t rare, and by now, with the tally over at the extrasolar planet encyclopedia poised to blast past 200, the announcement of a newly discovered run-of-the-mill Jupiter-sized planet barely raises the collective eyebrow.

The headline that everyone is anticipating is the discovery, or better yet, the characterization of a truly habitable world — a wet, Earth-sized terrestrial planet orbiting in the habitable zone of a nearby star. Who is going to get to this news first, and when?

299 million dollars of smart money says that Kepler, a NASA-funded Discovery mission currently scheduled for launch in June 2008, will take the honors. The Kepler spacecraft will fly in an Earth-trailing 377.5 day orbit, and will employ a 1-meter telescope to stare continuously (for at least four years straight) at a patchwork of 21 five-square-degree fields of the Milky Way in the direction of the constellation Cygnus. Every 15 minutes, the spacecraft will produce integrated photometric brightness measurements for ~100,000 stars, and for most of these stars, the photometric accuracy will be better than one part in 10,000. These specs should allow Kepler to detect transits of Earth-sized planets in front of Solar-type stars.

Kepler has a dedicated team, a solid strategy, and more than a decade of development work completed. It’s definitely going to be tough to cut ahead of Bill Borucki in line. Does anyone else stand a chance?

Practitioners of the microlensing technique have a reasonably good shot at detecting an Earth-mass planet before Kepler, but microlensing-detected planets are maddeningly ephemeral. There are no satisfying possibilities for follow-up and characterization. Doppler RV has been making tremendous progress in detecting ever-lower mass planets, but it seems a stretch that (even with sub-1 meter per second precision) the RV teams will uncover a truly habitable world prior to Kepler, although they may well detect a hot Earth-mass planet.

There is one possibility, however, whereby just about anyone could detect a habitable planet (1) from the ground, (2) within a year, and (3) on the cheap. Stay tuned…

In marveling at the avalanche of media attention during the last week, from the front pages of the New York Times and the New York Post, to NPR push notifications, to NASAwatch sleuthing out the story, to a co-opt of the front page of Google, I was struck by the fact that viewed externally, this is really just the massive amplification, complete with distortion — see the NASA/JPL go-to image — of an exceedingly faint signal. TRAPPIST-1 continually bathes the Earth with 14 Joules per second of energy. Over the course of the few weeks it took to detect the seven planets, its transits cumulatively decreased this share of the light by the energy equivalent of a single tic tac.

With the likes of an Earth-mass world orbiting Proxima Centauri and a staggeringly photorealistic better-than-the-real-thing rendering of Kepler 186f, it’s gotten increasingly difficult to mount a planet discovery press conference that achieves adequate signal-to-noise. Nonetheless, the new Gillon et al Nature paper detailing seven transiting, roughly Earth-sized, roughly Earth-mass planets orbiting a faint nearby red dwarf is a jaw-dropping document.

There’s a lot to like. The system is a pleasingly scaled-up version of the Jovian satellite systems and a pleasingly scaled-down version of the Kepler multiple-transit systems. It supports the empirical observation that the default satellite/planet formation process in the vicinity of objects ranging in mass from Uranus all the way up to the Sun tends to separate ~2×10^-4 of the system mass into a region large enough to delineate an average density of ~2×10^-5 g/cm^3. It’s not at all clear why this should be the case.

There’s a great deal of interest in planets that are more or less at room temperature. This means that, empirically speaking, the default planet-formation process selects (the Sun notwithstanding) the bottom of the main sequence as one’s best a-priori bet for Earth-mass planet with an Earth-like temperature. I’ll resist here the temptation to engage in holy hokey habitable zone talk. Chances of life, plate tectonics, proper ocean depths, etc. Let’s stick to the facts. What we do know is that if more than one of the Trappist-1 planets harbor advanced civilizations, and if the stock markets on those planets trade correlated securities with tight bid-offer spreads, then there will be excellent interplanetary latency arbitrage opportunities.

2MASS J20362926-0502285, now much better known as TRAPPIST-1, straddles the boundary between the lowest mass main sequence stars and the highest mass brown dwarfs. Depending on precisely what its mass and metallicity turn out to be, it could either be arriving at self-sustaining core hydrogen fusion, which would make it a main sequence star (about a 60% chance) or it could be currently achieving its peak brown dwarf luminosity and bracing for a near-eternity of cooling into obscurity (about a 40% chance). Let’s assume that TRAPPIST-1 is a full-blown star. If that’s the case, it’s got a twelve trillion year main-sequence life span ahead of it. Here’s what it’s evolution on the HR diagram will look like, in comparison to other low-mass objects:

An object with solar composition and 0.08 solar masses never turns into a red giant. As time goes on, it maintains a near-constant radius, and slowly burns nearly all of its hydrogen into helium. In roughly 10 trillion years, TRAPPIST-1 will reach a maximum temperature of ~4000K, pushing it briefly toward K-dwarf status for a few tens of billions of years, before eventually running out of fuel and fading out as a degenerate helium dwarf.

At the present moment, the spin angular momentum of TRAPPIST-1 is very close to the summed angular momentum of its seven known planets (both total, to one significant figure, 10^47 g cm^2 s^-1.). The planets, owing to their tight orbital radii, are safe from passing white dwarfs for quadrillions of years in the galactic potential, and are immune to the usual risk of red giant engulfment. A long, slow tidally mediated drama will unfold in which the planets will somehow act out, with resonances and tidal decay, punctuated by Roche-radius destructions and re-accretions, the dictate that the minimum energy configuration places all the system mass at the center and all the system angular momentum out at infinity.

A long-term buy.

Given the current situation, the destruction of planet Earth through an encounter with a black hole is a low-probability scenario that should elicit relatively little concern.

Nonetheless, the industry surrounding black holes and their various associated activities generates a non-negligible economic contribution. By way of setting scale, an article in this week’s New York Times points to the statistic that the total US commercial honeybee pollination industry has an annual value of order $500 million, with slim margins and the ongoing specter of colony collapse disorder. The movie Interstellar, by contrast, generated $675 million in receipts based on a $165 million production budget. Having seen the movie, I would hazard a guess that a significant, if not decisive, factor in the box office draw centered on the numerical calculation of ray bundle propagation through the curved spacetime of a spinning Kerr black hole, as described by James, Tunzelmann, Franklin & Thorne (2015).

Figure 16 from James et al. (2015)

Activities as diverse as the technology development and staffing of LIGO, the awarding of multi-million dollar prizes, and lurid television documentaries are all parts of the thriving Black-Holes-as-a-Business paradigm. Sure, I’m being a little facetious here, but not really… It’s a real phenomenon.

As far as planets are concerned, disasters associated with black hole encounters can be divided into three very distinct categories. Throughout the visible universe, over the course of cosmic time, a very large number of Earth-sized planets have come to untimely demise by crossing the event horizon of a supermassive black hole. Rather preposterously, this was the premise underlying a recent episode of History Channel’s The End. As a practical matter, we would have of order 500 million years of advance notice if a rogue M87-style supermassive black hole — presumably ejected during a 3-body encounter in a massive galaxy merger — were impinging on the Local Group. When an inhabited planet enters an isolated billion solar mass Schwarzschild black hole, there is a period measured in hours where one sails comfortably numbed through a bizarre GR-mediated light show. Things get bad only in the last thirty minutes or so before the encounter with the singularity.

A second genre of black hole disaster occurs whenever a planet encounters an ordinary Cygnus X-1 style black hole, or indeed, any black hole with a mass ranging from roughly planetary heft to millions of solar masses. In these events, a planet is generally tidally shredded before encountering the event horizon, and from an on-the-ground perspective, the histrionics fall broadly into the type experienced by the planet Theia ~4.51 billion years ago. In both the near term, as well as the extremely long term, Earth stands effectively zero chance of succumbing to black hole-mediated tidal destruction.

Primordial black holes might actually pose a non-absurdist threat. While still fully speculative, it has been proposed that density fluctuations in the early universe created black holes, and in the 10^17 to 10^26 gram mass range there is currently little actual constraint on their existence. Papers have been published that elucidate the seismic disturbances that would result, for example, from the collision of a 10^15 gram black hole traveling at 200 km/s through the Earth.

Generation of seismic waves in Earth following the passage of a 10^15 gram black hole with speed ~200 km/sec From Figure 2 of Luo et al. 2012.

In general, an encounter with a primordial black hole provides a hydrogen bomb-level of devastation at the entry and exit points, but no further consequences as the marauding black hole speeds away into interstellar space. In the early 1970s, a black hole encounter was briefly a credible model (at the got published in Nature level of credibility) for explaining the Tunguska impact.

A singularly unfortunate scenario results if Earth manages to capture a primordial black hole into an orbit with perigee inside Earth. This is hard, but not impossible, if the black hole is a member of a binary pair. The physics of the capture would be similar to the event that is thought to have given rise to Triton in orbit around Neptune. For those interested in details, I attach here some irresponsible order-of-magnitude notes that outline what I believe would happen if Earth were to collect an Enceladus-mass black hole in its thrall.

Mauna Kea from Mana Road

Time slips past. The discovery of 51 Pegasi b and the heady early days of planet detection are now more than two decades gone. The pulsar planets have been known for a full quarter century, and N=10,000 is the next milestone for the catalogs.

It’s fair to say that there have been amazing discoveries in twenty years, culminating with an Earth-mass planet in a temperate orbit around the closest star to the Sun. And there’s even significant funding to jump start the design of a probe that can go there.

Yet in the background, as the breakthroughs rolled in, the Keck I Telescope was gradually accumulating Doppler measurements of hundreds of nearby Sun-like stars with HD designations and magnitudes measured in the sevens and eights. This data is as important for what it shows (scores of planets) as for what it doesn’t show (a profusion of planets with Jupiter-like masses and orbits). There are several reasons why our Solar System is unusual, and Jupiter is one of them.

From Rowan+ 2016

The Lick Carnegie Exoplanet Survey has just released a uniformly reduced compendium of 60,949 precision Doppler Velocities for 1,624 stars that have been observed using the iodine cell technique with HIRES at the Keck-I telescope, with an accompanying paper to appear in the Astronomical Journal. The velocities are all freely available on line here, ready to be explored with the Systemic Console. They contain hundreds of intriguing, possibly planetary signals, including a strong hint of a super-Earth orbiting Lalande 21185, the fourth-closest stellar system.

Stay tuned…

A year ago, last January, Konstantin Batygin and Mike Brown lit up the Internet with their dossier of evidence for Planet Nine. Their conclusion was electrifying: An as-yet undetected super-Earth may be lurking a light week away in an eccentric orbit far beyond Neptune. Their article in the Astronomical Journal generated intense interest, including 311,371 (and counting) downloads of a .pdf containing a bracing dose of secular perturbation theory, along with push notifications from the likes of the New York Times and NPR to devices worldwide.

A solar system super-Earth would be extraordinary for a whole slew of reasons. Indeed, an astronomical problem of any stripe that is at once so compelling and potentially so dramatically resolvable comes along extremely rarely. The disparate clues that spurred development of the six-parameter lambda-CDM cosmological model form the only relatively recent example that I can think of. Planet Nine, however, does concordance cosmology one better by demanding six orbital elements plus a mass, and in addition, it’s not “big science”. At magnitude V~23, there are a whole range of telescopes that can potentially spot it. This low barrier to entry exerts a unique hold on one’s interest.

As 2017 gets underway, it’s a good time to review some of the Planet Nine developments that have occurred over the past year. In particular, what are the odds that it’s out there, and how close are we to establishing whether it actually exists? My feeling is that right now, the chance of a big announcement is peaking at a somewhat less than 1% per day.

The outer solar system is neither empty nor unsurveyed. Over two thousand trans-Neptunian bodies are now tracked and listed by JPL and by the Minor Planet Center. Many of these objects are minor indeed, with diameters no more than a few hundred kilometers across, despite being visible at distances out to roughly 100 AU. It thus seems counter-intuitive that a full-blown super-Earth could go undetected in the midst of such a crowd. Yet because we’re dealing with the Sun’s reflected light, the falloff in apparent brightness in the outer solar system with distance is severe, going as 1/r^4. If Neptune were lofted from 30 to 900 AU distance, its apparent brightness in our skies would decrease by a factor of 30^4=810,000, a near-millionfold hit that would place it near the 23rd magnitude. Last year, I wrote,

As for the planet itself? A frigid as-yet unseen world with ten times the mass of Earth. Its twenty thousand year orbit is eccentric, and at aphelion it languishes with 500 m/s speed, drifting slowly against the spray of background stars. Its cloud tops glow in the far infrared, a mere 40 Kelvin above absolute zero. At the far point of its orbit, it is invisible to WISE in all its incarnations, and far fainter than the 2MASS limits. Obscure. In the optical, it reflects million-fold diminished rays of the distant Sun to shine in the twenty fourth magnitude. Dim, indeed, but not impossibly dim… Traces of its presence might already reside on the tapes, in the RAID arrays, suspended in the exabyte seas, if one knows just where and how to look.

Or, more succinctly, its brightness depends on albedo (reflectivity), radius, and its current distance via

A handful of Kuiper Belt Objects have been found that are as dim or even dimmer than Planet Nine is expected to be. Trujillo and Sheppard’s discovery paper for 2012 VP 113 gives the details of how one such search was carried out. A wide-field camera on a large telescope takes repeated pictures of regions of the sky located “at opposition”, roughly 180 degrees away from the Sun. For VP 113, this was done using the DECam at CTIO, which has a 2.7 square-degree field of view and was exposed long enough so that 50% of the 24.5th magnitue objects present in the field would register on each image. Three images spanning about 3.5 hours in total were taken of each field and then inspected for moving objects by a computer. A fraction of the motion on the sky stems from the orbital trajectory of the distant object, but much more importantly, it also arises from the parallax shift generated by Earth’s motion. For an object at 100 AU, this amounts to 1.25 arc seconds per hour, whereas a body orbiting out at 1000 AU will move 0.125 arc seconds per hour. Planet Nine thus moves so slowly that many conventional KBO surveys, while sensitive enough to detect its reflected light, observe with a cadence that is too high to catch its motion. To find it using a wide-field camera, one is best-off taking images separated by at least a full night.

If Planet Nine is out there, it also produces its own infrared radiation. In this article, Jonathan Fortney and collaborators used their atmospheric modeling software to compute what Planet Nine might look like across a full range of wavelengths. The take-away is that with an intrinsic temperature of roughly 40K, Planet Nine’s atmosphere is likely cold enough for methane to condense out into layer of clouds. Rayleigh scattering from pristine hydrogen-rich air above the clouds would thus render the planet quite reflective at optical wavelengths, modestly boosting its detectability over a Neptune-clone at similar distance. Methane condensation also leads to a planet that is potentially twenty orders of magnitude brighter at 3.5 microns than a 40K black body would lead one to expect, generating daunting long-shot odds that it might be visible in the WISE satellite’s W1-band data sets. Aaron Meisner led an effort to very carefully sift the WISE data for a detection. And although their initial survey of 2,000 square degrees has turned up null, they report that they are in the process of extending the search to the full sky.

Planet Nine’s gravitational influence falls off less quickly with distance than does its reflected light. Neptune’s 1846 discovery, furthermore, presents an intriguing precedent. Neptune’s sky position was readily pinned down via its gravitational effects, despite the fact that its orbit was only roughly approximated. Perhaps something similar can be done to pinpoint the current direction to Planet Nine.

Any object orbiting beyond the Kuiper Belt is far enough away that over a time scale measured in years or even decades, its position is effectively static. As a result, Planet Nine would produce an essentially fixed tidal acceleration across the inner solar system. If it is 900 AU away and has ten Earth masses, the Earth experiences a component of acceleration toward it of 2×10^-11 cm/s^2, amounting to a displacement, d=1/2at^2 of roughly a football field per year. As far as our space situational awareness goes, 100 meters is quite a lot. The problem, however, is the entire solar system is being drawn toward planet Nine, and one needs to look for the differential — tidal — acceleration. For example, if Planet Nine currently lies in the direction of Saturn, then Saturn, being closer, will accelerate toward Planet Nine ~2% faster than they Earth does, and over time, sensitive measurements can potentially tease this out.

A few weeks after the appearance of the Batygin-Brown paper, Agnes Fienga and collaborators published a much-discussed paper that hinted at a possible sky position for Planet Nine. Their analysis used telemetry sent back over the years by the Cassini probe, which has been orbiting in the Saturnian system since 2004. Cassini’s ranging data give a very precise location for the spacecraft, and by extension, they transmit precise locations for Saturn. Saturn’s location, in turn, depends on how it is being accelerated by everything else in the solar system and beyond, including Planet Nine (if it’s out there). Fienga et al. discovered that they could get a modest yet tantalizing improvement in their model fit’s residuals to the Cassini probe’s ranging data if they added Planet Nine to their model at a location on the fiducial Batygin-Brown orbit at a current distance of ~622 AU from the Sun in the direction of the constellation Cetus:

In the weeks after the publication of the Fienga et al. paper, JPL issued a press release stating that “NASA’s Cassini spacecraft is not experiencing unexplained deviations in its orbit around Saturn.” In October, a JPL team led by William Folkner presented a poster paper at the Pasadena DPS meeting that made the case that the Cassini residuals show no signal from Planet Nine. They found that if it exists on the Batygin-Brown orbit, it needs to have either a mass lower than the 10 Earth mass value suggested by Batygin and Brown, or alternately, a current location near aphelion at a distance of 1,000 AU or more. A detailed paper from this group is rumored to be forthcoming.

In March, Renu Malhotra, Kathryn Volk, and Xianyu Wang posted a paper to arXiv that pointed out a remarkable, and until-then unnoticed fact:

The four longest period Kuiper belt objects have orbital periods close to integer ratios with each other. A hypothetical planet with orbital period ?17,117 years, semimajor axis ?665 AU, would have N/1 and N/2 period ratios with these four objects. The orbital geometries and dynamics of resonant orbits constrain the orbital plane, the orbital eccentricity and the mass of such a planet, as well as its current location in its orbital path.

This seemed like a critical, potentially breakthrough-level clue, and I have spent the last couple months working with Yale graduate student Sarah Millholland to see whether more detail — and in particular, a definitive sky location — can be teased out of the ideas presented in Malhotra et al.’s paper. Our own paper will appear soon in the Astronomical Journal, and is currently available on arXiv.

The real number line is dense with integer ratios, and the orbital periods of the most distant and most recently discovered Kuiper belt objects are not all that well determined. It thus seems possible that the period ratios of the known KBOs might simply have arisen by chance. We devised a Monte-Carlo simulation to determine the odds, and the answer is encouraging: there’s less than a 2% chance that we’re looking at a random distribution. It’s very plausible that Sedna is in 3:2, 2000 CR105 is in 5:1, 2012 VP113 is in 4:1, 2004 VN112 is in 3:1, and 2001 FP 185 is in 5:1 resonance with something having an orbital period of 16,725 years and a semi-major axis a~654 AU.

If this hypothesis is to work out, the unseen perturbing body needs to have the right orbit, the right location, and the right mass to maintain the resonances and keep the apsidal alignment of the distant KBO population intact. We carried out a sobering 3×10^17 ergs worth of integrations to pin down Planet Nine’s likely sky position, current distance, and visual magnitude. In short, if it’s out there, it’s probably just dimmer than V=23, 950 AU away, near the celestial equator, and at a right ascension of roughly 40 degrees. If asked for the odds that it’ll be found within 20 degrees of this spot, I would cite that most perfectly frustrating of percentages, 68.3.

Sarah has put together a manipulable 3D model of the orbit, along with more discussion. Until the real thing shows up, it’s the premiere Planet Nine destination.

It was like the opening pages of a thriller. In the gathering dusk of an early winter evening last year, the postman handed me a package with a Belgian postmark and a cryptic symbol.

Inside, wrapped in layers of translucent paper, were two books, both in French. Nothing else. Needless to say, I was intrigued…

Dialectique du Monstre by Sylvain Piron revealed itself (with the use of Google Translate) to be a close study of the life and work of Opicinus de Canistris (1296-c.1353), a mysterious, psychologically tormented clerical official attached to the Avignon Papacy. The book is filled with reproductions of Opicinus’ elaborate parchment diagrams, which are like figments of the fever dreams of Archimedes or Leonardo; bizarre maps and masses of symbols harboring intimations just out of reach, a taproot into unseen connections between individuals, cities, whole worlds.

A while back, I wrote of the Electra Hypothesis, the idea that as the planet binds itself and its bit processes into an ever more interconnected web of radio links and optical fiber, its computational edges and nodes will develop into something of a sequel to Lovelock’s Gaia. Although layered in ambiguity, and separated by a gulf of time and mindset, Canistris seemed to have been drawn toward a similar notion.

The second book, opaquely titled 6/5, vaults the web of interconnection squarely into the modern world. Written by Alexandre Laumonier, the Sniper in Mahwah, it is a history of modern electronic markets and the rise of machines. In contrast to Dialectique du Monstre, it connects not to the past but to the future. The changes, computerization, machine learning, algorithms, that have swept over the financial markets are now spreading ever more thoroughly into an ever-wider range of endeavor.

The title 6/5 is a compressed code for a set of developments that have unfolded mostly out of view. The first part of the book, 6, refers to the floored number of milliseconds for a signal to travel from Chicago to New York on the fastest optical fiber. The second section, 5, alludes to the faster-than-glass signaling over the same route by microwave, which has now dropped two notches below that round number, to 3.982, within a sliver of the vacuum latency on the great circle connecting the endpoints.

A node of Electra’s graph. Hundreds of billions of dollars in coded trades rush daily through the towers of this Appalachian ridgeline.

For nearly a year, I’ve left a latin phrase at the top of the site… Pythagoreorum quaestionum gravitationalium de tribus corporibus nulla sit recurrens solutio, cuius rei demonstrationem mirabilem inveniri posset. Hanc blogis exiguitas non caperet.

The translation of the phrase is connected to the pythagorean three-body problem, another obliquely related topic involving descending integers that has seen regular rotation on oklo.org. A remarkable feature of Burrau’s original version of the problem (masses of 3, 4, and 5 started from rest under Newtonian gravity at the vertices opposite the sides of a 3-4-5 right triangle) is that the solution is almost, but not quite periodic. At time, T~15.830, bodies 4 and 5 almost collide, while body 3 nearly comes to rest. In a paper from 1967, Szebeheley and Peters show that a slight adjustment of the initial positions is sufficient to transform the situation into one that repeats itself endlessly.

The integers 3, 4, and 5 are a single example drawn from the infinite set of Pythagorean triples, combinations of integers that correspond to the lengths of the the sides of right triangles. Each triple defines a variation on the original Pythagorean three-body problem, and I believe it’s the case that not a single member of this infinity of initial conditions will generate a periodic solution.

Scatter plot of the legs (a,b) of the first Pythagorean triples with a and b less than 6000. Negative values are included to illustrate the parabolic patterns. (Source: Wikipedia)

With a nod to Fermat, this assertion can be recast as a conjecture:

There exist no periodic solutions to any of the Pythagorean gravitational three-body problems. There may exist a truly marvelous demonstration of this proposition that this weblog has no space to contain.

Or at least it is true for every spot check that I’ve computed. For example, the tortured path of 20-21-29:

To place a tiny obstacle in the crush of progress, a translation into Latin beyond what Google can yet achieve seemed in order. I contacted Alexandre, who forwarded the request to Sylvain, who transmitted the following:

Pythagoreorum quaestionum gravitationalium de tribus corporibus nulla sit recurrens solutio, cuius rei demonstrationem mirabilem inveniri posset (could be found) /esse posset (could be). [Le verbe exstare (exister, être présent avec force) conviendrait mal àcette modalité.] Hanc blogis exiguitas non caperet.

Translation in English of “[Le verbe exstare (exister, être présentavec force) conviendrait mal à cette modalité]”: the verb “exist” would not be good here. inveniri posset seems to be the best solution.