Author Archive

Third-closest known transiting planet detected

March 7th, 2017 3 comments

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.

Categories: worlds Tags:

A signal amplified

February 25th, 2017 1 comment

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’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.

Categories: Electra, worlds Tags:

Not Fade Away

February 22nd, 2017 3 comments

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.

Categories: worlds Tags:

Black Hole Disasters

February 19th, 2017 Comments off

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.

Categories: worlds Tags:

60,949 Doppler Velocities of 1,624 Stars

February 13th, 2017 Comments off

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…

Categories: worlds Tags:

Planet Nine — A One Year Update

January 21st, 2017 4 comments

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.

Categories: worlds Tags:


December 22nd, 2016 Comments off

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 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.

Categories: Electra, worlds Tags:

A conjecture

February 14th, 2016 4 comments


Pythagoreorum quaestionum gravitationalium de tribus corporibus nulla sit recurrens solutio, cuius rei demonstrationem mirabilem inveniri posset. Hanc blogis exiguitas non caperet.

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Listening in

February 11th, 2016 Comments off


As with everyone else, LIGO made my day.

It’s interesting that transverse waves of spatial strain — ripples in spacetime — are consistently described as “sounds” in the media presentations. For example, the APS commentary accompanying the Physical Review Letter on GW150914 is entitled The First Sounds of Merging Black Holes.

Quite frankly, Python is a threat to the scientific guild. What used to require esoteric numerical skills — typing in recipes in Fortran and stitching them together, or licensed packages, “seats”, always priced to keep the riff-raff out, now comes completely for free with a one-click install of an Anaconda distribution. All this stuff places anyone just a few lines away from hearing the sound on Figure 1, which APS posted as a teaser while they scrambled to get servers on line to handle the crush of download demand:


Here’s what I did this morning to “hear” the signal while waiting for the servers to free up, so that I could download the full paper.

(1) Take a screen shot of the Hanford signal:

Screen Shot 2016-02-11 at 9.09.02 PM

(2) Upload the screenshot to WebPlotDigitizer, and follow the directions to sample the waveform. After a bit of fooling around with the settings, the web app gave me a .csv file that I named ligoDigitalData.csv. It contains containing 1712 x-y samples of the waveform. I added a header line listing “time” as the first column, and “amplitude” as the second column.


(3) Fire up an iPython notebook, import a few packages, import the file, and check that it looks right:


(4) The “wave” package packs integer samples into a .wav format file. A plain vanilla implementation at 4.41 kHz 16 bit sampling looks like this. Not exactly audiophile quality, but so cool nonetheless:


This produces a .wav file:

Now of course, one shouldn’t expect that a waveform that you can silkscreen onto a T-shirt is going to sound like the THX Deep Note

And how ’bout them prediction markets? Over at Metaculus, the consensus among 99 predictors was that there was a 68% chance that the Advanced LIGO Team would publicly announce a 5-sigma (or equivalent) discovery of astrophysical gravitational waves by March 31, 2016. According to the Phys Rev Letter, the significance of the GW150914 detection is 5.1 sigma, so just over the bar. The question is now closed, and some users are going to be racking up some points.

If you missed out, there’s plenty more markets to try your hand at. New boson at the LHC anyone?

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June 25, 2522

February 9th, 2016 1 comment


I remember the eclipse of February 26th, 1979 very clearly. In Urbana, Illinois, the moon covered 80% of the solar disk. It was a clear sunny day, and the crescent Sun projected magically through a pinhole into the 6th grade classroom.

Later, looking at a map, we noticed with considerable pride that a total eclipse will track over Southern Illinois on August 21, 2017. The date had an unreal, distant, science fiction feel to it.

Anthony recently posted a question on Metaculus that’s provocative, slightly creepy, and seems designed to transcend the day-to-day:

Will there be a total solar eclipse on June 25, 2522?

created by Anthony on Jan 28 2016

According to NASA, the next total solar eclipse over the U.S. will be August 14, 2017. It will cut right through the center of the country, in a swathe from South Carolina to Oregon.

A little over 500 years later, on June 25, 2522, there is predicted to be a nice long (longest of that century) solar eclipse that will pass over Africa.

In terms of astronomy, the 2522 eclipse prediction is nearly as secure at the 2017 one: the primary uncertainty is the exact timing of the eclipse, and stems from uncertainties in the rate of change of Earth’s rotation – but this uncertainty should be of order minutes only.

However, 500 years is a long time for a technological civilization, and if ours survives on this timescale, it could engineer the solar system in various ways and potentially invalidate the assumptions of this prediction. With that in mind:

Will there be a total solar eclipse on June 25, 2522?

For the question to resolve positively, the calendar system used in evaluating the resolution must match the Gregorian calendar system used in the eclipse predictions; the eclipse must be of Sol by a Moon with at least 95% of its original structure by volume unaltered, and must be observable from Earth’s surface, with “Earth” defined by our current Earth with at least 95% of its original structure by volume altered only by natural processes.

What do you think? Head over to Metaculus and make your prediction count.

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