## 3:45

I’ve written several times, most recently last year, about the Pythagorean Three-Body Problem, which has just marked its first century in the literature (See Burrau, 1913).

Assume that Newtonian Gravity is correct. Place three point bodies of masses 3, 4, and 5 at the vertices of a 3-4-5 right triangle, with each body at rest opposite the side of its respective length. What happens?

The solution trajectory is extraordinary in its intricate nonlinearity, and lends itself to an anthropomorphic narrative of attraction, entanglement and rejection, with bodies four and five exiting to an existential eternity of No Exit, and body three consigned to an endless asymptotic slide toward constant velocity.

This past academic year, I worked with Ted Warburton, Karlton Hester, and Drew Detweiler to stage an interpretive performance of the problem, along with several of its variations. The piece was performed by UCSC undergraduates and was part of the larger Blueprints year-end festival. Here is a video of the entire 17 minute program.

The first of the four segments is an enactment of the standard version of the problem (As set above), and was done with a ballet interpretation to underscore that this is the “classical” solution. Prior to joining the faculty at UCSC, Ted was a principal dancer at the American Ballet Theater, and so the cohoreography was in an idiom where he has a great deal of experience.

The score for the performance was performed live, and is based wholly on percussion parts for each of the three bodies. The interesting portion of the dynamics is mapped to 137.5 measures, which satisfyingly, last for three minutes and forty five seconds.

The nonlinearity of the Pythagorean Problem gives it a sensitive dependence to initial conditions. It is subject to Lorenz’s Butterfly Effect. For the second segment of the performance, we chose a version of the problem in which body three is given a tiny change in its initial position. Over time, the motion of the bodies departs radically from the classical solution, and the resolution has body three leaving with body five, while body four is ejected. A more free-flowing choreography was drawn on to trace this alternate version.

A fascinating aspect of the problem is that while the solution as posed is “elliptic-hyperbolic”, there exist nearby sets of initial conditions in which the motion is perfectly periodic, in the sense that the bodies return precisely to their initial positions, and the sequence repeats forever. In the now-familiar solution to the classical version of the problem, the bodies manage to almost accomplish this return to the 3-4-5 configuration at a moment about half-way through the piece. This can be seen just after measure 65, at which time body 4 (yellow), body 5 (green), and body 3 (blue) are nearly, but are not exactly, at their starting positions, and are all three moving quite slowly:

If the bodies all manage to come to rest, then the motion must reverse and retrace the trajectories like a film run backward. With this realization, one can plot the summed kinetic energy of the bodies, which is a running measure of the amount of total motion. Note the logarithmic y-axis:

The bodies return close to their initial positions at Time = 31, at which time there is a local minimum in the total kinetic energy.

Next, look at the effect of making a small change in the initial position of one of the bodies. To do this, I arbitrarily perturbed the initial x position of body 3 by a distance 0.01 (a less than one percent change), and re-computed the trajectories. The kinetic energy measurements of this modified calculation are plotted as gray. During the first half of interactions the motion is extremely similar, but that the second half is very different. Interestingly, the gray curve reaches a slightly deeper trough at Time = 31. The small change has thus created a solution that is slightly closer to the pure periodic ideal.

I next used a variational approach to adjust the initial positions in order to obtain solutions that have progressively smaller Kinetic energy at time 31. In this way, it’s easy to get arbitrarily close to periodicity. The motion in a case that is quite close to (but not quite exactly at) the periodic solution is shown just below. After measure 65, the bodies arrive very nearly exactly at their initial positions, and, for the measures shown in the plot below, they have started a second, almost identical run through the trajectories.

The perfectly periodic solution occurs when bodies 4 and 5 experience a perfect head-on collision at time ~15 (around measure 33). If this happens, bodies 4 and 5 effectively rebound back along their trajectory of approach, and the motion retraces, therefore repeating endlessly. Here’s the action which shows the collision:

Ted suggested that Tango and Rhumba could be the inspiration for the choreography of the perfectly periodic solution. I was skeptical at first, but it was immediately evident that this was a brilliant idea. The precision of the dancing is exceptional, and the emotion, while exhibiting passion, is somehow also controlled and slightly aloof. No jealousy is telegraphed by motion, allowing the sequence to repeat endlessly in some abstract plane of the minds eye.

Categories: worlds Tags:

## Malbolge

Fall quarter at UCSC has arrived, and with it, the latest iteration of my astrophysical fluid dynamics course.

This class covers the workings of bodies that are composed of gas, ranging from molecular clouds and accretion disks to stars and giant planets. These objects are complicated enough so that numerical calculations can often help to generate insight, so I’ve traditionally distributed some simple numerical routines for use in the class problem sets.

My first exposure to computers was in the mid-1970s, when several PLATO IV terminals were set up in my grade school in Urbana. My mid-1980s programming class was taught in standard Fortran 77. Somehow, these formative exposures, combined with an ever-present miasma of intellectual laziness, have ensured that Fortran has stubbornly remained the language I use whenever nobody is watching.

Old-style Fortran is now well into its sixth decade. It’s fine for things like one-dimensional fluid dynamics. Formula translation, the procedural barking of orders at the processor, has an archaic yet visceral appeal.

Student evaluations, however, tend to suggest otherwise, so this year, everything will be presented in python. In the course of making the sincere attempt to switch to the new language, I’ve been spending a lot of time looking at threads on stackoverflow, and in the process, somehow landed on the Wikipedia page for Malbolge.

Malbolge is a public domain esoteric programming language invented by Ben Olmstead in 1998, named after the eighth circle of hell in Dante’s Inferno, the Malebolge.

The peculiarity of Malbolge is that it was specifically designed to be impossible to write useful programs in. However, weaknesses in this design have been found that make it possible (though still very difficult) to write Malbolge programs in an organized fashion.

Malbolge was so difficult to understand when it arrived that it took two years for the first Malbolge program to appear. The first Malbolge program was not written by a human being, it was generated by a beam search algorithm designed by Andrew Cooke and implemented in Lisp.

That 134 character first program — which outputs “Hello World” — makes q/kdb+ look like QuickBasic:

(‘&%:9]!~}|z2Vxwv-,POqponl$Hjig%eB@@>}=m:9wv6wsu2t |nm-,jcL(I&%$#”CB]V?Txuvtt Rpo3NlF.Jh++FdbCBA@?]!~|4XzyTT43Qsqq(Lnmkj”Fhg${z@\> At first glance, it’s easy to dismiss Malbolge, as well as other esoteric programming languages, as a mere in-joke, or more precisely, a waste of time. Yet at times, invariably when I’m supposed to be working on something else, I find my thoughts drifting to a hunch that there’s something deeper, more profound, something tied, perhaps, to the still apparently complete lack of success of the SETI enterprise. I’ve always had an odd stylistic quibble the Arecibo Message, which was sent to M13 in 1974: It might have to do with the Bigfoot-like caricature about 1/3rd of the way from the bottom of the message. Is this how we present to the Galaxy what we’re all about? “You’ll never get a date if you go out looking like that.” Fortunately, I discovered this afternoon that there is a way to rectify the situation. The Lone Signal organization is a crowdfunded active SETI project designed to send messages from Earth to an extraterrestrial civilization. According to their website, they are currently transmitting messages in the direction of Gliese 526, and by signing up as a user, you get one free 144-character cosmic tweet. I took advantage of the offer to broadcast “Hello World!” in Malbolge to the stars. Categories: worlds Tags: ## Central Limit Theorem August 21st, 2013 Comments off We’re putting the finishing touches on a new research paper that deals with an old oklo.org favorite: HD 80606b. The topic is the Spitzer Telescope’s 4.5-micron photometry taken during the interval surrounding the planet’s scorching periastron passage, including the secondary eclipse that occurs several hours prior to the moment of closest approach (see the diagram just below). I’ll write a synopsis of what we’ve found as soon as the paper has been refereed. In writing the conclusion for the paper, we wanted to try to place our results in perspective — the Warm Mission has been steadily accumulating measurements of secondary eclipses. There are now over 100 eclipse depth measurements for over 30 planets, in bandpasses ranging from the optical to the infrared. A set of secondary eclipse measurements at different bandpasses amount to a low-resolution dayside emission spectrum of an extrasolar planet. When new measurements of secondary eclipse depths for an exoplanet are reported, a direct comparison is generally made to model spectra from model atmospheres of irradiated planets. Here is an example from a recent paper analyzing Warm Spitzer’s measurements of WASP-5: Dayside planet/star flux ratio vs. wavelength for three model atmospheres (Burrows et al. 2008) with the band-averaged flux ratios for each model superposed (colored circles). Stellar fluxes were calculated using a 5700 K ATLAS stellar atmosphere model (Kurucz 2005). The observed contrast ratios are overplotted as the black circles, with uncertainties shown. The model parameter kappa is related to the atmosphere’s opacity, while p is related to the heat redistribution between the day and night sides of the planet (Pn = 0.0 indicates no heat redistribution, and Pn = 0.5 indicates complete redistribution). As is certainly the case in the figure just above, the atmospheric models that are adopted for comparison often have a high degree of sophistication, and are informed by a substantial number of free parameters and physical assumptions. In most studies, some of the atmospheric parameters, such as the presence or absence of a high-altitude inversion-producing absorber, or the global average efficiency of day-to-night side heat redistributions are varied, whereas others, such as the assumption of hydrostatic equilibrium and global energy balance, are assumed to be settled. Invariably, the number of implicit and explicit parameter choices tend to substantially exceed the number of measurements. This makes it very hard to evaluate the degree to which a given, highly detailed, planetary atmospheric model exhibits any actual explanatory power. The central limit theorem states that any quantity that is formed from a sum of n completely independent random variables will approach a normal (Gaussian) distribution as n becomes large. By extension, any quantity that is the product of a large number of random variables will be distributed approximately log-normally. We’d thus expect that if a large number of independent processes contribute to a measured secondary eclipse depth, then the distribution of eclipse depth measurements should be either normally (or possibly log-normally) distributed. The “independent processes” in question can arise from measurement errors or from systematic observational issues, as well as from the presence of any number of physical phenomena on the planet itself (such as the presence or absence of a temperature inversion layer, or MHD-mediated weather, or a high atmospheric C/O ratio, etc.). The plot just below consolidates more than 100 existing secondary eclipse measurements onto a single diagram. Kudos to exoplanets.org for tracking the secondary eclipse depths and maintaining a parseable database! The observed systems are ordered according to the specific orbit-averaged flux as expressed by the planetary equilibrium temperaturs — the nominal black-body temperature of a zero-albedo planet that uniformly re-radiates its received orbit-averaged stellar energy from its full four-pi worth of surface area. The secondary eclipse depths in the various bands are transformed to flux ratios, F, relative to what would be emitted from a black-body re-radiator. If all of the measurements were perfect, and if all of the planets were blackbodies, all of the plotted points would lie on the horizontal line F=1. It’s somewhat startling to see that there is little or no systematic degree of similarity among the measurements. One is hard pressed to see any trends at all. Taken together, the measurements are consistent with a normal distribution of flux ratios relative to a mean value F=1.5, and with standard deviation of 0.65: This impression is amplified by the diagram just below, which is a quantile-quantile plot comparing the distribution of F values to an N(0,1) distribution. The nearly gaussian distribution of flux ratios suggests that the central limit theorem may indeed find application, and imparts a bit of uneasiness about comparing highly detailed models to secondary eclipse measurements. I think we might know less about what’s going on on the hot Jupiters than is generally assumed… Categories: worlds Tags: ## arrived August 10th, 2013 3 comments One prediction regarding exoplanets that did hold true was the Moore’s-Law like progression toward the detection of planets of ever-lower mass. More than seven years ago, not long after the discovery of Gliese 876 d, the plot of Msin(i) vs. year of discovery looked like this: With a logarithmic scale for the y-axis, the lower envelope of masses adhered nicely to a straight line progression, pointing toward the discovery of the first Earth-mass exoplanet sometime shortly after 2010. The honors went, rather fittingly, last year, to Alpha Cen B b. Here’s an update to the above plot. Planets discovered via Doppler velocity only are indicated in gray, transiting planets are shown in red… The data for the plot were parsed out of the very useful exoplanets.csv file published at exoplanets.org. And wait, what’s going on with that point in 1993? See http://en.wikipedia.org/wiki/Pollux_b. Categories: worlds Tags: ## Etymology August 6th, 2013 7 comments I think it’s worth making an attempt to coin a term for these “ungiant” planets that are, effectively by default, largely being referred to as super-Earths, a term which brings to mind Voltaire’s remark regarding the Holy Roman Empire. Planets in the category: 1. Have masses between ~1% and ~10% of Jupiter’s mass. 2. Have unknown composition, even if their density is known. Ideally, a term for such planets would: 3. Have a satisfying etymology springing from the ancient Greek. 4. Not be pretentious, or, much more critically, not be seen as being pretentious. Simultaneously satisfying conditions 3 and 4 is certainly not easy, and indeed, may not be possible. (See, e.g., http://arxiv.org/abs/0910.3989) I’ve noticed that the esoteric efforts to describe the interiors of these planets — in the absence of any data beyond bulk density — effectively boil down to Robert Fludd’s 1617 macrocosm of the four classical elemental spheres: This led me to look into Empedocles’ four elements themselves, see, e.g., here. Specifically, can a term of art for the planets of interest be constructed from the original Greek roots? The following table on p. 23 of Wright, M. R., Empedocles: The Extant Fragments, Yale University Press, 1981, contains various, possibly appropriate, possibilities: To get going, I had to refer to the rules for romanization of Greek. Initial attempts to coin names (while abundantly satisfying requirement #3 above) have so far failed miserably on requirement #4: chonthalaethian planets, ambroaethic planets, gaiapontic planets. Yikes! The Tetrasomia, or Doctrine of the Four Elements, alludes to the secure fact that these planets are unknown compounds of metal, rock, ices, and gas. Tetrian planets, maybe? Suggestions welcome… Categories: worlds Tags: ## The Frozen Earth April 20th, 2013 1 comment More than a decade ago, Fred Adams and I wrote a paper that wallowed into the slow motion disasters that can potentially unfold if another star or stars passes through the solar system. Here’s the abstract: Planetary systems that encounter passing stars can experience severe orbital disruption, and the efficiency of this process is enhanced when the impinging systems are binary pairs rather than single stars. Using a Monte Carlo approach to perform more than 200,000 N-body integrations, we examine the ramifications of this scattering process for the long-term prospects of our own Solar System. After statistical processing of the results, we estimate an overall probability of order 2×10^5 that Earth will find its orbit seriously disrupted prior to the emergence of a runaway greenhouse effect driven by the Sun’s increasing luminosity. This estimate includes both direct disruption events and scattering processes that seriously alter the orbits of the jovian planets, which force severe changes upon the Earth’s orbit. Our set of scattering experiments gives a number of other results. For example, there is about 1 chance in 2 million that Earth will be captured into orbit around another star before the onset of a runaway greenhouse effect. In addition, the odds of Neptune doubling its eccentricity are only one part in several hundred. We then examine the consequences of Earth being thrown into deep space. The surface biosphere would rapidly shut down under conditions of zero insolation, but the Earth’s radioactive heat is capable of maintaining life deep underground, and perhaps in hydrothermal vent communities, for some time to come. Although unlikely for Earth, this scenario may be common throughout the universe, since many environments where liquid water could exist (e.g., Europa and Callisto) must derive their energy from internal (rather than external) heating. As one might expect, our scholarly efforts generated only a middling interest from the astronomical community, which soon faded and froze altogether. Science writers, on the other hand sometimes run across the article and write with questions. I am doing a piece on rogue planets and the scenario that earth might become a rogue planet. I have found some stuff on this on the web and learned that you have done some research on rogue planets. 1. Why do you think rogue planets are so interesting? From an aesthetic standpoint, there’s something compelling about a world drifting cold and alone through the galaxy, or even through intergalactic space. From a more practical standpoint, if rogue planets are common (as it appears may possibly be the case from the micro-lensing results) it is possible that the nearest extrasolar planet is not orbiting a nearby star, but is rather travelling through the Sun’s immediate galactic neighborhood, say within a few light years of the solar system. 2. Could earth become a rogue planet, and is there any guess, how probable this is? Let’s assume it would happen, what would most probably be the reason for that? Earth could become a rogue planet if the solar system suffers a close approach by another star (or binary star). If another star passes within ~1 Earth-Sun distance from the Earth, then there is a good chance that the Earth would wind up being ejected into interstellar space. Fortunately, close encounters between stars are extremely rare. There is about a 1/100,000 chance that Earth will suffer this fate during the next five billion years. Those are very low odds, so in the grand scheme of things, we are in an extremely safe position. If we scale the galaxy down by a factor of ~10 trillion, then individual stars are like grains of sand separated by kilometers of empty space, and moving a meter or so per year. It’s clear that in such a system, a sand grain will drift for quite a long time before it comes close to another sand grain. 3. Could you speculate on how a human being on earth would experience the process of earth being kicked out of the solar system? There would be plenty of warning. With our current capabilities for astronomical observation, the interloping star would be observed tens of thousands of years in advance, and Earth’s dynamical fate would be quite precisely known centuries in advance. The most dramatic sequence of events would unfold over a period of about two or three years. Let’s assume that the incoming star is a red dwarf, which is the most common type of star. Over a period of months the interloping star would gradually become brighter and brighter, until it was bright enough to provide excellent near-daytime illumination with an orange cast whenever it is up the sky by itself. It’s likely that the size of its disk on the sky would become — for a few weeks — larger than the size of the full moon, and vastly brighter. Like the Sun, it would be too bright to look at directly. After several more months, one would start to notice that the seasons were failing to unfold normally. Both the Sun and the Red Dwarf would gradually draw unambiguously smaller and fainter in the sky. After a year, the warmth of the sun on one’s face would be gone, and it would be growing colder by the day… Over a period of several more years, the Sun would gradually appear more and more like a brilliant star rather a life-giving orb. A winter, dark like the Antarctic winter, but without end, and with ever-colder conditions would grip the entire Earth. 4. What do you expect, how long humans could survive such an incident? The Earth could not support its current population, but with proper planning, a viable population could survive indefinitely using geothermal and nuclear power. We would literally have a thousand years or more to get ready. Certainly, there are much worse things that could happen to humanity. 5. Would any life on earth survive? Earth would effectively become a large space-ship, and with proper planning, a controlled biosphere (like in a large space colony) could be maintained. Were there no intelligent direction of events, and the Earth was simply left to its own devices, then surface life would freeze away, but the deep biosphere (the oil field bacteria, the deep sea vents, and other other biomes not directly dependent on solar energy) would persist for millions, if not tens of millions of years. 6. What do you think are chances that we will find an earthlike rogue planet? This depends on what one means by “earthlike”. If one means a planet with Earth’s mass, at very large distance, say thousands of light years, the chances are very good that we will get micro-lensing detections within a decade or so. The data returned, however, will consist only of the likely masses of the planets. Nothing else. I would estimate that the chances of finding a rogue Earth-mass planet within a potentially reachable distance, say within a light year, are about 10%. The chances, however, that this planet will have an interesting frozen-out surface environment that would please a Hollywood screenwriter are effectively zero. Most rogue planets get ejected from their systems very early in their parent star’s history, long before really interesting things have had a chance to happen from an astrobiological perspective. Categories: worlds Tags: April 7th, 2013 2 comments This was no fruit of such worlds and suns as shine on the telescopes and photographic plates of our observatories. This was no breath from the skies whose motions and dimensions our astronomers measure or deem too vast to measure. It was just a colour out of space—a frightful messenger from unformed realms of infinity beyond all Nature as we know it; from realms whose mere existence stuns the brain and numbs us with the black extra-cosmic gulfs it throws open before our frenzied eyes. H.P. Lovecraft, The Colour out of Space Amazing Stories, Vol. 2, No. 6 (September 1927), 557–67. I’ve always thought that the Colour out of Space was H.P. Lovecraft’s best effort. One can argue about economy of expression, but the story is nearly unmatched in its attempt to confront — and imagine — the truly alien. I think we currently have substantially less understanding of the extrasolar planets than is generally assumed. Thousands of planets are known, but there is no strong evidence that any of them bear a particular resemblance to the planets within our own solar system. There’s always a tendency, perfectly encapsulated by the discipline of astrobiology, with its habitable zones and its preoccupation with water — to make wild extrapolations into the complete unknown. An interesting synopsis of much of what we do know can be gained by looking at the latest mass-radius diagram for the exoplanets. The number of planets with joint mass and radius determinations is growing rapidly, and the elastic virtue of a log-log plot fails to suppress the huge range in apparent planetary structures. To within errors, it appears that 6-Earth Mass planets range in radii by a factor of at least three. This is impressive, given that constant density implies R~M^{1/3}… On the figure, I’ve plotted three potential mass-radius relations for super-Earths. This first (in Earth units) is the standard-issue M=R^{2.06} fit that one gets from the solar system planets (excluding Jupiter). The second (again in Earth units) is the vaguely alarming M=3R relation suggested by Wu & Lithwick’s transit timing analysis. The third mass-radius relation is what one might expect if planets form in-situ and accumulate low-density hydrogen envelopes around rocky cores. (Evaporative mass loss makes this more of an upper limit). Frustratingly, all three relations remain plausible. It’s thus fantastic news that NASA’s TESS Mission has been selected for flight. TESS will find effectively all of the transiting Super-Earths orbiting the few million brightest stars, and with dedicated ground-based radial velocity follow-up, will — less than a decade from now — allow for a fantastically detailed version of the above plot. Categories: worlds Tags: ## The Tau Ceti Five December 31st, 2012 6 comments Tau Ceti has street cred. Lying only 11.9 light years away, it is the second-closest single G-type star. It’s older than the Sun, and photometrically quiet. It’s naked-eye visible from both hemispheres, ensuring VIP seating at any SETI fundraiser. And so what about planets? It’s been clear for a few years that Tau Ceti has a zeroth-order dissimilarity with the solar system. That is, if it had a Jovian-mass planet in a Jovian-like orbit, a press conference would have been dedicated to it several years ago. Indeed, because it is so bright and so quiet, Tau Ceti is among the handful of stars in the sky that are best suited to long-term high-precision monitoring via the Doppler velocity technique. It’s at or near the top of the list for all of the major Doppler surveys. Tau Ceti displays a marked excess luminosity in the far-infrared. Blotchy sub-millimeter images imply that this excess luminosity arises from a wide ring of cold dust at Pluto-like distances from the star. In this picture, the radiating dust arises from ongoing collisions within a Kuiper belt-like disk comprising roughly an Earth-mass worth of icy asteroidal bodies: Tau Ceti’s Kuiper belt seems to be about ten times more massive than our own Kuiper belt, despite the fact that Tau Ceti’s metallicity is only about one-third that of the Sun. There’s little risk in hypothesizing (read hand-waving) that the low metallicity of Tau Ceti’s protoplanetary disk meant slow growth for Tau Ceti’s retinue of proto-Jovian cores, which subsequently missed out on rapid gas accretion. The ensuing presence of Neptunes, and the concomitant absence of a Jupiter, generated a different dynamical history compared to the Solar System’s — namely one with more stuff left over at the end of the day in the icy outer reaches. Given this picture, the a-priori odds are excellent that Tau Ceti resembles tens of billions of ordinary, single Population I stars in the galaxy and also harbors multiple inner planets with masses between Earth and Neptune, on nearly circular, nearly co-planar orbits with periods of 100 days or less. Should such worlds exist in orbit around Tau Ceti, then it’s likely that sufficient radial velocity data now exist to dig them out… Readers surely noticed the paper by Mikko Tuomi and colleagues that was posted to astro-ph earlier this month. Tuomi and collaborators report on a joint analysis of three large-N data sets that comprise thousands of radial velocity measurements (from HARPS, KECK and AAT) spanning a total time base line in excess of 13 years. Ideally, one would like have a fully definitive conclusion emerge from such a massive data set, but frustratingly, Tau Ceti is holding its cards very close to the vest, and as radial velocity half-amplitudes inexorably drop below K=1 m/s, this will be an increasingly common behavior from other nearby high-value stars. In their arXiv preprint, Toumi et al. lay off their risk and remain ambiguous regarding actual detections of actual planets, providing only a fully hedged speculation at the end of the abstract, that these “periodicities could be interpreted as corresponding to planets…” The modeling strategy for Tau Ceti taken in the Tuomi et al. paper provides an alternative to the approach adopted by Dumusque et al. in digging the K=0.5 m/s Alpha Cen Bb out of a similarly challenging data set. For both systems, the authors adopt the stance that it is no longer sufficient to write off excess scatter in radial velocity fits as “stellar jitter”. Dumusque’s team developed a physical model for starspot activity migrating latitudinally on a differentially rotating star, and also modeled the convective blueshift arising from stellar activity. Application of these physical models spurred the removal of systematic “noise” from the time series, thereby revealing a candidate Earth-mass planet in a 3.2-day orbit. Tuomi et al. excavate five potential planets by exploring the use of ARMA(p,q) — AutoRegressive Moving Average — models which recognize that (in addition to a Keplerian signal) both the value of given velocity measurement as well as its accompanying error are potentially correlated with previous measurements. ARMA models and their generalizations, ARCH, GARCH, NGARCH, etc., are an old standby for modeling financial time series. Near-term VIX predictions anyone? Indeed, planet detection and trading have certain similarities. Noisy signals, non-stationary processes, cut-throat competition, and the opportunity to land yourself in the media spotlight when things go awry. And the possible planets? Should the signals isolated by Tuomi et al turn out to be both real and Keplerian, then Tau Ceti will join the legions of stars in the galaxy that harbor fully ordinary planetary systems. Categories: worlds Tags: ## The MMEN November 10th, 2012 4 comments Galileo’s unveiling of Io, Europa, Ganymede and Callisto is unarguably shortlisted for the most important astronomical discovery of all time. The Galilean satellites constitute a planetary system in miniature, and their clockwork presence is a centerpiece of Newton’s De mundi systemate. And indeed, if one bases one’s expectations for exoplanetary systems on the Jovian satellites (as well as the regular satellite systems of Saturn and Uranus) then the startling abundance of compact systems discovered by the Geneva Team and by Kepler are hardly startling at all. The Galaxy’s default planetary system — as expressed around many, if not most of its stars — has a handful of planets on near-circular orbits, with periods ranging from days to weeks, and masses of order one part in ten thousand of the central star. Out here in the sticks, near the Sun, we’ve got an Earth, yes, but unlike most stars, we have no super Earths. There is an intriguing, seemingly anti-Copernican disconnect between the solar system and the extrasolar planets. Much of the theoretical framework of planet formation is based on the paradigm provided by the Minimum Mass Solar Nebula (MMSN), the$\sigma \propto r^{-1.5}\$ disk of net solar composition that is required to account for the solar system’s planets. In the standard formulation, the MMSN holds its power-law form inward to about 0.5 AU, where it meets a murkily indistinct inner boundary that’s needed to account for the lack of anything interior to Mercury’s orbit.

Interestingly, the MMSN fades out just where the super-Earths really start to appear. This has led to the widespread assumption that planets somehow form at large radii and then migrate long distances in order to be found in their observed orbits. That seems rather odd.

Eugene Chiang and I have been exploring an alternative idea — namely that the solar system doesn’t present a good starting template for studying extrasolar planets, and that planets, in general, don’t migrate very far (if at all). Could it be that the huge population of super-Earths formed right where they are observed? If that’s the case, it makes life simpler, and it implies that the template we’re after is the Minimum Mass Extrasolar Nebula (MMEN), which can be defined by grinding up the planets that have been observed by Kepler, and which is not all that different from what one gets if one simply takes the MMSN and runs it all the way into the dust sublimation boundary at ~0.05 AU.

Our paper is available at arXiv.

Categories: worlds Tags:

## The Crescent Neptune

November 3rd, 2012 1 comment

A few weeks ago, I got an e-mail from a reporter related to a story that will feature favorite space photos:

We’re hoping some space-themed photo comes to mind, either a picture taken by a space telescope, or by yourself from your own backyard, or anything else that relates to space. We’d also welcome any comments about the photo’s meaning to you.

I think my favorite space photo is the Voyager image of the crescent Neptune and Triton.

For two reasons. First, there’s no false color, no artifice, no agenda. This photograph is calming, mysterious and aesthetically perfect.

Second, the image is dominated by the night side of Neptune. Implicit in the photograph is the amazing fact that it was taken from a vantage that was further than the Sun than the planets. Less than one Neptune orbit elapsed between its discovery in 1846 and the Voyager flyby in 1989. A crescent Neptune seems to me far more subtly profound than the iconic “pale blue dot” image taken by the same spacecraft not all that long thereafter.

Categories: worlds Tags: