Alpha Centauri B b

October 16th, 2012 12 comments

Image Credit: ESO.

I had the good fortune of being asked to sit in as an external commentator for the ESO’s media briefing on the Geneva Team’s discovery of Alpha Centauri B b. It was startling to see the amount of interest on the line. All of the familiar names from the science journalism community were logged in, and there was a very substantial representation from the mainstream media. The ESO officials remarked that it was the largest audience that they’d ever seen for a press briefing. It was very clear that Alpha Centauri and Earth-mass planet combine for a headline draw. The story was supposed to be held until 17 Oct. 19:00 CET, but the embargo was broken in rather disorderly fashion, and, according to ESO, b, by the end of the afternoon was officially out of the bag.

Paul Gilster, who leads the Centauri Dreams site asked me for a brief perspective for a piece that he’ll be writing tomorrow (Lee Billings has a very nice article on Centauri Dreams today). I was eager to oblige — Paul has played a clear, consistent role in getting the community’s attention focused squarely on or charismatic next-door neighbor. I wrote back:

I really like the particular way that the narrative is unfolding. The presence of the 3.2-day planet, taken in conjunction with the myriad Kepler candidates and the other results from the HARPS survey, quite clearly points to the possibility, and I would even say the likelihood, of finding additional planets at substantially more clement distances from the star. Alpha Cen A and B, however, are drawing closer together over the next several years, severely metering the rate at which high-precision measurements can be obtained. This builds suspense! It reminds me a bit of a mission like New Horizons, where the long coast to the destination serves to build a groundswell of excitement and momentum for the dramatic close encounter. I think that this is important for a society that is increasingly expectant of immediate interactivity and instant gratification… I hope that this detection of Alpha Cen Bb provides an impetus for the funding of additional radial velocity infrastructure, and also for space-based missions such as TESS, which can find and study the very best planets orbiting the very nearest stars.

With K=0.51 m/s, Alpha Cen B b has a RV half-amplitude that is over a third lower than the previous record-holder, HD10180b. The relative insignificance of an Earth-mass world in comparison to the great bulk of Alpha Cen B is immediately evident with a scale diagram of the star, the orbit and the planet. The planet resolves to ~6/10th of a pixel in this figure, barely visible as a faint gray speck.

As far as the faint gray speck itself goes, the ESO-produced artist’s impression (shown as the splash image for this post, and over the past few hours, splashed all over the Internet) is quite good for this genre. Granted, the apparent surface brightness of the Milky Way in the background is about 1,000,000,000 times too high, but the planetary crescent and the lighting geometry make the grade. And thankfully: No lava.

Alpha Cen B has a radius about 90% as large as the Sun. This means that transits, if they occur, would have a maximum photometric depth of ~0.01%, and would last up to three hours. These numbers make for a challenging, but by no means impossible, detection. HST (perhaps using the FGS instrument) should be able to reach a transit of this depth, and given that the phase, the depth, and the period are known in advance, I think that a purpose-engineered ground-based solution can be made to work as well. For example, see this post on orthogonal transfer arrays — Alpha Cen B delivers almost 5 megawatts to the Earth, and Alpha Cen A is a nice comparison star right next door.

During the press briefing, the “habitable zone” came up repeatedly. Put succinctly, Venus at B would be on the A-list.

The Nature News and Views commentary by Artie Hatzes draws on the extraordinary claims argument to imbue the detection with a question mark. “The researchers used 23 parameters related to the star’s rotation period to model the variation in stellar activity, and then filtered it out from the data, unveiling the planet’s signal.” Given the skepticism, it’s interesting to look in more detail at how the signal was dug out.

In the past, I’ve used this blog as a platform for urging that Alpha Cen receive the maximum possible allotment of Doppler-based attention. From August 2009:

Now nobody likes backseat drivers. As the saying goes, “theorists know the way, but they can’t drive”, and theorists have had a particularly dismal record in predicting nearly everything exoplanetary.

But nevertheless, I’m urging a factor-of-four increase to that data rate on Alpha Cen B. I would advocate two fully p-mode averaged velocities per night, 50 nights per year. I know that because Alpha Cen B is so bright, the duty cycle isn’t great. I know that there are a whole panoply of other interesting systems calling for time. It is indeed a gamble, but from the big-picture point of view, there’s a hugely nonlinear payoff in finding a potentially habitable planet around Alpha Centauri in comparison to any other star.

The current HARPS data set has an impressive 459 individual p-mode averaged velocities, with uncertainties in the range of 1 m/s. In a naive universe governed by featurelessly luminous stars and normal distributions, such a data set would allow planetary orbits with K<0.1 m/s to be probed. It was just such a universe that informed some of my earlier simulated data sets that modeled what one might expect from Alpha Centauri. For instance, here’s a plot from June 2009 that shows what I thought the HARPS data set of that time might have looked like.

With 459 points along such lines, Alpha Cen’s whole retinue of terrestrial planets would now be visible. Indeed, with just the synthetic data in the above figure, a simulated super-Earth in the habitable zone sticks out like a sore thumb.

With hindsight, it’s not surprising that the real data set is more complicated. Although Alpha Cen B is a very quiet star, it does have a magnetic cycle, and it does have starspots, which rotate at the ~37-day spin period of the star, and which come and go on a timescale of months.

The raw radial velocities are completely dominated by the binary orbit. The following figure is from the SI document associated with the Dumusque et al.’s Nature article.

Alpha Centauri B has a velocity component in our direction of more the 50,000 miles per hour, more than twice the speed attained by the Saturn V’s just after their trans-lunar injection burns. The AB binary orbit has a period of ~80 years, and is currently drawing toward a close approach on the planet of the sky. (Figure below is from Wikipedia.) The next periastron will be in 2035.

Strictly speaking, one needs five parameters (P, K, e, omega, and MA) to model a binary star’s effect on a radial velocity curve. However, because the HARPS data covers only 5% of a full orbit, it’s sufficient to model the binary’s contribution to the Doppler data with a 3-parameter parabola. When the binary is removed, the data look like this:

There’s a clear long-term multi-year excursion in the velocities (traced by the thick gray line), and there are almost 10 meters per second of variation within each observing season. That’s not what one would have ideally hoped to see, but it is an all too familiar situation for many stars that have years of accumulated radial velocity data. Browsing through the Keck database shows numerous stars with a vaguely similar pattern, for example, this one:

Many long-term trends of this sort are the product of stellar activity cycles that are analogues of the 11-year sunspot cycle on our own Sun. In the absence of sunspots, the surface of a sun-like star is uniformly covered by granulation — the pattern of upwelling convective cells.

Image Source.

Most of the surface area of the granules is composed of plasma moving up and away from the Sun’s center. The gas gushes upward, disgorges energy at the photosphere, and then spills back into the darker regions that delineate the granule boundaries. On the whole, the majority of the stellar surface is blueshifted by this effect. In the vicinity of sunspots, however, the granulation is strongly suppressed, and so when there are a lot of sunspots on the surface of the star, the net blueshift is reduced.

Sunspot activity is very tightly correlated with the strength of emission in the cores of the Calcium II H and K lines (for an accessible overview, see here). As a consequence, a time series of this so-called H&K emission is a startlingly good proxy for the degree to which the granulation blueshift is suppressed by sunspots. Figure 2 of the Dumusque paper charts the H&K emission. Its variation is seen to do an excellent job of tracking the erratic long-term Doppler RV signal displayed by the star (compare with the plot above). Hence, with a single multiplicative scale parameter, the variations measured by the H&K time series can be pulled out of the Doppler time series.

Can’t stop there, however. Starspots, which come and go, and which rotate with the surface of the star at the ~37-day stellar spin period, generate an additional signal, or rather sequence of periodic signals and overtones. Dumusque et al. handle the rotation-induced signals in conceptually the same way that one would handle a set of planets with variable masses, periods of several months: measure the strength of the periodogram peaks, and remove the signals year by year. This involves 19 free parameters, the moral equivalent of successively removing four planets to get down to the final brass (or more precisely iron) tack:

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The Pythagorean Problem

October 10th, 2012 1 comment

Image source: Drew Detweiler

The Pythagorean version of the gravitational three-body problem is very simple to state.

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?

This particular problem seems to have been first posed in the late 1800s by the German mathematician Meissel, who mysteriously asserted that the motion of the three bodies should be periodic. That is, he felt that they would come back to their exact starting positions after executing a complex of intermediate motions. A first attempt at the solution — using numerical integration with a variable stepsize — was published in 1913 by Carl Burrau. He was able to map out the intial trajectories through several close encounters, but he was unable to integrate far enough to determine what eventually happens.

The correct solution was found in 1967 by Szebeley and Peters, who used the technique of three-body regularization to resolve the succession of close encounters. Here’s one of their diagrams showing a segment of the complicated motion.

The Szebehely-Peters paper is fun to read. It emphasizes that this nonlinear problem is surprisingly tricky to solve, and that it shows the classic sensitive dependence on small variations in the initial conditions. For example, here’s a link to a recent, attractively rendered YouTube video that animates the trajectories and osculating orbits, as obtained via an implementation that uses Mathematica’s NDsolve.

Screenshot source.

Unfortunately, however, a careful analysis shows that the motion from 2:47 through the end of the video is completely incorrect…

I’ve always been struck by the fact that there’s a fascinating subtext to the trajectories of the three bodies if they are interpreted as a narrative of interpersonal relations. An initial value problem for a set of six coupled, first-order ordinary differential equations unfolds to telegraph a drama of attraction, betrayal, redemption, triumph and loss.

This summer, I had an opportunity to collaborate on the development of a scored, choreographed 3-minute 45-second performance of the problem which was premiered last month at the ZERO1 Biennial in San Jose. Our goal was to simultaneously convey the interpretive subtext while adhering to an fully accurate set of trajectories. It took a lot of work and was quite an intense experience. From the description at the ZERO1 site:

Three dancers in illuminated costumes create a live video visualization of the elliptic-hyperbolic solution to the classic Pythagorean three-body problem. A custom light tracing application detects light emitted from LEDs on the dancers’ soft circuitry costumes to create a visual model of their trajectories across the 2D plane of the stage. This realtime graphic visualization is projected on a large screen behind the stage in order to provide the audience with a birds eye perspective of their complex motion.

The use of digital technologies presents challenges for contemporary choreographic methods as data visualization guides movement through performative space on scientifically accurate trajectories. Live accompaniment from three musicians enhances physical performance as each body is interpreted through movement and sound. Feelings of longing, connection, and isolation are intertwined as the bodies are flung apart by the same gravitational forces that draw them together.

(That last sentence could more properly read, The bodies are flung apart despite feeling only attractive gravitational forces.)

To give a better sense, here are some notes and diagrams from mid-way through the process, as the choreography and the rehearsals were beginning to gel.

It’s particularly fascinating how the immediate outcome of the near-return to the pythagorean condition at the halfway mark is so different from how things unfold at the start of the piece. I like the interpretation that body 4 is somehow lazy at this point, or late to realize the import of the situation, and is marginalized as a result. This is the first real opportunity for bodies 3 and 4 to express emotion — shock for body 4, joy for body 3.

In the following measures, body 4 is marginalized, sulky, scheming, whereas body 3 is doing its best to impress, in the set of looping, private engagements. A reverie! The successive body 3-5 interchanges should _highlight_ the difference in masses between 3 and 5. Body 3 is light footed, fleet, body 5 glides smoothly, deliberately, (but not dully) as an anchor.

Body 4 must come back from its runout with a renewed sense of determination and purpose. The ensuing encounter between 4 and 5 must somehow convey 4’s charms and strengths. In a very real sense, this encounter is the tipping point that determines the outcome for all time. This is where the youtube video went off the rails.

As a consequence, there should be a sense of unfulfillment in the next body 3-5 encounter (a grasp, a gaze that fails to connect?) which sets up 3 to dive through on the way to its penultimate run-out. In this sequence, body 3 must somehow fail to live up to the expectations that it so brightly promised. The outcome is now determined, and the bodies know it, although the audience doesn’t.

While body 3 is at the arc of its final run-out, body 4 is weaving a spell on body 5, cementing the outcome ever more decisively. Indeed, Body 5 is only briefly engaged as 3 makes its final dramatic run through body 4 and 5’s orbit. The final sequence of encounters between body 4 should grow ever more identical, signaling the finality of the outcome.

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The latest anomalies

June 8th, 2012 3 comments

Photographed at UCSC Art Dept. Spring 2012 Open Studios

The announcement of new transiting hot-Jupiter type planets, such as WASP 79b or HAT-P-38b, by the ground-based surveys no longer generates press releases, but the march of discovery does give us an ever-clearer view of the planetary census.

Yesterday, Matteo Crismani turned in his UCSC Senior Thesis. In addition to the results that we published in our 2011 paper (described in this post and this post) he also took an updated look at the relationship between the radius anomaly (the fractional discrepancy between the theoretically predicted radius and the actual observed radius) and the insolation-derived effective temperature of the planet. With the large aggregate of hot Jupiter-class planets that now have good measurements for both planetary mass and planetary radius, the dependence of the radius anomaly on the planetary temperature has grown clearer.

The best fit power-law now has the radius anomaly scaling as T^2.9, with an uncertainty on the exponent of ~0.3. This is quite close to the T^2.6 relation that stems from the back-of-the-envelope arguments that invoke the Batygin-Stevenson Ohmic heating mechanism. In effect, these hot Jupiters are like Ball Park Franks…

11 December, 2117

June 6th, 2012 Comments off

Full-size (1536×2048) .gif images: one, two.

We went up to Mount Hamilton yesterday afternoon, and, as was the case for everyone who saw the transit, it was a unique conjunction of time, place, and circumstance.

The Lick Observatory staff deployed the historic 36-inch refractor to extraordinary advantage. Rather than project the image, which has the effect of divorcing the instrument from the event, they removed the eyepiece and stretched a cloth across the focal plane. The resulting effect, somehow, was to seamlessly integrate the transit into its surroundings and its historical context, 1639, 1761, 1769, 1874, 1882, 2004, 2012…

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once in a lifetime

June 5th, 2012 3 comments

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Headed out to Pluto

April 8th, 2012 5 comments

Time slips by. It’s now been more than six years since launch and more than five years since the New Horizons probe got its gravitational assist from Jupiter. I looked back through the archives and found a post covering the event.

One day, one hour, and nine minutes ago, the New Horizons spacecraft sailed flawlessly through its closest approach to Jupiter. A day later, Jupiter still looms large in New Horizon’s field of view, with an angular size more than five times greater than the size of the full moon in our sky.

That was on March 1st, 2007, a day after the 500 point drop in the DJIA that signaled the first shudder of unease portending the global financial crisis. and New Horizons have both been gradually slowing down over the past six years, with New Horizons passing the orbit of a planet at roughly the cadence of 100 additional oklo posts. New Horizons is currently 9 AU from Pluto, and will arrive in the system in July 2015.

I discovered from reading the wikipedia page that a third circumbinary satellite was recently found in orbit around Pluto and Charon.

The three small moons in the Plutonian system are surprisingly reminiscent of what we might expect a typical circumbinary extrasolar planetary system to look like: orbital periods measured in weeks, masses of order a part in ten thousand of the central binary, low eccentricities, and orbits that are close to, but not in mean-motion resonance. During the next few years, as the Kepler data continues to roll in, and as the eclipsing binary systems in Kepler’s field of view are carefully scrutinized, we’ll find out whether such properties are indeed the norm.

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line of sight

March 31st, 2012 5 comments

The ring of geosynchronous satellites and the global web of submarine cables constitute two of planet Earth’s most remarkable physical features. The moment I press Publish, the diagram just below will be sent — encoded in modulated light — on a profusion of undersea journeys from the Bluehost servers in Utah to Japan, Europe, Australia, South America and beyond. Optical wavelengths are small, the speed of light is fast, and the quantity of data that can be transmitted on optical fiber is impressive. A fairly recent lab-based data transport record involved multiplexing 155 channels, each carrying 100 Gbit/s over a 7000 km fiber.

For the impatient, however, the latencies of the long-haul international fiber connections are something of an issue. The index of refraction in glass is n~1.5, and the cable routes don’t adhere to the great circles. Using NTT’s Looking Glass service, one can run traceroute between far-flung nodes on the Internet. For example, right now, round-trip travel times between London and Tokyo are taking about 265 milliseconds, with routing that runs on the Atlantic and Pacific Ocean bottoms and (effectively) along Route 66:

A quarter of a second round-trip is pretty slow. Light traveling in vacuum along the 9602 km great circle connecting London and Tokyo would do the round-trip in 64 milliseconds, a factor-of-four improvement. Things should get better in 2013, however, when the Arctic Link cable connects Japan to Britain via the Northwest Passage. This line will run at 170 milliseconds round trip.

Even with global warming lending a helping hand, it’s a hassle to lay cables over the top of the planet. A more effective plan is to go straight through. The straight-line distance through the Earth from London to Tokyo is 8719km, implying a minimum round-trip of only 58 milliseconds.

It was thus rather interesting to read of the first actual demonstration of signaling by neutrinos posted to arXiv earlier this month. A team at Fermilab reports that they have established a neutrino communication link with a data rate of 0.1 bits/sec and a bit error rate of 1% over a distance of 1.035 km, along a path that includes 240 m of solid Illinois dolomite.

A one or a zero every ten seconds is very similar to the bit rate that one gets with smoke signals. It’s a staggeringly long way from the petabit-per-second transmission rates that one can currently achieve with a strand of freshly lit fibers. Nonetheless, it’s an exotically high-tech accomplishment, and so it’s fair to overlook the abysmal bandwidth and error rate. What I would like to criticize, however, is the completely lame initial message that was transmitted over the neutrino link: N-E-U-T-R-I-N-O.

Jeez. Did none of the 113 authors of Demonstration of Communication Using Neutrinos appreciate that style is paramount when one is performing expensive high-profile stunts?

In Stancil et al.’s defense, though, the contents of historic first messages have generally been sorely lacking in pizazz. Alexander Graham Bell’s first telephone call consisted of “Watson, come here! I want to see you!” Even worse, was the unreadably uncompressed purple prose transmitted (over the course of 19 hours) on August 16, 1858 as a part of the first transatlantic telegraph messages between Queen Victoria and President Buchanan:

“it is a triumph more glorious, because far more useful to mankind, than was ever won by conqueror on the field of battle. May the Atlantic telegraph, under the blessing of Heaven, prove to be a bond of perpetual peace and friendship between the kindred nations, and an instrument destined by Divine Providence to diffuse religion, civilization, liberty, and law throughout the world.”

Had I been part of the arXiv:1203.2847 author list, I would have agitated to turn the contents of that first message over to the inimitable Oscar Wilde:

“It is a very sad thing that nowadays there is so little useless information.”

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multiple transits

March 25th, 2012 2 comments

Enceladus, Dione, Titan, Mimas and Saturn.

On Tuesday, Venus reaches its maximum elongation of 46 degrees from the Sun. Thereafter, its angular separation from the Sun steadily decreases until June 6th, when it undergoes transit.

Transits of Venus are newsworthy because they are rare. Venus’ orbit is inclined by 3.4 degrees relative to the ecliptic, and so Earth must be near Venus’ nodal line if a transit is to be observed. The last one occurred in 2004, and the next one after June 6th will occur in December 2117. When talking transits-of-Venus in this day and age of astronomers flossing their “premium-platinum” frequent flyer status, it’s hard to resist that obligatory mention of Guillaume Le Gentil, whose unsuccessful expedition to observe the 1761 transit took 11 years, and had him returning to Paris in October 1771, only to find that he had been declared legally dead and been replaced in the Royal Academy of Sciences. His wife had remarried, and all his relatives had “enthusiastically plundered his estate.”

Nobody’s estate gets enthusiastically plundered on account of transits of the solar system’s Jovian planets by the solar system’s Jovian satellites. Many of the larger moons of Jupiter, Saturn and Uranus orbit with very small inclinations to the host-planet equatorial planes. As a result, it’s possible to get pictures such as the splash image for this post, with a whopping 4 moons transiting at once, without having to wait around for centuries.

Loosely speaking, eccentricities and inclinations are dynamical bruises acquired during the formation process. When the assembly of a system occurs in a quiescent, dissipative setting, then orbits wind up closer to circular and closer to co-planar. Violent interactions in the absence of dissipation produce systems that are more distended. To get a feeling for this, I gave a 3D-normal distribution of random impulsive kicks with standard deviation 0.003*v_circ to an aggregate of initially co-planar and circular orbits. The resulting distribution of inclinations and eccentricities, plotted as a locus of gray points, is reminiscent of the bulk of the Jovian satellites (blue points):

Cranking up the magnitude of the impulsive kicks by a factor of ten yields a distribution of eccentricities and inclinations that looks better suited to the actual planets in our solar system (green points). Note that Mercury and Iapetus fall outside the diagram.

The big surprise from the Kepler mission has been the large number of systems that display multiple transiting planets. Kepler sees plenty of set-ups that contain four, five, and even six individually transiting planets. This distribution is startling, however, only if one draws on the solar system as the template for expectations. Had the preconceived notions been drawn from the regular satellite systems of the Jovian planets, then the statistics would seem completely unsurprising.

A recent preprint by Figueira et al. describes a consistency analysis between the results of the HARPS and Kepler surveys. They find that the two distributions can be reconciled (and the large number of multiple-transiting planet systems accounted for) if planet-planet mutual inclinations are generally less than one degree.

This implies that the eccentricity measurements that have been published to date for low-mass planets are likely to contain a substantial number of overestimated e‘s…

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some real alpha

March 18th, 2012 Comments off

Kraftwerk will be playing eight shows in April at the MOMA, but all eight sold out well before I even found out about it. Getting clued in at this late date is a bit like finding out about a new hot Jupiter orbiting a 14th magnitude star — given that its already March 2012, it’s marginally (or not even) publishable on its own.

The ability to make good predictions prevents one from being perennially late to the game. A good prediction is one that has both accuracy and utility, and for the past two decades, the field of extrasolar planets has been sorely lacking in predictions that make good on either virtue. Yet it didn’t have to be that way! Like many others, in the early 1990s, I was perfectly well aware of Goldreich and Tremaine’s 1980 paper which lays out the essential principles of disk migration.

Even if one only read the abstract, it was very clear that the prospect of Jupiter-like planets on short-period orbits was well worth exploring further. Another example is provided by the Kozai mechanism, that relatively straightforward phenomenon first described in the early 1960s that derives directly from the physics and assumptions underlying the circular restricted three-body problem. With simple models for tidal dissipation thrown in, it could have been clear long ago that visual binary stars have the ability to produce Jupiter-like planets with orbital periods of order a week.

Admittedly, to hear such grousing and second-guessing is like sitting next to a losing bettor on the train back from the track. The productive approach is to keep an eye open to all the equally starting predictions that are yet to be made and which can potentially lead to substantial future profits.

With that lead-in in mind, its very interesting to read the recent abstract of Perets, Kratter and Kenyon (which, I’m told, will soon be followed-up by a substantial paper). Perets and collaborators run up the score with a basic point that definitely falls in the should have thought of that myself category: Mass loss in binary evolution alters the zero-velocity surfaces available to a planet that starts life stably in orbit about one member of a binary pair. As the system experiences stellar evolution, with one or both stars losing substantial mass to red giant winds, a planet is able to radically alter its trajectory, and indeed, can wind up orbiting the opposite member of the pair. Tidal friction can then be invoked to elicit a permanent capture.

There’s a cool paper by Elbert E.N Macau from 2000 which draws on a similar idea to put a spacecraft on a low-cost slow-boat trajectory to the Moon. In this case, the impetus is provided by a weak rocket rather than mass loss, but the principle is similar:

Figure 4 from Macau, E. E. N. Acta Astronautica 47, 12, 871-878: Starting from a circular parking orbit around the Earth, a thrust is applied to inject the spacecraft into a chaotic region. The spacecraft is then left to move freely in the chaotic region. The uncontrolled trajectory can eventually reach the Moon. In this example, it takes approximately 8 years to reach the vicinity of the Moon. However, after that, the spacecraft quickly leaves the Moon.

Perhaps the most interesting aspect of planetary orbital transfer in evolving binary systems is that it provides a plausible mechanism for delivering Earth-sized worlds to long-lived potentially habitable orbits in the vicinity of white dwarfs. As described in this post from last July, such worlds, when they transit, can be detected from the backyard…

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lights in the sky

March 11th, 2012 8 comments

It’s hard to miss Jupiter and Venus in the early evening sky right now, and later this week, on March 15th at 10:37 UT, they will reach an impressive conjunction, with Venus near maximum elongation (separated by 46 degrees from the Sun) and Jupiter only 3.3 degrees from Venus.

At the time of conjunction, Venus will have an apparent magnitude of V=-4.2 and Jupiter will be at V=-1.9. They are thus both brighter than Sirius, and the display is all the more impressive because the planets are still well above the horizon at the end of astronomical twilight.

The combination of the HARPS Survey and the Kepler data are indicating that the architecture of our solar system is — to at least a modest degree — somewhat unusual. If we were living in a run-of-the-mill planetary system, we could expect to have several planets with ~2x Earth’s radius orbiting with periods of 100 days or less, along with no Jupiter in a Jupiter-like orbit. A pair of standard-issue sub-Neptunes would appear substantially brighter than Venus in the dusk and dawn skies, but night-time displays as impressive as the one we’ve got now wouldn’t occur, since the maximum elongations would be ~30 degrees or less.

Jupiter’s distance from the Sun puts the regular motions of the Gallilean satellites just outside the reach of naked-eye observability, and in a similar vein, Venus’ size and semi-major axis leave it just on the threshold of displaying visible phases. If our eyes were just a little better, the “Copernican Revolution” wouldn’t be a cliche, and Archimedes would have come up with the Universal Law of Gravitation.

Our night sky does, however, give us one very nice order-of-magnitude foothold. The apparent brightness of the outermost visible planet, Saturn, falls exactly in the magnitude range populated by the brightest stars. For example, when Saturn’s rings are at a less-than-full opening angle, the planet has a nearly identical apparent brightness to Alpha Cen A. This means that if one knows the AU, has the telescopic ability to resolve the disk of Saturn, and makes the (shaky) assumption that the brightest stars are Sun-like, and the (less shaky) assumption that Saturn is highly reflective, the distances to the nearest stars can be estimated. Very roughly,

which is close to the true 4.4 light year distance. (A version of this argument was used in the late 1600s to get the first real estimate of the staggering separations between the stars.)

If one also assumes that stars travel at relative speeds that are similar to the velocities with which the planets orbit the Sun, then an extension of the ball-park argument indicates that the configuration of the night sky should be radically altered on a timescale of millions of years. This is indeed the case. There was a cool 1998 article in Sky and Telescope that used the (then-new) Hipparcos data to compute the brightest stars within the last and next five million years. At the dawn of the Pliocene era, Epsilon and Beta Canis Majoris were both of similar brightness to Venus.

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