oklo

Image source.

Last week, I wrote about a plan to send a tiny spacecraft on a trip to the vicinity of Alpha and Proxima Centauri. The idea is to employ a multi-stage rocket to boost a tiny payload toward the stellar system at high speed. When the destination is reached, the principle of gravity de-assist (in the form of successive close flybys of the stars) is used to haul the spacecraft into a bound orbit without using any on-board fuel. [This, of course, is an exercise in orbital dynamics, and not mission proposal. There are better ways to get to Alpha Centauri.]

The problem was tackled by UCSC graduate student Jeremy Wertheimer as his term project for my Astrophysical Dynamics class. Our initial plan was to use a multiparameter minimization scheme (such as the genetic algorithm or simulated annealing) to vary the incoming trajectory of the spacecraft until we found the largest arrival speed that allows for a final bound orbit. To do this requires us to have a precise orbital model for the Alpha AB — Proxima trio.

Amazingly, we discovered that the most recent papers in the literature (from the early 1990s) had arrived at the conclusion that Proxima Centauri is not bound to the Alpha Centauri binary, but rather is in the process of merely drifting past them like a ship in the night. The a-priori odds of Proxima being so close, and so nearly bound, are less than one in a million, but nevertheless, the best position and velocity measurements at that time suggested that this was indeed the case.

In the intervening years since the Matthews & Gilmore 1993 and Anosova et al. 1994 Proxima-Alpha papers were published, there has been a tremendous improvement in our knowledge of the positions, distances, and space velocities of the nearby stars. This improvement is largely due to the European Space Agency’s Hipparcos astrometric satellite, which flew between November 1989 and March 1993 (and whose data was published in June 1997). Hipparcos obtained excellent 3D positional and plane-of-the-sky velocity measurements for both Alpha and Proxima Centauri. When combined with mass and radial velocity measurements for the three stars, the Hipparcos data allows a much better determination of the orbit.

When Jeremy computed an orbit for Proxima using the updated Hipparcos data, he discovered that the measurements now suggest that Proxima Centauri is just barely bound to the Alpha AB pair. He found an enormous elliptical orbit with semi-major axis 272212 AU. (This works out to a whopping 4.3 light years, which is coincidently quite close to the current Sun-Proxima distance.) Clearly, this orbit is much too large, but it’s encouraging to see that the centroid kinematic measurements now indicate that Proxima is formally bound to Alpha.

Even the latest measurements for the Proxima-Alpha Centauri positions and velocities contain uncertainties. In particular, it turns out that the absolute radial velocity for Proxima Centauri has a (still surprisingly large) 1-sigma uncertainty of 200 meters per second. Proxima’s radial velocity in turn has an important effect on whether the three stars are gravitationally bound. Jeremy ran a Monte Carlo simulation in which he drew 10,000 models of the Proxima-Alpha system parameters from the Gaussian distributions implied by the uncertainties in the observations. When he plots the binding energy of these models against the value for Proxima’s radial velocity, he finds that 44% of the trial systems are bound (that is have total energy = gravitational energy + kinetic energy less than zero).

Many of the bound trial systems have energies very close to zero, and hence place Proxima in absurdly large orbits around Alpha. In these configurations, Proxima is currently at the periastron (that is, the near-point) of its orbit. An object in a highly eccentric orbit, on the other hand, spends most of its time near apastron (the orbital far-point).

To illustrate how objects tend to spend their time near apastron, here’s a home-made mpeg-4 stop-action animation of a peppercorn orbiting a kumquat in an ellipse with e=0.90. (Try this version if the other one loads a screen of gibberish).

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If Proxima is indeed bound to Alpha, then we would (a-priori) expect to find it near apastron. In the figure above, the Monte-Carlo generated orbits in which Proxima is close to apastron have been marked with stars. These orbits all fall in the part of the graph where Proxima’s radial velocity is in the vicinity of -22.1 kilometers per second. We thus have a prediction: if Proxima’s radial velocity is measured to high accuracy, then the value will be ~-22.1 kilometers per second, rather than the current value of ~-21.8 kilometers per second.

In the figure below, I’ve plotted two of the Monte-Carlo orbits for Proxima with respect to Alpha. The “two sigma” orbit is an example of a realization in which Proxima is slightly closer to apastron than periastron. The orbits are projected onto the plane of the sky, and superimposed on an actual photograph of Centaurus (with the full Moon digitally superimposed to give a sense of scale). If our analysis is correct, it should take Proxima about a million years to make one orbit of Alpha, and the semi-major axis of the orbit should be about 1/6th of a light year:

“It means nothing to me. I have no opinion about it, and I don’t care.”

–Pablo Picasso

(July 21, 1969, in the New York Times on the occasion of the first lunar landing by the astronauts of Apollo 11.)

“Now, let’s get off. Forget the camera. [Garbled]…” are among Eugene Cernan’s last words spoken on the moon. (In Apollo 7 astronaut Walter Cunningham’s 2003 book The All-American Boys, Cernan’s last words on the Moon are reported to be the far more colorful, “Let’s get this mother out of here!”)

The 1972 splashdown of Apollo 17 marked the end of the last manned foray into deep space. The space age recordings and artifacts — We came in peace for all mankind! — uttered in a swell of self-conscious foresight now touch millions as quaint samples on dance and techno records. “We have loss of signal”, “Apollo 8, You are go.”

The horizons have moved both outward and inward in thirty four intervening years. Robot emissaries have filtered through the solar system. There exists a photograph of a crescent Neptune.

1976: the Viking probes landed on Mars, and photographed stunning panoramic views while their orbiting mother ships circled the planet. The landers dug trenches, analyzed the soil, searched for life.

2004: the Spirit and Opportunity probes accomplish much the same feat. These new probes dispatch small rovers, which crawl carefully, arduously, across the barren Martian surface, transmitting information about the surface rocks, informing us that Mars really had a world-encircled sea.

1976: Pong was closing its heyday as one of the first popular computer-based arcade games. Microprocessor controlled, the innards of the Pong machines were a marvel of technical sophistication and miniaturization.

2004: Grand Theft Auto (San Andreas version) features realistic characters, high speed urban driving, gun fighting, a pulsing soundtrack, and cameo appearances from larger-than-life characters such as Snoop Dogg.

And therein lies the resolution of Fermi’s paradox.

One of the systemic project’s most important goals is to reach the point where we can have genuinely realistic and aesthetically satisfying simulated images and animations of extrasolar planets. This will serve the scientific purpose of allowing us to get better comparisons with infrared observational data, and will ultimately allow us to embark on vicarious missions to the new-found worlds of our Galaxy.

In the interim, there’s a lot of coding and computing to do.

As described in this post from earlier this month, I’m advising UCSC Physics graduate student Jonathan Langton on a Ph.D. thesis geared to simulating the atmospheres of irradiated extrasolar planets. Jonathan finally graded his way through a horrific stack of lab reports and final exams, and has now been able to put full focus on the research. Progress is evident in a heavy stream of e-mails containing increasingly detailed animations.

Jonathan is using a numerical technique known as the pseudo-spectral method to do his simulations. The key idea is that the flow pattern on the surface of the simulated planetary sphere can be broken down into a superposition of Fourier modes. For example, as one moves around the planet, the longitudinal variations in the flow can be described in terms of a superposition of sinusoidal patterns. Sinusoids have analytically computable derivatives, which allow one to make a highly accurate representation of changes in the flow without resorting to a cripplingly large amount of computation.

Spectral methods have their drawbacks, however, in the form of high-frequency numerical noise. This noise was evident in the earlier simulations in the form of transient ribbed patterns within the flow. Over the past few days, Jonathan has designed an elegant filtering scheme which seems to be working very well in supressing these spurious features without killing the actual structures in the flow.

The snapshot above is from a test-calculation that implements the new filtering scheme. It’s part of an animation that simulates the development of an initially random vortical flow on the surface of a planet with the radius, mass, and rotation period of HD 209458 b. (Potential vorticity is the quantity plotted, resolution is 256×128, and the simulation runs for 5 rotational periods).

Here is a link to Jonathan’s latest (7MB) animation. It’s hot off the computer.

Proxima and Alpha Centauri are the Sun’s closest stellar neighbors. As they drift through the void, a mere twenty four trillion miles beneath the Earth, they exert a special fascination. Do worlds orbit these stars? Will we ever reach them? Will anyone ever stand on the surface of Alpha Centauri A “b” to witness the double sunrises that occur every 39.5-odd years?

Tiny Proxima lies a mere 15,000 AU from the Alpha Centauri AB binary pair, and moves with them through the Galaxy in a very similar direction and with a nearly identical speed. The likelihood of such a stellar configuration occurring purely by chance is less than one in a million, and based on this incredibly improbable arrangement, it has been suspected (since 1917) that the three stars constitute a bound triple system.

It’s too bad our solar system doesn’t have a companion like Proxima. If the nearest red dwarf lay 15,000 AU instead of 260,000 AU away, it would shine with the 3rd magnitude. It would be easily visible to the naked eye, and its parallax would amount to 1/120th the diameter of the full Moon. If Proxima belonged to us, rather than to Alpha Centauri, then the distances to the external stars would have likely been first measured directly by someone like Robert Hooke in the 1600s rather than Friedrich Bessel in the 1800s. We would now be avidly searching Proxima for possible terrestrial planets, and the prospects for interstellar travel would not seem quite so daunting.

Every other academic year, I teach a graduate course on astrophysical dynamics at UCSC, and one of the requirements for completion of the class is a piece of original research. During the Summer prior to the start of the class, I design a set of projects, and then we collaborate to see them through to completion.

The Alpha-Proxima Centauri system is an excellent source of projects. All three stars are extremely well characterized, and it’s interesting to look at the sorts of orbits that the configuration can support. In the Fall 2004 course, first-year UCSC graduate student Jeremy Wertheimer started to work on the following problem:

Let’s imagine that we want to send a probe to the Alpha-Proxima Centauri system, and for the sake of concreteness, let’s assume that the probe can be accelerated to a large speed by a multi-stage rocket, but that it carries no fuel of its own for the purposes of orbital insertion. Using only the principles of gravity assist and de-assist [see diagram below], and employing the gravitational fields provided by the three stars, what is the largest speed with which the probe can approach the system, and be brought into a bound orbit about any of the three stars? This maximum speed of approach serves to define a characteristic travel time to the system. How long is this time?

[Note that this is a dynamics problem falling under the general topic of multi-parameter minimization, and is not a mission proposal! There are certainly better, more effective ways to visit Alpha Centauri.]

The strategy for solving the problem is as follows: First, set up a model of the orbits of the three stars about their common center of mass. Then, define a “population” of trajectories involving possible approaches (parameterized by velocity, impact parameter, and angle of attack). Use a genetic algorithm to breed promising trajectories, and after
many generations, arrive at one that is (hopefully) near-optimal.

As soon as we set to work on the problem, we found a remarkable result in the literature. In order to optimize the trajectory, we needed to know the most accurate available orbital parameters for Alpha Centauri A, B, and Proxima. To our surprise, we discovered that the most recent papers which study the system dynamics Anosova et al. 1994, and Matthews & Gilmore 1993, both suggest that Proxima is not gravitationally bound to Alpha Centaui AB. The results in the literature imply that the three stars are independent of one another, and just happen to be experiencing a close encounter while moving in the same general direction, despite the approximately million-to-one odds.

We were astonished by this result. It just didn’t seem to make sense. In addition to having the same kinematics, Proxima also seems to have an age and metallicity consistent with those of Alpha Centauri AB. Furthermore, in order to solve our minimum travel time problem, we needed to know whether the literature result is correct. The two extant papers, written in 1993 and 1994, were published prior to the release of the highly accurate Hipparcos data. Surprisingly, as far as we can tell, nobody has attempted to use the modern measurements to see whether Proxima really is unbound from the AB pair.

We therefore realigned Jeremy’s research project to provide an updated anlaysis of Proxima’s dynamical situation. Is it bound to Alpha or not? In an upcoming post, I’ll talk about what we’ve found.

Last week, we had a one-day seminar on planets and planet formation at UC Santa Cruz that brought together researchers from both UCSC and NASA Ames. One of the talks was by Jonathan Fortney, who is currently a post-doctoral researcher in the Planetary Systems Branch of the Space Science Division at NASA Ames.

Fortney and his NASA Ames collaborator Mark Marley have a state-of-the-art radiative transfer code which can compute the emergent (and reflected) spectrum from a hot Jupiter. (See this recent post.) They’ve recently applied their code to compute how the intrinsic radiation from the flow pattern on the surface of the planet would look if you could resolve it with a pair of night-vision goggles. Jonathan writes:

Here’s an MPEG of the full 360 orbit of HD 209458b, in 36 10-degree increments. This is as seen from Earth. It’s the Cooper & Showman (2006) dynamical simulation, run through our radiative transfer solver.

Red is 5 microns, green is 3.3 microns, and blue is 2.2 microns. The Cooper and Showman model predicts a day side that is very similar to a blackbody, leading to a whitish appearance. On the night side, which is fairly cool, strong methane absorption knocks out the blue and green, leaving only red. I have artificially pumped up the red on the night side so that you can actually see it on the monitor. If you don’t, it’s a dark red which is hard to see compared to black–the night side has little flux compared to bright (hot) day.

Here’s a sequence of frames from the movie:

The animation draws on calculations described by Fortney, J. J., et al., 2006, “The Influence of Atmospheric Dynamics on the Infrared Spectra and Light Curves of Hot Jupiters”, which has been submitted to the Astrophysical Journal.

This is a big step forward for the “computational imaging” of extrasolar planets, and I’m really excited about the future directions that Fortney is planning to take these calculations. For starters, it will be very interesting to see the movie with the reflected light component added in. It will also be cool to place the point of view above a particular spot on the planet and animate the time-dependant flow pattern (the above movie rotates a single snapshot model of the planet, but it does not show the actual time evolution that is computed in Cooper and Showman’s hydrodynamical simulations). Animations of the time-dependant flow will start to bring exoplanets into the territory covered by the cloud-pattern movies that the Voyager and Cassini probes radioed to Earth as they flew past Jupiter. Finally, by using John Moore’s integrating sphere to produce the actual visible colors corresponding to individual computed spectra, it will also be possible to produce true visible light (rather than night-vision-goggle infrared) animations of the simulated surface of HD 209458 b and other hot Jupiters. (In particular, HD 80606!)

Regular oklo readers all know that HD 209458 b is a lot bigger than it’s supposed to be.

In a previous post, we saw that the theoretical models that provide reasonable matches for the other 9 transiting planets predict that HD 209458 b’s radius should be slightly larger than Jupiter’s radius. The observations, on the other hand, make it clear that the planet is actually has a diameter about 1.35 times larger than Jupiter. HD 209458 b is by far the best-studied exoplanet, so it’s of more than passing interest to understand why it’s so large.

There’s general agreement that HD 209458 b must be privately tapping an unusual source of internal heat. Somehow, a lot of extra energy is being generated in the planetary interior. The surplus heat allows the planet to maintain an expanded outer envelope, and hence endows the planet with a larger overall size. The big question is: what is the anomalous extra heat source? A few years ago, I was enthusiastic about the idea that there might be a second companion planet that is gravitationally perturbing HD 209458 b, forcing it to maintain a slightly eccentric orbit. This eccentricity would be continually damped as a result of tidal interactions with the parent star, which would generate a sufficient amount of interior heating. Such a state of affairs is analogous to the heating of the inner Jovian satellites. The heat generated by tidal friction lends Io its off-the-hook volcanism, and maintains Europa as the Astrobiology poster world.

Unfortunately, however, the perturbing companion model no longer seems to be a viable explanation of HD 209458 b’s large size. That is, if you use the systemic console to fit to the HD 209458 b radial velocities, you’ll find that there is very little latitude for inserting significant extra planets. Try it and see, and upload your fits to the systemic back-end.

Josh Winn (MIT) and Matthew Holman (Harvard-Smithsonian CfA) have written a paper that presents an interesting hypothesis for resolving the HD 209458 b radius dilemma. Winn and Holman propose that the planet is caught in a so-called Cassini state, which is a resonance between spin precession and orbital precession. In a future post, I’ll give a heuristic discussion of the dynamics of how this situation can arise, and how the Cassini states work, but in short, if HD 209458b is trapped in the “Cassini state 2”, then its spin axis will lie almost in the orbital plane. Like all hot Jupiters, the planet will spin once per orbit, but it will literally be lying on its side as it orbits the parent star. A synchronous planet in Cassini state 2 will experience a large amount of tidal heating, even in the complete absence of any other planets in the system.

I like the Winn-Holman hypothesis because it’s potentially testable. If the planet is in Cassini state 2, then the pattern of illumination on the surface, and hence the time-dependant global infrared signature, will be very different than if it is locked into the standard upright configuration. In the standard scenario, a hot Jupiter has a fixed substellar point on its equator that does not wander significantly as the planet executes its orbit. One hemisphere of the planet is in perpetual day, while the other hemisphere experiences an endless night. Hydrodynamic calculations by James Cho and his collaborators (link), and by Adam Showman and his students (link), suggest that hot Jupiters should have a single strong equatorial jet that advects heat from the hot dayside to the cool night side. The oklo splash image has been adapted from Cho’s calculations, and shows this jet in action (see this post for more discussion).

I’ve been advising UCSC Physics graduate student Jonathan Langton, who has recently begun a study of what the flow pattern on a hot Jupiter should look like if the planet is caught in Cassini state 2. If the planet’s rotation axis lies in the orbital plane, and if the planet spins on its axis once per orbit, then the play of light and shadow across the planetary orb has a pattern that is totally unlike our seasons here on Earth. At the north and south poles, of a spin-synchronous Cassini-state-2 planet, the parent star rises, passes directly overhead, and then sets once per orbit. At one special spot on the equator, on the other hand, the star is always visible, and additionally passes directly overhead once per orbit. At the opposite spot on the equator (which we’ll call the anti-stellar point), the star never fully rises, but rather peeks half of its diameter above opposite horizons once per orbit.

Jonathan has made two short .avi format animations that help to illustrate the situation. In the first animation, we hover above the point on the equator that receives maximum illumination. In the second animation, we hover above the point on the equator that receives the least illumination. The mythology on such a world would likely be pretty interesting.

When a spin-synchronous planet is illuminated in this bizarre manner, the flow pattern on its surface should be very different than the flow pattern that would occur if the planet is in the standard upright configuration. Jonathan has finished a preliminary set of simulations using the so-called shallow water approximation which indicate that this is indeed the case. (The shallow water approximation is a 2-dimensional method for simulating atmospheric dynamics on the surface of the planet under the assumption that the depth of the fluid is much smaller than the horizontal scales of interest. Use of this approximation doesn’t require us to assume that the red-hot Jupiter is actually covered with water!)

Here are two of Jonathan’s .avi format animations that show the (still very) preliminary results. The first animation [11 MB, modem users watch out!] shows the evolution of the temperature distribution (on the anti-stellar hemisphere) for a planet in Cassini state 2. Here’s a snapshot at a particular moment in time:

The second animation [12 MB] shows the distribution of vorticity across the planet surface. The vorticity at a particular spot in a fluid flow can be thought of as the ability of the flow to cause a tiny imaginary paddle-wheel to spin. Here’s a snapshot from the animation. The high-vorticity orange structure is a giant fiery hurricane-like storm on the surface of the planet:

I think readers sometimes wonder why oklo.org posts about extrasolar planets tend to be illustrated with seemingly random photographs from my house, my yard, and my neighborhood that bear (at best) a distant relation to the topic at hand. I’m liable to get written up on charges of pseudo-artistic pretentiousness for attempting to run the Jones Soda of Astronomy.

We’re taking this approach for several reasons. The oklo blog is designed to recruit users for the systemic collaboration, and and I’m paying the ISP out-of-pocket. An idiosyncratic format makes it easier to keep the posts flowing in the midst of the chronically never caught up academic routine of teaching, research, qualifying exams, topic defenses, homework grading, proposal writing, committee meetings, undergraduate theses, graduate advising, editing, visiting speaker hosting, etc.

Another reason is to drive home my belief that the real interest, the real fascination with extrasolar planets will ultimately lie in their tiny ephemeral details. It’s one thing to gain an abstract, theoretical understanding of the growth of planetary cores through the accretion of small bodies — it’s quite another to see a one-kilometer bolide succumb to the fiery tendrils of fluid instability as it slams into the toxic atmospheric murk of an unsettled four Earth-mass world. It’s one thing to know Neptune’s orbital eccentricity to five significant figures, it’s quite another to swoop in close to see the flow of feathered cirrus outline the turbulent core of the great dark spot.

I can’t get over the fact that I can photograph the subtly intricate details of a habitable planet available in my own backyard, and later that afternoon have them transmitted digitally across the globe. Just think, if we had an autonomous lander with a 5-megapixel camera engaged in a small-scale survey of a hectare-sized region of an Earth-mass terrestrial planet in the habitable zone of any G2V star other than the Sun, then this site would be getting a lot more than 400 visitors per day.

Finally, in resorting to the use of photographs of familiar objects to illustrate unfamiliar things, I want to underscore the urgent need for scientifically correct visualizations of extrasolar planets.

New planet-related discoveries are the subject of numerous NASA and NSF press releases and press conferences, and because these dicoveries generally report information obtained by indirect observational techniques, there’s a need for illustrations to accompany the releases “to capture the public interest”.

During the last ten years, these images conveying scientific results have generally been supplied by artist’s impressions. On occasion, some of these have veered toward the bizarre, the lurid, and the just plain wrong. Consider, for example, the above two illustrations of HD 209458 b. Both appeared in fairly recent HST press releases, and both are riddled with profound misconceptions. First look at that painting on the left. At the time when the image was most recently released, it was well-known that satellite orbits around the planet are highly dynamically unstable (e.g. Barnes and O’Brien 2002). The shadowing of the planet only makes sense if the star is much smaller than the planet and is somehow orbiting just above the planetary surface. The number of cloud bands and zones indicate that the planet is rotating with a ~12 hour period (like Jupiter) whereas in reality, the planet must be spin-synchronous with its P=3.5257 day orbit, and should probably have a single prominent equatorial jet. The panel on the right, which attempts to show that the planet is surrounded by an optically thin, yet Lyman-alpha-absorbing hydrogen cloud, is dramatically inconsistent with the laws of perspective, illumination, and radiative transfer. Literally billions of dollars are being spent on extrasolar planets. Surely, at the pinnacle of our dispatches, we can do better than this.

Genuinely realistic visualizations of extrasolar planets that incorporate all known information in a self-consistent way are going to become an oklo.org rallying cry over the coming year. We won’t be flying to Upsilon Andromedae any time soon. Radial velocities, photometric light curves, stellar and planetary models, dynamical integrators, radiative transfer routines, hydrodynamic codes, and Maya are what we have to work with. If it is the destiny of the extrasolar planets to truly inspire, then we must strive to see them as they really are.