Good news for the systemic user base! Eugenio has posted a new version of the downloadable systemic console. This most recent update fixes several bugs, and offers a better graphical interface for those working at limited display resolutions. Progress overall has been rapid during the past several days, and next week we’re planning to roll out both a fully threaded version of the console as well as the Systemic Jr. catalog of synthetic radial velocity data sets. Systemic Jr. will be a testbed for the full Systemic simulation, and will allow us to answer a number of interesting questions regarding the fidelity of planetary models as a function of orbital parameters and observational sampling. Put oklo.org on your bookmark list and tell your friends to drop by. We’ve manufactured plenty of consoles to hand out.

By tomorrow I’ll be back on the extrasolar planets beat, but I thought it would be interesting to show a few more results connected to the strange orbits detailed in the previous post.

It’s clear from the sample of eight orbits that were charted that the m=1 singular isothermal disk potential supports an extensive variety of orbital families: tube orbits, box orbits, chaotic orbits, resonant orbits, and Enron orbits just to name a few. Is it possible to design a map that shows the regions of parameter space that are delineated by the different kinds of orbits?

The best method that I’ve been able to devise consists of what I’ll call an “excursion map”. We can clasify orbits by the total angle that they accumulate over time. For example, a loop orbit (such as the first trajectory shown in the previous post) experiences a steady accumulating of total angle — 360 degrees worth per orbital period. A box-type orbit on the other hand (like the seventh and eighth trajectories shown in the previous post) oscillates back and forth across the x-axis and never accumulates more than 90 degrees or so of total angular excursion. Chaotic orbits (such as the sixth example trajectory) execute a random walk in angular excursion, and on average accumulate a total absolute angular excursion (either positive or negative) which is proportional to the square root of the time.

We can thus pack information about the orbital structure of the potential function into a single diagram. We choose intial starting conditions parameterized by position on the x-axis and the e-parameter of the potential function. A given starting condition corresponds to a point on a two-dimensional diagram, and also defines an orbit. The orbit can be integrated for a characteristic time (t=1000, say) and the total angular excursion or the orbit can be logged. A color code can then be assigned: white for orbits that accumulate positive angle in direct proportion to the time, gray for orbits that accumulate angle in proportion to the square root of the time, and dark gray for orbits that never get beyond plus or minus 90 degrees. With this coding, the excursion map looks like this:

The numbers label the locations of the 8 different orbits shown in the previous post.

Take orbit 3, for example. It corresponds to a loop-type orbit within an island of similar loop orbits surrounded by a sea of chaotic orbits. If we zoom in on the island with a magnification factor of ten, we see structure emerge. Tiny changes in the initial conditions determine whether an orbit is stable (white) or chaotic (gray). Two jagged fingers of box orbits jut up into the map.

Zooming in by another factor of ten shows that the map has a fractal structure, with detail emerging on every level of magnification:

It’s strange to realize how so much bizarre structure is inherent within such a simple potential function. Somehow, encapsulated into one simple formula, the dynamics are all folded up like an inifinite series of orgami cranes, waiting patiently to be observed…

Image Source.

The academic quarter is pulling up to the half-way point at UCSC, and it’s getting tougher to keep up with everything that I’m supposed to do. I’ve been spending a lot of time getting the lectures together, so in the interest of sticking to a schedule of posts, I thought I’d veer from the all-planets-all-the-time approach and show some scanned transparencies from Friday’s class.

The orbit of an idealized planet around an idealized star is a keplerian ellipse. A planet on an elliptical trajectory conserves its eccentricity, orbital period, longitude of periastron, inclination, and line of nodes. The only orbital element that changes over time is the mean anomaly. We can thus say that the Keplerian orbit contains five integrals (constants) of the motion.

If the potential arises from a mass distribution that’s not a perfect point mass, then in general we won’t have five integrals of motion. It’s interesting to look at a subsection of the weird variety of planar orbits that occur in a two-dimensional potential distribution that looks like this:

The ln(r) term causes this potential goes to negative infinity at the origin while remaining unbounded at large radii. The second term, in which e is specified to take on values between 0 and 1, lends a modulation that makes the force law non-axisymmetric with respect to the origin. One can roughly think of orbits in this potential as the motion of a marble rolling in a funnel-shaped, lopsided bowl.

The potential function does not change with time, and so the energy of an orbiting particle is conserved. Further, because of the self-similarity of the potential, the structure of orbits at one energy will be an exact copy of the orbit structures at all other energies. Thus, there’s no loss in generality by sampling orbits having only a single total energy (kinetic + potential). In the following sampler of pictures, I integrate the trajectories of single particles launched from the long (+x) axis of the potential with initial velocities always perpendicular to the long axis. The magnitude of the velocities are determined by the total energy choice: particles starting closer to the origin must have a higher initial kinetic energy to offset their more negative gravitational potential energy. I also vary the parameter e.

For e=0.2 and an initial position x=0.78, the orbit is reasonably circular, and steady precession smears the excursion of the particle over many orbits into a thin annular region centered on the origin.

For e=0.42, x=0.55, the particle starts fairly close to its zero velocity curve. It thus falls inward almost to the origin before making a second loop, and then a second approach to the origin which sends it rocketing back up close to its initial position.

Taking e=0.2, x=0.78 leads to a single loop orbit that dives in very close to the origin.

Here’s the result of taking e=0.42 and x=0.55. It’s a good thing the Earth isn’t orbiting in this potential with these particular starting conditions.

This one, which arises from e=0.81 and x=0.85 is pretty cool. Most of the time it runs counterclockwise as viewed from above, but before close approach to the origin it switches to clockwise. One’s tempted to classify it as an Enron orbit. From all appearances it appears to be clocking a steady increase in angle, whereas in reality, when the books are finally audited, it’s accumulating -2pi radians every period.

Choosing e-0.30, x=0.03 launches the particle on a highly chaotic trajectory. This orbit is uniformly sampling the entire area allowed to it by its total energy constraint.

For larger values of the e parameter the orbits often show a fundamentally different behavior. Choosing e=0.72, x=0.22 leads to a motion which is restricted to oscillations of a narrow angular range centered on the long-axis of the potential. The particle is basically rolling back and forth in the narrow valley provided by the high-e potential.

e=0.90, x=0.20 gives an orbit with a similar quality:

Image Source.

Last week, Joseph Harrington and his collaborators published a paper in Science that announced the results of a very interesting set of observations of Upsilon Andromedae with the Spitzer Space Telescope.

As console users know, Upsilon Andromedae is accompanied by three Jovian planets. The innermost body (officially known as “Dinky“) has at least 70% of Jupiter’s mass and orbits with a period of 4.6 days. Observers have checked to see whether Dinky passes directly in front of the parent star. They found that transits don’t occur, and so the orbital geometry likely looks something like this (as seen from Earth, with the planet grossly not to scale):

Harrington and collaborators made careful measurements of the infrared brightness of the star in the 24-micron band at five known phases during the planetary orbit. These phases are marked with small yellow circles in the above plot.

When the data were reduced, it was found that the brightness of the star was varying in phase with the orbital period of the planet. The brightness is lower when Dinky is in front of the star (near “inferior conjunction”) and higher when more of the planet’s illuminated surface is in view.

The difference in brightness during the course of the orbit is consistent with a temperature difference of order 1000 K between the illuminated dayside and the dark night side. The planet should be spin-synchronized, so that one side always faces the star and the other face is always pointed away. Harrington et al. showed that the data could be understood if it is assumed that the planet transfers very little heat to the night-side, thus allowing the large temperature difference to be maintained. In fact, they were able to get a good model of the brightness variations by assuming that the night-side was not radiating at all. Such a model curve looks like this:

Intuitively, this result seems to make perfect sense. You’d expect a spin-synchronized planet to be hottest at the subsolar point, and coldest at the antistellar point, and this picture is fully consistent with the five observed fluxes. The results are surprising, however, when we take into account the fact that there should be hellacious winds on the planetary surface which should disgorge heat onto the night side.

UCSC graduate student Jonathan Langton has been studying the surface flows on hot Jupiters using a hydrodynamic technique known as the shallow water approximation. A often-seen feature of his models is that the hottest point on the surface of a synchronously rotating planet is well eastward from the substellar point. (A similar state of affairs is predicted by Cooper and Showman, who use a full 3D GCM-type model.)

Similarly, the coldest spot on the night side, is also displaced eastward from the anti-stellar point:

These models predict a smaller day-night temperature difference than the no-redistribution model that Harrington et al. fitted to the data. A smaller day-night temperature difference can indeed be accomodated by the observations, but the predicted phase shift seems highly inconsistent at first glance. Eastward-displaced hot and cold spots give a (edge-on inclined) lightcurve that is clearly out of phase:

Taken at face value, the observations thus seem to suggest that the flows on the planet are very effective at radiating heat. That is, the upper layers that we can actually observe seem to have a short radiative time constant. In a set of upcoming posts, we’ll have a closer look at the interpretation of this very interesting new result.

Image Source.

SETI and the idea of alien life are the stuff of endlessly fascinating speculation. I remember wild late-night conversations with my freshman dorm mates when we should have been writing lab reports and studying for chemistry exams. To date, however, the SETI hasn’t turned up anything, and the Fermi Paradox seems as perplexing as ever. Proponents of the conventional SETI approach argue that this is because we’ve barely scratched the surface in terms of the number of stars that we’ve observed. Build a bigger telescope, they argue, scan more stars, and success will come.

If I look at my own behavior, the trend has been toward increasingly frequent correspondence with more and more people. The cell phone rings many times a day. I send a lot of e-mails via wireless internet. I look on flickr to see if my photos have accumulated views or comments. My life revolves around connectivity. I rarely send letters through regular post, and I have little interest in conversations with a response time of 8.78 years. I’m not inclined to beam coded messages to the sky, and I don’t shine high-power collimated lasers at nearby stars. My behavior is similar in aggregate to many, many others here on Earth.

It seems reasonable, then, that the most promising strategy for a succesful SETI is to look for behaviors that resemble our own. I think it’s much more likely to detect another civilization through their signal “leakage” rather than through reception of a directed message. If I knew that it was going to take at least 8.78, and in all likelihood millions of years for my photos to accumulate views, I’d soon start neglecting to post them.

When I was at the CfA last week, I had an interesting conversation with Avi Loeb, who pointed out that at present, the largest sources of artificial terrestrial radio emission are military radars, FM radio broadcasts, and television broadcasts, all of which emit their power in the frequency range between about 40 and 800 Mhz. SETI searches, on the other hand, have focused in the frequency range above 1 Ghz.

Loeb is involved in the Mileura Wide Field Array (MWA), which is a low-frequency radio telescope designed to study highly redshifted 21 centimeter emission from hydrogen. By mapping the spatial distribution and redshift distribution of 21 centimeter emission, the Mileura project will be able to make a 3-dimensional map of the distribution of atomic hydrogen in the early universe.

The MWA will provide an enormous increase in sensitivity at exactly the frequencies that we here on Earth broadcast. Loeb recently received a grant from the FQXi foundation to run a SETI-search on data obtained during the course of MWA survey observations. The cosmic signals received will be combed for telltale artificial emissions from nearby stars. The array will be sensitive enough to detect Earth-like leakage from more than 1000 of the nearest stars, a list that includes oklo.org Southern Hemisphere favorites such as Alpha Centauri B, Beta Hyi, GJ 780, and Tau Ceti.

Loeb informs me that he’s posted an overview paper on astro-ph. Look for it on Sunday night, 5PM PST.

While we’re on the topic, I recently participated in a panel discussion on SETI that closed up the AIAA Space 2006 meeting in San Jose. I argued that the resolution of the Fermi Paradox lies in the fact that we’re inward bound. My understanding is that the video of the discussion will go up on the web at some point, but for the moment, here’s a .pdf (4MB) file with the transparencies that I showed in my 10 minute summary.