October 17, 2005

This Week's Finds in Mathematical Physics (Week 222)

John Baez

Last week there was a big conference on quantum gravity at the Albert Einstein Institute near Berlin:

1) Loops '05, http://loops05.aei.mpg.de

The focus was loop quantum gravity and spin foams, but there were also talks about other approaches, so it was much bigger than last year's get-together in Marseille. Last year about 100 people attended; this time about 160 did! It was strange seeing old pals like Ashtekar, Lewandowski, Loll, Rovelli and Smolin almost lost in a sea of new faces. But, it was great to talk to everyone, both old and new.

I'll say more about this conference, but first let's talk about γ ray bursters, a black hole without a host galaxy, the newly discovered moon of planet Xena, and lots of other transneptunian objects.

Actually, just for fun, let's start with this science fiction novel I picked up in Heathrow en route to Berlin:

2) Charles Stross, Accelerando, Ace Books, New York. Also available at http://www.accelerando.org/book/

This is one of the few tales I've read that does a good job of fleshing out Verner Vinge's "Singularity" scenario, where the accelerating development of technology soars past human comprehension and undergoes a phase transition to a thoroughly different world. This is a real possibility, and it's been discussed a lot:

3) Wikipedia, Technological singularity, http://en.wikipedia.org/wiki/Technological_singularity

Ray Kurzweil, The Singularity, http://www.kurzweilai.net/meme/frame.html?m=1

Anders Sandberg, The Singularity, http://www.aleph.se/Trans/Global/Singularity/

However, it's not an easy subject for fiction - at least not for mere human readers! Stross makes it gripping: sometimes goofy, sometimes thrilling, and sometimes rather sad. Characters include a robot cat with ever-growing powers and some space-faring uploaded lobsters.

The hero, Manfred Macx, starts out as a freeware developer, futurist and all-purpose wheeler-dealer. Here's a scene from the beginning of the book, before all hell breaks loose:

Manfred's mood of dynamic optimism is gone, broken by the knowledge that his vivisectionist stalker has followed him to Amsterdam to say nothing of Pamela, his dominatrix, source of so much yearning and so many morning-after weals. He slips his glasses on, takes the universe off hold, and tells it to take him for a long walk while he catches up on the latest on the tensor-mode gravitational waves in the cosmic background radiation (which, it is theorized, may be waste heat generated by irreversible computational processes back during the inflationary epoch; the present-day universe being merely the data left behind by a really huge calculation). And then there's the weirdness beyond M31: according to the more conservative cosmologists, an alien superpower - maybe a collective of Kardashev Type Three galaxy-spanning civilizations - is running a timing channel attack on the computational ultrastructure of space-time itself, trying to break through to whatever's underneath. The tofu-Alzheimer's link can wait.
An idea a minute - and the book is free online: what more could you want?

But right now, the big news in astronomy is not about a type III civilization lurking beyond M31 (otherwise known as the Andromeda Galaxy). It's some evidence that short γ ray bursts are caused by collisions involving neutron stars and black holes!

Γ ray bursts are among the most energetic events known in the heavens. They happen in galaxies throughout the universe; we see about one a day, and each releases somewhere between 1045 and 1047 joules of energy. The larger figure is what you'd get by turning the entire mass of the Sun into energy.

There could be several kinds of γ ray bursts, but there seem to be at least two: short and long. Short bursts last between 40 milliseconds and 10 seconds - imagine the whole Sun turning into energy that fast! Long ones last between 10 and 100 seconds. The two kinds seem to be qualitatively different: for example, the short ones consist of higher-frequency γ rays. The big news is that they happen in different kinds of galaxies!

In "week204", I described how people caught a long γ ray burst in the act in March 2003. A γ ray detector aboard a satellite relayed information to telescopes in Australia and Japan, allowing them to spot a visible afterglow right after the burst. The details of this glow fit the "hypernova" theory of long γ ray bursts.

The hypernova theory says that when a star more than 25 times heavier than the Sun runs out of fuel and collapses, it forms a black hole that sucks down the star's iron core before a normal supernova explosion can occur. In just a few seconds, about a solar mass of iron spirals into the black hole, forming a pancake-shaped disk as it goes down. In the process, this disk becomes incredibly hot and shoots out jets of radiation in the transverse directions. As they plow through the star's outer layers, these jets create beams of γ rays.

The short bursts have been harder to catch. By the time a telescope on Earth could be aimed at the spot where the γ rays were seen, no afterglow could be seen!

So, in October 2004 NASA launched Swift: a γ-ray detecting satellite equipped with an X-ray telescope and an ultraviolet/optical telescope that can respond quickly whenever a burst is seen:

4) Official NASA Swift homepage, http://swift.gsfc.nasa.gov/docs/swift/swiftsc.html

5) Gamma-ray burst real-time sky map, http://grb.sonoma.edu/

On May 9th, 2005, Swift detected a short burst and caught 11 photons of the burst's X-ray afterglow. Another short burst detected by HETE-II had its X-ray afterglow caught by the Chandra X-ray satellite. Analysis of these and two more short bursts has convinced some scientists that they're caused by collisions between neutron stars and/or black holes:

6) D. B. Fox et al, The afterglow of GRB050709 and the nature of the short-hard γ-ray bursts, Nature 437 (October 2005), 849-850. Also available at http://www.nasa.gov/pdf/135397main_nature_fox_final.pdf

Despite what the news media are saying, I don't see that this paper "proves" the short γ-ray bursts are caused by such collisions. Instead, I see some good pieces of evidence.

The faintness of the afterglows suggests some mechanism other than a hypernova. But as far as I can tell, the best evidence is that short γ ray bursts tend to happen near the edges of old galaxies, while the long ones happen near the centers of young galaxies.

The center of a young galaxy is where you'd expect to find a really huge Wolf-Rayet star, the sort that dies in a hypernova. The edge of an old galaxy is where you'd expect to see black holes and neutron stars collide. Why? Because such collisions can only happen long after stars are first formed. First you need an orbiting pair of giant stars to go supernova and collapse into neutron stars and/or black holes. Then you need plenty more time for this pair to spiral down thanks to gravitational radiation, and eventually collide. By then the pair may sail off to the edge of the galaxy, thanks to the "kick" delivered by the supernova explosions.

I hope astronomers can clinch the case for the collision theory of short γ ray bursts. After all, these collisions involving neutron stars and black holes are precisely what gravitational wave detectors like LIGO and VIRGO are hoping to see! If we know to look for gravitational waves precisely when we see short γ ray bursts, and we know where they're coming from, we'll have a better chance of finding them.

(Of course, we'll also have a better chance of fooling ourselves into thinking we found them, until we do some double-blind tests.)

By the way, LIGO is already analysing data to look for gravitational waves. I talked about this in "week189", but here's something new: now you can help them by running a cool screensaver called Einstein@Home on your computer! Check it out:

7) Einstein@Home, http://einstein.phys.uwm.edu/

Speaking of black holes, last month the Hubble Space Telescope and the Very Large Telescope in Chile detected a quasar that seems to have no host galaxy:

8) European Southern Observatory, Black hole in search of a home, http://www.eso.org/outreach/press-rel/pr-2005/pr-23-05.html

HubbleSite, Quasar without host galaxy compared with normal quasar, http://hubblesite.org/newscenter/newsdesk/archive/releases/2005/13/image/a

Quasars are thought to be super massive black holes; they're usually found in the centers of galaxies, where they devour stars and shoot out enormously powerful jets of radiation. However, the quasar HE0450-2958 is surrounded only by a blob of ionized gas. Nearby, a wildly disturbed spiral galaxy can be seen.

Compare HE0450-2958 (at left) with a normal quasar (at right):

The quasar HE0450-2958 is in the middle of the left-hand picture; the disturbed galaxy is above and a completely irrelevant foreground star is below. For more details on what this image means, click on it.

Did this quasar begin life in the middle of a galaxy and then get kicked out when that galaxy collided with something containing a super-massive black hole? What could that something be?

Puzzles, puzzles, in the sky....

Closer to home, astronomers at the Keck Observatory in Hawaii have discovered that planet Xena has a moon!

They nicknamed it Gabrielle, after this famous TV character's sidekick:

9) Michael E. Brown, The moon of the 10th planet, http://www.gps.caltech.edu/~mbrown/planetlila/moon/index.html

If you hadn't heard about planet Xena, or you don't like the idea of naming a planet after a TV character - even a "warrior princess" - don't get worked up just yet. Xena's official name is currently 2003 UB313, and though she's larger than Pluto, the International Astronomical Union has not decided whether she'll officially be considered a planet.

If Xena becomes a planet, she'll probably be renamed Persephone, after the reluctant queen of the underworld in Greek mythology. But, she may have to settle for the status of a mere "transneptunian object", like Quaoar and Sedna. Indeed, if Pluto had been discovered more recently, folks probably wouldn't have called him a planet either.

If you haven't even heard of Quaoar and Sedna... well, you must be too absorbed by mundane concerns to keep track of the burgeoning population of our Solar System. But it's not too late to mend your ways! Impress your friends by casually dropping some of this jargon:

For a great introduction to the Kuiper belt and related topics, try this:

10) David C. Jewitt, Kuiper belt, http://www.ifa.hawaii.edu/faculty/jewitt/kb.html

For transneptunian objects in general, try:

11) William Robert Johnston, Transneptunian objects, http://www.johnstonsarchive.net/astro/tnos.html

Also check out this newsletter:

12) Distant EKOs: the Kuiper Belt Electronic Newsletter, http://www.boulder.swri.edu/ekonews/

Quaoar was discovered in 2002 by Chad Trujillo and Mike Brown of Caltech:

13) Chad Trujillo, Quaoar, http://www.gps.caltech.edu/~chad/quaoar/

For evidence of crystalline water ice on Quaoar, see:

14) David C. Jewitt and Jane Luu, Crystalline water ice on the Kuiper belt object (50000) Quaoar, Nature 432 (2004), 731-733. Also available at http://www.ifa.hawaii.edu/faculty/jewitt/quaoar.html

Xena was discovered in 2003 by Trujillo, Brown and a colleague of theirs at Yale University:

15) Michael E. Brown, Chad A. Trujillo and David L. Rabinowitz, Discovery of a planetary-sized object in the scattered Kuiper belt, submitted to ApJ Letters, available at http://www.gps.caltech.edu/%7Embrown/papers/ps/xena.pdf

Brown has a nice webpage about Xena and Gabrielle:

16) Michael E. Brown, The discovery of UB313, the 10th planet, http://www.gps.caltech.edu/~mbrown/planetlila/

The same gang of three also discovered Sedna in 2003:

17) Michael E. Brown, Chad A. Trujillo and David L. Rabinowitz, Discovery of a candidate inner Oort cloud planetoid, to appear in ApJ Letters, available at http://www.gps.caltech.edu/%7Embrown/papers/ps/sedna.pdf

... and Brown has a fun Sedna webpage too:

18) Michael E. Brown, Sedna (2003 VB12), http://www.gps.caltech.edu/~mbrown/sedna/

How all these transneptunian objects got where they are is a wonderful puzzle in celestial mechanics, but you can read more about that in the references above, especially Jewitt's Kuiper belt webpage.

Now I want to talk about Loops '05!

Instead of trying to review all the talks - a hopeless task, since there were 86 - I'll just mention the two strands of work I find most exciting.

First, there's new evidence that a quantum theory of pure gravity (meaning gravity without matter) makes sense in 4-dimensional spacetime.

To understand why this is exciting, you have to realize that in some quarters, the conventional wisdom says a quantum theory of pure gravity can't possibly make sense, except as a crude approximation at large distance scales, because this theory is "perturbatively nonrenormalizable".

Very roughly, this means that as we zoom in and look at the theory at shorter and shorter distance scales, it looks less and less like a "free field theory" where gravitons zip about without interacting. Instead, the interactions get stronger and more complicated!

So, in the jargon of the trade, we don't get a "Gaussian ultraviolet fixed point".


Well, roughly, an "ultraviolet fixed point" is a quantum field theory that keeps looking the same as you keep viewing it on shorter and shorter distance scales. A "Gaussian" ultraviolet fixed point is one that's also a free quantum field theory: one where particles don't interact.

If quantum gravity approached a Gaussian ultraviolet fixed point as we zoomed in, we could calculate what gravitons do at arbitrarily high energies (at least perturbatively, as power series in Newton's constant - no guarantee that these series converge). Particle physicists would then be happy and say the theory was "perturbatively renormalizable".

But, it's not.

The conventional wisdom concludes that to save quantum gravity, we must include matter of precisely the right sort to make it perturbatively renormalizable. This is the quest that led people first to supergravity and ultimately to superstring theory - see "week195" for more of this story.

But, as far back as 1979, the particle physicist Weinberg raised the possibility that pure quantum gravity is "nonperturbatively renormalizable", or "asymptotically safe". This means that as we zoom in and look at the theory at shorter and shorter distance scales, it approaches some theory other than that of noninteracting gravitons.

In other words, Weinberg was suggesting that pure quantum gravity approaches a non-obvious ultraviolet fixed point - possibly a "non-Gaussian" one.

The big news is that this seems to be true!

Even cooler, in this theory spacetime seems to act 2-dimensional at very short distance scales.

This idea has been brewing for a long time - I talked about it extensively back in "week139". But now there's more solid evidence for it, coming from two quite different approaches.

First, people doing numerical quantum gravity in the "causal dynamical triangulations" approach are seeing this effect in their computer calculations. This is what Renate Loll explained at Loops '05. The best place to read the details is here:

19) Jan Ambjørn, J. Jurkiewicz and Renate Loll, Reconstructing the universe, Phys. Rev. D72 (2005) 064014. Also available as hep-th/0505154.

but if you need something less technical, try this:

20) Jan Ambjørn, J. Jurkiewicz and Renate Loll, The universe from scratch, available as hep-th/0509010.

The titles of their papers are a bit grandiose, but their calculations are solid stuff - truly magnificent. I described their basic strategy in my report on the Marseille conference in week206. So, I won't explain that again. I'll just mention their big new result: in pure quantum gravity, spacetime has a spectral dimension of 4.02 ± 0.1 on large distance scales, but 1.80 ± 0.25 in the limit of very short distance scales!

Zounds! What does that mean?

The "spectral dimension" of a spacetime is the dimension as measured by watching heat spread out on this spacetime: the short-time behavior of the heat equation probes the spacetime at short distance scales, while its large-time behavior probes large distance scales. Spectral dimensions don't need to be integers - for fractals they're typically not. But, Loll and company believe they're seeing spacetimes that are exactly 2-dimensional in the limit of very small distance scales, exactly 4-dimensional in the limit of very large scales, with a continuous change in dimension in between. The error bars in the above figures come from doing Monte Carlo simulations. They're just using ordinary computers, not supercomputers. So, with more work one could shrink their error bars and test their result.

My main worry about their work is that it uses a fixed slicing of spactime by timelike slices. So, there's a danger that their procedure breaks Lorentz-invariance, even in the continuum limit which they are attempting to compute. I would like to find a way around this problem!

Luckily, some other people are getting similar results from a second procedure that definitely does not break Lorentz invariance:

21) Oliver Lauscher and Martin Reuter, Fractal spacetime structure in asymptotically safe gravity, available as hep-th/0508202.

Reuter spoke about all this work at Loops '05. The idea is to investigate Weinberg's original idea in excruciatingly precise detail using "renormalization group flow" ideas. The above paper is a review of lots of others, and you need to read a bunch to get what's really going on. The upshot, however, is that they find evidence for a non-Gaussian ultraviolet fixed point in pure quantum gravity. Moreover, the spectral dimension of spacetime approaches 2 in the limit of very short distance scales.

Suppose this is all true. What does it mean?

Nobody knows yet; there are lots of attitudes one could take.

Ambjørn, Jurkiewicz and Loll could probably just plunge ahead and use computers to calculate lots of things about quantum gravity. (Right now they want to test their results in lots of ways.) One good thing would be to include matter of various sorts and see how it affects the conclusions.

Similarly, Lauscher and Reuter could just plunge ahead and compute, if they wanted.

This is excellent. But personally, I'd like to find a beautiful theory in which spacetime is 2-dimensional at short distance scales, which reduces to general relativity at large scales. In other words, to redo all these calculations "from the bottom up".

Unsurprisingly, I hope this beautiful theory is a spin foam model, since spin foams are 2-dimensional and I like them a lot. I presented some rough ideas on how one might invent such a model:

22) John Baez, Towards a spin foam model of quantum gravity, talk at Loops '05, available at http://math.ucr.edu/home/baez/loops05/

But, these ideas are very tentative and only time will tell if they amount to anything. What's more important is that pure quantum gravity seems to exist - as a theory, that is - and people seem to be learning actual facts about it, instead of just arguing endlessly about it. That's progress!

The second most exciting thing at Loops '05, in my biased opinion, was the work of John Barrett, Laurent Freidel, Karim Noui and others on "matter without matter" in 3d quantum gravity. Simply by carving a Feynman-diagram-shaped hole in 3d spacetime and doing quantum gravity on the spacetime that's left over, you get a good theory of quantum gravity coupled to matter! You can even take the limit as Newton's gravitational constant goes to zero and get ordinary quantum field theory on flat spacetime!

Check these out:

23) John Barrett, Feynman diagams coupled to three-dimensional quantum gravity, available as gr-qc/0502048.

John Barrett, Feynman loops and three-dimensional quantum gravity, Mod. Phys. Lett. A20 (2005) 1271. Also available as gr-qc/0412107.

24) Laurent Freidel and David Louapre, Ponzano-Regge model revisited I: gauge fixing, observables and interacting spinning particles, Class. Quant. Grav. 21 (2004) 5685-5726. Also available as hep-th/0401076.

Laurent Freidel and David Louapre, Ponzano-Regge model revisited II: equivalence with Chern-Simons, available as gr-qc/0410141

Laurent Freidel and Etera R. Livine, Ponzano-Regge model revisited III: Feynman diagrams and effective field theory, available as hep-th/0502106.

25) Laurent Freidel, Daniele Oriti, and James Ryan, A group field theory for 3d quantum gravity coupled to a scalar field, available as gr-qc/0506067.

26) Karin Noui and Alejandro Perez, Three dimensional loop quantum gravity: coupling to point particles, available as gr-qc/0402111.

This is mindblowingly beautiful, especially because lots of it is already mathematically rigorous, and we can easily make more so. It's even related to n-categories: my student Jeffrey Morton presented a poster on this aspect.

Together with my student Derek Wise, Jeffrey Morton and I plan to have a lot of fun studying this stuff. So, I won't talk about it more now - I'll probably get around to saying more someday, especially about how the whole story generalizes to 4 dimensions.

There's a lot more to say about Loops '05, but this will have to do. In a while, a bunch of the talks should be visible on the conference homepage.... that should give you a better idea of what happened.

Addendum: Here are some comments on this Week's Finds by Gene Partlow, Phillip Helbig and Robert Helling, and my replies - as well as a replies by Jonathan Thornburg and Arnold Neumaier.

Gene Partlow writes:

In a recent John Baez post he mentions discovery of probably the first known quasar found without a host galaxy. He says:
    Did this quasar begin life in this galaxy and then get kicked 
    out when the galaxy collided with something containing a 
    super-massive black hole?
I suggest that a fairly ordinary explanation may be that the nearby "wildly disturbed" galaxy may have contained several supermassive black holes which interacted via a gravitational slingshot scenario. This would be like a larger version of the effect where smaller mass stars can be flung out of globular clusters when encountering larger mass stars near the cluster center.
Sounds like a possibility worth exploring. I'm no expert, so I can't tell how likely this is. I agree that a collision with some other object containing a super-massive black hole sounds a little odd, given that this object - most plausibly another galaxy - has not been seen. I wouldn't have ventured such a guess myself. But, it's mentioned on the European Southern Observatory webpage I cite above. To quote:
The absence of a massive host galaxy, combined with the existence of the blob and the star-forming galaxy, lead us to believe that we have uncovered a really exotic quasar, says team member Frederic Courbin (Ecole Polytechnique Federale de Lausanne, Switzerland). "There is little doubt that a burst in the formation of stars in the companion galaxy and the quasar itself have been ignited by a collision that must haven taken place about 100 million years ago. What happened to the putative quasar host remains unknown."

HE0450-2958 constitutes a challenging case of interpretation. The astronomers propose several possible explanations, that will need to be further investigated and confronted. Has the host galaxy been completely disrupted as a result of the collision? It is hard to imagine how that could happen. Has an isolated black hole captured gas while crossing the disc of a spiral galaxy? This would require very special conditions and would probably not have caused such a tremendous perturbation as is observed in the neighbouring galaxy.

Another intriguing hypothesis is that the galaxy harbouring the black hole was almost exclusively made of dark matter. "Whatever the solution of this riddle, the strong observable fact is that the quasar host galaxy, if any, is much too faint", says team member Knud Jahnke (Astrophysikalisches Institut Potsdam, Germany).

Phillip Helbig writes:
John Baez writes:
     Pluto is quite different than anything else we call a planet: it 
     has an eccentric orbit that ranges between 30 and 50 AU, and its 
     orbit is tilted 17 degrees to the ecliptic.  
Also, for a period of several years during each orbit, it is closer to the Sun than Neptune ever is. Until relatively recently, in fact, it was in this phase and Neptune was farther from the Sun than Pluto.
Yeah! The US may still send a mission to Pluto before its atmosphere freezes, despite delays and indecision. The "Pluto Express" satellite was scheduled for launch in December 2004, but in 2000 NASA ordered a stop-work order on the project, due to lack of money and rising cost estimates. In 2003, Congress gave NASA money to proceed with the project:

27) The Planetary Society, Pluto and Europa Campaign Page, http://www.planetary.org/html/UPDATES/Pluto/pluto_europa_action.html

and I guess it's now called "New Horizons":

28) New Horizons: NASA's Pluto-Kuiper Belt Mission, http://pluto.jhuapl.edu/

This webpage gives a timetable of:

I wonder if they have any plans to study the Pioneer anomaly?

Robert Helling writes:

John Baez wrote:
     o Twotino - A twotino is a Kuiper belt object whose orbit is in 2:1
       resonance with Neptune.  These are rare compared to plutinos, and
       they're smaller, so they're stuck with boring names like 1996 TR66.
       There are also a couple of Kuiper belt objects in 4:3 and 5:3
       resonances with Neptune.
Is there an easy way to see why these resonance orbits come about? Why do three body systems with a large central object, an intermediate planet and a small probe happen to get the probe in resonance with the planet? Is this just "frequency locking happens in chaotic systems" or is there an easy but more quantitative way to understand this?
I'm shamefully ignorant of this, so ten minutes' research on the web was able to double my knowledge. I got ahold of this paper online:

29) B. Garfinkel, On resonance in celestial mechanics: a survey, Celestial Mech. 28 (1982), 275-290, also available at http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1982CeMec..28..275G

and while not easy to understand - I guess there's a huge body of work on this subject - it uses Hamiltonian perturbation theory and continued fractions to study resonance, and talks about a difference between "shallow" and "deep" resonances.

The orbits of Jupiter and Saturn are almost in 5:2 resonance, but this is a "shallow" resonance, and Saturn wiggles back and forth around this resonance with a period of about 880 years - an effect called the "Great Inequality". The first person to study this was Laplace. I read elsewhere that:

The dynamics of the Sun-Jupiter-Saturn system was recognized as problematic from the beginnings of perturbation theory. The problems are due to the so-called Great Inequality (GI), which is the Jupiter-Saturn 2:5 mean-motion near-commensurability.
This is from:

30) F. Varadi, M. Ghil, and W. M. Kaula, The great inequality in a planetary Hamiltonian theory, available as chao-dyn/9311011.

This shallow 5:2 resonance is related to the continued fraction

1/(2 + 1/(2 + 1/(14 + 1/(2 + .... ))))
which is close to 2/5.

The Pluto-Neptune 3:2 resonance, on the other hand, is a "deep resonance" and related to the continued fraction

1/(2 - 1/(2 + 1/(10 + .... )))
which starts out close to 2/3.

I wish I understood the connection between continued fractions and dynamical systems better! I know it gives rise to the role of the Golden Ratio in chaos theory, which I tried to explain in "week203". But, I don't understand it very deeply.

Robert Helling also asked:

There are people doing numerical long term stability analysis of the solar system. From what I know, they are not just taking F=ma and Newton's law of gravity, replacing dt by Δt and then integrating but use much fancier spectral methods. Could somebody please point me to an introduction into these methods?
I referred him to the work of Gerald Sussman, Jack Wisdom and others in "week107", but Jonathan Thornburg posted this reply:
I don't do this sort of work myself, but the buzzwords you want are "symplectic ODE integrator". The basic idea is to use an ODE integration scheme which conserves energy, angular momentum, and maybe other nice things, up to floating-point roundoff error, rather than just up to finite differencing error like a standard ODE integrator would do.
prompting Arnold Neumaier to give a nice list of references, which I will take the liberty of numbering:
31) Tetsuharu Fuse, Planetary perturbations on the 2:3 mean motion resonance with Neptune, Publ. Astron. Soc. Japan 54 (2002), 494-499. Also available at http://astronomy.nju.edu.cn/~xswan/reference/Fuse_PASJ54_493.pdf

uses symplectic integration to study 2:3 resonances numerically.

The thesis:

32) Luz Vianey Vela-Arevalo, Time-frequency analysis based on wavelets for Hamiltonian systems, Caltech, 2002. Also available at http://www.cds.caltech.edu/~luzvela/th2s.pdf

contains in Chapter 4 interesting numerical information about chaos, resonances, and stability in the restricted 3-body problem. Other interesting papers include:

33) George Voyatzis and John D. Hadjidemetriou, Symmetric and asymmetric librations in planetary and satellite systems at the 2/1 resonance, available at http://users.auth.gr/~hadjidem/Asymmetric1.pdf

Mihailo Cubrovic, Regimes of stability and scaling relations for the removal time in the asteroid belt, available as astro-ph/0501004.

Ryszard Gabryszewski and Ireneusz Wlodarczyk, The resonant dynamical evolution of small body orbits among giant planets, available as astro-ph/0203182.

Luz V. Vela-Arevalo and Jerrold E. Marsden, Time-frequency analysis of the restricted three-body problem: transport and resonance transitions, Class. Quant. Grav. 21 (2004), S351-S375. Also available at http://cns.physics.gatech.edu/~luzvela/VelaArevaloMarsdenCQG_2004.pdf

Harry Varvogli, Kleomenis Tsiganis, and John D. Hadjidemetriou, The "third" integral in the restricted three-body problem revisited, available at http://www.astro.auth.gr/~varvogli/varv5.ps

So many worlds, so much to do,
So little done, such things to be.
      - Tennyson, In Memoriam

© 2005 John Baez