One More Epicycle

[Currently playing: "Love is All Around", a version of which was used as the Mary Tyler Moore theme song.  Turns out to have been written by Sonny Curtis, who also wrote "I Fought the Law".  The things you learn from listening to the Diner!]

This week’s full of a lot of talks to launch EPSTAR, the Experimental Particle, String Theory and Astronomy Research consortium — although by astronomy they have in mind cosmology and not planetary dynamics.  Professor Sir Roger Penrose, Fellow of the Royal Society, Order of Merit, as he’s formally introduced, is the Very Special Guest.

On Monday he gave a public lecture on “Before the Big Bang”.  I tried to record it but something went wrong, which is too bad: think what a bootleg copy of a Penrose talk could bring on the black market!  [nothing --ed.  spoilsport! --me]  He presented various ideas of different levels of plausibility about what (if anything) happened before, and whether or not there were ways to mathematically continue the models past/through/around the initial singularity, assuming there was one.  Much of his talk was actually focused on what happens at the hypothetical heat death of the universe, and considered whether in a massless and radiation-dominated regime, the inability to construct a clock (because you don’t have any mass out of which to build anything) means that the standard picture of spacetime will break down in a way similar to the way we expect it breaks down at the earliest period of the universe.  (Richard was also there, and he wondered if Penrose’ ideas required proton decay or not.  Penrose was ambiguous on this point.)

The hall was packed — the cult of (scientific) celebrity, as someone said — and either the British education system is better than I’ve been led to believe or he may have aimed high in assuming that the audience was comfortable with conformal transformations.  (Mappings which preserve angles, both magnitude and direction, but can let other things like size vary.)

Yesterday he gave another talk, a longer and more technical lecture, on his pet idea, twistor theory.  Room was also packed: people were standing.  The algebra was pretty rough slogging unless you’re more used to working with spinors than I am, but you could follow the overall argument well enough if you paid attention.  I think I’d have been completely lost if I hadn’t reread chapter 33 of his book “The Road to Reality” beforehand: he stayed pretty close to his presentation there, just with more of the details.

He had a cute line about first cohomology, also from the book: it’s a precise, nonlocal measure of the degree of physical impossibility of an object, and gave the impossible triangle as an example.  Locally, the triangle’s perfectly constuctible — there’s nothing weird about it.   And if you remove a corner, you can build it.  But you can’t build the object as a whole, and that impossibility of globally satisfying certain connection constraints is what first cohomology measures.  (The triangle case is a particularly simple case of this.)  He noted that he didn’t know of an equivalent easy way to explain second cohomology, but that just as twistors are naturally useful for dealing with 1-particle wavefunctions because of their first cohomological properties, to handle such things as EPR entanglement second cohomology may be important.

He intermittently returned to the GR/QM unification theme, and suggested that his twistor program (in which the lightcone stays fixed but the idea of an “event” gets fuzzy) has advantages over some other proposals (in which the lightcone gets fuzzed up) as a road to quantum gravity.  I’ll go back on Thursday for part two.

In an hour or so the lectures start up again, but no Penrose today.  Instead it’s Peter Coles, once of Queen Mary and now of Nottingham, on dark matter and dark energy; Michael Green, once of Queen Mary and now of Cambridge, on string theory (yeah, that Michael Green); and Bryan Webber, never of Queen Mary and now of Cambridge, on the frontiers of particle physics.

Should be fun!

Yet another installment in my numerical adventures.

Had a meeting with Richard yesterday where I showed him the results of many of my test suites designed to show that my new two-timestep code was behaving pretty well. Ran some followup tests before I left for the day: he’d expressed interest in knowing exactly where it was the integrator broke down, and that sounded like a good idea.

Originally I’d planned to use a smooth transition function from the outer to the inner regions, but the analytics became a nightmare almost immediately so I’d have to have switched from the Duncan-style recursion to Chambers-style numerical integration.. and I wanted to put that off for a while. If the method worked pretty well except for transiting objects, then switching to a smooth transition would probably solve that, but if it didn’t work at all, then smoothing the transition wouldn’t help.. so it made sense to try the simple thing first.

I managed to show that the code works surprisingly well in most situations, with a handful of exceptions which I suspect are due to numerical resonances. I think this is because the forces are admittedly becoming more inaccurate on the inside but they’re also less important, and an averaged behaviour is a better approximation. However, when objects cross the boundary between inner and outer zones repeatedly — for example, an object with a respectable eccentricity with semimajor axis at the transition point — then they don’t behave so nicely. They become artificially eccentric.

I can only think of two ways to get around this. The first is to forbid objects from switching back to an outer zone timestep even if they move into the outer zone, but that’d cause problems with the way I’m handling my close encounters. The second, unfortunately, is to put in a smooth transition, which means that I have to finally finish my Chambers code. Since the boss is away for a few days, now’s the time.

On the bright side, given the subtleties of the problem, I can probably release a short paper on the technique aside from the science we’re going to generate, which is a plus..

Today Steven Balbus (late of the University of Virginia, now of ENS-Paris) was the speaker at the astronomy seminar. He’s famous for the Balbus-Hawley instability, usually just called the magneto-rotational instability (MRI) these days.. he was modest and used the generic name. The MRI was a clever solution to the puzzle of how accretion discs can act as if they’re viscous when the densities and temperatures of the discs mean that the usual molecular viscosity is far too weak to be of any use.

Loosely speaking, viscosity is the ’self-stickiness’, or resistance to pouring, of a fluid: it’s what maple syrup has a lot of that water doesn’t. The problem for accretion discs is that when you estimate the timescale on which molecular viscosity would let the disc viscously accrete onto the star (time of accretion ~ R*R/nu, where R is the size of the disc and nu is the viscosity), it takes five or six orders of magnitude too long. So what’s providing the viscosity if not the disc’s syrupy goodness?

Enter Balbus and Hawley, who noticed that when you introduced magnetic fields into the picture things improve. The magnetic field can act like a spring connecting different parts of the disc together. For example, when two elements of the fluid above and below the disc midplane are knocked a little bit to the side, the tension of the magnetic spring tries to increase the angular momentum of the now-inner element (which is orbiting faster) and decrease the momentum of the now-outer one (orbiting slower).. which only increases the tension.. and we have positive feedback, the process runs away, and the result is turbulence. The magnetic coupling allows different parts of the disc to talk to each other quickly which couldn’t if they could only communicate via purely molecular interactions, and the turbulence allows for much greater angular momentum transport than you’d have otherwise. If you cross your fingers, you can pretend the resulting transport is due to a viscosity. (Cross them hard.)

If you have the time, you can check out Jim Stone’s webpage for his gallery of magnetohydrodynamic animations. Impressive stuff. But a word of advice: if you ever meet him, don’t call the viscosity-like term in the Shakura-Sunyaev alpha-parameter model a viscosity.. he takes his terminology seriously! (The alpha model is what you get if you wave your hands madly and try to guess how the angular momentum transport is going to behave based on simple dimensional and physical arguments which leave a lot of important stuff out but often work well enough in practice.) Stone spoke at the Kingston-in-Kingston meeting this summer — a.k.a. the Henriksenfest! — and more than once I saw him grimace and object loudly when this came up.

Balbus was talking about the low-luminosity regime of accretion around black holes, and discussed magnetothermal and magnetoviscous analogies to the MRI instability which may play a role in diffuse plasmas. The physical analogies he provided were helpful, even if the seminar still required amateurs like me to pay quite a lot of attention to follow along.

Oh, and earlier today I found out that he’s married to protoplanetary disc expert Caroline Terquem! Sometimes it takes forever to learn certain things because everyone assumes you already know..