Sunday, 27 November 2011


I've been interested in chaos since reading Gleick's book back in the 1980s. I was first introduced to fractals when I was a summer student at the Rutherford Labs in the late 1980s; this was when colour printers were rare and expensive and I spent a lot of time convincing the guardian of the printer that printing out large colour fractals for my bedroom wall was essential for my studies of proton-anti-proton scattering.

But that's another story.

A little while ago, I caught an excellent documentary called "The Secret Life of Chaos" by the equally excellent presenter, Jim Al-Khalili. This linked a lot of topics, including chaos and complex systems, which is when a group of things following simple rules results in complicated (and sometimes difficult to predict) behaviour.

One of the key things I learnt was the importance of this man
in some of the earliest work in the field. I'm sure a number of you recognise him as Alan Turing. I first came across him in his work on cracking the Enigma code in World War II, and then discovered his founding work on modern computing.

But it turns out that in 1952 (and this is what I learnt from the Chaos doco), he published this paper
and it's an amazing look by a mathematician into the world of biology. I am not an expert in this area, or a historian of science, but I know of Schrodinger's attempt to talk about information in genes in his book What is Life?; Schrodinger was wrong in many aspects, but I like the fact that he was thinking about these questions.

Turing's paper addresses a very interesting question, namely how does a complex being like a human, arise from a pretty uniform and symmetric sack of cells soon after conception. How do cells know to become liver, heart, nerve and skin cells? What triggers the process?

My reading of Turing's paper (and I need to read it in more depth) is that the feeling of the day was that any inhomogeneities in the sack of cells would diffuse out, and the sack would remain essentially featureless. What Turing did, however, was to create a mathematical model for such a sack. It was not intended to be strictly biological, and much of the paper discusses an unrealistic ring of cells, but the fact that he could write down equations means he could start to predict what happens to the cells.

His basic model was that the local chemistry decides what a cell will do, and chemical can diffuse through the ring, from higher to lower concentration. So far, so standard, as you would expect the net chemistry of the sack of cells to be effectively uniform.

But then the stroke of genius. Get the chemicals to interact with one another, and get the reaction rates to depend on the amount of chemical locally. What happens? As noted by Turing, things do not just settle down to a uniform, but you get waves and pulses of chemicals moving through the cells.

This is taken from Turing's paper
which is the 2D distribution of chemicals in one of his simple models. As pointed out, this patten doesn't look too different to
and perhaps cows patterns, or that of a leopard, zebra or my old Jack Russell, are simply the indication of the chemical mathematics in the skin of the forming animal.

Anyway, it was pouring with rain yesterday morning, and so I thought I would try and make a simple Turing model. I decided to make a ball of cells, laying them out using HealPIX to put them on a sphere (this means each cell occupies the same area and so I don't have to worry about correcting for that). I chose three chemicals (let's call them Red, Green and Blue) and allowed each cell to have effectively the same amount at the start, except for a few overdoped cells.

After this, I allowed diffusion between nearest neighbour cells, and also put in interaction terms at each point, depending on the amount of chemical in that cell. Of course, this is just a simple set of differential equations, which takes little time to code up in MatLab.

Without the interaction terms, the ball of cells becomes a uniform bland colour, but once we have the interaction terms turned on, cool things start to happen. I actually dumped the output of MatLab into a rendering format (PovRay, fiendishly difficult - but free) and make some pictures. So, this is just one output,
Pictures are nice, but we want to see some action. So, here's a couple of movies I made
I must admit that I cheated slightly and used the Lorenz equations to ensure I got chaotic behaviour, but with the diffusion terms as well we get to see the chemical signatures pulsing through the cells. Well, I like it.

Now, some might say "why bother?" To you, my monomath cousins, I say that the answer is that it is interesting and continues to teaching me things beyond my area of expertise, and new skills that I might need to use one day. Anyway, there are worse ways to spend a rainy Saturday morning :)

Wednesday, 23 November 2011

The Star Formation History and Dust Content in the Far Outer Disc of M31

You wait for a bus, and then two come along at once.

Postdoctoral researcher, Edouard J. Bernard, working with Annette Ferguson at Edinburgh's Institute for Astronomy, and myself, has had his paper on the star formation history in a couple of fields observed with the Hubble Space Telescope. Before continuing, I want to say that the IfA has the best, thickest custard in the entire world!

The focus of the paper is deep Hubble Space Telescope fields in the outer parts of the Andromeda galaxy. The absolutely wonderful thing about Hubble is that being above the atmosphere, we can accurately measure the brightness of faint stars, but the annoying thing about Hubble is that the field o view is tiny. Here's the fields we got
The grey area is the sky that we've observed as part of the PAndAS program with Canada-France-Hawaii Telescope, whereas the tiny squares are the bit covered by Hubble; all telescope time is quite competitive, but getting lots of Hubble time is difficult (although the PHAT team is getting much of the disk of M31).

So, what can you do with these observations? Well, you can make a fantastic picture like this

These are colour-magnitude diagrams, extremely powerful tools in astrophysics. I should note that the F606W and F814W are the filters on the Hubble Space Telescope, and these deep images get way down into the stellar populations. Here we can clearly see the Red Giant Branch (RGB) and the Red Clump. Overlaid are theoretical isochrones which allows us to unravel lots, like the star formation history of each of the regions looked at.

After a significant amount of work, you end up with pictures like this
that chart the star formation history and element enrichment over time. Great stuff! What is so fantastic is that in this history we see a burst of star formation when M33 and M31 last interacted, a few billion years ago. Isn't astrophysics simply wonderful?

Well done Edouard & Annette!

The Star Formation History and Dust Content in the Far Outer Disc of M31

Edouard J. Bernard, Annette M. N. Ferguson, Michael K. Barker, Sebastian L. Hidalgo, Rodrigo A. Ibata, Michael J. Irwin, Geraint F. Lewis, Alan W. McConnachie, Matteo Monelli, Scott C. Chapman
We present a detailed analysis of two fields located 26 kpc (~5 scalelengths) from the centre of M31. One field samples the major axis populations--the Outer Disc field--while the other is offset by ~18' and samples the Warp in the stellar disc. The CMDs based on HST/ACS imaging reach old main-sequence turn-offs (~12.5 Gyr). We apply the CMD-fitting technique to the Warp field to reconstruct the star formation history (SFH). We find that after undergoing roughly constant SF until about 4.5 Gyr ago, there was a rapid decline in activity and then a ~1.5 Gyr lull, followed by a strong burst lasting 1.5 Gyr and responsible for 25% of the total stellar mass in this field. This burst appears to be accompanied by a decline in metallicity which could be a signature of the inflow of metal-poor gas. The onset of the burst (~3 Gyr ago) corresponds to the last close passage of M31 and M33 as predicted by detailed N-body modelling, and may have been triggered by this event. We reprocess the deep M33 outer disc field data of Barker et al. (2011) in order to compare consistently-derived SFHs. This reveals a similar duration burst that is exactly coeval with that seen in the M31 Warp field, lending further support to the interaction hypothesis. The complex SFHs and the smoothly-varying age-metallicity relations suggest that the stellar populations observed in the far outer discs of both galaxies have largely formed in situ rather than migrated from smaller galactocentric radii. The strong differential reddening affecting the CMD of the Outer Disc field prevents derivation of the SFH. Instead, we quantify this reddening and find that the fine-scale distribution of dust precisely follows that of the HI gas. This indicates that the outer HI disc of M31 contains a substantial amount of dust and therefore suggests significant metal enrichment in these parts, consistent with inferences from our CMD analysis.

Conservation Laws

With exam marking and committee meetings, it has been a slow week, but here's a pop quiz.

What does this woman
have to do with this video?
(the video has a nasty crack at the end of it, and so don't watch if squeamish). The answer is not that the woman in the photograph is the girl in the video.

Of course, what we are looking at here is a collision, and due to the use of the yoga balls, it is a pretty elastic collision, and so energy is almost conserved (and if you account for the energy that goes into heat and noise, it's completely conserved).

But what we all remember from our high school physics is that the thing called momentum, the sum of mass times velocity, is always conserved in collisions, and so if, before the collision, we take the mass of the boy and multiply it by his velocity, and then do the same for the girl, and then add the two quantities together, this sum is the total momentum. If we do the same after the collision (assuming no external forces), the total momentum remains the same (this is why it is a conservation law).

But you knew that. It still, however, leaves a question. Why is the sum of mass times velocity conserved? Why isn't the sum of mass divided by velocity conserved? Some of you will undoubtedly think this is a silly question, and the answer is "because a teacher told me so".

This brings us back to the woman in the top picture. Some of you will recognise her as Emmy Noether, and she is one of the most important (and yet unknown) people in physics (even though in reality she was a mathematician).

What's the link? Well, Noether discovered something amazing. She discovered that symmetries in physics give us conserved quantities. Without getting into the technical details, what she discovered is that if the laws of physics here are the same as the laws of physics over there (translational symmetry) then the quantity we call momentum (strictly linear momentum) must be conserved.

Also, if the laws of physics remain the same if I rotate myself around (rotational symmetry), then we have a quantity known as angular momentum conserved. And, if the laws of physics are the same in the past and future as they are now (temporal symmetry) then we have a quantity called energy that is conserved.
Ohh!! You might say, that's interesting. But then you might say that isn't energy conservation a bit obvious. I mean, clearly "energy cannot created or destroyed" is obviously true. After all, you were told it in school, and they would never lie to you.... would they.

This is where it starts to hurt a little. Emmy Noether's original work on conservation laws considered a Newtonian universe, and Newtonian laws of mechanics. Things were pushed into 4-D space-time to accommodate special relativity and electromagnetism. This is cool as we now have quantities such as relativistic energy (and kinetic energy) which are conserved. Furthermore, there is a symmetry you can pull out, there is no change in the laws of physics is you shift the phase of the quantum mechanical wave-function, which implies that charge is conserved.

The last one will, I'm sure, annoy some. Charge being conserved is obvious, electrons are little balls and charge is a little minus side painted to their point-like side. But alas, no. Without the appropriate symmetry, charge would not have to be conserved.

In fact, symmetries in the quantum groups give us lot of other quantities which are conserved in interactions.

What about non-conserved things? Well, there are. Mass does not have an associated symmetry and is not conserved in interactions (i.e. an electron and a positron annihilate to give two photons - mass is not conserved, but momentum, spin, energy etc are.)

Things get really hairy when we consider general relativity. Space-time in special relativity is the same everywhere and everywhen, and so we have the outcomes of physics experiments being the same and symmetries and conserved quantities we are happy with. But in general relativity, the underlying geometry of space-time changes with position and time and so the symmetries we were happy with previously are gone, and so are the conserved quantities.

The one that gets most people is the expanding universe and the loss of the time-symmetry; the space-time in the future will be different to the space-time now. No t-symmetry, no conservation of energy. Energy is not conserved in an expanding universe.

Photons lose energy as they travel through the universe. Where does that energy go? The equations of relativity tell us; nowhere! Don't worry, energy is not conserved in an expanding universe.

High speed masses slowly grind to a halt, as the universe expands, with respect to the the local environment. Where did all that energy go? Nowhere! Don't worry, energy is not conserved in an expanding universe.

Don't just listen to me. Listen to Sean Carroll over at Cosmic Variance. Energy is not conserved in an expanding universe.

I know some of you will not like this, as the conservation of energy is clearly obvious, and makes perfect common-sense. All I can say is that common-sense is a pretty poor guide to the last 100 years of physics.

Wednesday, 9 November 2011

Black hole noms: planetary treats for the galactic monster

I didn't write the title, and had to check the dictionary on what a nom is (and was surprised that it was actually a word), but I wrote a brief article for The Conversation

It summarizes a paper by Kastytis Zubovas of the University of Leicester on the continual burps of energy from the black hole at the centre of the Milky Way. You can read the original paper here

Sgr A* flares: tidal disruption of asteroids and planets?

Kastytis Zubovas, Sergei Nayakshin, Sera Markoff
It is theoretically expected that a supermassive black hole (SMBH) in the centre of a typical nearby galaxy disrupts a Solar-type star every ~ 10^5 years, resulting in a bright flare lasting for months. Sgr A*, the resident SMBH of the Milky Way, produces (by comparison) tiny flares that last only hours but occur daily. Here we explore the possibility that these flares could be produced by disruption of smaller bodies - asteroids. We show that asteroids passing within an AU of Sgr A* could be split into smaller fragments which then vaporise by bodily friction with the tenuous quiescent gas accretion flow onto Sgr A*. The ensuing shocks and plasma instabilities may create a transient population of very hot electrons invoked in several currently popular models for Sgr A* flares, thus producing the required spectra. We estimate that asteroids larger than ~ 10 km in size are needed to power the observed flares, with the maximum possible luminosity of the order 10^39 erg s^-1. Assuming that the asteroid population per parent star in the central parsec of the Milky Way is not too dissimilar from that around stars in the Solar neighbourhood, we estimate the asteroid disruption rates, and the distribution of the expected luminosities, finding a reasonable agreement with the observations. We also note that planets may be tidally disrupted by Sgr A* as well, also very infrequently. We speculate that one such disruption may explain the putative increase in Sgr A* luminosity ~ 300 yr ago.
Basically, what he is saying is the energy is released as small objects, planets and asteroids, as ripped apart on their final death plunge into the black hole.

This is not a new idea, as I remember people suggesting that such flares could be due to comets etc being ripped apart, but he has a little twist on the story that some of these objects are effectively recycled from objects that were smashed in the frenetic orbits close to the black hole.

Anyway, you can read my article here - questions and comments below :)