Sunday, 24 November 2013

Seeing length contraction

It's the 50th anniversary of both the assassination of John F. Kennedy and the first episode of Doctor Who, and yes, I did get up at 6:50 and watched The Day of the Doctor. So, today's post is about Time and Relative Dimensions in Space.

Everyone loves a bit of relativity, even though its consequences can be quite mind-bending. Of course, one of the things that happens is that when things are moving relative to each other you get length contraction, and people disagree on how long something is. Length contraction is responsible for some cool physical effects, including explaining why two parallel currents attract one another.

Check out this excellent video by Derek Muller on Veritasium which explains this.
While the video is mostly correct, there is something which is not quite right. Notice the bit where he drives passed himself in a car, he sees the car squeezed due length contraction. So, the question I want to look at is "What do things look like moving at relativistic speeds?"

To answer the question, we need to think about two things. Firstly, we need to consider how we transform coordinates from the "moving" object into the coordinates of the observer. Let's assume the moving object is a sphere. In its own coordinate system is is sitting there, doing nothing, with its clock ticking. To the observer, the sphere is moving, and so its coordinates are changing, and its clock is ticking at a different rate to the observer.

We've known how to transform between the coordinates of the two for more than 100 years, and what we use is the Lorenz transformation. Using this we can work out where the points on the sphere are in the observer's coordinate system as a function of the observer's clock - easy!

Here's the Lorenz transformation (from wikipedia).

where 

and
Most of you will recognise this as being nothing but a matrix multiplication - the mathematics of special relativity is really not that difficult.

But remember the question is "What do we see?" And what we need to consider is the time that light rays take to travel from the sphere into the eye of the observer. 

How do we work out the path that light will follow? That's quite easy, as light rays travel along what is known as "null geodesics" and it is simple to trace out the path through space-time as it must follow.


Again, nothing complicated. Just a bit of algebra. Any high school student can do this :)

So, all you need to do is pick a time at the observer, as we want to consider all light rays that arrive at a certain time, and fine out at what time did photons leave the sphere to arrive at the observers eye. This is a bit of algebra, but it's a little "non-linear" so I use a small root finding algorithm to find the solution (for the curious, I use brentq function in python).

So, what do we get? To start with, let's just throw some random points onto the surface of the sphere and get it moving passed the observer 90% of the speed of light across the sky. Here's an image of the points on the sky as seen by the observer.
 Eermmmm. It looks round. Why isn't squashed in one of the directions?

Ah! you say, perhaps 90% of the speed of light is not fast enough? What if we go at 99.99%? Here's what we get.
Hmmmmm. It still looks like a circle on the sky. Just where has the length contraction gone?

Let's investigate this a little more. Instead of throwing down points at random on a sphere, let's make the moving object a cube, so we'll gave eight points at the corners of the cube and one at the centre.
Here's the cube at rest, as seen by the observer.
Lovely! Now, let's get this cube moving. Here's what we see if the cube is moving at 10% the speed of light.
Hmmm. The cube is not contracted, but appears to have rotated slightly. What if we up the speed to 50% the speed of light?
Still no contraction, but the cube definitely appears to be rotated! OK - let's up the speed to 90% of the speed of light.
And 99%!
And may as well pull out all stops and get up to 99.99%!
Wow. How cool is that - we don't have "length contraction" along the direction of motion (from left to right), but the cube has rotated and got skinny. So, actually seeing "length contraction" is more complicated than you think.

Now, none of this is new. The effect, known as Terrell rotation, was published by Terrell and separately by Roger Penrose, in 1959, although it was noted as far back as 1924 by Anton Lampa, although it doesn't appear in too many textbooks. There are some great articles out there on the interwebs about the optical effects of seeing things moving at relativistic speeds; I heartedly recommend you have a read. If I get some time of Christmas, I'll write about another favourite of mine, namely Bell's spaceship paradox.

Before I go, however, I was quite amazed how the ABC on both TV and radio were clammering to interview whovians to ask them what they thought of the new episode, and who their favourite Doctor was. However, some of us have a long memory and remember how the ABC, through the Chaser, presented fans of Doctor Who.
How things have changed :)

Saturday, 9 November 2013

Major Substructure in the M31 Outer Halo: the South-West Cloud

Another week has flown by and I don't know where the time went. But another good week in terms of research with a new paper accepted.

This one is led by postdoctoral researcher, Nick Bate, with newly minted doctor, Anthony Conn, and PhD student, Brendan McMonigal. The focus of the study, substructure in the halo of the Andromeda Galaxy from, you guessed it, the rather fantastic Pan-Andromeda Archaeological Survey (PAndAS). The focus this time is a particularly prominent blob, known as the South-West Cloud (or, more colloquially to us, Japan). Here's a map of the substructure again.
OK - I'll admit that the SW-Cloud doesn't look a lot like Japan, but the name stuck.

As you can see, it's a reasonably big chunk of stuff, but we want to know what it is. And that's the focus of the present paper. This is my favourite picture from the paper, presenting the density of stars in and around the SW-Cloud.
As you can see, it's a bit of a mess, but the SW-Cloud is clearly visible, as is a small dwarf galaxy (And XIX). The little stars labelled PA-7, PA-8 and PA-14 are really interesting as they are globular clusters, balls of roughly a million stars that I wrote about in the previous post. As I've written about before, it looks like a lot of these globulars were brought in on galaxies that have now been disrupted.

So, it looks like the SW-Cloud used to be a dwarf galaxy, with some of its own globular clusters, that has fallen into Andromeda and is being tidally torn apart. Now, that's interesting, but what else can we learn?

A while ago, I talked about the work Anthony was doing to measure the distances to M31 by locating the tip of the red giant branch, a very useful distance indicator. Once we have isolated the stars in the SW-Cloud, getting rid of all those annoying stars within our own Galaxy, we can search for the brightest red giant stars, which are the ones that define the tip. Here's the luminosity function.
Slightly hard to see (this stuff isn't easy!) but there is a set at around 21 which is the location of the tip.

We chopped the SW-Cloud into pieces, and calculated the distance to each bit, and this is what we find;
The black points are the individual fields, and you can see that the measurements are a bit noisy, but the red is the average of the three, which puts the SW-Cloud at almost the same distance and Andromeda itself. But the cool thing is that two of the globulars are at the same distance, slowing that they are all part of the same system.

But, as they say, there is more! We were also to work out the metallicity of the SW-Cloud, which means we are working out how chemically enriched the stars are. Remembering that the first stars were purely hydrogen and helium, the amount of chemical enrichment is a measure of how many generations of stars a population has gone through; in every generation, heavier chemical elements are produced and they pollute the gas clouds which make the next generation of stars.

Big galaxies like the Milky Way have lots of gas and are constantly producing stars, but dwarfs, which have a much smaller gravitational pull, can easily lose their gas and so only have a very limited number of stellar generations. This means that the smallest are usually metal-poor. What's the chemical make up of the SW-Cloud? This figure tells us all;
The left-hand panels are the colour-magnitude diagrams, and the thick black smudge up the middle of the top one is the red giant branch of the SW-Cloud. The middle panel is the "background" which is all the stars in the Milky Way (and some mis-classified background galaxies) which we subtract off, leaving the nice piccy at the bottom.

The right-hand panels are the the distribution of the metallicity of the stars in each of the colour-magnitude diagrams, the bottom is that for the SW-Cloud. We can see that the metallicity is about -1.3, which is not really metal-poor, and not really metal-rich. This tells us that what ever the SW-Cloud was originally, it was not a small dwarf galaxy, but would be amongst the largest that we know. Looking at other similar dwarfs, we can see that the SW-Cloud has lost about 25% of its mass, meaning that we must be looking at a very recent disruption.

How cool is that? Right, that's one substructure down, quite a few more to go!

Well done Nick, Anthony and Brendan!


Major Substructure in the M31 Outer Halo: the South-West Cloud

We undertake the first detailed analysis of the stellar population and spatial properties of a diffuse substructure in the outer halo of M31. The South-West Cloud lies at a projected distance of ~100 kpc from the centre of M31, and extends for at least ~50 kpc in projection. We use Pan-Andromeda Archaeological Survey photometry of red giant branch stars to determine a distance to the South-West Cloud of 793 +/- 45 kpc. The metallicity of the cloud is found to be [Fe/H] = -1.3 +/- 0.1. This is consistent with the coincident globular clusters PAndAS-7 and PAndAS-8, which have metallicities determined using an independent technique of [Fe/H] = -1.35 +/- 0.15. We measure a brightness for the Cloud of M_V = -12.1 mag; this is ~75 per cent of the luminosity implied by the luminosity-metallicity relation. Under the assumption that the South-West Cloud is the visible remnant of an accreted dwarf satellite, this suggests that the progenitor object was amongst M31's brightest dwarf galaxies prior to disruption.

Saturday, 2 November 2013

Dynamical Modeling of NGC 6809: Selecting the best model using Bayesian Inference

Science has been in the news over the last week, and it's been quite a successful research week for me. But while science has been in the news, I'm not 100% impressed by the way it has been presented.

Firstly, there was the lack of a dark matter detection by the Lux experiment. The reports around the web on this have been generally OK, but some have indicated that this is somehow a failure. But what is important, and is often not appreciated, is that in science the lack of a detection is as important as a detection. Negative results like this rule out possibilities and so are vital in cutting down the possibilities for what dark matter is. In fact, a lot of dark matter searches basically following the Holmes adage "eliminated the impossible, whatever remains, however improbable, must be the truth". Not seeing something increases our knowledge.

The second made me a little unhappy. The article in question appeared in the Conversation and was titled "Is it possible to add statistics to science? you can count on it". The target of the article is the award of the Prime Minister's Prize for Science to Terry Speed at the Walter+Eliza Hall Institute of Medical Research.

Now, don't get me wrong! I'm not unhappy about the existence of the award (I think it is great when science is recognised) and Terry Speed is a very worth recipient whose expertise in statistical analysis and bioinformatics has advanced cancer research.

No, the thing that annoys me is the title of the article - "Is it possible to add statistics to science? You can count on it". I have mentioned this before, and I will say it again, but the situation is not that science is over here and statistics is over there, but statistics (or more accurately inference) is absolutely and utterly central to science. Often people think science is observation and experimentation on one side, and theoretical study on the other. But the meat of science is the interface between the two, and to do that you need to be statistically (and also mathematically) adept.

It depresses me that we struggle to convince our students of this during their undergraduate years :(

But that brings me to the good news, the acceptance of a new paper by PhD student Foivos Diakogiannis. The question is a seemingly simple one, namely do globular clusters contain dark matter. Globulars are balls of roughly a few million stars whizzing about together and they orbit galaxies. They are a bit weird, and people are unsure of their formation mechanism.

If they collapsed from gas clouds in the very early universe then they should not have lots of dark matter in them, but if they formed like dwarf galaxies they would have formed from gas pooling in a dark matter halo and they would be dark matter dominated.

The focus of this paper is a particular cluster, NGC 6809, also known as M55. Here's a lovely picture of it.
A few years ago we took spectra of the stars in this globular cluster, and we used the Doppler shift to measure their speeds. Here's a picture of the speeds that we saw, as a function of the distance away from the centre of the cluster.
So, the stars seen towards the centre zip around the fastest, and they go more slowly at the edge.

How do we work out the mass? It is actually a very tricky problem as we only know the velocity along the line of sight, and don't know the velocity on the sky. But we also know how the light is distributed, and what we want to do is make a "self consistent" so that we have we can predict the observed light distribution and observed velocity profile, and that the globular cluster is stable.

How we do this is quite mathematical, and so read the paper if you are interested, but the we used inference, and in particular Bayesian analysis (I'll say it again, this kind of thing is central to science). But here's an example of the fits that we get
Notice that the velocity data is a bit ratty, but we get good fits.

And the result? We have shown conclusively that this particular cluster is not dominated by the presence of dark matter, and so they were not formed in the same way as objects like our own Milky Way. How did they form? We don't know, but our results are giving us some more clues.

Well done Foivos!

Dynamical Modeling of NGC 6809: Selecting the best model using Bayesian Inference
The precise cosmological origin of globular clusters remains uncertain, a situation hampered by the struggle of observational approaches in conclusively identifying the presence, or not, of dark matter in these systems. In this paper, we address this question through an analysis of the particular case of NGC 6809. While previous studies have performed dynamical modeling of this globular cluster using a small number of available kinematic data, they did not perform appropriate statistical inference tests for the choice of best model description; such statistical inference for model selection is important since, in general, different models can result in significantly different inferred quantities. With the latest kinematic data, we use Bayesian inference tests for model selection and thus obtain the best fitting models, as well as mass and dynamic mass-to-light ratio estimates. For this, we introduce a new likelihood function that provides more constrained distributions for the defining parameters of dynamical models. Initially we consider models with a known distribution function, and then model the cluster using solutions of the spherically symmetric Jeans equation; this latter approach depends upon the mass density profile and anisotropy β parameter. In order to find the best description for the cluster we compare these models by calculating their Bayesian evidence. We find smaller mass and dynamic mass-to-light ratio values than previous studies, with the best fitting Michie model for a constant mass-to-light ratio of Υ=0.90+0.140.14 and Mdyn=6.10+0.510.88×104M. We exclude the significant presence of dark matter throughout the cluster, showing that no physically motivated distribution of dark matter can be present away from the cluster core.