Spin correlations, part I

This is the first of the two blogs that are dedicated to our latest paper, that we already advertised here. The topic is fascinating, but quite tough to understand for the first time. Have patience and enjoy.

Every single high-school textbook that I come across has the spinning ice-skater story hidden somewhere in the section on the angular momentum. The main question is, how do ice-skaters manage to spin-up to these dazzling spinning velocities? The answer lies in carefully observing what they do: they spin up a little bit, by pushing with their legs and with their arms outstretched and then they spin-up by bringing their arms towards their bodies. The physics behind it is the conservation of angular momentum. You can watch an edifying Youtube video that incidentally include an ice-skater here.

This principle states that, in absence of external torques (think of torques as actions that make things spin, for example by pulling the side of a globe to make it spin, you’re exerting a torque on it), the angular momentum of the system is conserved. Angular momentum is simply the product of the ice-skater’s angular velocity, i.e. the rate at which it is spinning, and its moment of inertia, which is a measure of how far from the axis of rotation its mass is distributed. Since the ice-skater skates with virtually with no friction, the external forces are very small. By pulling her arms towards herself, she decreases her moment of inertia and voila, she must spin up as a result of angular momentum conservation. For scientifically minded amongst you, this beautiful principle can be solely derived from the isotropy of our space: if I perform an experiment and then rotate it by any angle and perform it again, I should get the same result (in absence of the nagging Earth’s gravity).

Galaxies do something very similar. After the initial kick (more about that later), the hydrogen gas cools, forms stars and eventually contracts. This contraction creates beautiful disks of spinning matter that we observe as spiral galaxies. The spiral arms of these galaxies are most likely to be formed by higher order effects, such as density waves. In most (96%) of galaxies, these spiral arms trace the sense of rotation of the matter in the galaxy, which effectively coincides with the sense of rotation of the dark matter halo hosting the galaxy. And this is where the Galaxy Zoo comes in. By classifying spiral galaxies into clockwise and anti-clockwise spirals, we get some information about sense in which galaxies are spinning. Why should this matter? It is a complicated issue that we will eventually get to, but let us first carefully think about what are we really measuring.

Think of a spiral galaxy (or draw one on a piece of paper) and put your right hand (left hand will do as well, but make sure you use the same hand throughout!) down, so that the fingers follow the spiral arms from the outside towards the centre and the thumb is outstretched as illustrated below (tiny hands courtesy of my office mate Reiko Nakajima):

Your thumb now defines two things: it defines an axis around which the galaxy is spinning, but it also defines a unique ‘sense’, a direction from the centre. This is called a vector in mathematics, it is line with an arrow on it. When describing spins of galaxies, we call this a “spin vector”. You can convince yourself that regardless of how you rotate your piece of paper, your thumb will always point in the same direction. See below:

You will notice, that if you use left hand instead of the right one, you will get results which are consistent, but always point in the opposite direction compared to using the right hand. We use world “chiral” to describe things that are sensitive to left-hand vs right-hand orientation.

However, the sky gives us another problem: we see galaxies in projection. We don’t know whether the top or the bottom end of the galaxy is closer to us. Perfectly circular objects end up pancaked into ellipses.

Imagine seeing one such ring. If one has no information about spiral arms, one has two options for the axis of the galaxy as illustrated below (see what is happening to the word “GALAXY”:

Since we discovered that there are two possible sense of rotations for each axis, there are actually four possible directions for our thumb direction. Now, let us bring in the information on the sense of rotation from the galaxy zoo. Imagine we see a clock-wise spiral. Out of the four options above, we can immediately discard two. But we are still left with two distinct possibilities. These are the two posibilites for a clock-wise galaxy (note S-like shapes in the circle):

and these are for the anti-clockwise galaxy (note Z-like shapes in the circle):

Now comes the important bit: in the case of the clock-wise galaxy, thumbs always point away from the observer and in the case of anti-clockwise galaxy, thumbs point towards the observer. So we do get some information after all! In mathematical speak, we get the sign (+1 or -1) of the spin vector projected along the line of sight!

But, why should this matter? If we are in no special position in the Universe and if there is no special direction in the universe, as we have shown in our previous paper, shouldn’t we see clockwise galaxies with the same frequency as the anti-clockwise galaxies? That is all true. However, we are asking a far more cunning question: if I see a galaxy vector pointing in on direction, is the spin vector of a neighbouring galaxy spinning in a similar direction? In a roughly opposite direction? Or is it equally likely to spin in paraller or anti-parallel direction but unlikely to spin in a perpendicular direction? Answer to these questions remain unknown until we attempt to answer them using real data. In physics, we are talking about correlations: are the spin vectors of neighbouring galaxies correlated?

We will answer these and other question in our next blog entry!