Yesterday I came across a very interesting paper on the arXiv by Sebastian Hutschenreuter et al. entitled The Galactic Faraday rotation sky 2020 which contains this stunning map of Faraday Rotation across the sky (presented in Galactic coordinates, so the plane of the Milky Way appears across the middle of the map):
The abstract of the paper is here:
If you’ll pardon a short trip down memory lane, this reminds me of a little paper I did back in 2005 with a former PhD student of mine, Patrick Dineen (which is cited in the Hutschenreuter et al. paper).
What we had back in 2005 was a collection of Faraday Rotation measurements of extragalactic radio sources dotted around the sky. Their distribution is fairly uniform but I hasten to add that it was not a controlled sample so it would be not possible to take the sources as representative of anything for statistical purposes and there weren’t so many of them: we had three samples, with 540, 644 and 744 sources respectively.
Faraday rotation occurs because left and right-handed polarizations of electromagnetic radiation travel at different speeds along a magnetic field line. The effect of this is for the polarization vector to be rotated as light waves travel and the net rotation angle (which can be either positive or negative) is related to the line integral of the component of the magnetic field along the line of sight travelled by the waves. The picture below shows the distribution of sources, plotted in Galactic coordinates and coded black for negative and white for positive.
Some radio galaxies have enormously large Faraday rotation measures because light reaches us through regions of the source that have strong magnetic fields. However, for most sources in our sample the rotation measures are smaller and are thought to be determined largely by the propagation of light not through the emitting galaxy, near the start of its journey towards us, but through our own Galaxy, the Milky Way, which is near the end of its path.
If this is true then the distribution of rotation measures across the sky should contain information about the magnetic field distribution inside our own Galaxy. Looking at the above picture doesn’t give much of a hint of what this structure might be, however.
What Patrick and I decided to do was to try to make a map of the rotation measure distribution across the sky based only on the information given at the positions where we had radio sources. This is like looking at the sky through a mask full of little holes at the source positions. Using a nifty (but actually rather simple) trick of decomposing into spherical harmonics and transforming to a new set of functions that are orthogonal on the masked sky we obtained maps of the Faraday sky for the different samples. Here is one:
(The technical details are in the paper, if you’re interested.) You probably think it looks a bit ropey but, as far as I’m concerned, this turned out surprisingly well!
The most obvious features are a big blue blob to the left and a big red blob to the right, both in the Galactic plane. What you’re seeing in those regions is almost certainly the local spur (sometimes called the Orion Spur; see below), which is a small piece of spiral arm in which the local Galactic magnetic field is confined. The blobs show the field coming towards the observer on one side and receding on the other. The structure seen is relatively local, i.e. within a kiloparsec or so of the observer.
I was very pleased to see this come out so clearly from an apparently unpromising data set, although we had to confine ourselves to large-scale features because of instabilities in the reconstruction of high-frequency components.
Now, 15 years later we have the beautiful map produced by Hutschenreuter et al.
You’ll see the vastly bigger data set (almost a hundred times as many sources) and way more sophisticated analysis technique has produced much higher resolution and consequently more detail, especially near the Galactic plane, but we did at least do a fairly good job at capturing the large-scale distribution: the blue on the left and red on the right is clearly present in the new map.
There’s something very heartening about seeing scientific progress in action! This also illustrates how much astrophysics has changed over the last 15 years: from hundreds of data points to more than 50,000 and from two authors to 30!
. Their (Eulerian) positions at some later time 
are taken to be![vec{r}(vec(q),t) = a(t) vec{x}(vec{q},t) = a(t) left[ vec{q} + b(t) vec{s}(vec{q},t) right].](https://s0.wp.com/latex.php?latex=vec%7Br%7D%28vec%28q%29%2Ct%29+%3D+a%28t%29+vec%7Bx%7D%28vec%7Bq%7D%2Ct%29+%3D+a%28t%29+left%5B+vec%7Bq%7D+%2B+b%28t%29+vec%7Bs%7D%28vec%7Bq%7D%2Ct%29+right%5D.&bg=000000&fg=B0B0B0&s=0&c=20201002)
coordinates are comoving, i.e. scaled with the expansion of the Universe using the scale factor 
. The displacement 
between initial and final positions in comoving coordinates is taken to have the form
is a kind of velocity potential (which is also in linear Newtonian theory proportional to the gravitational potential).If we’ve got the theory right then the gravitational potential field defined over the initial positions is a Gaussian random field. The function 
is the growing mode of density perturbations in the linear theory of gravitational instability.
In the Dark 