Today’s post is also from Dr Enno Middelberg and is the second part of two explaining in more detail about radio interferometry and the techniques used in producing the radio images in Radio Galaxy Zoo.

In a previous post I have explained how the similarity of the electric field at two antennas’ locations is related to the Fourier transform of the sky brightness. Unfortunately, we’re not quite there (yet). You may have heard about sine and cosine functions and know that they are one-dimensional. Images, and the sky brightness distribution, however, are two-dimensional. So how can we imagine a two-dimensional Fourier transform? In this case, we have to combine 2D waves with various frequencies, amplitudes, and orientations into one image. We can make a comparison with waves on a lake. Just like a sine or cosine wave, a water wave has an amplitude and a frequency, but in addition it also has an orientation, or a direction in which it travels. Now let us think of a few people sitting around a pond or lake. Everyone kneels down to generate waves which then propagate through the water. Let us further assume that the waves are not curved, but that the crests and valleys are parallel lines. Now all these waves, with properly chosen frequencies, amplitudes, and directions will propagate into the center of the pond, where the waves interfere. With just the right parameters, the interference pattern can be made to look like a 2D image. In a radio interferometer, every two telescopes make a measurement which represents the properties of such a wave, and all waves combined then can be turned into an image. Let me point out that the analogy with the lake is taking things a little bit too far: since the water waves keep moving across the lake, a potential image formed by their intererence will disappear quite quickly, but I hope you get the point about interfering 2D waves.

To illustrate this further I have made a little movie. Let us assume that the radio sky looks just like Jean Baptiste Joseph Fourier (top left panel in the movie). I have taken this image from Wikipedia, cropped it to 128×128 pixels, and calculated its Fourier transform. The Fourier transform is an image with the same dimensions, but the pixels indicate the amplitude, phase and frequency of 2D waves which, when combined, result in an image. Then I have taken an increasing number of pixels from this Fourier transform (which ones is indicated at the top right), calculated which 2D waves they represent (bottom right), and incrementally added them into an image (bottom left). At the beginning of the movie, when only few Fourier transform pixels are used, the reconstructed Mr. Fourier is barely recognizable, with 50 Fourier pixels added, one begins to identify a person, and with an increasing number of waves added, the image more and more resembles the input image. You should play it frame by frame, in particular at the beginning, when the changes in the reconstructed image are large. In radio interferometry, Mr. Fourier’s image is what we want (how does the sky look like?), but what we get is only the pixels shown in the upper right image. Each of these pixels, all by itself, provides information as illustrated in the bottom  right, but all together, they yield an image such as in the bottom left image. And the more pixels we measure, the more accurate the image becomes.

So in summary: a radio interferometer makes measurements of the similarity of the electric field at two locations, and the degree of similarity represents the Fourier transform of the sky radio brightness for the two antennas in that instant. Astronomers then reconstruct the sky brightness from all these measurements taken together – that’s also why the technique is called “synthesis imaging”, or “aperture synthesis”. And if you’ve kept reading until here without having your brain turn to mush – congratulations! This is typically the subject of lectures for advanced physics students. I’ve been learning about radio interferometry now for more than 15 years and am still discovering new and interesting bits.

Today’s post comes from Dr Enno Middelberg and is the first part of two  explaining in more detail about radio interferometry and the techniques used in producing the radio images in Radio Galaxy Zoo.

I have written in an earlier post about the basic idea of how to increase the resolution of a radio telescope: use many telescopes, separated by kilometers, and observe the same object with all. Here is a little more information about how this works.

At the very heart of an interferometer is the van Cittert-Zernike theorem: it essentially states that the degree of similarity of the electric field at two locations is a measure of the Fourier transform of the sky brightness distribution. Now that’s a big bite to swallow, but let me explain it in less confusing words: the electric field is all we can measure – radio waves are electromagnetic waves, and radio telescopes are sensitive to the electric field. Now we can build a radio telescope in a way that it produces as its output a voltage which is proportional to the electric field which the antenna receives from, e.g., a galaxy. Much of the signal will be noise from our own Milky Way, the atmosphere and the electronics which amplify the feeble signals, but a tiny little bit of the signal will be caused by radio waves from space, and both antennas will receive a little bit of these. Now suppose we have two telescopes separated by 1 km or so, and both telescopes produce such voltages which contain a little bit of this signal. The voltages are digitised and the two data streams are fed into a correlator. The correlator is a computer which takes the two data streams and calculates their correlation coefficient, which is an indicator for their similarity. If the two data streams have nothing in common (for example, because an unexperienced PhD student pointed the two antennas in different directions 🙂 ) then the correlation coefficient will be zero, which is to say that they are not similar at all. However, if the two telescopes point at the same source, the data streams will have a few bits in common, and the correlator spits out a correlation coefficient which is not zero. This is our measurement!

Now that we’ve that out of the way, we need to talk about Fourier transforms. The van Cittert-Zernike theorem states that the correlation coefficient is a measure for the Fourier transform of the sky brightness. Now what is a Fourier transform? The Fourier transform is an ingenious way of representing a mathematical function with a sum of sine and cosine functions. That is, if I take a large number of sine and cosine functions with various (but carefully selected!) frequencies and amplitudes, then their sum will be an accurate representation of another function, for example a square wave or a sawtooth. Check out the Wikipedia page on Fourier series (which are related to Fourier transforms, but easier to understand), which has a number of nice animations to illustrate this, such as this one:

You can also play with Paul Falstad’s Java applet to see how to construct functions using sine and cosine waves interactively – very instructive! In part 2 of this post I will explain how astronomers use 2D Fourier transforms to assemble images of the radio sky.

We’re pleased to announce that Radio Galaxy Zoo has been 
translated to traditional character Chinese. Many thanks 
to the Zooniverse’s Chris Snyder for getting all the technical 
things set up for the translation to go live and Mei-Yin Chou 
at Academia Sinica’s Institute of Astronomy & Astrophysics 
(ASIAA) for helping verifying the translation. What follows 
is an announcement describing Radio Galaxy Zoo and the translation 
in traditional character Chinese and then in English:

電 波星系動物園[中 文版]歡 迎你的加入！在此我們欣然宣佈本計畫中文版開始啟用。
感謝中央研究院天文及天文物理研究所 Dr. Meg Schwamb (Meg是 參與Planet 
Hunters 和Planet Four計 畫的科學家)以 及天文推廣團隊成員黃珞文協助，
將英文內容翻譯轉換為繁體中文。在 許多星系核心深處其實隱匿著一些超大質量的黑洞，
其質量往往為太陽的幾億倍，這些大黑洞雖然無法直接看得到，不過有 
時仍可看得到從星系核心噴出的巨大噴流。由 電波波段看到的天空和在「可見光」
波段所看到的（可見的意思就是人類肉眼所看得到的），此兩種景象有時大異其 趣。
譬如，有些星系中心根本沒有電波輻射發出，但卻會向外噴發電波噴流，
或有時這些噴流的外觀是筆直的直線，有時卻 是一團、只有單邊而非雙邊對稱、
甚或是彎曲弧線等。藉由大型全天電波普查觀測計畫，我們取得了各式各樣多達幾十萬個 
電波噴流和團塊影像，它們需要和所屬的宿主星系做成匹配。因 此，我們邀請你加入我們的行列，
來認識一個「從未曾見過」的宇宙，也借用您眼所見的，協助辨識電波波段之噴流（或細絲），
再將它 們和紅外圖像進行比較及匹 配，這麼一來，在你的協助下，
噴流和宿主星系間本來付之闕如的關聯 性，未來將可建立成形。
 http://radio.galaxyzoo.org/?lang=zh_tw



Welcome to Radio Galaxy Zoo (Chinese)!  It is with great 
pleasure that we announce the launch of the Chinese version 
of our project.  We are very grateful to  Dr Meg Schwamb 
(from Planet Hunters and Planet Four) and  Lauren Huang from 
Academia Sinica’s Institute of Astronomy & Astrophysics 
(ASIAA) for their help with translating our project from 
English to traditional Chinese characters.

Supermassive black holes (~several hundred million times 
the mass of our Sun) lie deep in the cores of many galaxies.  
And though we cannot directly see these black holes, we do 
sometimes see the huge radio jets originating from the galaxy 
cores.

Galaxies in the radio sky can look quite different from 
the one seen in the optical wavelengths by instruments such 
as our own eyes.  Some galaxies do not have any central radio 
emission but only radio jet(s) emanating outwards. Sometimes 
these jets are straight  but at other times, they can be blobby, 
one-sided or bent.  With very large all-sky radio surveys, we 
have many hundreds of thousands of radio jets and blobs that 
need to be matched to their host galaxies.

Therefore we invite you to see the Universe as you have never 
seen before and help us map the radio sky by matching the radio 
jets and filaments to the galaxies (as seen in the infrared images) from 
whence they came.

 http://radio.galaxyzoo.org/?lang=zh_tw