Mapping the Radio Universe

ICMS

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So the central problem in radio astronomy is:

How does one form an image of a radio source?

Radio Dishes are used in pairs, and the two outputs of the dishes are combined.

- This combined output, from many different pairs of radio telescopes, is then transformed by the mathematical operation of the Fourier Transform, to yield the image of the source i.e. the intensity distribution at a particular wavelength over the sky.

What is a Fourier Transform?

The idea behind a (1-dimensional) Fourier Transform is that any complicated 1-dimensional function can be treated as the sum of an infinite number of simpler functions.

Let us take a vibrating string: The complicated vibrating pattern on the string can be constructed by adding a certain number of sine waves of different harmonic frequencies. Harmonic frequencies are all multiples of a certain fundamental frequency. So instead of knowing the vertical displacement of every point on the string, we need to only determine the frequency and amplitude of the component sine waves. The amplitude-the vertical height to which the vibrations rise, and phase-the position of the sine wave, are together represented as a complex amplitude. Note that the mathematical function describing the complicated wave is restricted to the length of the string and will repeat after that length (i.e. it is a periodic function).

In the case of an arbitrary or non-periodic function (e.g. an infinitely long string) the waveform can be decomposed into an infinite no. of sine waves, of all possible frequencies from 0 to ¥.

This is called the Fourier Transform of that function. The complex amplitude of a component is called its Fourier coefficient, and tells us how much of that component is present in the sum.

Coming back to our complicated image: By using the Fourier Transform in two dimensions (for a 2-dimensional function) we can construct the image by adding up simpler images. For a component image of a certain frequency (i.e. frequencies along both dimensions), its shape is known but its amplitude needs to be found. This amplitude of a component image is its Fourier coefficient. So by finding the values of these Fourier coefficients and adding up component images multiplied by their appropriate coefficients we can construct the complicated image. When the Fourier Transform is computed, a 2-D Fourier Transform is easily reduced to 1-D Fourier Transforms.

Thus the Fourier Transform enables us to construct an image not element by element but Fourier coefficient by Fourier coefficient.

The importance of the Fourier Transform lies in its invertibility: Given an image one can get its Fourier coefficients and given a set of Fourier coefficients one can construct the corresponding image.

The last piece of the problem is solved by the van Cittert-Zernike Theorem. This tells us that the required Fourier coefficients of the radio image are nothing but the outputs of radio telescopes combined pairwise! This is why we use a pair of radio telescopes to measure the Fourier coefficients and then use a Fourier Transform to construct an image of the radio source. The details of combining radio telescope outputs, applying the van Cittert-Zernike Theorem and the mathematics of the Fourier Transform are now explained in greater detail.

As we saw, a single radio dish has hopelessly poor resolution. To a single dish, even a large galaxy will look like a featureless spot. So a pair of radio dishes, placed far apart for better resolution, is used and the two outputs multiplied together. Each dish points towards the same source in the sky, which is of a finite angular size, measuring the amplitude and phase (together called the complex voltage) of the electromagnetic wave that is incident on it. Phase is just a measure of how much one wave is ahead or behind another.