Aperture Synthesis

As we saw in the previous section, the spatial correlation of the electric field in the U-V plane is related to the source brightness distribution. Further, for the typical radio array the relationship between the measured visibility and the source brightness distribution is a simple Fourier transform. Correlation of the voltages from any two radio antennas then allows the measurement of a single Fourier component of the source brightness distribution. Given sufficient number of measurements the source brightness distribution can then be obtained by Fourier inversion. The derived image of the sky is usually called a ``map'' in radio astronomy, and the process of producing the image from the visibilities is called ``mapping''.

The radio sky (apart from a few rare sources) does not vary^2.8. This means that it is not necessary to measure all the Fourier components simultaneously. Thus for example one can imagine measuring all required Fourier components with just two antennas, (one of which is mobile), by laboriously moving the second antenna from place to place. This method of gradually building up all the required Fourier components and using them to image the source is called ``aperture synthesis''. If for example one has measured all Fourier components up to a baseline length of say 25 km, then one could obtain an image of the sky with the same resolution as that of a telescope of aperture size 25 km, i.e. one has synthesized a 25 km aperture. In practice one can use the fact that the Earth rotates to sample the U-V plane quite rapidly. As seen from a distant cosmic source, the baseline vector between two antennas on the Earth is continuously changing because of the Earth's rotation (see Figure 2.3). Or equivalently, as the source rises and sets the Fourier components measured by a given pair of antennas is continuously changing. If one has an array of N antennas spread on the Earth's surface, then at any given instant one measures ${}^NC_2$ Fourier components (or in radio astronomy jargon one has ${}^NC_2$ samples in the U-V plane). As the Earth rotates one samples more and more of the U-V plane. For arrays like the GMRT with 30 antennas, if one tracks a source from rise to set, the sampling of the U-V plane is sufficiently dense to allow extremely high fidelity reconstructions of even complex sources. This technique of using the Earth's rotation to improve ``U-V coverage'' was traditionally called ``Earth rotation aperture synthesis'', but in modern usage is usually also simply referred to as ``aperture synthesis''.

Figure 2.3: The track in the U-V plane traced out by an east-west baseline due to the Earth's rotation.

From the inverse relationship of Fourier conjugate variables it follows that short baselines are sensitive to large angular structures in the source and that long baselines are sensitive to fine scale structure. To image large, smooth sources one would hence like an array with the antennas closely packed together, while for a source with considerable fine scale structure one needs antennas spread out to large distances. The array configuration hence has a major influence on the kind of sources that can be imaged. The GMRT array configuration consists of a combination of a central 1x1 km cluster of densely packed antennas and three 14 km long arms along which the remaining antennas are spread out. This gives a combination of both short and long spacings, and gives considerable flexibility in the kind of sources that can be imaged. Arrays like the VLA on the other hand have all their antennas mounted on rails, allowing even more flexibility in determining how the U-V plane is sampled.

Other chapters in these notes discuss the practical details of aperture synthesis. Chapter 3 discusses how one can use radio antennas and receivers to measure the electric field from cosmic sources. For an N antenna array one needs to measure ${}^NC_2$ correlations simultaneously, this is done by a (usually digital) machine called the ``correlator''. The spatial correlation that one needs to measure (see equation 2.4.6) is the correlation between the instantaneous fields at points $P_1$ and $P_2$. In an interferometer the signals from antennas at points $P_1$ and $P_2$ are transported by cable to some central location where the correlator is - this means that the correlator has also to continuously adjust the delays of the signals from different antennas before correlating them. This and other corrections that need to be made are discussed in Chapter 4, and exactly how these corrections are implemented in the correlator are discussed in Chapters 8 and 9. The astronomical calibration of the measured visibilities is discussed in Chapter 5, while Chapter 16 deals with the various ways in which passage through the Earth's ionosphere corrupts the astronomical signal. Chapters 10, 12 and 14 discuss the nitty gritty of going from the calibrated visibilities to the image of the sky. Chapters 13 and 15 discuss two refinements, viz. measuring the spectra and polarization properties of the sources respectively.

Footnotes

... vary^2.8

Or, in the terminology of random processes cosmic radio signals are stationary, i.e. their statistical properties like the mean, auto and cross-correlation functions etc. are independent of the absolute time.