Introduction

It has been shown in Chapter 2 that the visibility measured by the interferometer, ignoring the phase rotation, is given by

$$V(u,v,w)=\int\int I(l,m)B(l,m)e^{-2\pi\iota (ul+vm+w(\sqrt{1-l^2-m^2}))} {dldm \over \sqrt{1-l^2-m^2}},$$ (14.1.1)

where $(u,v,w)$ defines the co-ordinate system of antenna spacings, $(l,m,n)$ defines the direction cosines in the ($u,v,w$) co-ordinates system, $I$ is the source brightness distribution (the image) and $B$ is the far field antenna reception pattern. For further analysis we will assume $B=1$, and drop it from all equations (for typing convenience^14.1!)

Eq. 14.1.1 is not a Fourier transform relation. For a small field of view ($l^2+m^2 << 1$) the above equation however can be approximated well by a 2D Fourier transform relation. The other case in which this is an exact 2D relation is when the antennas are arranged in a perfect East-West line. However often array configurations are designed to maximize the $uv$-coverage and the antennas are arranged in a `$Y$' shaped configuration. Hence, Eq. 14.1.1 needs to be used to map full primary beam of the antennas, particularly at low frequencies. Eq. 14.1.1 reduces to a 2D relation also for non-EW arrays if the time of observations is sufficiently small (snapshot observations).

In the first part of this chapter we will discuss the implications of approximating Eq. 14.1.1 by a 2D Fourier transform relation and techniques to recover the 2D sky brightness distribution.

The field of view of a telescope is limited by the primary beams of the antennas. To map a region of sky where the emission is at a scale larger than the angular width of the primary beams, mosaicing needs to be done. This is discussed in the second part of this lecture.

Footnotes

... convenience^14.1

The same assumptuin has been made in Chapter 2