An array like the
GMRT measures the visibility function along baselines which
move along tracks in the
plane as the earth rotates, For simplicity,
let us assume that these measurements have been transferred onto a
discrete grid and
baselines are measured in units of the wavelength. The sky brightness
distribution
in the field of view is a function of
which
are direction cosines of a unit vector to a point on the celestial sphere
referred to the
and
axes.
The basic
relationship between the measured visibility function
and the sky
brightness
is a Fourier transform.
Many things have been left out in this expression, such as the proper units, polarisation, the primary beam response of the individual antennas, the non-coplanarity of the baselines, the finite observing bandwidth, etc. But it is certainly necessary to understand this simplified situation first, and the details needed to achieve greater realism can be put in later.
Aperture synthesis, as originally conceived, involved filling in the
plane without any gaps upto some maximum baseline
which would
determine the angular resolution. Once one accepts this resolution limit,
and writes down zeros for visibility values outside the measured circle,
the Fourier transform can be inverted. One is in the happy situation of
having as many equations as unknowns. A point source at the field
centre.(which has constant visibility)
would be reconstructed as the Fourier transform of a
uniformly filled circular disk of diameter
. This is the famous
Airy pattern with its first
zero at
. The baseline
is already measured in
wavelengths, hence the missing
in the numerator. But even in
this ideal situation, there are some problems. Given an array element of
diameter
(in wavelengths again!), the region of sky of interest could
even be larger than a circle of angular diameter
. A
Fourier component describing a fringe going through one cycle over
this angle corresponds to a baseline of
. But
measuring such a short baseline would put two dishes into collision, and
even
somewhat larger baselines than
run the risk of one dish shadowing the
other. In addition,
the really lowest Fourier component corresponds to
,
the total flux in the primary beam. This too is not usually measured in
synthesis instruments Thus, there is an inevitable ``short and zero
spacings problem'' even when the rest of the
plane is well sampled.