Many astronomers will want images with the highest angular resolution
which will still give adequate brightness sensitivity in the amount of
time the scheduling committee has granted them to observe their
source. (There are exceptions to this statement, researchers looking
for microwave background fluctuations of a certain scale, for
example.) There is a natural tradeoff between resolution and
brightness sensitivity. This tradeoff can be made continuously by
tapering a single array configuration, or more efficient from an
operational standpoint but less efficient from a scientific
standpoint, discretely by switching among arrays of different
size.
The combined effect of tapering and switching among different
array configurations is shown schematically in Figure 6
for the cases of uniform and centrally
condensed Fourier plane distributions.
The solid straight line represents the brightness
sensitivity as a function of resolution for some constant amount of
observing time which would result if the MMA antennas could be
continuously reconfigured to any resolution. The letters A, B, C, and
D indicate where on this ``optimal'' line the resolution and
brightness sensitivity of the actual arrays lie (assuming the
resolution of each is separated by a factor of 4.0). The dashed,
discontinuous lines indicate the result of tapering each array, ie,
trading resolution for brightness sensitivity. The top dashed lines
illustrate how a uniform Fourier plane distribution responds to
tapering, and the lower dashed lines illustrate how a centrally
condensed Fourier plane distribution respond to tapering. We do not
consider tapering to a resolution lower than that of the next smaller
array. As each smaller array enters in, we get a large improvement in
surface brightness sensitivity as no tapering is used at full
resolution. These lines represent the limits of detection, and sources
with 
below the lines will not be detected. The heavy curves
represent the surface brightness as a function of resolution for three
Gaussian sources of the same flux but of sizes 0.2, 0.5, and 1.0
arcseconds. To the right, each source is unresolved. As each source
is observed at higher resolution, it eventually becomes undetectable,
or ``resolved out'', by the arrays.
We assume that there is no a priori source size which is more likely or more important than other source sizes. Under this assumption, we would desire that our discrete array configurations be able to come as close to the solid straight line as possible. We can do this by maximizing the number of configurations which are financially and operationally feasible, and by optimizing the way in which the array sensitivity degrades as we taper to lower resolution.
We now compare the brightness sensitivity of the uniform and centrally condensed Fourier plane distributions for the same amount of observing time. Observing in a ``C'' configuration (resolution of about 1.1 arcsec), all three sample sources are firmly detected at full resolution and there is little difference between the two Fourier plane distributions. When the 1 arcsec Gaussian source is observed with the uniform ``B'' configuration (resolution of about 0.28 arcsec), it is basically undetected at all tapers. However, the 1 arcsec source is detected by the centrally condensed ``B'' array at beams larger than 0.6 arcsec. Either ``B'' configuration will detect the 0.5 arcsec source at full resolution. Both the uniform and condensed ``A'' configurations require some tapering to detect the 0.2 arcsec source, but the centrally condensed configuration requires less tapering, and can therefore image the 0.2'' source at somewhat higher resolution than the uniform configuration.