The argument which drives the D configuration antenna layout is to maximize the surface brightness sensitivity, which requires a centrally condensed Fourier plane distribution, achieved by placing the antennas as close together as possible. No change in the current design is suggested for the D configuration. Until now, the antenna configuration for the larger arrays has been driven by the desire to provide uniform Fourier plane coverage, a desire which has not been supported by any analysis. I argue that the larger configurations should have moderately centrally condensed Fourier plane coverages to provide better shaped naturally weighted beams and better noise performance with respect to tapering. One possible exception is the largest array, which might be a ring-like array to get the most sensitivity at the resolution of the longest baselines.
The argument which drives the D configuration antenna layout is to maximize the surface brightness sensitivity, which requires a centrally condensed Fourier plane distribution, achieved by placing the antennas as close together as possible. No change in the current design is suggested for the D configuration. Until now, the antenna configuration for the larger arrays has been driven by the desire to provide uniform Fourier plane coverage, a desire which has not been supported by any analysis. I argue that the larger configurations should have moderately centrally condensed Fourier plane coverages to provide better shaped naturally weighted beams and better noise performance with respect to tapering. One possible exception is the largest array, which might be a ring-like array to get the most sensitivity at the resolution of the longest baselines.
Does the MMA still need four configurations? Absolutely! Even for the centrally condensed (linearly decreasing) Fourier plane distribution studied here, the C array will have a factor of 3 better sensitivity than the B array tapered to the C array's full resolution. An even more highly centrally condensed coverage would not lose sensitivity so quickly, but will result in a very bad synthesized beam with a wide plateau at its base, indicating that there is not enough sensitivity on the long baselines, and is undesirable.
Antennas randomly distributed over a circular region with a uniform deviate will give (u,v) coverage which approximates the centrally condensed distribution which we've been using. Some of the inner antenna pads for a given array may be common with the outer pads of the next smaller array. Even with tricks like this, the configurations with centrally condensed (u,v) coverage will require much more road and cables than the ring-like arrays currently under consideration. Tim Cornwell once suggested that the MMA arrays be built out of concentric rings of antenna stations, with two adjacent rings being populated each with half the antennas to produce a tapered (u,v) coverage. To obtain a monotonically decreasing Fourier plane density as a function of baseline, each array would have to be 3.0 times larger than its adjacent smaller array, which would require 5 different arrays to span from the smallest to the largest array. Such an array falls right in between the uniform and the centrally condensed linearly decreasing distribution for sensitivity loss as a function of taper, but due to sharp corners in it's three tiered ``wedding cake'' Fourier distribution, the image fidelity suffers a great deal in simulations. To what extent can the sharp edges be rounded off by a careful layout of the antennas on the two rings? We need to study this further.
For the future, we need to debate the merits of uniform and centrally condensed Fourier plane coverage for the MMA, clarify the shape of the optimal beam, clarify the use of multiple configurations in the MMA's imaging, further explore the various options for centrally condensed Fourier plane distributions, and design antenna configurations which result in the desired centrally condensed Fourier plane distribution and permit economical road, power, and communications layouts. Furthermore, we need to investigate the possibility of flexible layout of antenna pads which would permit arrays with several Fourier plane distributions.
Figure: Radial Fourier plane distributions from actual ring and
filled circle antenna configurations
Figure: The uniform and centrally condensed (linearly decreasing)
Fourier plane distributions used in this study.
The radial cuts have been plotted such that the integral over the 2-D
Fourier plane is the same for both distributions. Just arguing from
geometry, trading a little bit of baseline density in the expansive
outer part of the plane buys us a lot of baseline density in the cozy
inner plane.
Figure: How does point source sensitivity degrade as we taper the
Fourier plane coverage? We plot the point source sensitivity, normalized
to full array sensitivity, as a function of resolution for uniform
and linearly decreasing Fourier plane distributions. The upraised spurs
at the highest resolutions are for robust and uniform weighting.
Figure: But will these Fourier plane distributions give me a nice
beam? We plot the fractional difference of the beam integral and the
integral of the fit Gaussian beam as a function of tapered beam size
for the uniform and centrally condensed (linearly decreasing) Fourier
plane distributions. Uniform Fourier coverage gives us a highly
non-Gaussian beam at full natural resolution, due to its excess of long
baselines. Uniformly weighted beams for both uniform and centrally
condensed distributions are highly non-Gaussian.
Figure: What's wrong with the uniform distribution's beam?
Here are the radial profiles of the naturally weighted uniform
Fourier distribution's beam and its best fit Gaussian. The
uniform Fourier distribution has much too many long baselines to yield
a good Gaussian beam. Conversely, in order to get a good Gaussian
beam, the long baselines need to be tapered down, resulting in
a loss of about 30% of the sensitivity.
Figure: Schematic Log-Log plot of brightness sensitivity as a
function of resolution for an infinitely reconfigurable array
(solid straight line), uniform Fourier plane distribution
versions of A, B, C, and D arrays subject to tapering (top
dashed line), and centrally condensed Fourier plane distribution
versions of A, B, C, abd D arrays (lower dashed line).
The heavy curves represent surface brightness as a function of
resolution for Gaussian sources of different sizes. The sources
are detected when their curves lie above the array sensitivity
lines.