Angular Resolution, Image Complexity, Observing Time

A synthesis telescope offers a great deal of flexibility when deciding how to image an object. In particular, you need to consider trade-offs between:

• observing time $versus$ sensitivity;
• angular resolution $versus$ brightness sensitivity;
• maximum resolution $versus$ sampling interval in the visibility plane.

You should consider initially making a low resolution image: this might expose confusing sources or extended structure in your source. Your angular resolution should be chosen to match the expected brightness of the object. Choosing optimal $uv$ plane coverage is difficult, and in general one takes whatever one can get.

The amount of independent information needed to describe the image must not exceed the number of independent $uv$ plane samples. Unfortunately, neither of these independent quantities are easily defined. At one extreme, consider making an image of a complex source filling the entire primary beam area. The primary beam limits the size of the structure that can be imaged and the Nyquist sampling interval corresponds (approximately) to the 15m increment (above the shortest physical baseline of 30.6m) obtained with the full set of 25 configurations. Alternatively, if the structure is smaller than the primary beam then the Nyquist sampling interval is larger and the number of $uv$ samples required is reduced. The observing time can then be reduced either by decreasing the number of configurations or the amount of hour angle coverage. For the ATCA, reducing the number of configurations is the more practical approach. If the source is large but partly empty then it can be considered to have a size corresponding to its area.

As a guide to the observing time needed, the table below gives the maximum size of structures that can be reliably imaged for some typical sets of observing configurations. This table is only a rough guide as the actual coverage needed depends on details of the 2D brightness distribution, on the actual configuration and the type of deconvolution used.

Table 3.3: Largest well-imaged structure for different arrays, based on Nyquist sampling for the size tabulated. However as the shortest spacing is 30.6m, the largest smooth structure may need the addition of single dish data.

	LARGEST WELL-IMAGED STRUCTURE
	AT 6cm WAVELENGTH[a]
OBSERVING TIME	6 or 3km array	1.5km array	750m array
25 days	6'	-	-
12 days	4.5'	6'	-
4 days	160"	4'	8'
2 days	115"	160"	5'
1 day	80"	115"	230"
$6\times \sim 1-10$ minutes[b]	30"	50"	100"
$\sim 1-10$ minutes[c]	20"	40"	80"

¹ For other wavelengths, scale sizes by $\lambda/6$cm

Notes:
(a) For other wavelengths, scale sizes by $\lambda/6$cm.
(b) Distributed in hour angle.
(c) 1-dimensional information only.

A further consideration is the minimum physical spacing available. This is never less than 30.6m and for a given configuration can be much larger. Finite-sized minimum spacings act as a high pass filter removing all Fourier components less than the minimum spacing. If this is a serious problem, short baseline information (e.g., from a single dish) can be added separately during data processing. Mosaicing can partially recover missing spacings (down to approximately 15m).