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In an array telescope configuration, like VLA and ATA, what dictates the distance between each dish?

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Physically there is nothing dictating the distances between dishes.

These telescope arrays are interferometers. By combining the signal of the two dishes you can achieve a spacial resolution of a giant virtual dish that is about as large as the distance (called the baseline) between the two real telescopes.

A nice example of the power of this technique is the VLBI, which is essentially such an array only the individual dishes are actually just observatories all over the world.

For more detail on how this all works exactly take a look at the wiki pages on aperture synthesis and astronomical interferometry.

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Increasing the separation of the dishes makes the effective aperture of the array larger, so it increases the spatial resolution. However, the overall collection area of the aperture has not increased, so the signal-to-noise ratio will not increase.

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