To make a radio image as sharp as possible, using many small antennas spread out, what fundamental distance between these antennas must be made as large as possible?

The fundamental distance between these antennas that must be made as large as possible is the baseline. Specifically, for the sharpest possible radio image, it is the maximum baseline of the array that needs to be maximized. The baseline refers to the physical separation, including both distance and orientation, between any two individual antennas within the interferometer array. When discussing the sharpness of an image, we are referring to its angular resolution, which is the ability to distinguish fine details or separate two closely spaced objects in the sky. A smaller angular resolution value indicates a sharper image. Radio interferometry uses multiple small antennas spread out to synthesize the observing capabilities of a much larger, single antenna. The angular resolution achievable by an interferometer is inversely proportional to the maximum baseline and directly proportional to the wavelength of the radio waves being observed. Therefore, by making the maximum baseline as large as possible, a smaller angular resolution is achieved, resulting in a significantly sharper radio image. This is because a larger separation between antennas allows for the collection of more detailed spatial frequency information, which is crucial for reconstructing an image with finer detail through a process called aperture synthesis.