ALMA Memo 354 Choices of Antenna Size and Number for the Atacama Compact Array

The lack of large scale structure in array maps is the result of the hole in the center of the visibility plane that arises because the smallest spacing between antennas is limited to one antenna diameter. Visibility data may be extracted for a region in the center of the hole from a single antenna map made with one array antennas. This can be accomplished by Fourier transforming the map and dividing out the transform of the gain function to produce the visibility. If a mosaic of pointings is obtained with the array in its interferometric mode, this data set allows extrapolation of the visibilities inward from the edge of the hole. This can be done by a similar procedure in which the same gain function is divided from the observed visibilities to obtain visibilities within the edge of the hole. From the overlap, a complete map may be constructed. Pointing errors spoil this procedure. The e ect of the pointing errors is to produce phase and amplitude errors in the visibilities that increase toward the overlap region from both the origin and the edge of the hole. This is doubly bad, because the transforms of the gain functions also tend toward zero in the overlap region and the data errors are ampli ed there. For the homogeneous array, the e ects of even small errors in pointing are severe. The use of an array of smaller antennas provides a better overlap in the central hole, and reduces the e ects of pointing errors. The smaller the compact array antennas, the better is the overlap, but more antennas are required and calibration becomes more di cult. A reasonable compromise would be 6m, half the 12m diameter diameter and a traditional choice. For the compact array to contribute a point source sensitivity that matches that of the more closely spaced 12m antennas, the necessary number of