Gain determination in heterogeneous arrays

  Assume we have a heterogeneous array with N antennas, of diameters . The r.m.s. visibility rms on baseline ij is:

In the least square analysis one should then weigh each baseline by where (assuming the system temperatures equal). One may normalize the weights by

Now the least squares analysis gives:

which may be rewritten:

We define

Then

after multiplying by and summation over i one may compute :

where

and obtain a closed form for :

For the r.m.s. fluctuations we get:

However if we note the flux sensitivity of a baseline connecting antenna i with an identical antenna:

Then

Now if N is large enough, and . Neglecting all terms in 1/N:

which shows than when the number of elements N is large, the precision of the gain measurement for one element is only determined by the size of this element and the total collecting area.