3 Closure phase

  The closure phase $\Phi$ is a combination of phases $\psi$ around a triangle of baselines, defined as

 

$$\Phi_{ijk}= \psi_{ij} + \psi_{jk} - \psi_{ik} ~.$$ (2)

When a telescope is quite displaced from the other two telescopes (i.e., an unbalanced triangle), the two longest baselines become nearly parallel give essentially identical phases, but with opposite sign. Their contributions therefore almost cancel in the closure equation. The only remaining contribution is that of the shortest baseline; when the source is unresolved on that baseline, the result is a closure phase whose magnitude is only a few degrees.

As shown by Massi (1989), the contribution of the longest baselines to the closure phase is a function of the ratio s/L between the shortest (s) and the longest (L) leg of the closure phase triangle. For a very small ratio (i.e., $s/L \ll 0.3$)the contribution may drop to a few degrees. This can be seen, for example, in the plots of closure phase shown in Fig. 8. The triangle at the top of the figure makes use of the intermediate length baseline MK to KP (the one flagged in Test-2 of Sect. 2) and has $s/L\simeq 0.3$. The closure phase for this triangle reaches a value of about $40^\circ$,significantly larger than the $12^\circ$ and $8^\circ$ reached for the other two triangles shown, with ratios of $s/L\simeq 0.07$ and 0.05 respectively.

  

Figure 8: Observed closure phase for selected triangles

In conclusion in the case of missing intermediate spacing information the observed closure phase $\Phi_{\rm o}$ is a small quantity. While at first glance, this would imply a quite slow convergence, instead it gives rise, as shown in Sect. 2, to a very serious problem when the adopted starting model is a point source and Loop "A'' is followed. In the next section we show why for cases of uniform u-v coverage Loops "A'' and "B'' are equivalent, but for cases where only low closure phase values are present (associated with unbalanced triangles), differences become apparent.