Field-dependent averaging losses in radio interferometry ($\delta \nu B\theta /c\ll 1,\Delta t{\Omega }_{E}B\theta /\lambda \ll 1$)

After delay tracking and fringe stopping at a phase center, an off-axis direction retains delay of order $B\theta /c$ and rotation-induced fringe-rate offset of order ${\Omega }_{E}B\theta /\lambda$. Averaging across a wide channel or long integration then decorrelates that direction. The displayed inequalities give order-of-magnitude small-smearing conditions; exact coefficients depend on baseline geometry, averaging window and allowed loss. A larger field or longer baseline therefore demands finer frequency and time sampling.

A point source away from the tracked direction has residual geometric delay $\tau$. Averaging its frequency-dependent phase over a top-hat band gives the displayed sinc attenuation times its central-frequency complex phase. The first bandwidth null is complete cancellation, not a small-loss operating limit. Fine frequency channels preserve the information needed for later frequency-aware imaging.