Image Sensitivity

Image sensitivity is the RMS thermal noise (ΔIm) expected in a single-polarization image.   Image sensitivities for the 10-station VLBA, for typical observing parameters, are listed in column [7] of the "Receiver Frequency Ranges & Performance" table.

Alternatively, the image sensitivity for a homogeneous array with natural weighting can be calculated using the following formula (Wrobel 1995; Wrobel & Walker 1999).

\[\Delta I_m = {\rm SEFD} / [\eta_s \cdot ( N \cdot (N-1) \cdot \Delta \nu \cdot t_{\rm int} ) ^{1/2} ] \; \rm{Jy\; beam^{-1}}\]

Parameters SEFD, \(\eta_s\), and \(\Delta\nu\) are the same as those used in computing baseline sensitivity, \( N \) is the number of observing stations, and \(t_{\rm int}\) is the total integration time on source in seconds.

The expression for image noise becomes rather more complicated for a heterogeneous array such as the HSA, and may depend quite strongly on the data weighting that is chosen in imaging.   The EVN sensitivity calculator provides a convenient estimate.   For example, the RMS noise at 22 GHz for the 10-station VLBA in a 1-hr integration is reduced by a factor between 4 and 5 by adding the GBT and the phased VLA.

If simultaneous dual polarization data are available with the above value of ΔIm per polarization, then for an image of Stokes I, Q, U, or V,

\[\Delta I = \Delta Q = \Delta U = \Delta V = \frac{\Delta I_m}{\sqrt{2}}\]

For a polarized intensity image of \(P = \sqrt{Q^2 + U^2}\)

\[\Delta P = 0.655 \times \Delta Q = 0.655 \times \Delta U\]

It is sometimes useful to express \(\Delta I_m\) in terms of an RMS brightness temperature in Kelvins (\(\Delta T_B\)) measured within the synthesized beam.  An approximate formula for a single-polarization image is

\[\Delta T_b \sim 320 \times \Delta I_m \times (B^{\rm km}_{\rm max})^2 \; {\rm K}\]

where \(B^{\rm km}_{\rm max}\) is as in Table 5.

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