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I'm reading about radio astronomy and array configurations on the Very Large Array's (VLA) website. They state that the longest baselines provide the best angular resolution, but have very limited sensitivity to surface brightness, and vice-versa for the shortest baselines. Can someone explain this tradeoff?
Source: VLA's website
The surface brightness of a source is by definition the flux density per solid angle; the surface brightness sensitivity of a telescope is, analogously, its point source sensitivity divided by the beam solid angle.$^{\dagger}$ Interferometers have smaller beam solid angles than dishes with the same area (Frayer 2017). Given that the solid angle is $\Omega\sim\theta^2$, the ratio of the surface brightness sensitivity between an interferometer with longest baseline $b$ and a single-dish telescope of diameter $D$ is proportional to $$f\sim\left(\frac{\Omega_{\text{int}}}{\Omega_{\text{dish}}}\right)\approx\left(\frac{\theta_{\text{int}}}{\theta_{\text{dish}}}\right)^2=\left(\frac{\lambda/b}{\lambda/D}\right)^2=\left(\frac{D}{b}\right)^2$$ which is a filling factor of sorts. For a single dish, you could say that $D=b$ because the longest "baseline" is the telescope diameter; for an interferometer, we have $b\gg D$, and so the ratio is quite small.
This means that long baselines can be both a blessing and a curse, and it's why you shouldn't always opt for the longest baselines when conducting interferometry. As an example, the Very Large Array has four different configurations, with $b$ ranging from about 1 kilometer (D) to 36 kilometers (A). You might want to observe when the VLA is in the A configuration if you just care about angular resolution; you'd want the D configuration if you care about surface brightness. The D configuration's resolution is worse by a factor of 1/36, but its surface brightness sensitivity is better by a factor of (1/36)$^2$=1296.
$^{\dagger}$The point source sensitivity tells you how dim an object the telescope can detect; the surface brightness sensitivity tells you how low a surface brightness a source can have before it becomes undetectable.
