Mapping

The rms brightness temperature is related to the point source sensitivity by

$$\delta T_m = \frac{\rho \lambda^{2}}{ \theta _{1} \theta _{2} } \delta S$$ (2)

where $\lambda$ is the wavelength in millimeters, $\theta _1$ and $\theta _2$ are the beam major and minor axis in arcseconds, $\delta S$ is the point source sensitivity in Jy, and $\rho \simeq 15$ for untapered maps with natural weighting.

For a typical experiment with configuration BC or CD, tracking the source for 8 hours (but spending 2 hours on calibration) on each configuration, we obtain $\delta T_{BC}$ = 0.11 K and $\delta T_{CD}$ = 0.03 K, for 0.625 MHz channel spacing with the spectral correlator, with the same assumptions as above at 90 GHz. The factor $\rho$ is increased by uniform weighting and tapering: the amount depends on the uv-coverage and on the phase noise as a function of baseline lengths (1 to 3), but tapering also increases the synthesized beam size.

An equivalent way to look at noise in a map is by comparison to the single-dish sensitivity (given here for position switching)

$$\delta T_s = \frac{2 T_{sys}}{\sqrt{\delta\nu \delta t}} \cdot \frac{1}{\sqrt{N_{\rm pol}}}$$ (3)

which is typically $\delta T_s = 4.5$mK in one hour (for a channel spacing of 0.625 MHz and a frequency below 110GHz).

The mapping sensitivity is roughly related to $\delta T_s$ by

$$\delta T_m = \frac{B^2}{(\theta _1 \theta _2)} \frac{\delta T_s} {\eta _{j} \eta _{p} \sqrt{N(N-1)} }$$ (4)

where $B$ the size of the antenna primary beam. Then $\delta T_{CD} = 40 \delta T_s$, $\delta T_{BC} = 190 \delta T_s$, and $\delta T_{AB} = 590 \delta T_s$. In practice, a source that is well detected but unresolved at the 30-m telescope can easily be imaged at the Plateau de Bure interferometer.