From the temperature power to the flux

For a point source, it is more natural to express the signal, and thus the sensitivity, in unit of flux. The flux ( $F_\ensuremath{\mathrm{}}$) of a point source is linked to its brightness temperature $T_\ensuremath{\mathrm{A}}^\star$ through

$\displaystyle \ensuremath{F_\ensuremath{\mathrm{}}}= \ensuremath{J_\ensuremath{...
...rm{ant}}^\ensuremath{\mathrm{sd}}} \ensuremath{T_\ensuremath{\mathrm{A}}^\star}$   with$\displaystyle \quad
\ensuremath{J_\ensuremath{\mathrm{ant}}^\ensuremath{\mathrm...
...remath{F_\ensuremath{\mathrm{eff}}}}{\ensuremath{A_\ensuremath{\mathrm{eff}}}},$ (4)

where $k$ is the Boltzman constant, and $A_\ensuremath{\mathrm{eff}}$ is the effective area of the antenna (eq. 3-113 in Kraus , 1982), and $J_\ensuremath{\mathrm{ant}}^\ensuremath{\mathrm{sd}}$ the conversion factor for a typical interferometer antenna. The effective area depends on the observing wavelength when the surface rms accuracy becomes a significant fraction of the wavelength. For NOEMA, the effective area is close to the antenna geometrical surface at 3 mm but significantly lower than it at 1 mm. Using the same conversion factor for the sensitivity, we yield

$\displaystyle \ensuremath{\sigma_\ensuremath{\mathrm{Jy}}} = \frac{\ensuremath{...
...}}\,\sqrt{2\,\ensuremath{d\nu}\,\ensuremath{\Delta t_\ensuremath{\mathrm{}}}}}.$ (5)

We have the same subtlety for $J_\ensuremath{\mathrm{ant}}^\ensuremath{\mathrm{sd}}$ as for $T_\ensuremath{\mathrm{sys}}$, ie, $J_\ensuremath{\mathrm{ant}}^\ensuremath{\mathrm{sd}}$ here characterizes one baseline between, eg., antennas $i$ and $j$. Instead of just $J_\ensuremath{\mathrm{ant}}^\ensuremath{\mathrm{sd}}$, we should write in all generality $\ensuremath{J_\ensuremath{\mathrm{ant}}^\ensuremath{\mathrm{sd,ij}}}$ with

$\displaystyle \ensuremath{J_\ensuremath{\mathrm{ant}}^\ensuremath{\mathrm{sd,ij...
...{sd,i}}}\,\ensuremath{J_\ensuremath{\mathrm{ant}}^\ensuremath{\mathrm{sd,j}}}},$ (6)

where $\ensuremath{J_\ensuremath{\mathrm{ant}}^\ensuremath{\mathrm{sd,i}}}$ and $\ensuremath{J_\ensuremath{\mathrm{ant}}^\ensuremath{\mathrm{sd,j}}}$ are the single dish conversion factors for antenna $i$ and $j$. For simplicity, we will also keep the notation $J_\ensuremath{\mathrm{ant}}^\ensuremath{\mathrm{sd}}$ hereafter.