Sensitivity for mosaicking

Note: No provision is made to estimate the sensitivity of a mosaic in frequency cycling mode. This is to avoid mixing the complexity of both modes...

Mosaicking is a particular case of wide-field imaging: The user wishes to observe a given field of view larger than the primary beam size with a sensitivity as uniform as possible. The targeted field (which area is $A_\ensuremath{\mathrm{map}}$, define by the user) can be divided in a number of independent resolution elements or independent (primary) beams $n_\ensuremath{\mathrm{beam}}$. We have:

$\displaystyle \ensuremath{n_\ensuremath{\mathrm{beam}}}= \frac{\ensuremath{A_\ensuremath{\mathrm{map}}}}{\ensuremath{A_\ensuremath{\mathrm{beam}}}}$ (33)

where $A_\ensuremath{\mathrm{beam}}$is the area of the primary beam. It is linked to the telescope full width at half maximum ( $\theta_\ensuremath{\mathrm{}}$) by

$\displaystyle \ensuremath{A_\ensuremath{\mathrm{beam}}}= \frac{0.8\,\pi\,\ensuremath{\theta_\ensuremath{\mathrm{prim}}}^2}{4\,\ln(2)},$ (34)

The 0.8 factor represents the truncation of the beam at 20% of its maximum, which is performed during the imaging process.

For the sensitivity estimation we assume a standard sampling of targeted field and the on-source time is equally divided between the independent primary beams $n_\ensuremath{\mathrm{beam}}$ in the targeted field of view. We thus yield at the center of a Nyquist sampled mosaic

$\displaystyle \ensuremath{\sigma_\ensuremath{\mathrm{Jy}}} = \frac{\ensuremath{...
...{n_\ensuremath{\mathrm{pol}}}\,\ensuremath{\Delta t_\ensuremath{\mathrm{on}}}}}$ (35)

   with$\displaystyle \quad
\ensuremath{\Delta t_\ensuremath{\mathrm{on}}}= \ensuremath...
...point/cycle}}}}+\ensuremath{\ensuremath{\Delta t_\ensuremath{\mathrm{slew}}}}},$ (36)

where where $\ensuremath{\Delta t_\ensuremath{\mathrm{point/cycle}}}$ is the integration time per pointing during each single mosaic cycle, and $\ensuremath{\Delta t_\ensuremath{\mathrm{slew}}}$ is the time to slew between two consecutive pointings. There are several subtleties in these computations.

In summary, correctly setting up the sensitivity estimate for a mosaic requires

  1. To compute the number of independent primary beams in the targeted field of view

    $\displaystyle \boxed{%
\ensuremath{n_\ensuremath{\mathrm{beam}}}= \frac{\ensur...
...\frac{0.8\,\pi\,\ensuremath{\theta_\ensuremath{\mathrm{prim}}}^2}{4\,\ln(2)}.
}$ (50)

  2. To compute the number of pointings in the mosaic and the number of pointings per track

    $\displaystyle \boxed{%
\ensuremath{n_\ensuremath{\mathrm{point}}}= \ensuremath...
...math{\mathrm{point}}}}{\ensuremath{n_\ensuremath{\mathrm{track}}}} \right) }.
}$ (51)

  3. To compute the optimal integration time per scan with

    $\displaystyle \boxed{%
10\,\mbox{sec} \le \ensuremath{\ensuremath{\Delta t_\en...
...{A_\ensuremath{\mathrm{map}}}}} \right) }
\quad \mbox{with} \quad
\eta = 0.5.
}$ (52)

    The minimum integration time per scan of 10 seconds implies the maximum dynamic of scales that can be observed in a single track.
  4. To compute the number of points per track that define the limit between small and large mosaics with

    $\displaystyle \boxed{%
\ensuremath{n_\ensuremath{\mathrm{point/track}}^\ensure...
...math{\Delta t_\ensuremath{\mathrm{cal max}}}}= 25\ensuremath{\mathrm{\,min}}.
}$ (53)

  5. To compute the maximum number of repeats per pointing and per cycle, depending on the mosaic size.
  6. To check that the maximum number of pointings per track is not yet reached with

    $\displaystyle \boxed{%
\ensuremath{n_\ensuremath{\mathrm{point/track}}}\le \en...
...th{\Delta t_\ensuremath{\mathrm{cycle max}}}}= 60\ensuremath{\mathrm{\,min}}.
}$ (55)

    If the PI wishes to observe an area that will require more that 150 pointings per independent track, the estimator will ask to either increase the requested elapsed telescope time or to decrease the requested field-of-view area.
  7. To adapt the optimal integration time per scan to ensure that all pointings per track will be cycled in $\ensuremath{\Delta t_\ensuremath{\mathrm{cycle max}}}$ only for a large mosaic with

    $\displaystyle \boxed{%
\mbox{if} \quad
\ensuremath{n_\ensuremath{\mathrm{point...
...remath{\ensuremath{\Delta t_\ensuremath{\mathrm{slew}}}} \right) } \right\}}.
}$ (56)

  8. To compute the actual mosaic efficiency and time to complete a full cycle with

    $\displaystyle \boxed{%
\ensuremath{\eta_\ensuremath{\mathrm{mos}}}= \frac{\ens...
...rm{max}}}\,\ensuremath{\ensuremath{\Delta t_\ensuremath{\mathrm{int/scan}}}},
}$ (57)

    and

    $\displaystyle \boxed{%
\ensuremath{\ensuremath{\Delta t_\ensuremath{\mathrm{cy...
...}}}}+\ensuremath{\ensuremath{\Delta t_\ensuremath{\mathrm{slew}}}} \right) }.
}$ (58)

    These two quantities should be given as feedback to the user.
  9. To finally compute the sensitivity with

    $\displaystyle \boxed{
\ensuremath{\sigma_\ensuremath{\mathrm{Jy}}} =
\frac{\ens...
...emath{\mathrm{setup}}}}{\ensuremath{n_\ensuremath{\mathrm{beam}}}} \right) }.
}$ (59)