Relation between \ensuremath{t_\ensuremath{\mathrm{onoff}}} and \ensuremath{\ensuremath{t_\ensuremath{\mathrm{sig}}}^\ensuremath{\mathrm{beam}}}

By construction

Both \ensuremath{\ensuremath{t_\ensuremath{\mathrm{sig}}}^\ensuremath{\mathrm{beam}}} and \ensuremath{t_\ensuremath{\mathrm{onoff}}} are proportional to $\ensuremath{n_\ensuremath{\mathrm{cover}}}\,\ensuremath{t_\ensuremath{\mathrm{submap}}}$ (cf. Eqs. [*] and [*]). It is thus easy to derive that
  $\displaystyle \ensuremath{t_\ensuremath{\mathrm{onoff}}}= \ensuremath{\ensurema...
...ystyle\left( 1+\sqrt{\ensuremath{n_\ensuremath{\mathrm{on/off}}}} \right) }^2.
$ (34)
Using Eq. [*], we can replace \ensuremath{n_\ensuremath{\mathrm{on/off}}} and obtain
  $\displaystyle \ensuremath{t_\ensuremath{\mathrm{onoff}}}= \ensuremath{\ensurema...
...athrm{submap}}}}+\sqrt{\ensuremath{n_\ensuremath{\mathrm{beam}}}} \right) }^2.
$ (35)
Using Eqs. [*], [*] and [*], we obtain
  $\displaystyle \ensuremath{\sigma_\ensuremath{\mathrm{psw}}} =
\frac{\ensuremat...
...h{\eta_\ensuremath{\mathrm{tel}}}\,\ensuremath{t_\ensuremath{\mathrm{tel}}}}}.
$ (36)
The last equation in theory enables us to find the rms noise as a function of the elasped telescope time (sensitivity estimation) and vice-versa (time estimation). However, it is not fully straightforward because we must enforce that \ensuremath{n_\ensuremath{\mathrm{cover}}} and \ensuremath{n_\ensuremath{\mathrm{submap}}} have an integer value.