Relation between and

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

By construction

The number of submaps is the area of the map divided by the area of a submap

$\begin{displaymath} \ensuremath{n_\ensuremath{\mathrm{submap}}}= \frac{\ensurem... ...athrm{map}}}}{\ensuremath{A_\ensuremath{\mathrm{submap}}}}. \end{displaymath}$ (25)
The number of on per off is the number of independent resolution elements in each submap

$\begin{displaymath} \ensuremath{n_\ensuremath{\mathrm{on/off}}}= \frac{\ensurem... ...thrm{submap}}}}{\ensuremath{A_\ensuremath{\mathrm{beam}}}}. \end{displaymath}$ (26)
The number of independent resolution elements in the map is the product of number of submaps by the number of on per off

$\begin{displaymath} \ensuremath{n_\ensuremath{\mathrm{beam}}}= \ensuremath{n_\e... ...hrm{submap}}}\,\ensuremath{n_\ensuremath{\mathrm{on/off}}}. \end{displaymath}$ (27)
The submap area is the product of the area velocity by the time to cover it

$\begin{displaymath} \ensuremath{A_\ensuremath{\mathrm{submap}}}= \ensuremath{v_... ...{\mathrm{}}}\, \ensuremath{t_\ensuremath{\mathrm{submap}}}. \end{displaymath}$ (28)
The time to scan a submap is the sum of the $\ensuremath{n_\ensuremath{\mathrm{on/off}}}$ independent on integration time

$\begin{displaymath} \ensuremath{t_\ensuremath{\mathrm{submap}}}= \ensuremath{n_... ...h{t_\ensuremath{\mathrm{on}}}^\ensuremath{\mathrm{cover}}}. \end{displaymath}$ (29)
The relations between times per coverage and times integrated over all the coverages are

$\begin{displaymath} \ensuremath{\ensuremath{t_\ensuremath{\mathrm{on}}}^\ensure... ...ath{t_\ensuremath{\mathrm{on}}}^\ensuremath{\mathrm{cover}}}. \end{displaymath}$ (30)
Using the last two points, it is easy to derive

$\begin{displaymath} \ensuremath{\ensuremath{t_\ensuremath{\mathrm{sig}}}^\ensur... ...off}}}+\sqrt{\ensuremath{n_\ensuremath{\mathrm{on/off}}}}}. \end{displaymath}$ (31)
Finally, the total time spent on and off is given by

$\begin{displaymath} \ensuremath{t_\ensuremath{\mathrm{onoff}}}= \ensuremath{n_\... ...h{t_\ensuremath{\mathrm{off}}}^\ensuremath{\mathrm{cover}}}). \end{displaymath}$ (32)

Using Eqs. and , we derive

$\begin{displaymath} \ensuremath{t_\ensuremath{\mathrm{onoff}}}= \ensuremath{n_... ...rt{\ensuremath{n_\ensuremath{\mathrm{on/off}}}}} \right) }. \end{displaymath}$ (33)

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

$\begin{displaymath} \ensuremath{t_\ensuremath{\mathrm{onoff}}}= \ensuremath{\en... ...sqrt{\ensuremath{n_\ensuremath{\mathrm{on/off}}}} \right) }^2. \end{displaymath}$

(34)

Using Eq.

, we can replace $\ensuremath{n_\ensuremath{\mathrm{on/off}}}$ and obtain

$\begin{displaymath} \ensuremath{t_\ensuremath{\mathrm{onoff}}}= \ensuremath{\en... ...sqrt{\ensuremath{n_\ensuremath{\mathrm{beam}}}} \right) }^2. \end{displaymath}$

(35)

Using Eqs.

and

, we obtain

$\begin{displaymath} \ensuremath{\sigma_\ensuremath{\mathrm{psw}}} = \frac{\ens... ...th{\mathrm{tel}}}\,\ensuremath{t_\ensuremath{\mathrm{tel}}}}}. \end{displaymath}$

(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.

Next: Time/Sensitivity estimation Up: Position switched Previous: Two key points: 1) Contents

Gildas manager 2014-07-01