Time/Sensitivity estimation

This paragraph describes the algorithm to do a time/sensitivity estimation for a position-switched On-The-Fly observation.

Step #1: Computation of and

is just computed as the ratio $\ensuremath{A_\ensuremath{\mathrm{map}}}/\ensuremath{A_\ensuremath{\mathrm{beam}}}$. Using Eqs.  and , we obtain

$$\ensuremath{n_\ensuremath{\mathrm{submap}}}= \frac{\ensurem... ...\mathrm{}}}\, \ensuremath{t_\ensuremath{\mathrm{submap}}}}.$$ (37)

Using this equation, we start to compute for $\ensuremath{t_\ensuremath{\mathrm{submap}}}= \ensuremath{t_\ensuremath{\mathrm{stable}}}$ and $\ensuremath{v_\ensuremath{\mathrm{area}}^\ensuremath{\mathrm{}}}= \ensuremath{v_\ensuremath{\mathrm{area}}^\ensuremath{\mathrm{max}}}$. We want to enforce the integer character of in a way which decreases the product $\ensuremath{t_\ensuremath{\mathrm{submap}}}\,\ensuremath{v_\ensuremath{\mathrm{area}}^\ensuremath{\mathrm{}}}$. To do this, we use

$$\ensuremath{n_\ensuremath{\mathrm{submap}}}= 1+\texttt{int(}\ensuremath{n_\ensuremath{\mathrm{submap}}}\texttt{)}.$$ (38)

Eq.  ensures that $\ensuremath{t_\ensuremath{\mathrm{submap}}}\,\ensuremath{v_\ensuremath{\mathrm{... ...{stable}}}\,\ensuremath{v_\ensuremath{\mathrm{area}}^\ensuremath{\mathrm{max}}}$. The value of must be decreased so that Eq.  is enforced.

Step #2: Computation of or

We use the following equations in descending order to compute the elapsed telescope time and in ascending order to compute the rms noise level:

$$1. \ensuremath{\ensuremath{t_\ensuremath{\mathrm{sig}}}^\ensuremath{... ...th{\mathrm{}}}^2\,\ensuremath{d\nu}\,\ensuremath{n_\ensuremath{\mathrm{pol}}}},$$ (39)

$$2. \ensuremath{t_\ensuremath{\mathrm{onoff}}}= \ensuremath{\ensurema... ...mathrm{submap}}}}+\sqrt{\ensuremath{n_\ensuremath{\mathrm{beam}}}} \right) }^2,$$ (40)

$$3. \ensuremath{\eta_\ensuremath{\mathrm{tel}}}\,\ensuremath{t_\ensuremath{\mathrm{tel}}}= \ensuremath{t_\ensuremath{\mathrm{onoff}}}.$$ (41)

Step #3: Computation of derived quantities

$$1. \ensuremath{n_\ensuremath{\mathrm{on/off}}}= \frac{\ensuremath{n_\ensuremath{\mathrm{beam}}}}{\ensuremath{n_\ensuremath{\mathrm{submap}}}},$$ (42)

$$2. \ensuremath{n_\ensuremath{\mathrm{cover}}}= \frac{\ensuremath{\en... ...\mathrm{on/off}}}+\sqrt{\ensuremath{n_\ensuremath{\mathrm{on/off}}}} \right) },$$ (43)

$$3. \ensuremath{\ensuremath{t_\ensuremath{\mathrm{on}}}^\ensuremath{\... ...h{t_\ensuremath{\mathrm{submap}}}}{\ensuremath{n_\ensuremath{\mathrm{on/off}}}}$$ (44)

$$3. \ensuremath{\ensuremath{t_\ensuremath{\mathrm{off}}}^\ensuremath{... ...ensuremath{\mathrm{beam}}}\,\sqrt{\ensuremath{n_\ensuremath{\mathrm{on/off}}}}.$$ (45)

Step #4: Is an integer number?

The interpretation of the above equation to compute has two cases.

1. $\ensuremath{n_\ensuremath{\mathrm{cover}}}< 1$. This means that either the user tries to cover a too large sky area in the given telescope elasped time (sensitivity estimation) or the telescope need a minimum time to cover at the maximum velocity possible with the telescope and this minimum time implies a more sensitive observation than requested by the user (time estimation).
2. $\ensuremath{n_\ensuremath{\mathrm{cover}}}\ge 1$. will generally not be an integer, we can think to decrease from to obtain an integer value. However, this must be done at constant $\ensuremath{A_\ensuremath{\mathrm{submap}}}(= \ensuremath{v_\ensuremath{\mathrm{area}}^\ensuremath{\mathrm{}}}\, \ensuremath{t_\ensuremath{\mathrm{submap}}})$. Decreasing thus implies increasing . It is not clear that this is possible because of the constraint $\ensuremath{v_\ensuremath{\mathrm{area}}^\ensuremath{\mathrm{}}} < \ensuremath{v_\ensuremath{\mathrm{area}}^\ensuremath{\mathrm{max}}}$. Another way to deal with this is to keep to its maximum value and to adjust (sensitivity estimation) or and thus (time estimation) to obtain an integer value of . However, this implies a change in the wishes of the user. The program can not make such a decision alone and we will only warn the user. Indeed, the worst case is when is changing from 1 to 2 because this can decrease the sensitivity by a factor 1.4 (sensitivity estimation) or double the elapsed telescope time (time estimation). The larger the value of the less harm it is to enforce the integer character of .