## Additional notions and notations

The On-The-Fly (OTF) observing mode is used to map a given region of the sky. The time/sensitivity estimator will have to link the elapsed telescope time to cover the whole mapped region to the sensitivity in each independent resolution element. To do this, we need to introduce

- and , which are respectively the area of the map and the area of the resolution element. The map area is a user input while the resolution area is linked to the telescope full width at half maximum ( ) by
  (12)
  where comes from the fact that the OTF data is gridded by convolution. When the convolution kernel is a Gaussian of FWHM equal to (the default inside the GILDAS/CLASS software), it is easy to show that
  (13)
- The number of independent measurement ( ) in the final map which is given by
  (14)
- The on and off time spent per independent measurement, and . The associated can then be written
  (15)
- The on and off time spent to map the whole map, and . is deduced from and in a way which depends on the switching scheme.

In addition, we must ensure that the user does not try to scan faster than the telescope can slew. To do this, we need to introduce

- The linear scanning speed, , and its maximum value, .
- The area scanning speed, , and its maximum value, . When the scanning pattern is linear, then and are linked through
  (16)
  where is the separation between consecutive rows. To avoid nasty signal and noise aliasing problems, we must ensure a Nyquist sampling, i.e.
  (17)