### 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 . Using Eqs. and , we obtain
  (37)
  Using this equation, we start to compute for and . We want to enforce the integer character of in a way which decreases the product . To do this, we use
  (38)
  Eq. ensures that . 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:

  |  |  |  | (39)
  |  |  |  | (40)
  |  |  |  | (41)

Step #3: Computation of derived quantities

  |  |  |  | (42)
  |  |  |  | (43)
  |  |  |  | (44)
  |  |  |  | (45)

Step #4: Is an integer number?
  The interpretation of the above equation to compute has two cases.
  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. . will generally not be an integer, we can think to decrease from to obtain an integer value. However, this must be done at constant . Decreasing thus implies increasing . It is not clear that this is possible because of the constraint . 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 .