### Relation between and

By construction

- The number of submaps is the area of the map divided by the area of a submap
  (25)
- The number of on per off is the number of independent resolution elements in each submap
  (26)
- The number of independent resolution elements in the map is the product of number of submaps by the number of on per off
  (27)
- The submap area is the product of the area velocity by the time to cover it
  (28)
- The time to scan a submap is the sum of the independent on integration time
  (29)
- The relations between times per coverage and times integrated over all the coverages are
  (30)
- Using the last two points, it is easy to derive
  (31)
- Finally, the total time spent on and off is given by
  (32)
  Using Eqs. and , we derive
  (33)

Both and are proportional to (cf. Eqs. and ). It is thus easy to derive that
(34)
Using Eq. , we can replace and obtain
(35)
Using Eqs. , and , we obtain
(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 and have an integer value.