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.