Two key points: 1) Sharing OFF among many ONs and 2) system stability timescale

When the stability of the system is long enough, we can share the same off for several independent on-positions measured in a row (e.g. ON-ON-ON-OFF-ON-ON-ON-OFF...). The first key point here is the fact that the on-positions must be independent. The OTF is an observing mode where the sharing of the off can be used because the goal is to map a given region of the sky made of independent positions or resolution elements. When sharing the off-position between several on, Ball (1976) showed that the optimal off integration time is

$$\ensuremath{t_\ensuremath{\mathrm{off}}}^\ensuremath{\mathr... ...mathrm{on/off}}}} \, \ensuremath{t_\ensuremath{\mathrm{on}}}$$ (23)

where is the number of on measurements per off. Replacing by its optimal value in eq. , we obtain

$$\ensuremath{t_\ensuremath{\mathrm{sig}}}= \frac{\ensuremath... ...rac{1}{\sqrt{\ensuremath{n_\ensuremath{\mathrm{on/off}}}}}}.$$ (24)

We thus see that the rms noise decreases when the number of independent on per off increases. It seems tempting to have only one off for all the on positions of the OTF map. However, the second key point of the method is that the system must be stable between the first and last on measurement. To take this point into account we must introduce

• The concept of submap, which is a part of a map observed between two successive off measurements.
• , which is the area covered by the telescope in each submap.
• the number of such submaps needed to cover the whole map area.
• , the typical time where the system is stable. This time will be the maximum time between two off measurements, which is noted .
• , the number of coverages needed either to reach a given sensitivity or to exhaust the acquisition time.
• and are the times spent respectively on and off per independent measurement and per coverage.

We note that the number of on per off ( ) is a purely geometrical quantity. This implies that the time spent off is linked to the time spent on by Eq.  both in each individual coverage and when averaging all the coverages.