### Area-based estimate for quick TSE calculations

We first derive the mathematics of a “simple” sensitivity estimation when the mosaic is correctly sampled at the observing frequency. This computation is fast, and correct for single band observations. This is why it is used in the stand-alone sensitivity estimator.

Mosaicking is a particular case of wide-field imaging: The user wishes to observe a given field of view larger than the primary beam size with a sensitivity as uniform as possible. The targeted field (which area is , define by the user) can be divided in a number of independent resolution elements or independent (primary) beams . We have:

| (33)

where is the area of the primary beam. It is linked to the telescope full width at half maximum ( ) by

| (34)

The 0.8 factor represents the truncation of the beam at 20% of its maximum, which is performed during the imaging process.

For the sensitivity estimation we assume a standard sampling of targeted field and the on-source time is equally divided between the independent primary beams in the targeted field of view. We thus yield at the center of a Nyquist sampled mosaic

| (35)

with | (36)

where where is the integration time per pointing during each single mosaic cycle, and is the time to slew between two consecutive pointings. There are several subtleties in these computations.

- must be larger than 2 times . Below this threshold we advise to use the track sharing mode with two independent fields.
- Eq. is only valid at the center of a mosaic, not at its edges! Indeed, the processing (imaging and deconvolution) of a mosaic implies a division by the primary beam of the interferometer. As the primary beam is to first order a Gaussian decreasing to zero, this implies that the noise of the mosaic will vary over the field of view. In particular it increases sharply at the edges of the field of view.
- In these equations, is smaller that , the number of pointed positions that are observed in the mosaic. The cycling of the pointings of the mosaic should ensure Nyquist sampling of the observed field of view. This implies that there is an important redundancy between the pointings, contrary to track sharing where the sources are supposed to be fully independent on the sky. For instance, when mosaicking with a hexagonal compact pattern, each line of sight will be observed by 7 contiguous pointings, except at the mosaic edges. It can thus been shown that the number of mosaic pointings, , is related to the number of independent elements through

  | (37)

  for a correctly sampled mosaic. Equation is thus only valid inside a correctly sampled mosaic.
- Degradation of the observing efficiency The pointings of a mosaic must be observed in relatively short time cycles to ensure that all pointings are observed with similar weather conditions and that they share similar coverage. This will minimize the shift-variant part of the interferometer wide-field imaging response. This calls for the shortest possible integration time per pointing. However, the interferometer takes time to slew from one pointing to the next one without integrating. As a result, the observing efficiency is degraded in the cases of mosaics. We model it as a mosaic efficiency factor, . Having a large integration time per pointing during one cycle compared to will decrease the mosaicking overhead. This requirement is in sharp contrast with the previous one, namely the need to homogenize the interferometer wide-field response. The best compromise comes from two different considerations.
  1. The smallest integration time per scan is set by the acquisition system (for instance, the maximum achievable data rate). In pratice, we enforce that

     withsec | (38)

  2. The distance covered by a visibility in the -plane during an integration should always be smaller than the distance associated to tolerable aliasing (see Pety and Rodríguez-Fernández 2010 for more details). This can be written as the following condition (Eq. C.3 in this article)

     | (39)

     where is the map angular size, and the angular resolution. For a given angular resolution, the interferometer minimum integration time corresponds to

     | (40)

     where is a ad-hoc number set to 0.5 to ensure that the condition defined in Eq. is met.
  3. For operational reasons, the duration of a scan is limited to 45 .
  4. In summary,

     sec | (41)

  As the typical slew time between two pointings is sec, we yield that

  | (42)

- Improving the observing efficiency for small mosaics In A configuration, the integration time per scan should be less than 22 sec to meet the criteria of Eq. . However, this implies a mosaic efficiency of only ! It is possible to greatly increase this efficiency for small mosaics by observing several time the same pointing in time-contiguous scans. Let's assume that we use an integration time of per scan, and that we repeat it before slewing to another pointing position. This gives

  and | (43)

  with

  | (44)

  As stated above, we wish to cycle as quickly as possible through the different pointings in a single cycle to homogeneize the observing conditions over the mosaic. This implies to introduce a maximum time to cycle all the mosaic pointings. Let's assume that we want to homogeneize the conditions during the maximum time allowed between two gain calibrations, ie., min. This yields the maximum number of repeats per pointing and per cycle and thus the mosaic efficiency as

  and | (45)

  The maximum number of repeats must be larger than 1. This sets the limit in number of pointings per track between small and large mosaics.

  | (46)

- Checking the feasability of a large mosaic For large mosaics, it is advised to cycle all the pointings of a given mosaic in less than 1 hour. This ensures that the uv plane will be sampled at least 8 times per standard track of 8 hours. Setting the maximum time to cycle all the pointings, , to 60 minutes, we yield that the maximum number of pointing per track is

  | (47)

  Hence, if the PI wishes to observe an area that will require more that 150 pointings per independent track, the estimator will ask to either increase the requested elapsed telescope time or to decrease the requested field-of-view area.

  Moreover, we need to adapt the optimal integration time per scan to ensure that all pointings per track will be cycled in . This gives

  | (48)

  In the case of large mosaics, the minimum mosaic efficiency is

  | (49)

In summary, correctly setting up the sensitivity estimate for a mosaic with this fast but approximate method requires

1. To compute the number of independent primary beams in the targeted field of view

   | (50)

2. To compute the number of pointings in the mosaic and the number of pointings per track

   | (51)

3. To compute the optimal integration time per scan with

   | (52)

   The minimum integration time per scan of 10 seconds implies the maximum dynamic of scales that can be observed in a single track.
4. To compute the number of points per track that define the limit between small and large mosaics with

   | (53)

5. To compute the maximum number of repeats per pointing and per cycle, depending on the mosaic size.
   - For a small mosaic , it is

     | (54)

   - For a large mosaic , it is just set to .
6. To check that the maximum number of pointings per track is not yet reached with

   | (55)

   If the PI wishes to observe an area that will require more that 150 pointings per independent track, the estimator will ask to either increase the requested elapsed telescope time or to decrease the requested field-of-view area.
7. To adapt the optimal integration time per scan to ensure that all pointings per track will be cycled in only for a large mosaic with

   | (56)

8. To compute the actual mosaic efficiency and time to complete a full cycle with

   | (57)

   and

   | (58)

   These two quantities should be given as feedback to the user.
9. To finally compute the sensitivity with

   | (59)