## Elapsed telescope time vs on-source time

The goal of a sensitivity estimator is to find the rms noise obtained when observing during the elapsed telescope time, . The total integration time spent on-source is shorter than the elapsed telescope time due to several factors. The actual on source time is then computed taking into account the following points:

1. Instrumental setup time: At the beginning of an observing track a significant time ( minutes according to history of observations) is spent in receiver tuning and calibration observations before observing the actual astronomical target. This means that even for a very short ON source time, a project cannot be shorter than . The setup time is computed as

   | (23)

   where is the minimum setup time when only one frequency is observed (40 minutes), is the additional setup time per additional frequency, and is the number of frequency observed in frequency cycling mode. is for the moment also set to 40 minutes.
2. Number of tracks per science goal: Also, for long projects observed in several ( ) tracks the time spent for tuning and calibration is . We thus define the time spent for observations (i.e. without instrumental setup) as:

   | (24)

   The number of tracks is computed as

   | (25)

   where is the typical time when the source is visible from Bure, which depends on the source declination:
   - Sources above are observed 8 hours at most (for instrumental management),
   - 8.2 hours for a declination of (truncated to 8 hours),
   - 6.5 hours for a declination of ,
   - 3.9 hours for a declination of ,
   - 0.0 hours for a declination of .
   - Sources below can not be observed.
   A linear interpolation with the declination is performed in the appropriate range between and .
   For short projects ( ), the number of tracks is set to 1. Otherwise, the floating value of is used in the computation of . Since is constant whatever the length of a track the use of a floating value for is somehow unnatural but it ensures that the conversion from to is a monotonic function, without regular threshold effects.
3. Observing efficiency: After the initial phase of instrumental setup, the observing mode does not dedicate 100% of the time to the astronomical target. Part of the time is spent for calibration (pointing, focus, atmospheric calibration,...) and to slew the telescopes between useful integrations. The time actually spent on source is defined as

   | (26)

   where is the observing efficiency. The exact computation of depends on the observing mode. The observing mode can be split into two categories that can be combined:
   1. Detection vs Imaging projects: From the sensitivity estimation viewpoint, the main difference between these observing modes is the number of gain calibrators regularly observed: for detection, and 2 for mapping projects.
   2. Frequency cycling mode: In this case, the gain calibrators have to be observed at each frequency of the cycle.
   These results in a computation of the observing efficiency as

   | (27)

   Note that the exact value of and actually depend on several parameters such as the distance between the source and the calibrator(s). In “standard” mode, we obtain the usual and 1.9, for detection and mapping projects, respectively. In frequency cycling mode with 2 frequencies, we yield and 2.5 for detection and mapping projects, respectively.
4. From to on-source time: Finally, the distribution of observing time into the time spent on-source, , actually used to estimate the sensitivity depends on three main observation kinds that are assumed exclusive from each other.
   Single-source, single-field observations
     where the telescope tracks a single source during the full integration time. This mode is used when the signal-to-noise ratio is the limiting factor.
   Track-sharing, single-field observations
     where the telescope regularly cycles between a few close-by sources. This mode is used when the sources are so bright that the limiting factor is the Earth synthesis, not the signal-to-noise ratio.
   Single-source mosaicking
     where the telescope regularly cycles between close-by pointings that usually follows a hexagonal compact pattern whose side is , where is the diameter of the interferometer antennas. This modes is used to image sources wider than the primary beam field of view.
   In the following, we will work out the equations needed by the sensitivity estimator for each of these observing modes.
5. Overall efficiency Finally, the overall observing efficiency, , is evaluated by computing the sum of the on-source time over the total telescope time

   | (28)

   The sum of the on-source time allows us to take into account the fact that 1) the telescope time may be shared between sources or frequencies, and 2) dual band observations are more efficient than single band ones. We also stress that when frequency cycling is combined with dual band observations, both receiver bands are affected by the efficiency loss of the frequency cycling even though one of the two bands could not require frequency cycling at all! A warning is raised when .