# Summary of the formulas for a single pixel receiver

We summarize here the relations between the rms noise ( ) and the elapsed telescope time ( ) derived by Pety et al. (2009) in the case of a single-pixel heterodyne receiver. The results depends on a combination of

- The observation kind:
  Tracked observations
    where the telescope track the source, i.e. it always observes the same position in the source referential. The result is a single spectra.
  On-The-Fly observations
    where the telescope continuously slew through the source with time to map it. The result is a cube of spectra.
- and of the switching mode:
  Position switch
    where the off-measurement is done on a close-by sky position devoid of signal. Wobbler switching is a particular case.
  Frequency switch
    where the telescope always points towards the the source and the switching is done in the frequency (velocity) space. In this case,

The formulas are

- for tracked observations
  (1)
- for OTF observations
  (2)

In these formulas

- is the efficiency of the telescope. It includes the time needed 1) to do calibrations (e.g. pointing, focus, temperature scale calibration), and 2) to slew the telescope between useful integrations, etc... Its value is decided by IRAM: It should not be changed by the PI.
- is the spectrometer efficiency.
- is the frequency resolution.
- is the number of polarizations tuned at the same frequency (1 or 2).
- is the system temperature, which is a summary of the noise added by the system. It is usual to approximate it (in the scale) with
  (3)
  where is the receiver image gain, the telescope forward efficiency, the airmass, the atmospheric opacity in the signal band, the mean physical atmospheric temperature, the ambient temperature in the receiver cabine and the noise equivalent temperature of the receiver and the optics.
- is the number of independent measurement in the map observed in the OTF mode. It is given by
  (4)
  where is the map area, is the area of the resolution element in the map, is the smoothing factor due to gridding and is the telescope full width at half maximum.
- the number of submaps needed to cover the whole map area, a submap being the area covered between two successive off measurements. is computed with
  (5)
  where is the telescope scanning speed and is the typical timescale of stability of the observing system.

The demonstrations and additional subtleties for the OTF case are fully described in Pety et al. (2009).