We will consider the rms, (K), for observations with a receiver of system temperature, T, frequency resolution, per polarization, and total integration time per observing cycle, . Note that , where is the total bandwidth of the spectrometer per polarization, and is the number of independent points in the computed spectrum (e.g. the number of spectrometer channels for an unsmoothed spectrum.) If the spectrum has been Hanning smoothed, then the effective frequency resolution is broadened by a factor of about 1.67. The sensitivity calculations given below represent the analog case, and it should be remembered that 9-level operation of our spectrometer achieves 96% of the signal-to-noise of analog correlation, whereas 3-level operation achieves 81%. For the following observing modes, the theoretical sensitivities are;
For ``in-band'' frequency switching, the line under investigation is always in the observing band. For each of the two positions where the line falls, only one half of the time is spent looking at the line, with noise being present all the time. Using a ``flip, shift and average'' operation on the raw frequency-switched spectrum, gives, for a single polarization, or if both polarizations are averaged to obtain the final spectrum.
For simple ON/OFF position switching on a target possessing significant continuum radiation, the standing-wave pattern due to the continuum emission from the target source is not at all cancelled by subtracting the source-free OFF data from the ON. A standing-wave residual whose amplitude is proportional to the source intensity remains to degrade the spectrum. To minimize the effects of this residual standing wave when observing a strong continuum source, another (reference) continuum source, (preferably of different redshift to avoid it having an emission/absorption line near the line frequency of the target), is also observed in ON/OFF mode. The azimuth-zenith angle track followed during this observation should be as near as possible that for the target source. Division of the (ON OFF) target spectrum by that of the reference source then cancels the residual standing wave, and yields a spectrum whose magnitude is proportional to the ratio of the target and reference flux densities across the observing band, including any spectral-line component that may be present in the target.
If equal time is spent on the target and reference cycles, the line is
observed for one quarter of the time, but noise is observed all the
time. This gives, to first approximation,
per polarization. Note that T here is
not just the ``blank-sky'' system temperature, but should allow for the
contribution due to the continuum emission of the target and reference
sources, i.e. if both have a flux density, SJy, then T should
be increased by
is the telescope gain (in K/Jy) at zenith angle, .
For full details of the technique and sensitivity considerations, see;
http://www.naic.edu/~astro/aotms/performance.shtml,, then click on
Robert Minchin 2017-10-30