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;

**Total-Power Observations:**Here, all the observing time is spent looking at the target (``Point-and-Shoot''). This gives, per polarization, or if both polarizations are averaged to obtain the final spectrum. Pure total-power observations are found to be adequate for many Arecibo observation, especially if narrow total bandwidth observing is to be used (say less than 1 MHz total), and has even been used for relatively short integrations on Galactic HI. Note that this is also the case for total-power continuum observations.**``In-Band'' Frequency switching:**NOTE: Traditional frequency switching is NOT presently supported, although its efficacy at Arecibo has been investigated.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.

**Position Switching or ``Out-of-Band'' Frequency Switching:**Here the line is only observed for one half of the time, with noise being present all of the time. This gives, per polarization, or if both polarizations are averaged to obtain the final spectrum. Note that this is also the case for simple Dicke-switched continuum observations.**Position Switching on a Target Source, and a Band-Pass Continuum Calibrator:**When simple position-switching is used to measure the emission or absorption line spectrum of a source that also has significant continuum emission, problems can arise due to residual standing waves, especially for any telescope with a partially blocked aperture. The Arecibo telescope represents an extreme case of aperture blockage, with the suspended platform and a good fraction of its support cables being situated within the volume traversed by the incoming rays focused on the telescope feed. This structure also scatters significant amounts of radiation from the surrounding hills, and other sources of radiation arising outside the telescope main beam.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 , where 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 report ``2001-02.ps''.

Robert Minchin 2017-10-30