The purpose of this document is to compare the sensitivity of single-dish observations, used to observe to measure the "short spacing" sky data, which is complementary to interferometric observations. The goal is to answer - or at least give a range of answers - to the question "how many ALMA single dish antennas are needed to be outfitted for total power, single dish observations."

This is not a precise estimate. There are so many "ifs and buts" and unknown factors, depending on the exact observing technique, the characteristics of the source, the precise science that is being investigated, the level of systematic errors (pointing, tracking, spurious ground radiation, sidelobes etc. etc.) that the estimate becomes reminscent of the Drake Equation.

1. It is first assumed that we do indeed want to equalize sensitivities of the shortest interferometer spacing and of the higher spatial frequencies that can usefully be obtained from the single dish observations; later a factor is estimated for equalizing s/n ratios rather than noise. The "equalize the noise" case is a big assumption, which may be wrong. Arguably, what counts is singal/noise ratio rather than just noise. Since, for the sources where the short spacing information is important, the flux at a given spatial frequency increases rapidly with decreasing spatial frequency, this assumption may result in a large error in the final conclusion. The choice - s/n equalization or noise equalization - is undoubtedly at some level dependent on exactly what science is being investigated; studies of large-scale halos around galaxies will have different requirements from studies of small hot-spots. A very rough estimate is made below of the different implications of the 2 alternative assumptions.
2. Collecting area, correlation
   The sensitivity of a single 2-element interferometer, compared to a single dish in total power mode with no on-source off-source switching, is higher by a factor of 2./sqrt(2). The numerator comes from the doubled total collecting area of a 2-element interferometer, the sqrt(2) comes from the cross-correlation process with a single 2-element interferometer. This gives a gain in sensitivity of 2/sqrt(2) in favor of the interferometer, or a relative factor of 2 in observing time.
3. SD switching
   In practice single dish observations always have to switch against something, to be able to allow for (1) the background system noise and (2) the background atmospheric noise. For conventional position or beam switching, the loss in sensitivity is a full factor of 2. For frequency switching, the loss in sensitivity of the SD system is sqrt(2). For rapid OTF observing, the loss in sensitivity is reduced to a little more than 1 - because, in effect, many "on source" measurements are made for each "off source" measurement and the observing efficiency increases, in the optimum case and ignoring telescope overhead, to approach 100%. (The degradation factor is (1 = 1/sqrt(M)), where M is the number of "ons" to "offs," and provided more integration time is spent on the "offs" by the precisely optimum factor of sqrt(M).) For 1 on and 1 off, this degradation factor is precisely 2; it's the straightforward Dicke switching case. For M=100 the SD degradation factor is 1.1 in sensitivity, or 1.21 in observing time.
4. Antenna spatial frequency response
   For a homogeneous array, we will probably be trying to retrieve spatial frequencies that are attenuated by a relative factor of ~3 by the spatial frequency response of the SD primary beam. The precise factor depends on the maximum packing density of the array, and on the dish feed illumination taper. This sensitivity degratation factor is of course a factor of 9 in observing. Note that this factor is an inevitable consequence of the choice of a homogeneous configuration, with the same size of antenna used for single dish and for interferometric observations. If the array ever becomes heterogeneous - for example by including, say, 6 or 7-meter antennas as part of the Japanese contribution - then this homogeneous penalty factor would be very substantially reduced.
5. Redundant array baselines.
   Depending on the total number of antennas and on the antenna configuration layout, there will be more or less redundant short spacings. It will be necessary to observe longer, or to have more single dishes, in order to match sensitivity with the multiple short spacing interferometer baselines. This factor is probably between 2 and 4 - it will be higher for the close packed configuration and less for the ring configurations.
6. Systematic effects.
   In simulations of the homogeneous vs. heterogeneous arrays, both Holdaways's simulation and Emerson's (MMA Memo #62) showed that by averaging many single dish osbervations with different antennas, the systematic errors, which result in lower dynamic range and lower fidelity, are substantially reduced. The main parameter studied in these earlier simulations was the effect of pointing errors. If a single dish observation set consists of many observations repeated with different antennas (or, equally, many observations repeated over time with the same antenna) then pointing errors tend, statistically, to an apparent increase in the antenna beamwidth. This is much less deleterious than systematic pointing errors that would be left at full amplitude if only one observation were made. A similar argument applies to sidelobes. Errors in the dish surfaces are likely to be, at least in part, uncorrelated from antenna to antenna. So, by using many different antennas to make a SD observations, the sqrt(N) factor on such systematic effects will improve the quality of the data. There are undoubtedly other systematic effects that would be reduced by the use of many independent of antennas. Admittedly there will also be some systematic effects (e.g. spillover) that are fairly constant between antennas. This factor is very hard to quantify.
7. What fraction of time for projects where SD is important?
   Obviously if only a tiny fraction of ALMA's projects required the large scale structure information that is supplied by SD observations, we could afford to dedicate a tiny number of antennas to the task. Is 0.25 a reasonable estimate? Very uncertain.

   Summary of sensitivity requirement, Single Dish to Interferometric observations
   Mechanism	SD degradation factor	Implied factor for # SD antennas or implied SD observing time factor
   Aperture, and correlation	2./sqrt(2)	2
   Switching	1.1 to 2	1.2 to 4
   SD spatial frequency response	~3	9
   Redundant array baselines	2 - 4? Say 3 typical	4 - 9
   Systematic effects	1 - 2??	1 - 4??
   Total, assuming noise equalization at short SD/interferometer baselines:	9.3 to 68	86 to 2600
   Additional Factors reducing # dishes needed
   Equalize s/n vs. equalizing noise	.2 - 0.5 ???	0.04 - 0.25???
   Fraction of SD projects		0.25??
   Estimate assuming s/n equalization not noise equalization		1 to 160 ???

   
   

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   Conclusions

   This estimate has a lot in common with the Drake Equation; most of the factors are frankly just best guesses. It derives an estimate of needed # of single dishes operating in total power mode that varies from ~100 to ~2500, or if you assume that s/n rather than noise has to be equalized and that .25 of the array operation will need SD data, from 1 to ~160 antennas.