Continuum Sensitivity of Submillimeter Telescopes

A surprising implication of the table above is that the SPST is faster at submillimeter detection of protogalaxies than the ALMA and FIRST. Is this plausible, given that these instruments are at least an order of magnitude more expensive?   Suppose we are conducting a large-scale survey at a wavelength of 450µm  for point-source objects at an rms flux density F_limit. The time required for a  detection in one pixel of a map of the sky is given by the radiometer equation

where

is the atmosphere-corrected effective system temperature,  A is the total collecting area of the telescope, is telescope efficiency, is the atmospheric opacity at the elevation angle of the observation, and B is the pre-detection bandwidth. T_atmosphere ~ T_ambient ~ 200K at the Pole and  ~ 260K at mid-latitude sites.  Here we neglect polarization, relative sideband gain, the distinction between telescope efficiency and aperture efficiency, digitization and data processing corrections: these introduce factors of order unity which have been considered in the table above but which can be neglected in the present plausibility argument. We assume that the radiometer is switched rapidly enough to filter out sky noise. The radiometer equation is then approximately correct regardless of whether the system is heterodyne or bolometric, or whether the collecting area, A, is arranged in multiple antennas or a single dish. All telescopes will be designed for reasonably high efficiency, ~ 0.9. 

Future submillimeter-wave systems will be background-limited, meaning that the T_receiver term will be smaller than the atmospheric (T_atmosphere) and telescope background (T_ambient) terms. When > 1, as it often is for ground-based submillimeter-wave observations, the opacity-correction term, , dominates all other effects: 

and

so in this case relatively small improvements in make a large difference. As a point of comparison, we can refer to the 25% winter opacity curves at 450µm and 45^o elevation in the opacity model, where we see that  < 0.36 at least 25% of the time at the Pole as opposed to < 0.79 at least 25% of the time at Chajnantor. When is small or zero, for example in space or on the ground at centimeter wavelengths, then the telescope background (1- )T_ambient  dominates. For the SPST, T_sys ~ 1000K (a value that is often surpassed on AST/RO), for the ALMA, T_sys ~ 2500K (this high system temperature is due mostly to atmospheric opacity), and for FIRST, T_sys ~ 50K (the low background of a cooled telescope in space). For the SPST B ~ 100 GHz, for FIRST, B ~ 200 GHz, whereas for the ALMA B ~ 16 GHz. We therefore conclude that the time required to achieve a F_limit = 1 mJy noise level, t_mJy, is about four hours for the SPST, forty minutes for FIRST, and one minute for the ALMA. To reach an rms noise of, say, 0.1 mJy would require 100 times longer integration in each case. ALMA and FIRST are both faster than the SPST at detecting individual sources. 

Now we note that the area of sky covered by the field of view during this integration time, the "instantaneous sky coverage" (S in the above Table, see also the figure showing the image sizes of submillimeter-wave telescopes), is very different for the three telescopes. For the SPST, S is over an order-of-magnitude larger than FIRST and nearly four orders-of-magnitude larger than the ALMA. An efficient configuration of n diffraction-limited detectors (bolometer pixels or heterodyne receivers) on a telescope of total area A will yield an instantaneous sky coverage of 

The speed at which the sky can be mapped is S/t_limit. A figure of merit for blank sky surveys can therefore be defined:

The trade-offs in survey speed between a single-dish telescope like the SPST and an array instrument like the ALMA can be estimated by evaluating this figure of merit. Suppose the single dish telescope has a focal plane array containing  N × N  detectors, and the interferometer consists of M antennas, each with the same diameter as the single-dish telescope, and each having one heterodyne receiver. Then

S_dish = N ² S_{interferometer}  ,

A_{interferometer} = M A_dish  ,

n_{interferometer} = M ,

and

n_dish = N ²  .

If the two instruments have the same T_sys , the single-dish telescope will be faster in survey mode by a factor

Comparing the SPST with the ALMA,  N ~ 100,  M ~ 80, and B_dish/B_{interferometer}  ~  8 .

Even discounting the superior sky at the South Pole and the quantum noise superiority of bolometers, the SPST will be an order-of-magnitude faster than the ALMA at submillimeter-wave sky surveys.    

It might be argued that N ² detectors represents a lot of complex electronics, and that for large numbers it may be easier to build M whole antennas than N ² detectors---however, any interferometer will necessarily have M ² correlators; these correlators are likely to be more complex than the photolithographically-produced detector plus amplifier plus multiplex needed for each of the N ² pixels of a bolometer array. This table shows the number of detectors, n, in some current and proposed millimeter and submillimeter instruments. Compared to single-dish maps, maps made by the interferometric instrument have much higher resolution and positional accuracy, but the rate at which the maps are made is at least an order-of-magnitude slower and the initial capital cost is at least an order-of-magnitude higher. It is a waste of scarce resources to map large areas of blank sky with the ALMA. Note too that at a survey speed of ~ 2.4 × 10³ hours per square degree, the ALMA will not be able to survey more than a hundred square degrees during its operating lifetime, but there are many thousands of square degrees containing potentially interesting sources. The best strategy to detect and study protogalaxies is to survey the sky quickly with the relatively inexpensive SPST and then study the detected sources in detail with the ALMA. 

The above figure of merit can also be used to compare the SPST to a hypothetical submillimeter-wave instrument with the configuration of CBI---an interferometric array where the entire array is contained within a single 10 m diameter mount. (The CBI itself was not designed as a submillimeter-wave instrument and currently operates at wavelengths near 1 cm.)  In this case, n = 13,  A = 8.3 m², and  B = 10 GHz  for the "sub-mm CBI", compared to  n ~ 10,000,   A = 79 m², and  B ~ 100 GHz  for the SPST.  The single dish is therefore seen to have a raw mapping speed more than four orders-of-magnitude greater than the interferometer.  Construction of a submillimeter-wave CBI might nevertheless be worth considering because of its entirely different systematic noise characteristics. 

The trade-offs between the SPST and FIRST are straightforward: the two instruments have almost equivalent NEFD in the radiometer equation, because the vastly superior opacity in space is almost balanced by the large collecting area possible at the Pole. In mapping speed, as given by the figure of merit, the SPST is faster because of its large field of view. FIRST could improve its survey speed by increasing the size of the field of view, S---this would, however, cause an approximately proportional increase in evaporation of liquid Helium and shorten the mission lifetime by a proportional amount. The detector dewar at the Pole can either be refilled with liquid Helium (20,000 liters of LHe are used at the Pole each year) or cooled by a cryogenic refrigerator powered by South Pole Station's powerplant.

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Last modified: April 22, 2000