The Mass/Volume Sensitivity Function

The mass/volume sensitivity function characterizes HI surveys according to the total volume in which a survey can detect a galaxy of a given mass. It is usually based on simple assumptions about the line shape of galaxies and the effects of noise. I will argue later that this description is insufficient to generate an accurate luminosity function, but it is a useful starting point.

Consider a galaxy with an HI mass (in solar masses). Rotation of the galaxy (and to a much lesser extent turbulence) will Doppler-broaden the HI line to a width w km . For all practical purposes the 21 cm line is optically thin and its emission strength is independent of temperature, so the integrated power in the line is directly proportional to the total mass of HI. If this galaxy is at a distance D Mpc, its integrated emission is

where is the mean flux density (in mJy) within the line profile. I will use distances based on km / Mpc throughout to translate from redshifts to distances.

This flux must be detected against a noise arising from the receiver and background emission. If the spectrum has a velocity resolution of (in km ), the noise can be reduced by averaging or ``smoothing'' the spectrum to a lower resolution, with the noise improving as . However, if the smoothing is too great, the line itself is averaged with the background; the best result is achieved when the smoothing just equals the line width. Starting with a spectrum having a measured rms noise (at the resolution ), and smoothing it to a resolution matching the galaxy line width gives a signal-to-noise ratio of:

More realistically, the 21 cm spectrum should allow the line to be detected over several resolution elements to identify a signal, so this signal-to-noise ratio is not achieved in practice. We must also contend with occasional large noise fluctuations, baseline irregularities, imperfect pointing, and interference. Given these problems and the extra free parameter introduced by searching over a range of line widths, it is not certain how much advantage is gained by smoothing. However, if we simply require the nominal smoothed signal-to-noise ratio to be >5, the maximum distance to which a galaxy could be detected is:

in Mpc, for a mass in solar masses, in mJy, and velocities in km . Of course, the maximum distance is limited by the observed bandpass if it does not reach as high a redshift as this distance implies.

To determine the survey volume, we also need the survey area. Radio telescopes have beam diameters approximately inversely proportional to the telescope diameter d, so larger telescopes require about more observations to cover the same area. For unresolved sources the integration time needed to reach a given flux sensitivity goes as , while for resolved sources the integration time goes as . Therefore, the time required to observe the same area of the sky to the same depth is inversely proportional to the square of the telescope diameter for unresolved sources and approximately the same for resolved sources (ignoring time spent moving the telescope, etc.). Because larger-telescope surveys have normally taken advantage of the sensitivity dependence for unresolved sources to cut their integration times and cover more area, one special niche the Parkes surveys can fill is through their sensitivity to very extended, low surface brightness HI emission.

The distance sensitivity also depends on where the source is within the beam. In fact some surveys (Sorar 1994) have used sidelobes as a further probe of lower sensitivity but larger area. For comparisons' sake, I ignore sidelobes and suppose that the beam is uniform within the half-power beam width. I also do not correct here for how the relative sensitivity as a function of offset from beam center (see, for example, Shostak 1977) reduces the volume sensitivity. Depending on the amount of overlap between observations the effective volume searched may be only half of what is quoted here. Thus surveys that are more contiguous and have more uniform integrated sensitivities, as is planned for the Parkes surveys, can have a significantly better volume sensitivity than a simple comparison indicates.

  
Table 1: Major ``Blind'' Extragalactic HI Surveys

Table 1 lists the vital statistics for several recent HI surveys along with a set of projected values for the Parkes Multibeam Surveys (Staveley-Smith 1997). The velocity range and beam size are listed in columns 2 and 3. There are two entries for surveys that covered two velocity ranges. Not all of the necessary data were always included, but I have attempted to place all the surveys on a common scale. Thus, surveys carried out in drift scan mode are given a number of observation points (column 4) that would generate an equivalent search area, and the integration time and spectral resolution (column 5) were sometimes used to estimate rms noise levels (column 6). Arecibo noise values are based on an average of the frequency-dependent noise across the spectra.

The corresponding volume sensitivities for different HI masses are shown in Figure 1. I assume a galaxy with a line width of 100 km , which is reasonable for the lower HI masses; at higher masses, the effective search volumes would be smaller than shown. For the volume calculations I also limit the minimum velocity to >300 km , because of confusion with high velocity clouds in the Milky Way. Most of the curves show a characteristic (flux-limited) rise with mass as at low masses until a mass is reached which the survey is sensitive to at its maximum redshift. Surveys with high minimum redshifts (Weinberg et al. 1991; Krumm & Brosch 1984) have a sharp cutoff at low masses. The Arecibo surveys have a more complex sensitivity roll-off with frequency that steepens the curves slightly.

  
Figure: The volume/mass-sensitivity relationships of major HI surveys listed in Table 1. The curves show the total volume within which each survey would have been sensitive to a hypothetical galaxy with a line width of 100 km of the given HI mass. Green Bank surveys are shown by dotted lines, Arecibo by short dashes, VLA by long dashes, and Parkes by solid lines.

Many observers have pointed out that in the course of standard HI observations, calibration (``off'') scans have been collected that could potentially represent an enormous survey volume. I limit consideration to the deliberate blind surveys because much more effort was put into these surveys to identify possible signals. Most observers, pressed by the exigencies of their particular project, dismiss small negative features in their spectra as interference because they are usually correct in this assumption. By contrast, the blind surveys devote most of their attention to identifying real signals and discriminating them from interference or other instrumental problems.

© Copyright Astronomical Society of Australia 1997