How do we extract information from a map calibrated in Jy/beam

Analyzing maps calibrated in Jy/beam is easy; especially if we want to deduce flux densities for point sources or compact sources even when the source is embedded in a cloud with strong extended emission. For a point source the peak flux of the source is the same as the total flux corrected for any background emission. For an extended source we need to measure the FWHM and correct it for the measured HPBW of the telescope. We normally do this by fitting a double Gaussian, one for the source and one for the background. At 850 $\mu$m the fitted peak signal minus background, S$_{\rm peak}$, is now the peak flux density measured in Jy/beam. From the fitted Full Width at Half Maximum (FWHM) we can derive the true FWHM, $\theta_s$, by deconvolving with the measured HPBW, $\theta_{A}$. This is trivial, because now we can assume a Gaussian source and a Gaussian beam. After we know the source size, $\theta_s$ we multiply the peak flux with the correction factor we derive from the size, i.e. for a spherically symmetric source with the source size, $\theta_{s}$, the total flux, S$_{\rm tot}$ is simply

$$S_{\rm tot} = S_{\rm peak} \times (1 +(\theta_{s}/\theta_{A})^2)$$ (9)

.

For 450 $\mu$m the error beam amplitude is no longer negligible, but when we fit a double Gaussian, the error beam will blend in with the extended cloud emission, i.e. it adds into the background level, or we may fit the source with a single Gaussian plus a second order surface, or whatever best approximates the background in a limited area around the source. From our analysis of the 450 $\mu$m beam maps of Uranus, we find that the combined error beam amplitude is of the order of 5% of the peak amplitude, and we should therefore multiply the peak signal by 1.05 before applying a source size correction (see e.g. Weintraub et al. [17]).

To find integrated intensities over large areas is more complicated, because now we need to correct for the error beam pickup, which now depends on the area we integrate over. This is equivalent to the varying the FCF as a function of aperture that one has to account for if the map is calibrated in Jy/pixel, but with the map calibrated in Jy/beam it is much easier to separate compact sources and extended emission. To determine the excess emission from the error beam, we again have to go back to our beam map. If we calibrate our 850 $\mu$m map in Jy/beam and then integrate over 120'' circular aperture, we find that the flux we derive is 86.8 Jy, while we know that the total flux of Uranus is only 67.9 Jy. We therefore have to scale our derived total flux density by the ratio of true flux density over measured flux density (for our calibrated planet map), which in this case is 0.78. At 450 $\mu$m the situation is much worse. Even though the amplitude of the error lobe is still low, the area is large, and if we integrate over our calibrated 450 $\mu$m beam map we now derive 415.6 Jy, if we integrate over the same 120" circular aperture, while the total flux density from FLUXES is only 179.3 Jy. In this case our scaling factor is 0.43, i.e. we pick up more emission in the extended error beam than we do in the main beam.

For careful work, you may therefore want to deconvolve your SCUBA maps. This becomes especially important if you want to ratio the 450 and the 850 $\mu$m maps, because if you want to smooth the 450 $\mu$m map to the same resolution as the 850 $\mu$m map, you first have to remove the error beam. For example of how this can be done, see e.g. Hogherheijde and Sandell [13] or Sandell and Weintraub [16].