Measuring angular diameters of extended sources

When measuring diameters of partially resolved sources like planetary nebulae, H II regions or galaxies, often a technique called Gaussian deconvolution is used. This technique yields a Gaussian diameter, which subsequently has to be multiplied by a conversion factor to obtain the true angular diameter of the source. This conversion factor is a function of the FWHM of the beam or point spread function, and also depends on the intrinsic surface brightness distribution of the source. In this paper, conversion factors are presented for a number of simple geometries: a circular constant surface brightness disc and a spherical constant emissivity shell, using a range of values for the inner radius. Also, more realistic geometries are studied, based on a spherically symmetric photoionization model of a planetary nebula. This enables a study of optical depth effects, a comparison between images in various emission lines, and the use of power-law density distributions. It is found that the conversion factor depends quite critically on the intrinsic surface brightness distribution, which is usually unknown. The uncertainty is particularly large if extended regions of low surface brightness are present in the nebula. In such cases the use of Gaussian or second-moment deconvolution is not recommended. As an alternative, a new algorithm is presented which allows the determination of the intrinsic FWHM of the source using only the observed surface brightness distribution and the FWHM of the beam. Hence no assumptions concerning the intrinsic surface brightness distribution are needed. Tests show that this implicit deconvolution method works well in realistic conditions, even when the signal-to-noise ratio is low, provided that the beamsize is less than roughly 2/3 of the observed FWHM and the beam profile can be approximated by a Gaussian. A code implementing this algorithm is available