Measuring angular diameters of extended sources

When measuring diameters of partially resolved sources like planetary nebul ae, H II regions or galaxies, often a technique called Gaussian deconvoluti on is used. This technique yields a Gaussian diameter, which subsequently h as to be multiplied by a conversion factor to obtain the true angular diame ter 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 br ightness distribution of the source. In this paper, conversion factors are presented for a number of simple geom etries: a circular constant surface brightness disc and a spherical constan t emissivity shell, using a range of values for the inner radius. Also, mor e realistic geometries are studied, based on a spherically symmetric photoi onization model of a planetary nebula. This enables a study of optical dept h effects, a comparison between images in various emission lines, and the u se of power-law density distributions. It is found that the conversion fact or depends quite critically on the intrinsic surface brightness distributio n, which is usually unknown. The uncertainty is particularly large if exten ded regions of low surface brightness are present in the nebula. In such ca ses the use of Gaussian or second-moment deconvolution is not recommended. As an alternative, a new algorithm is presented which allows the determinat ion of the intrinsic FWHM of the source using only the observed surface bri ghtness distribution and the FWHM of the beam. Hence no assumptions concern ing the intrinsic surface brightness distribution are needed. Tests show th at this implicit deconvolution method works well in realistic conditions, e ven when the signal-to-noise ratio is low, provided that the beamsize is le ss than roughly 2/3 of the observed FWHM and the beam profile can be approx imated by a Gaussian. A code implementing this algorithm is available.