Measuring angular diameters of extended sources - I. Theory

Abstract

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 with a conversion factor to obtain the true angular diameter of the source. This conversion factor is a function of the Full Width at Half Maximum (FWHM) of the beam. It also depends on the intrinsic surface brightness distribution of the source. In this paper the theory behind this technique will be studied. This study will be restricted to circularly symmetric geometries and beams. First an implicit equation will be derived, from which the conversion factor for a given surface brightness distribution and beam size can be solved. Explicit expressions for the conversion factor will be derived from this equation which are valid in cases where the beam size is larger than the intrinsic size of the source. A more detailed discussion will be given for two simple geometries: a circular constant surface brightness disk and a spherical constant emissivity shell with arbitrary inner radius. The theory is subsequently used to construct a new technique for determining the FWHM of an arbitrary observed surface brightness distribution. Usually the FWHM of the source and beam are measured using gaussian fits, but second moments can also be used. The alternative use of second moments in this context is studied here for the first time and it is found that in this case the conversion factor has a different value which is independent of the beam size. In the limit for infinitely large beam sizes, the values of the conversion factors for both techniques are equal.