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The unprecedented quality of the observations available from the Atacama Large Millimetre/sub-millimetre Array (ALMA) calls for analysis methods making the best of them. Reconstructing in space the morphology and kinematics of radio sources is an underdetermined problem that requires imposing additional constraints for its solution. The hypothesis of rotational invariance about a well-defined star axis, which is a good approximation to the description of the gas envelopes of many evolved stars and protostars, is particularly efficient in this role. In the first part of the article, a systematic use of simulated observations allows for identifying the main problems and for constructing quantities aimed at solving them. In particular the evaluation of the orientation of the star axis in space and the differentiation between expansion along the star axis and rotation about it are given special attention. The use of polar rather than Cartesian sky coordinates is shown to better match the morphology and kinematics of actual stars. The radial dependence of the gas density and temperature and the possible presence of velocity gradients are briefly considered. In the second part, the results obtained in the first part are applied to a few stars taken as examples with the aim of evaluating their usefulness when applied to concrete cases. A third part takes stock of what precedes and formulates some guidelines for modelling the radio emission of axisymmetric radio sources, limited however to the mathematics and geometry of the problem, physics considerations being ignored.
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We analyse ALMA observations of the 12CO(3-2) emission of the circumstellar envelope (CSE) of the Mira variable binary star W Aql. These provide, for the first time, spatially resolved Doppler velocity spectra of the CSE up to angular distances to the central star of ~ 5 (meaning some 2000 AU). The exploratory nature of the observations (only five minutes in each of two different configurations) does not allow for a detailed modelling of the properties of the CSE but provides important qualitative information on its morphology and kinematics. Emission is found to be enhanced along an axis moving from east/west to north-east/south-west when the angular distance from the central star projected on the plane of the sky increases from zero to four arcseconds. In parallel, the Doppler velocity distribution displays asymmetry along an axis moving from east/west to north-west/south-east. The results are discussed in the context of earlier observations, in particular of the dust morphology.
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