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The gas turbulence in planetary nebulae: quantification and multi-D maps from long-slit, wide-spectral range echellograms

Context. This methodological paper is part of a short series dedicated to the long-standing astronomical problem of de-projecting the bi-dimensional, apparent morphology of a three-dimensional distribution of gas. Aims. We focus on the quantification and spatial recovery of turbulent motions in planetary nebulae (and other classes of expanding nebulae) by means of long-slit echellograms over a wide spectral range. Methods. We introduce some basic theoretical notions, discuss the observational methodology, and develop an accurate procedure disentangling all broadening components (instrumental resolution, thermal motions, turbulence, gradient of the expansion velocity, and fine structure of hydrogen-like ions) of the velocity profile in all spatial positions of each spectral image. This allows us to extract random, non-thermal motions at unprecedented accuracy, and to map them in 1-, 2- and 3-dimensions. Results. We discuss general and specific applications of the method. We present the solution to practical problems in the multidimensional turbulence-analysis of a testing-planetary nebula (NGC 7009), using the three-step procedure (spatio-kinematics, tomography, and 3D rendering) developed at the Astronomical Observatory of Padua (Italy). In addition, we introduce an observational paradigm valid for all spectroscopic parameters in all classes of expanding nebulae. Conclusions. Unsteady, chaotic motions at a local scale constitute a fundamental (although elusive) kinematical parameter of each planetary nebula, providing deep insights on its different shaping agents and mechanisms, and on their mutual interaction. The detailed study of turbulence, its stratification within a target and (possible) systematic variation among different sub-classes of planetary nebulae deserve long-slit, multi-position angle, wide-spectral range echellograms containing emissions at low-, medium-, and highionization, to be analyzed pixel-to-pixel with a straightforward and versatile methodology, extracting all the physical information (flux, kinematics, electron temperature and density, ionic and chemical abundances, etc.) stored in each frame at best.