We use the results of magnetohydrodynamic (MHD) simulations to emulate spectroscopic observations and use maps of centroids to study their statistics. In order to assess under which circumstances the scaling properties of the velocity field can be retrieved from velocity centroids, we compare two point statistics (structure functions and power-spectra) of velocity centroids with those of the underlying velocity field and analytic predictions presented in a previous paper . We tested a criterion for recovering velocity spectral index from velocity centroids derived in our previous work, and propose an approximation of the early criterion using only the variances of "unnormalized" velocity centroids and column density maps. It was found that both criteria are necessary, however not sufficient to determine if the centroids recover velocity statistics. Both criteria are well fulfilled for subsonic turbulence. We find that for supersonic turbulence with sonic Mach numbers M s 2.5 centroids fail to trace the spectral index of velocity. Asymptotically, however, we claim that recovery of velocity statistics is always possible provided that the density spectrum is steep and the observed inertial range is sufficiently extended. In addition, we show that velocity centroids are useful for anisotropy studies and determining the direction of magnetic field, even if the turbulence is highly supersonic, but only if it is sub-Alfvénic. This provides a tool for mapping the magnetic field direction, and testing whether the turbulence is sub-Alfvénic or super-Alfvénic.

In a previous work, Lazarian and Pogosyan suggested a technique to extract velocity and density statistics, of interstellar turbulence, by means of analysing statistics of spectral line data cubes. In this paper, we test that technique by studying the effect of correlation between velocity and density fields, providing a systematic analysis of the uncertainties arising from the numerics, and exploring the effect of a linear shear. We make use of both compressible magnetohydrodynamics simulations and synthetic data to emulate spectroscopic observations and to test the technique. With the same synthetic spectroscopic data, we have also studied anisotropies of the two point statistics and related those anisotropies with the magnetic field direction. This presents a new technique for magnetic field studies. The results show that the velocity and density spectral indices measured are consistent with the analytical predictions. We have identified the dominant source of error with the limited number of data points along a given line of sight. We decrease this type of noise by increasing the number of points and by introducing Gaussian smoothing. We argue that in real observations the number of emitting elements is essentially infinite and that the source of noise vanishes.

Context. Some PNe and PPNe show compact knots, travelling at high velocities away from the central sources.Aims. We compute a number of models from which we obtain predictions of the emission-line spectrum, which can be compared with the spectra of the observed knots. Methods. We completed a series of 11 axisymmetric simulations of an initially spherical cloudlet, travelling away from a photoionizing source, into a uniform medium. The simulations included a multi-frequency transfer of the ionizing radiation, and a 33 species non-equilibrium ionization network. Results. From our simulations, we computed emission maps and spatially-integrated emission-line spectra. The predictions show a transition from spectra similar to those of shock wave models (for simulations with lower photoionization rates) to spectra similar to those of photoionized regions (for simulations with higher photoionization rates). Conclusions. The spectra from our photoionized cloudlet models have a range of line ratios that agree approximately with the observed spectra when shown in two-line ratio diagnostic diagrams. The predicted and observed spatial distributions of the emission (with high ionization lines extending more towards the source than lower ionization lines) agree in a qualitative way.

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