Context. X-ray observations of galaxy clusters provide insights into the nature of gaseous turbulent motions, their physical scales, and the fundamental processes to which they are related. Spatially-resolved, high-resolution spectral measurements of X-ray emission lines provide diagnostics on the nature of turbulent motions in emitting atmospheres. Since they are acting on scales comparable to the size of the objects, the uncertainty on these physical parameters is limited by the number of observational measurements, through sample variance.
Aims. We propose a different and complementary approach to repeating numerical simulations for the computation of sample variance (i.e. Monte-Carlo sampling) by introducing new analytical developments for lines diagnosis.
Methods. We considered the model of a “turbulent gas cloud”, consisting in isotropic and uniform turbulence described by a universal Kolmogorov power-spectrum with random amplitudes and phases in an optically thin medium. Following a simple prescription for the four-term correlation of Fourier coefficients, we derived generic expressions for the sample mean and variance of line centroid shift, line broadening, and projected velocity structure function. We performed a numerical validation based on Monte-Carlo simulations for two popular models of gas emissivity based on the β-model.
Results. Generic expressions for the sample variance of line centroid shifts and broadening in arbitrary apertures are derived and match the simulations within their range of applicability. Generic expressions for the mean and variance of the structure function are provided and verified against simulations. An application to the Athena/X-IFU (Advanced Telescope for High-ENergy Astrophysics/X-ray Integral Field Unit) and XRISM/Resolve (X-ray Imaging and Spectroscopy Mission) instruments forecasts the potential of sensitive, spatially-resolved spectroscopy to probe the inertial range of turbulent velocity cascades in a Coma-like galaxy cluster.
Conclusions. The formulas provided are of generic relevance and can be implemented in forecasts for upcoming or current X-ray instrumentation and observing programmes.
On the characterisation of the normal population by the independence of the sample mean and the sample variance.
