Towards mapping turbulence in the intra-cluster medium -- I. Sample
  variance in spatially-resolved X-ray line diagnostics
  X-ray observations of galaxy clusters provide insights on the nature of
gaseous turbulent motions, their physical scales and on the fundamental
processes they are related to. 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. We propose a different and complementary approach for the
computation of sample variance to repeating numerical simulations (i.e.
Monte-Carlo sampling) by introducing new analytical developments for lines
diagnosis. We consider 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 4-term correlation of Fourier
coefficients, we derive generic expressions for the sample mean and variance of
line centroid shift, line broadening and projected velocity structure function.
We perform a numerical validation based on Monte-Carlo simulations for two
popular models of gas emissivity based on the beta-model. 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 and XRISM/Resolve instruments forecasts the potential of
sensitive, spatially-resolved spectroscopy to probe the inertial range of
turbulent velocity cascades in a Coma-like galaxy cluster.
