# Towards mapping turbulence in the intra-cluster medium -- I. Sample variance in spatially-resolved X-ray line diagnostics

## TL;DR

This paper develops analytical methods to quantify sample variance in X-ray line diagnostics of turbulence in galaxy clusters, enhancing interpretation of high-resolution spectral data.

## Contribution

It introduces new analytical formulas for sample mean and variance of turbulence diagnostics, validated against Monte-Carlo simulations, for use in spatially-resolved X-ray spectroscopy.

## Key findings

- Analytical expressions match Monte-Carlo simulations within applicable ranges.
- Forecasts demonstrate potential of future instruments to probe turbulence scales.
- Validated formulas for line centroid shift, broadening, and structure function variance.

## Abstract

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.