Spherically Averaging Ellipsoidal Galaxy Clusters in X-Ray and Sunyaev-Zel'dovich Studies: I. Analytical Relations

TL;DR

This paper derives analytical formulas to understand how assuming spherical symmetry affects measurements of ellipsoidal galaxy clusters in X-ray and SZ studies, revealing biases and providing tools for more accurate cluster analysis.

Contribution

It introduces analytical relations for triaxial ellipsoidal galaxy clusters, quantifies biases from spherical averaging, and generalizes the onion peeling method for ellipsoidal deprojection.

Findings

Mass bias from spherical averaging is shape and orientation independent for certain potentials.

The ratio of SZ to X-ray pressure depends only on shape, orientation, and H_0.

Y_SZ and Y_X biases differ, causing offsets in their relation.

Abstract

This is the first of two papers investigating the deprojection and spherical averaging of ellipsoidal galaxy clusters. We specifically consider applications to hydrostatic X-ray and Sunyaev-Zel'dovich (SZ) studies, though many of the results also apply to isotropic dispersion-supported stellar dynamical systems. Here we present analytical formulas for galaxy clusters described by a gravitational potential that is a triaxial ellipsoid of constant shape and orientation. For this model type we show that the mass bias due to spherically averaging X-ray observations is independent of the temperature profile, and for the special case of a scale-free logarithmic potential, there is exactly zero mass bias for any shape, orientation, and temperature profile. The ratio of spherically averaged intracluster medium (ICM) pressures obtained from SZ and X-ray measurements depends only on the ICM intrinsic shape, projection orientation, and H_0, which provides another illustration of how cluster geometry can be recovered through a combination of X-ray and SZ measurements. We also demonstrate that Y_SZ and Y_X have different biases owing to spherical averaging, which leads to an offset in the spherically averaged Y_SZ - Y_X relation. A potentially useful application of the analytical formulas presented is to assess the error range of an observable (e.g., mass, Y_SZ) accounting for deviations from assumed spherical symmetry, without having to perform the ellipsoidal deprojection explicitly. Finally, for dedicated ellipsoidal studies, we also generalize the spherical onion peeling method to the triaxial case for a given shape and orientation.