solid angle of the cluster is an observable quantity that is often used for this purpose, $$Y = \int{y(\Omega)\rmn{d}\Omega},$$ where $Y$ is calculated within a given projected radius. However, for the purposes of this work, we choose to use an intrinsic measure of $Y$ that is independent of the distance to the cluster, and hence will allow comparison of clusters at different redshifts. We therefore assume spherical symmetry and use the following definition $$Y\,D_\rmn{A}^{2} = 4\pi\int{{P_\rmn{e}\sigma_\rmn{T}\over m_\rmn{e}c^{2}}r^{2}\rmn{d}r},$$ where $D_\rmn{A}$ is the angular diameter distance to the cluster and $Y\,D_\rmn{A}^{2}$ is calculated over the cluster volume, enclosed by a physical radius typically equal to the virial radius.

The observed continuum X-ray surface brightness from galaxy clusters is caused primarily by bremsstrahlung radiation from scattered electrons in the hot ICM, with some emission also from high energy transition lines. The continuum X-ray emission is proportional to the projected square of the electron density, multiplied by the emissivity along the line-of-sight. If all of the emission were completely dominated by bremsstrahlung radiation then the emissivity would go as the square root of the temperature. While in practise this is not the complete case, in general the emissivity is a weak function of the 