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