Spherically Averaging Ellipsoidal Galaxy Clusters in X-Ray and
  Sunyaev-Zel'dovich Studies: II. Biases
  This is the second of two papers investigating the spherical averaging of
ellipsoidal galaxy clusters in the context of X-ray and Sunyaev-Zel'dovich (SZ)
observations. In the present study we quantify the orientation-average bias and
scatter in observables that result from spherically averaging clusters
described by ellipsoidal generalizations of the NFW profile or a nearly
scale-free logarithmic potential. Although the mean biases are small and mostly
<1%, the flattest cluster models generally have a significant mean bias; i.e.,
averaging over all orientations does not always eliminate projection biases.
Substantial biases can result from different viewing orientations, where the
integrated Compton-y parameter (Y_SZ) and the concentration have the largest
scatter (as large as sigma ~10% for Y_SZ), and the emission-weighted
temperature (T_X) has the smallest (sigma < ~0.5%). The very small scatter for
T_X leads to Y_X and M_gas having virtually the same orientation biases.
Substantial scatter is expected for individual clusters (up to sigma ~8%) in
the correlation between Y_SZ and Y_X in comparison to the small mean bias
(sigma < ~1%) applicable to a random sample of clusters of sufficient size. For
ellipsoidal NFW models we show that the orientation bias for the total cluster
mass attains a minimum near the radius r_2500 so that the spherically averaged
mass computed at this radius is always within ~0.5% of the true value for any
orientation. Finally, to facilitate the accounting for orientation bias in
X-ray and SZ cluster studies, we provide cubic polynomial approximations to the
mean orientation bias and 1-sigma scatter for each cluster observable as a
function of axial ratio for the ellipsoidal NFW models.
