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 either ellipsoidal generalizations of the Navarro-Frenk-White (NFW) profile or a nearly scale-free logarithmic potential. Although the mean biases are small and mostly <1 per cent, 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 similar to 10 per cent for YSZ), and the emission-weighted temperature (T-X) has the smallest (sigma less than or similar to 0.5 per cent). 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 similar to 8 per cent) in the correlation between Y-SZ and Y-X in comparison to the small mean bias (sigma <= 1 per cent) 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 approximate to 0.5 per cent 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.