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 σ∼ 10 per cent for Y_SZ), and the emission-weighted temperature (T_X) has the smallest (σ≲ 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 σ∼ 8 per cent) in the correlation between Y_SZ and Y_X in comparison to the small mean bias (σ≲ 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 r₂₅₀₀ so that the spherically averaged mass computed at this radius is always within ≈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 1σ scatter for each cluster observable as a function of axial ratio for the ellipsoidal NFW models.