The Distribution of Two‐dimensional Eccentricity of Sunyaev‐Zel'dovich Effect and X‐Ray Surface Brightness Profiles

With the triaxial density profile of dark matter halos and the correspondingequilibrium gas distribution, we derive two-dimensional Sunyaev-Zeldovich (SZ)effect and X-ray surface brightness profiles for clusters of galaxies. It isfound that the contour map of these observables can be well approximated by aseries of concentric ellipses with scale-dependent eccentricities. Thestatistical distribution of their eccentricities (or equivalently axial ratios)is analyzed by taking into account the orientation of clusters with respect tothe line of sight and the distribution of the axial ratios and theconcentration parameters of dark matter halos. For clusters of mass$10^{13}h^{-1}{M}_{\odot}$ at redshift $z=0$, the axial ratio is peaked at$\eta \sim 0.9$ for both SZ and X-ray profiles. For larger clusters, thedeviation from circular distributions is more apparent, with $\eta$ peaked at$\eta \sim 0.85$ for $M=10^{15}h^{-1}{M}_{\odot}$. To be more close toobservations, we further study the axial-ratio distribution for mass-limitedcluster samples with the number distribution of clusters at different redshiftsdescribed by a modified Press-Schechter model. For a mass limit of value$M_{lim}=10^{14}h^{-1}{M}_{\odot}$, the average axial ratio is $<\eta > \sim0.84$ with a tail extended to $\eta \sim 0.6$. With fast advance of highquality imaging observations of both SZ effect and X-ray emissions, ouranalyses provide a useful way to probe cluster halo profiles and therefore totest theoretical halo-formation models.Comment: 28 pages, 6 figures. Accepted for publication in the Astrophysical Journa