The distribution of two-dimensional eccentricity of Sunyaev-Zeldovich Effect and X-ray surface brightness profiles

With the triaxial density profile of dark matter halos and the corresponding equilibrium gas distribution, we derive two-dimensional Sunyaev-Zeldovich (SZ) effect and X-ray surface brightness profiles for clusters of galaxies. It is found that the contour map of these observables can be well approximated by a series of concentric ellipses with scale-dependent eccentricities. The statistical distribution of their eccentricities (or equivalently axial ratios) is analyzed by taking into account the orientation of clusters with respect to the line of sight and the distribution of the axial ratios and the concentration 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, the deviation 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 to observations, we further study the axial-ratio distribution for mass-limited cluster samples with the number distribution of clusters at different redshifts described 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 $<\eta > \sim 0.84$ with a tail extended to $\eta \sim 0.6$. With fast advance of high quality imaging observations of both SZ effect and X-ray emissions, our analyses provide a useful way to probe cluster halo profiles and therefore to test theoretical halo-formation models.