Resolving the beta-discrepancy for clusters of galaxies

Previous comparisons of optical and X-ray observations of clusters of galaxies have led to the so-called β-discrepancy that has persisted for the last decade. The standard hydrostatic-isothermal model for clusters predicts that the parameter β_spec = σ_r²(kT/μm_p), which describes the ratio of energy per unit mass in galaxies to that in the gas, should equal the parameter β_fit, (where p_gas(r) ∝ ρ_gal(r)^βfit) determined from the X-ray surface brightness distribution. The observations suggest an apparent discrepancy: β_spec ∼ 1.2 (i.e., the galaxies are "hotter" than the gas), while β_fit ∼ 0.65 (i.e., the gas is "hotter" and more extended than the galaxies). Here we show that the discrepancy is resolved when the actual observed galaxy distribution in clusters is used, ρ_gal(r)∝ r ^-2.4±0.2 instead of the previously assumed steeper King approximation, ρ_gal(r) ∝ r^-3. Using the correct galaxy profile in clusters, we show that the standard hydrostatic-isothermal model predicts β_spec = β_fit^c ≃ (1.25 ± 0.1)β_fit, rather than β_spec ≃ β_fit (where β_fit is the standard parameter using the King approximation, and β_fit^c is the corrected parameter using the proper galaxy distribution). Using a large sample of clusters, we find best-fit mean values of β_spec = 0.94 ± 0.08 and β_fit^c = 1.25β_fit = 0.84 ± 0.1. These results resolve the β-discrepancy and provide additional support for the hydrostatic cluster model.