Constraints on the asymptotic baryon fractions of galaxy clusters at large radii

While X‐ray measurements have so far revealed an increase in the volume‐averaged baryon fractions f b ( r ) of galaxy clusters with cluster radii r , f b ( r ) should asymptotically reach a universal value f b (∞)= f b , provided that clusters are representative of the Universe. In the framework of hydrostatic equilibrium for intracluster gas, we have derived the necessary conditions for f b (∞)= f b . The X‐ray surface brightness profile described by the β model and the temperature profile approximated by the polytropic model should satisfy γ ≈2(1−1/3 β ) and γ ≈1+1/3 β for β <1 and β >1, respectively, which sets a stringent limit to the polytropic index: γ <4/3. In particular, a mildly increasing temperature with radius is required if the observationally fitted β parameter is in the range 1/3< β <2/3. It is likely that a reliable determination of the universal baryon fraction can be achieved in the small β clusters because the disagreement between the exact and asymptotic baryon fractions for clusters with β >2/3 breaks down at rather large radii (≳30 r c ) where hydrostatic equilibrium has probably become inapplicable. We further explore how to obtain the asymptotic value f b (∞) of the baryon fraction from the X‐ray measurement made primarily over the finite central region of a cluster. We demonstrate our method using a sample of 19 strong lensing clusters, which enables us to place a useful constraint on f b (∞): 0.094±0.035≤ f b (∞)≤0.41±0.18, corresponding to a cosmological density parameter 0.122±0.069≤Ω M ≤0.53±0.28 for H 0 =50 km s −1  Mpc −1 . An optimal estimate of f b (∞) based on three cooling flow clusters with β <1/2 in our lensing cluster sample yields 〈 f b (∞)〉=0.142±0.007 or Ω M =0.35±0.09.