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 $f_b(\infty)=f_b$, provided that clusters are representative of the Universe. In the framework of hydrostatic equilibrium for intracluster gas, we have derived the necessary conditions for $f_b(\infty)=f_b$: The X-ray surface brightness profile described by the $\beta$ model and the temperature profile approximated by the polytropic model should satisfy $\gamma\approx2(1-1/3\beta)$ and $\gamma\approx1+1/3\beta$ for $\beta1$, respectively, which sets a stringent limit to the polytropic index: $\gamma<4/3$. In particular, a mildly increasing temperature with radius is required if the observationally fitted $\beta$ parameter is in the range $1/3<\beta<2/3$. It is likely that a reliable determination of the universal baryon fraction can be achieved in the small $\beta$ clusters because the disagreement between the exact and asymptotic baryon fractions for clusters with $\beta>2/3$ breaks down at rather large radii (\ga30r_c) where hydrostatic equilibrium has probably become inapplicable. We further explore how to obtain the asymptotic value $f_b(\infty)$ of 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 $f_b(\infty)$: $0.094\pm0.035 \leq f_b(\infty) \leq 0.41\pm0.18$. An optimal estimate of $f_b(\infty)$ based on three cooling flow clusters with $\beta = 0.142\pm0.007$ or $\Omega_M = 0.35\pm0.09$.Comment: 6 pages + 4 figures; accepted for publication in MNRA

The silicate model and carbon rich model of CoRoT-7b, Kepler-9d and Kepler-10b

Possible bulk compositions of the super-Earth exoplanets, CoRoT-7b, Kepler-9d, and Kepler-10b are investigated by applying a commonly used silicate and a non-standard carbon model. Their internal structures are deduced using the suitable equation of state of the materials. The degeneracy problems of their compositions can be partly overcome, based on the fact that all three planets are extremely close to their host stars. By analyzing the numerical results, we conclude: 1) The iron core of CoRoT-7b is not more than 27% of its total mass within 1 $\sigma$ mass-radius error bars, so an Earth-like composition is less likely, but its carbon rich model can be compatible with an Earth-like core/mantle mass fraction; 2) Kepler-10b is more likely with a Mercury-like composition, its old age implies that its high iron content may be a result of strong solar wind or giant impact; 3) the transiting-only super-Earth Kepler-9d is also discussed. Combining its possible composition with the formation theory, we can place some constraints on its mass and bulk composition.Comment: 20 pages, 8figures, accepted for publication in RAA. arXiv admin note: text overlap with arXiv:0707.289