Galaxy clusters, type Ia supernovae and the fine structure constant
  As is well known, measurements of the Sunyaev-Zeldovich effect can be
combined with observations of the X-ray surface brightness of galaxy clusters
to estimate the angular diameter distance to these structures. In this paper,
we show that this technique depends on the fine structure constant, $\alpha$.
Therefore, if $\alpha$ is a time-dependent quantity, e.g., $\alpha=\alpha_0
\phi(z)$, where $\phi$ is a function of redshift, we argue that current data do
not provide the real angular diameter distance, $D_{\rm{A}}(z)$, to the cluster
but instead $D_A^{data}(z) = \phi(z)^2 D_{\rm{A}}(z)$. We use this result to
derive constraints on a possible variation of $\alpha$ for a class of dilaton
runaway models considering a sample of 25 measurements of $D_A^{data}(z)$ in
redshift range $0.023 < z < 0.784$ and estimates of $D_{\rm{A}}(z)$ from
current type Ia supernovae observations. We find no significant indication of
variation of $\alpha$ with the present data.
