The prediction of stellar ($gamma$,$alpha$) reaction rates for heavy nuclei is based on the calculation of ($alpha$,$gamma$) cross sections at sub-Coulomb energies. These rates are essential for modeling the nucleosynthesis of so-called $p$-nuclei. The standard calculations in the statistical model show a dramatic sensitivity to the chosen $alpha$-nucleus potential. The present study explains the reason for this dramatic sensitivity which results from the tail of the imaginary $alpha$-nucleus potential in the underlying optical model calculation of the total reaction cross section. As an alternative to the optical model, a simple barrier transmission model is suggested. It is shown that this simple model in combination with a well-chosen $alpha$-nucleus potential is able to predict total $alpha$-induced reaction cross sections for a wide range of heavy target nuclei above $A gtrsim 150$ with uncertainties below a factor of two. The new predictions from the simple model do not require any adjustment of parameters to experimental reaction cross sections whereas in previous statistical model calculations all predictions remained very uncertain because the parameters of the $alpha$-nucleus potential had to be adjusted to experimental data. The new model allows to predict the reaction rate of the astrophysically important $^{176}$W($alpha$,$gamma$)$^{180}$Os reaction with reduced uncertainties, leading to a significantly lower reaction rate at low temperatures. The new approach could also be validated for a broad range of target nuclei from $A approx 60$ up to $A gtrsim 200$.
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