  In the supercritical range of the polytropic indices $\gamma\in(1,\frac43)$
we show the existence of smooth radially symmetric self-similar solutions to
the gravitational Euler-Poisson system. These solutions exhibit gravitational
collapse in the sense that the density blows-up in finite time. Some of these
solutions were numerically found by Yahil in 1983 and they can be thought of as
polytropic analogues of the Larson-Penston collapsing solutions in the
isothermal case $\gamma=1$. They each contain a sonic point, which leads to
numerous mathematical difficulties in the existence proof.
