With the R$^{3}$B-LAND setup at GSI we have measured exclusive relative-energy spectra of the Coulomb dissociation of $^{18}$C at a projectile energy around 425~AMeV on a lead target, which are needed to determine the radiative neutron-capture cross sections of $^{17}$C into the ground state of $^{18}$C. Those data have been used to constrain theoretical calculations for transitions populating excited states in $^{18}$C. This allowed to derive the astrophysical cross section $sigma^{*}_{mathrm{n}gamma}$ accounting for the thermal population of $^{17}$C target states in astrophysical scenarios. The experimentally verified capture rate is significantly lower than those of previously obtained Hauser-Feshbach estimations at temperatures $T_{9}leq{}1$~GK. Network simulations with updated neutron-capture rates and hydrodynamics according to the neutrino-driven wind model as well as the neutron-star merger scenario reveal no pronounced influence of neutron capture of $^{17}$C on the production of second- and third-peak elements in contrast to earlier sensitivity studies.
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