We study the effect of the instantaneous gas expulsion on star clusters wherein the residual gas has a density profile shallower than that of the embedded cluster. This is expected if star formation proceeds with a given SFE per free-fall time in a centrally-concentrated molecular clump. We perform direct N-body simulations whose initial conditions are generated by the program mkhalo falcON adapted for our models. Our model clusters initially have a Plummer profile and are in virial equilibrium with the gravitational potential of the cluster-forming clump. The residual gas contribution is computed based on the model of Parmentier&Pfalzner(2013). Our simulations include mass loss by stellar evolution and the tidal field of the Galaxy. We find that a star cluster with a minimum global SFE of 15% is able to survive instantaneous gas expulsion and to produce a bound cluster. Its violent relaxation lasts no longer than 20 Myr, independently of its global SFE and initial stellar mass. At the end of violent relaxation the bound fractions of surviving clusters with the same global SFEs are similar regardless of their initial stellar mass. Their subsequent lifetime in the gravitational field of the Galaxy depends on their bound stellar masses. We therefore conclude that the critical SFE needed to produce a bound cluster is 15%, which is twice smaller than earlier estimates of 33%. Thus we have improved the survival likelihood of young clusters after instantaneous gas expulsion. Those can now survive instantaneous gas expulsion with global SFEs as low as those observed for embedded clusters of Solar Neighbourhood (15-30%). This is the consequence of the star cluster having a density profile steeper than that of the residual gas. However, in terms of the effective SFE, measured by the virial ratio of the cluster at gas expulsion, our results are in agreement with previous studies.
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