Density profiles at the time t = 0.118 s for three cases of S = 1,  1.4, and 2 related to the images at the top, middle, and bottom, respectively. These snapshots illustrate the density structure of the envelope when the ring-like gravitational instability becomes possible. However, the case with S = 2, i.e. the last profile, shows no axisymmetric growing mode. Models shown from top to bottom are A05-S10-SG-R12, A05-S14-SG-R10, and A05-S20-SG-R12, as listed in Table A.1.