Accepted for publication in The Astrophysical Journal
We have modelled planetary nebulae (PNe) in the context of the interacting stellar winds model. If the two interacting winds have constant properties, the velocity of the PN shell tends towards a constant with time and the shape becomes self-similar. Additionally, if the velocity of the fast wind is much higher than the expansion velocity of the shell, the interior of the hot shocked bubble becomes isobaric. Using semi-analytical methods, complemented by hydrodynamic simulations, we have calculated the shapes of PNe in the self-similar stage. An asymmetric density profile is assumed for the slow outer wind. The asymmetry is modelled using different functions, which depend on the degree of asymmetry and the steepness of the density profile in the angular direction. We include the effects of the ambient wind velocity, which has not received much attention since the work of Kahn \& West (1985). The fact that typical PNe velocities (10-40 ${\rm km}\, {\rm s}^{-1}$) are only marginally greater than typical red giant wind velocities (5-20 ${\rm km}\, {\rm s}^{-1}$) indicates that this is an important parameter. The morphological appearance is a consequence of the density contrast, steepness of the density profile and velocity of the ambient medium; classification of PNe purely on the basis of the first two factors may be misleading. Moderate values of the density contrast result in a cusp at the equator. A higher density contrast coupled with a low velocity for the external medium gives rise to extremely bipolar nebulae. For large density contrasts and a significant value of the slow wind velocity, the surface density maximum of the shell shifts away from the equator, giving rise to peanut-shaped structures with pronounced equatorial bulges. If the external wind velocity is small compared to the expansion velocity of the nebula, the PNe tend to be more bipolar, even with a moderate density contrast. If the PN velocity is close to that of the external wind, the shape is relatively spherical. However, a velocity asymmetry in the external wind can lead to a bipolar shape if the equatorial velocity is sufficiently low. Our numerical simulations show that asymmetric PN shells are corrugated because of Kelvin-Helmholtz instabilities. They also indicate that several doubling times are needed to approach the self-similar state. A ratio of interior sound speed to shell velocity $\gtrsim 10$ is found to yield nebulae whose shapes match those given by the isobaric approximation.
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