Analytical beginnings

Balick (1987, 1988) suggested that the interacting-winds model for planetary nebulae, if generalized to two dimensions, might explain the morphologies of nearly all PNs. He supposed that the fossil red giant envelope (RGE) is cylindrically symmetric and that the density is higher at the equator than at the poles. The PN morphology and its evolution is then purely a consequence of the mass distribution in the RGE and the properties of the fast stellar wind.

Analytical models for aspherical PNs have been quite successful in describing the propagation of the outer shock of a two-wind configuration. One uses either a snowplow-type approximation (Kahn & West 1985, Soker & Livio 1989) or a generalization of the work of Kompaneets (1960; Balick et al. 1987, Icke 1988, Icke et al. 1989), in which the wind is supposed to generate a uniform pressure inside the expanding bubble. The method is easy to apply and shows the generic features of the outer shocks clearly. It works as follows.