The formation of very narrow waist bipolar planetary nebulae

We discuss the interaction of the slow wind blown by an asymptotic giant branch (AGB) star with a collimated fast wind (CFW) blown by its main-sequence or white dwarf companion, at orbital separations in the range of several AU less than or similar to a less than or similar to 200 AU. The CFW results from accretion of the AGE wind into an accretion disk around the companion. The fast wind is collimated by the accretion disk. We argue that such systems are the progenitors of bipolar planetary nebulae and bipolar symbiotic nebulae with a very narrow equatorial waist between the two polar lobes. The CFW wind will form two lobes along the symmetry axis and will further compress the slow wind near the equatorial plane, leading to the formation of a dense slowly expanding ring. Therefore, contrary to the common claim that a dense equatorial ring collimates the bipolar flow, we argue that in the progenitors of very narrow waist bipolar planetary nebulae, the CFW, through its interaction with the slow wind, forms the dense equatorial ring. Only later in the evolution, and after the CFW and slow wind cease, does the mass-losing star leave the AGE and blow a second, more spherical, fast wind. At this stage the how structure becomes the one that is commonly assumed for bipolar planetary nebulae, i.e., collimation of the fast wind by the dense equatorial material. However, this results in the broadening of the waist in the equatorial plane and cannot by itself account for the presence of very narrow waists or jets. We conduct a population synthesis study of the formation of planetary nebulae in wide binary systems which quantitatively supports the proposed model. The population synthesis code follows the evolution of both stars and their arbitrarily eccentric orbit, including mass loss via stellar winds, for 5 x 10(4) primordial binaries. We show the number of expected systems that blow a CFW is in accord with the number found from observations, to within the many uncertainties involved. Overall, we find that similar to 5% of all planetary nebulae are bipolars with very narrow waists. Our population synthesis not only supports the CFW model but more generally supports the binary model for the formation of bipolar planetary nebulae.