Bubbles in Planetary Nebulae and Clusters of Galaxies: Precessing Jets

I derive the condition for narrow jets with varying axis, e.g., precessing jets, to inflate more or less spherical (fat) bubbles in planetary nebulae and clusters of galaxies. This work follows a previous work dealing with wide jets, i.e., having a wide opening angle. The expressions derive here are qualitatively and quantitatively similar to the conditions for inflating fat bubbles by non-precessing wide jets. This follows the similar physical cause of inflating fat bubbles, which is that the jet deposits energy inside the bubble. Fat bubbles in planetary nebulae (and similar stellar systems) and in clusters of galaxies, are likely to be formed by wide jets, precessing jets, or other jets whose axis is not constant relative to the medium they expand into.