The distribution of misalignment angles in multipolar planetary nebulae

We measure the projected angle on the plane of the sky between adjacent symmetry axes of tens of multipolar planetary nebulae and find that the distribution of these misalignment angles implies a random three-dimensional angle distribution limited to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mo>≲</mml:mo> <mml:msup> <mml:mn>60</mml:mn> <mml:mo>∘</mml:mo> </mml:msup> </mml:mrow> </mml:math> . We identify a symmetry axis as a line connecting two opposite lobes (bubbles) or clumps. We build a cumulative distribution function of the projected angles <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>α</mml:mi> </mml:math> and find that an entirely random distribution of the three-dimensional angles <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>δ</mml:mi> </mml:math> between adjacent symmetry axes, namely, uncorrelated directions, does not fit the observed one. A good fit to the observed distribution is a limited random distribution of the three-dimensional angle between adjacent symmetry axes, i.e., random distribution in the range of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msup> <mml:mn>20</mml:mn> <mml:mo>∘</mml:mo> </mml:msup> <mml:mo>≲</mml:mo> <mml:mi>δ</mml:mi> <mml:mo>≲</mml:mo> <mml:msup> <mml:mn>60</mml:mn> <mml:mo>∘</mml:mo> </mml:msup> </mml:mrow> </mml:math> . We assume that a pair of jets along the angular momentum axis of an accretion disk around the companion shape each symmetry axis. The limited random distribution might result from two sources of angular momentum to the accretion disks with comparable magnitude: one with a fixed direction and one with a stochastic direction variation. We discuss a scenario where the fixed-axis angular momentum source is the binary orbital angular momentum, while the stochastic source of angular momentum is due to the vigorous envelope convection of the mass-losing giant progenitor.