Symmetry in halo displays and symmetry in halo-making crystals

The relation between the symmetry in halo displays and crystal symmetry is investigated for halo displays that are generated by ensembles of crystals. It is found that, regardless of the symmetry of the constituent crystals, such displays are always left-right symmetric if the crystals are formed from the surrounding vapor. Left-right symmetry of a halo display implies here that the cross sections for formation of a halo arc on the left-hand side of the solar vertical and its right-hand side mirror image are equal. This property leaves room for two types of halo displays only: a full symmetric one (mmm-symmetric) and a partial symmetric one (mm2-symmetric) in which halo constituents lack their counterparts on the other side of the parhelic circle. Partial symmetric halo displays can only occur for point halos. Its occurrence implies that certain symmetry elements are not present in the shape of the halo-making crystals. These elements are: a center of inversion, any rotatory inversion axis that is parallel to the crystal spin axis P, a mirror plane perpendicular to the P-axis, a 2-fold rotation axis perpendicular to the P-axis. A simple conceptual method is presented to reconstruct possible shapes of the halo-generating crystals from the halos in the display. Halos that may occur on the Saturnian satellite Titan are discussed. The possibilities for the Huygens probe to detect these halos during its descent in 2005 through the Titan clouds are detailed.