Celestial polarization patterns sufficient for Viking navigation with the naked eye: detectability of Haidinger's brushes on the sky versus meteorological conditions

If a human looks at the clear blue sky from which light with high enough degree of polarization d originates, an 8-shaped bowtie-like figure, the yellow Haidinger's brush can be perceived, the long axis of which points towards the sun. A band of high d arcs across the sky at 90 degrees from the sun. A person can pick two points on that band, observe the yellow brushes and triangulate the position of the sun based on the orientation of the two observed brushes. This method has been suggested to have been used on the open sea by Viking navigators to determine the position of the invisible sun occluded by cloud or fog. Furthermore, Haidinger's brushes can also be used to locate the sun when it is below the horizon or occluded by objects on the horizon. To determine the position of the sun using the celestial polarization pattern, the d of the portion of the sky used must be greater than the viewer's degree of polarization threshold d* for perception of Haidinger's brushes. We studied under which sky conditions the prerequisite d > d* is satisfied. Using full-sky imaging polarimetry, we measured the d-pattern of skylight in the blue (450 nm) spectral range for 1296 different meteorological conditions with different solar elevation angles. and per cent cloud cover rho. From the measured d-patterns of a given sky we determined the proportion P of the sky for which d > d*. We obtained that P is the largest at low solar elevations. theta approximate to 0 degrees and under totally or nearly clear skies with cloud coverage rho = 0%, when the sun's position is already easily determined. If the sun is below the horizon (-5 degrees <= theta <= 0 degrees) during twilight, P = 76.17 +/- 4.18% for d*(min) = 23% under clear sky conditions. Consequently, the sky-polarimetric Viking navigation based on Haidinger's brushes is most useful after sunset and prior to sunrise, when the sun is not visible and large sky regions are bright, clear and polarized enough for perception of Haidinger's brushes.