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

Gábor Horváth,Szilvia Szilasi,András Barta,Péter Takács,Dénes Száz,Balázs Kretzer,Alexandra Farkas

doi:10.1098/rsos.160688

Abstract

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° 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 θ and per cent cloud cover ρ. 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 θ ≈ 0° and under totally or nearly clear skies with cloud coverage ρ = 0%, when the sun's position is already easily determined. If the sun is below the horizon (−5° ≤ θ < 0°) during twilight, P = 76.17 ± 4.18% for 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.

Highlights

If the human eye is stimulated by linearly polarized light, two 8-shaped bowtie-like figures are perceived [1,2]: (i) blue brushes parallel to the direction of polarization of the stimulus and (ii) yellow brushes perpendicular to the direction of polarization
If somebody looks at the clear blue sky from which light with high enough degree of polarization d originates, the yellow Haidinger’s brushes can be perceived, while the blue brushes are practically invisible, because they fade into the blue sky
We calculated the 12 × 9-size sun elevation θ–cloud cover ρ matrices of the average proportion of the sky usable for navigation with Haidinger’s brushes for the selected 1296 meteorological situations clear sky colour picture A

Summary

Introduction

If the human eye is stimulated by linearly polarized light, two 8-shaped bowtie-like figures are perceived [1,2]: (i) blue brushes parallel to the direction of polarization of the stimulus and (ii) yellow brushes perpendicular to the direction of polarization (figure 1a) These two 8-shaped figures combine to form a Maltese cross shape. The blue brushes could be perceived only if the sky is not blue, i.e. around sunset and sunrise when large portions of the sky are red and orange, or when the sky is covered by grey or white clouds or fog In the latter cases the d of skylight can be so low [8,9,10,11] that it falls below the perception threshold d* of Haidinger’s brushes. The brushes have the best contrast when d approaches 100%, for instance looking at a white area on a liquid crystal computer monitor, which employs a linear polarizer as part of the image-forming technology [2,12]

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