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° 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 [Formula: see text] 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.

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. These are the so-called Haidinger's brushes, the angular extension of which is about 5° centred at the fovea of the human retina, and are mediated by dichroic carotenoids in the macula lutea [1-7]. If somebody looks at the clear blue sky from which light with high enough degree of polarization d originates (for example, at 90° from the sun, where d is maximal and can reach 75% [2,8]), the yellow Haidinger's brushes can be perceived, while the blue brushes are practically invisible, because they fade into the blue sky (figure 1a). 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. However, in the latter cases the d of skylight can be so low [8-11] 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].

Figure 1.

Principle of the sky-polarimetric Viking navigation on the basis of Haidinger's brushes. (a) Clear sky. (b) Cloudy sky. Yellow bars: local direction of polarization of skylight. Red double-headed arrow: axis of mirror symmetry of the yellow Haidinger's brush being perpendicular to the direction of polarization. The long axis of the blue Haidinger's brush is parallel to the direction of polarization. Owing to Rayleigh scattering of sunlight in the atmosphere, the celestial great circle passing through the observed sky point parallel to the yellow Haidinger's brush crosses the sun.

Recently, Temple et al. [13] measured the degree of polarization threshold d* at which humans could just detect Haidinger's brushes. They presented gratings in polarization-only contrast at varying degrees of polarization d in order to measure the lower limits of human polarized light detection. Participants were, on average, able to perform the task down to a threshold of , and there was a normal continuous distribution of threshold values extending down to as low as and up to as high as [13].

Owing to the first-order Rayleigh scattering, at first approximation, the direction of polarization of skylight is perpendicular to the scattering plane determined by the relative position of the observer, the sun and the celestial point observed (figure 1). Using full-sky imaging polarimetry, Suhai & Horváth [14] and Hegedüs et al. [9,10] showed that depending on the cloud conditions, a considerable part of the sky is characterized by directions of polarization described well by the Rayleigh model. Moreover, those sky areas where the Rayleigh model can be considered as a good approximation of the real direction of skylight polarization have the highest d-values. Thus, if a human looks at the sky and perceives the yellow Haidinger's brush, the celestial great circle passing through the observed sky point parallel to this yellow brush crosses the sun (figure 1). Consequently, if two sky points are selected with high enough d-values to perceive Haidinger's brushes, the sun position can be determined, which is at the intersection of these two great circles (figure 1).

Ropars et al. [15,16] suggested that, during their voyages on the open sea, Viking navigators might have been able to determine the position of the invisible sun occluded by cloud or fog in this way (figure 1b). The determination of the position of the occluded sun is an essential part of the famous sky-polarimetric Viking navigation hypothesized, cited and disputed by many researchers [10,11,14-38].

The navigation method with Haidinger's brushes is a sound idea and has the advantage that Viking navigators did not need any additional optical instrument (e.g. calcite, cordierite or tourmaline sunstones) to analyse the sky polarization, because due to the perceived yellow Haidinger's brush they had their own analyser embedded in their eyes (figure 1). The weakness of this hypothesis is that the Haidinger's brushes are very faint and fade quickly within a few seconds due to visual adaptation. But they can be perceived again for some seconds if the fovea of the eye is restimulated by polarized skylight after winking or moving or rotating the position or orientation of gaze in the sky.

The atmospheric optical prerequisite of this Viking navigation hypothesis based on the observation of Haidinger's brushes [15,16] is that the degree of polarization d of skylight stimulating the human eye should be higher than the perception threshold d* of Haidinger's brushes. The aim of this work was to study 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 in Hungary (at latitude 47°15′29.83′′ N). From the measured d-patterns of a given sky we determined the proportion P of the sky for which d > d*, that is for which the Haidinger's brushes can be perceived. The entoptic phenomenon of Haidinger's brushes is mediated by the absorption of short-wavelength light by the macular carotenoids (lutein, zeaxanthin and meso-zeaxanthin) and is not visible at wavelengths above approximately 500 nm [39]. Therefore, our study was restricted to the blue colour band because only this is useful for perception of Haidinger's brushes.

Material and methods

Full-sky imaging polarimetry and selection of different skies

The patterns of the degree of polarization d of skylight were measured by imaging polarimetry, the method of which was described in detail by Barta et al. [24]. Data on sky polarization have been collected with an automatic full-sky imaging polarimeter set-up in the Gothard Astronomical Observatory of the Eötvös University, Szombathely, Hungary (47°15′29.83′′ N, 16°36′15.67′′ E). In the past 3 years this polarimeter functioned continuously and measured several tens of thousands of sky polarization patterns. We grouped these skies on the basis of the following two parameters: (i) the solar elevation angle θ above the horizon ranged from −5° to +55°. This range of θ was divided into 12 equal intervals with an increment of 5° as follows: −5° ≤ θ1 < 0°, 0° ≤ θ2 < 5°, 5° ≤ θ3 < 10°, 10° ≤ θ4 < 15°, 15° ≤ θ5 < 20°, 20° ≤ θ6 < 25°, 25° ≤ θ7 < 30°, 30° ≤ θ8 < 35°, 35° ≤ θ9 < 40°, 40° ≤ θ10 < 45°, 45° ≤ θ11 < 50°, 50° ≤ θ12 ≤ 55°. (ii) The per cent cloud cover ρ (% of the full sky covered by clouds) was determined with the use of the cloud detection algorithm kNN (k-nearest neighbour) described by Barta et al. [40]. The range 0% ≤ ρ ≤ 100% was divided into nine categories, as in meteorology oktas (or eighths) are common units of cloud coverage of the visible sky estimated by visual examination [41]. Oktas 0–8 refer to ever-increasing cloud coverage based on the division of eight equal intervals [42]: 0 okta refers to the totally clear sky (ρ0 = 0%); there is a category for few clouds (1–2 oktas), another for scattered clouds (3–4 oktas), for broken clouds (5–7 oktas) and the last one for an overcast sky (8 oktas). Based on this generally accepted method, our first category corresponds to the totally clear sky with ρ0 = 0%, and the further categories are composed of eight equal intervals with an increment of 12.5% in the following way: 0% < ρ1 < 12.5%, 12.5% ≤ ρ2 < 25%, 25% ≤ ρ3 < 37.5%, 37.5% ≤ ρ4 < 50%, 50% ≤ ρ5 < 62.5%, 62.5% ≤ ρ6 < 75%, 75% ≤ ρ7 < 87.5%, 87.5% ≤ ρ8 ≤ 100% (nearly and totally overcast sky). For example, if the cloud coverage is 3 oktas, this means that approximately 3/8 part of the visible sky is covered by clouds (in our measurement, it falls into the range of 25% ≤ ρ3 < 37.5%). It is not worth separating the totally overcast (ρ = 100%) case, because in our measurement all the sky conditions in the 8 okta range (87.5% ≤ ρ8 ≤ 100%) seemed totally overcast, which could not be separated visually. The cloudless case (ρ0 = 0%) is easy to recognize, therefore, its separation is logical in our measurements.

Thus, we created 12 × 9 = 108 (θ, ρ) groups. In each group, we selected 12 different sky samples (with θ- and ρ-values falling into the θ- and ρ-interval of the given group) from our polarimetric sky archives. Finally, we obtained 108 × 12 = 1296 different sky situations differing in θ and ρ, but in a given group their θ- and ρ-values were similar. Further on we used the d-patterns of these skies measured by imaging polarimetry in the blue (450 nm) spectral range.

Proportion of the sky with degrees of polarization higher than the perception threshold of Haidinger's brushes

Using the d-pattern of a given sky measured in the blue spectral range, we calculated the proportion P of the sky for which , and as these d* values were measured by Temple et al. [13]. The resulting values of , and give the probability that a viewer (Viking navigator) with high, average and low polarization sensitivity, respectively, finds celestial points where Haidinger's brushes can be perceived by the naked eye and used for sky-polarimetric Viking navigation. These Pmin-, Pave- and Pmax-values were calculated and visualized for the investigated 1296 different sky situations. In the 180° field-of-view circular images of the full sky, the under- and overexposed points as well as landmarks (vegetation, building parts) on the horizon were not taken into consideration. These sky points were masked out during calculations.

Results

When skies are clear (figure 2), the proportion of the celestial hemisphere P suitable for human polarimetric navigation based on perception of Haidinger's brushes is high (figure 3): P(d*=23%) = 78%, P(d* = 56%) = 35%, P(d* = 87%) = 0%. Thus, the most sensitive Viking navigators with 23% ≤ d* ≤ 56% could have perceived Haidinger's brushes in the clear sky of figure 2. However, when the sky is overcast (figure 2), P approaches 0%, making it infeasible for navigation by means of this (figure 3).

Figure 2.

(a)(A) A clear sky photographed on 7 June 2014 at 18.36.29 (=GMT + 2 h, where GMT means Greenwich Mean Time) with solar elevation θ = 0° and per cent cloud cover ρ = 0%. (B) Same as A where the white areas are under- and/or overexposed regions as well as landmarks on the horizon, which are not taken into consideration. The white sky regions were masked out during calculations. (C) Pattern of the degree of linear polarization d of skylight measured with full-sky imaging polarimetry in the blue (450 nm) spectral range. (D) Points of the clear sky are marked with black where . (b) Same as on the left for a cloudy sky photographed on 11 January 2015 at 15.48.59 (=GMT + 1 h) with solar elevation θ = 0° and per cent cloud cover ρ = 100%. (D) The circular area is blank, because there is no celestial point with d > 23%.

Figure 3.

Proportion P (%) of the clear and overcast skies in figure 2 with degrees of skylight polarization not lower than d in the blue (450 nm) spectral range. The vertical lines show the values d* = 23, 56, 87% being the three characteristic perception thresholds of Haidinger's brushes.

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 in the blue (450 nm) spectral range for the perception thresholds of Haidinger's brushes and (figure 4a). In every cell of a given matrix there are 12 different skies. Each cell contains the average