Accuracy of sun localization in the second step of sky-polarimetric Viking navigation for north determination: a planetarium experiment

Abstract

It is a widely discussed hypothesis that Viking seafarers might have been able to locate the position of the occluded sun by means of dichroic or birefringent crystals, the mysterious sunstones, with which they could analyze skylight polarization. Although the atmospheric optical prerequisites and certain aspects of the efficiency of this sky-polarimetric Viking navigation have been investigated, the accuracy of the main steps of this method has not been quantitatively examined. To fill in this gap, we present here the results of a planetarium experiment in which we measured the azimuth and elevation errors of localization of the invisible sun. In the planetarium sun localization was performed in two selected celestial points on the basis of the alignments of two small sections of two celestial great circles passing through the sun. In the second step of sky-polarimetric Viking navigation the navigator needed to determine the intersection of two such celestial circles. We found that the position of the sun (solar elevation θ(S), solar azimuth φ(S)) was estimated with an average error of +0.6°≤Δθ≤+8.8° and -3.9°≤Δφ≤+2.0°. We also calculated the compass direction error when the estimated sun position is used for orienting with a Viking sun-compass. The northern direction (ω(North)) was determined with an error of -3.34°≤Δω(North)≤+6.29°. The inaccuracy of the second step of this navigation method was high (Δω(North)=-16.3°) when the solar elevation was 5°≤θ(S)≤25°, and the two selected celestial points were far from the sun (at angular distances 95°≤γ(1), γ(2)≤115°) and each other (125°≤δ≤145°). Considering only this second step, the sky-polarimetric navigation could be more accurate in the mid-summer period (June and July), when in the daytime the sun is high above the horizon for long periods. In the spring (and autumn) equinoctial period, alternative methods (using a twilight board, for example) might be more appropriate. Since Viking navigators surely also committed further errors in the first and third steps, the orientation errors presented here underestimate the net error of the whole sky-polarimetric navigation.

North error estimation based on solar elevation errors in the third step of sky-polarimetric Viking navigation

“…Because this downwards movement of the arms is completely random, this effect was different in the five measurement sessions, causing the increasing standard deviation of the elevation errors. The systematic overestimation of some test persons (which was also described by Bernáth et al [15,31] and Farkas et al [26]) can also be explained by the same effect. When the fists of the test person move downwards, he systematically reported a higher elevation value than the true one, as seen for test person 7 in the electronic supplementary material, figure S2, for example.…”

“…In a following paper, we will study how the errors of the three steps of sky-polarimetric Viking navigation add up under different sky conditions. For that study, we will use the error functions of the first and second steps measured earlier [26,27] along with the polarization patterns of numerous (more than 1000) different skies measured by full-sky imaging polarimetry.…”

“…According to the theory, sunstones could possibly be dichroic cordierite, tourmaline and andalusite, for instance, or birefringent calcite (Icelandic spar) [18][19][20][21][22][23][24][25], through which the observer (navigator, always a male in the Viking age) can see the changes in the intensity of the transmitted linearly polarized skylight while he rotates the crystal in front of his eyes. The steps in this sky-polarimetric Viking navigation method are described in detail elsewhere [7,17,26,27].…”

“…Previously, we have measured by imaging polarimetry the atmospheric optical prerequisites of this navigation method under skies with different cloud and fog coverage [7,17,[28][29][30]. The accuracy of the first step has been measured in a laboratory experiment [27] and that of the second step in a planetarium experiment [26]. Now, we have measured the accuracy of the third step in a psychophysical experiment in a planetarium.…”

Accuracy of the hypothetical sky-polarimetric Viking navigation versus sky conditions: revealing solar elevations and cloudinesses favourable for this navigation method

“…1b). According to our earlier field experience with sunstones (Bernáth et al 2013bFarkas et al 2014), point m 2 cannot be too close to (0° < τ < 45°) or too far (90° < τ ≤ 180°) from m 1 , otherwise the accuracy of the 2nd step of sky-polarimetric navigation decreases considerably.  Using the measured degrees of polarization p 1 and p 2 in sky points m 1 and m 2 , we calculated the uncertainties e 1 = e(p 1 ) and e 2 = e(p 2 ) of sunstone adjustment, where e(p) is the uncertainty function of the 1st step measured in psychophysical laboratory experiments for cordierite, tourmaline and (best-performing) calcite crystals (Száz et al 2016a).…”