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

Speedy bearings to slacked steering: Mapping the navigation patterns and motions of Viking voyages

Viking sailors ruled the North Atlantic Ocean for about three hundred years. Their main sailing route was the 60° 21' 55'' latitude between Norway and Greenland. Although they did not have a magnetic compass, in sunshine they used a sun-compass to determine the geographical north (solar Viking navigation: SVN). It has been hypothesized that when the Sun was invisible, Viking navigators determined the direction of polarization of skylight with sunstones (dichroic/birefringent crystals), and then estimated the geographical north using the sun-compass (sky-polarimetric Viking navigation: SPVN). Many details of the hypothetical SPVN have been thoroughly revealed in psychophysical laboratory and planetarium experiments. Combining these results with measured celestial polarization patterns, the success of SPVN was obtained as functions of sailing, meteorological and navigation parameters (sunstone type, sailing date, navigation periodicity, night sailing, cloudiness conditions). What was so far lacking in this experimental and computational archeological approach is the study of the success of SVN and a combined navigation using solar cues in sunshine (SVN) and sky polarization at invisible Sun (SPVN), the latter being the most realistic method. In this work we determine the success of the sole SVN and the combined SVN-SPVN relative to the mere SPVN for three navigator types (determining the intended sailing direction with large, medium or small frequencies) at spring equinox and summer solstice, with and without night sailing. We found that to maximize the sailing success, navigators had to choose different navigation methods depending on the navigation frequency. Using sky polarization with very frequent navigation, resulted in the highest chance to survive a three-week voyage from Norway to Greenland.

Sensitivity and robustness of sky-polarimetric Viking navigation: Sailing success is most sensitive to night sailing, navigation periodicity and sailing date, but robust against weather conditions

Although Viking sailors did not have a magnetic compass, they could successfully navigate with a sun-compass under a sunny sky. Under cloudy/foggy conditions, they might have applied the sky-polarimetric Viking navigation (SPVN), the high success of which has been demonstrated with computer simulations using the following input data: sky polarization patterns measured with full-sky imaging polarimetry, and error functions of the navigation steps measured in psychophysical laboratory and planetarium experiments. As a continuation of the earlier studies, in this work we investigate the sensitivity of the success of SPVN to the following relevant sailing, meteorological and navigational parameters: sunstone type, sailing date, navigation periodicity, night sailing, dominance of strongly, medium or weakly cloudy skies, and changeability of cloudiness. Randomly varying these parameters in the simulation of Viking voyages along the latitude 60° 21’ 55’’ N from Norway to Greenland, we determined those parameters which had strong and weak influences on the success of SPVN. The following intrinsic parameters of the simulation were also randomly changed: sailing speed, visibility distance of Greenland’s southeast coastline and start time of diurnal sailing. Our results show that the sailing success is sensitive to the night sailing, navigation periodicity and sailing date, while it is robust against the sunstone type, dominance of strongly, medium or weakly cloudy skies, and changing cloudiness.

Psychophysical study of the moon illusion in paintings and landscape photos

The moon illusion is a visual deception when people perceive the angular diameter of the Moon/Sun near the horizon larger than that of the one higher in the sky. Some theories have been proposed to explain this illusion, but not any is generally accepted. Although several psychophysical experiments have been performed to study different aspects of the moon illusion, their results have sometimes contradicted each other. Artists frequently display(ed) the Moon/Sun in their paintings. If the Moon/Sun appears near the horizon, its painted disc is often exaggeratedly large. How great is the magnitude of moon illusion of painters? How different are the size enlargements of depicted lunar/solar discs? To answer these questions, we measured these magnitudes on 100 paintings collected from the period of 1534–2017. In psychophysical experiments, we also investigated the moon illusion of 10 test persons who had to estimate the size of the lunar/solar disc on 100 paintings and 100 landscape photographs from which the Moon/Sun was retouched. Compared to the lunar/solar disc calculated from reference distances estimated by test persons in paintings, painters overestimated the Moon's size on average Q  = 2.1 ± 1.6 times, while the Sun was painted Q  = 1.8 ± 1.2 times larger than the real one, where Q  =  r painted / r real is the ratio of the radii of painted ( r painted ) and real ( r real ) Moons/Suns. In landscape photos, test persons overestimated the Moon's size Q  = 1.6 ± 0.4 times and the Sun was assumed Q  = 1.7 ± 0.5 times larger than in reality, where Q  =  r test / r real is the ratio of the radius r test estimated by the test persons and the real radius r real of Moons/Suns. The majority of the magnitude of moon illusion Q  = 1.6, 1.7, 1.8, 2.1, 2.8, 2.9 measured by us are larger than the Q -values 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.8 obtained in previous psychophysical experiments due to methodological differences.

A Bio-Inspired Polarization Sensor with High Outdoor Accuracy and Central-Symmetry Calibration Method with Integrating Sphere

A bio-inspired polarization sensor with lenses for navigation was evaluated in this study. Two new calibration methods are introduced, referred to as "central-symmetry calibration" (with an integrating sphere) and "noncontinuous calibration". A comparison between the indoor calibration results obtained from different calibration methods shows that the two proposed calibration methods are more effective. The central-symmetry calibration method optimized the nonconstant calibration voltage deviations, caused by the off-axis feature of the integrating sphere, to be constant values which can be calibrated easily. The section algorithm proposed previously showed no experimental advantages until the central-symmetry calibration method was proposed. The outdoor experimental results indicated that the indoor calibration parameters did not perform very well in practice outdoor conditions. To establish the reason, four types of calibration parameters were analyzed using the replacement method. It can be concluded that three types can be easily calibrated or affect the sensor accuracy slightly. However, before the sensor is used outdoors every time, the last type must be replaced with the corresponding outdoor parameter, and the calculation needs a precise rotary table. This parameter, which is mainly affected by the spectrum of incident light, is the main factor determining the sensor accuracy. After calibration, the sensor reaches an indoor accuracy of ±0.009° and a static outdoor accuracy of ±0.05° under clear sky conditions. The dynamic outdoor experiment shows a ±0.5° heading deviation between the polarization sensor and the inertial navigation system with a ±0.06° angular accuracy.

Success of sky-polarimetric Viking navigation: revealing the chance Viking sailors could reach Greenland from Norway

According to a famous hypothesis, Viking sailors could navigate along the latitude between Norway and Greenland by means of sky polarization in cloudy weather using a sun compass and sunstone crystals. Using data measured in earlier atmospheric optical and psychophysical experiments, here we determine the success rate of this sky-polarimetric Viking navigation. Simulating 1000 voyages between Norway and Greenland with varying cloudiness at summer solstice and spring equinox, we revealed the chance with which Viking sailors could reach Greenland under the varying weather conditions of a 3-week-long journey as a function of the navigation periodicity Δt if they analysed sky polarization with calcite, cordierite or tourmaline sunstones. Examples of voyage routes are also presented. Our results show that the sky-polarimetric navigation is surprisingly successful on both days of the spring equinox and summer solstice even under cloudy conditions if the navigator determined the north direction periodically at least once in every 3 h, independently of the type of sunstone used for the analysis of sky polarization. This explains why the Vikings could rule the Atlantic Ocean for 300 years and could reach North America without a magnetic compass. Our findings suggest that it is not only the navigation periodicity in itself that is important for higher navigation success rates, but also the distribution of times when the navigation procedure carried out is as symmetrical as possible with respect to the time point of real noon.