PMC

The navigators used in the pointing task. In a semi-circular arc along the top of the navigator, pictures of different views of the actual town are shown, as seen from different angles at the starting location at either one end or the other end of the town. At the tip of the compass pointer (red) an image was shown of the target object for the current trial. It could be moved by moving the pointer. A. Navigator from Mike’s Restaurant point of view; B. Navigator from House of Pizza point of view.

Instruments for solar navigation possibly used by Viking navigators to maintain course in open sea. (a) A horizon board bears holes on its perimeter coding the azimuth angles ψ_S of the rising and setting Sun at a given latitude. Shift of the rising and setting Sun during the whole navigation season can be followed by choosing appropriate holes along the perimeter. Pegs inserted into the holes help to read the bearings. The navigator can derive true directions at sunrise and sunset by aiming at the Sun with the central and the peripheral pegs, and also at noon by recording the direction of the shortest shadow. Seasonal change of ψ_S can be followed by using different holes. The accuracy of orientation is limited by the exact timing of readings, which is difficult at northern latitudes because of the small angles γ between the celestial path of the Sun and the horizon. (b) A Sun compass bears a thin and high central gnomon and one or more matching gnomonic lines. During the day, the navigator can derive true directions by fitting the shadow tip to the gnomonic line. At low solar elevation angles θ_S, the shadow tip falls off the board, thus orientation is not possible. (c) A twilight board is a combination of a horizon board and a Sun compass bearing a short conical gnomon and a matching gnomonic line (thick solid line). The gnomon base is tangential to the asymptotes (dashed arrows) of the gnomonic line. At low solar elevations θ_S, the edge of the shadow lies on an asymptote of the gnomonic line. The navigator can derive true directions by fitting the shadow edge to the outer section of the gnomonic line. The ‘cast’ shadow (dash-dotted line) of an imaginary down-pointing gnomon (dashed circle) when ‘illuminated’ by the set Sun from below the horizon would act likewise.

(a) Fitted polynomial functions representing the optimal base diameters of conical gnomons for a twilight board at the 46th, 51st, 56th, 61st and 66th latitudes providing 7.9 mm length of the cast shadow on the day of the Equinox. The difference between a single base diameter of the gnomon and the actual optimal base diameter was required to be less than 0.55 mm (straight dashed lines) defining the longest possible season of operation. At the 61st latitude (solid line), which is the traditional Viking sailing route to Greenland, the above requirement defines a vernal navigation season between 24 February and 15 April in the Gregorian calendar system. The season is significantly extended at southern latitudes. (b) Characteristic sizes of the hypothesized shadow-stick to be used with the Uunartoq artefact dial shown in 1:1 scale. (c) A Viking bone pendant of a circular design from Estonia, from the tenth–eleventh century AD, shown in 1:1 scale (source: http://www.icollector.com/Viking-bone-pendant-of-a-circular-design-and-wolf-tooth-pendant-from-Estonia-10th-11th-century-AD_i8646474). Its markings do not form a pattern coding evenly distributed elevation angles, but naive imitations of ‘navigator pendants’ and genuine shadow-sticks are supposed to take the form of pendants with similar shape, size and decorations.

Steps of determination of the North uncertainty Δω_N. (a) Celestial square grid from which sky point m₁ is chosen, where the first sunstone is rotated. (b) Polar grid from which point m₂ is chosen, where the second sunstone is rotated at an angular distance γ from m₁. (c) The first step of sky-polarimetric navigation. (d) Adjusting the orientation of sunstones at sky points m₁ and m₂ has uncertainties e₁ and e₂, which determine a spherical rectangle (grey) containing the real Sun position S and all possible estimated Sun positions E. (e) By finding the intersection of the great celestial circles where the estimated Sun positions E are located, the navigator commits elevation uncertainty Δθ and azimuth uncertainty Δϕ, and thus E has to be shifted with a vector of (Δθ, Δϕ) getting the shifted estimated Sun position E_sh (f). The third step of sky-polarimetric navigation with a North uncertainty Δω_N. The elevation of the Sun is estimated with an elevation uncertainty Δθ_E, and thus the estimated Sun position P_θ from step 3 can be found in the inaccuracy interval of E_sh– Δθ_E and E_sh + Δθ_E. (g) In the fourth step, the navigator had to align the shadow-stick of the sun compass parallel to the meridian (vertical azimuth plane) of the invisible Sun. The uncertainty of this alignment was characterized by the measured e_4th(θ) function and for each Sun position P_θ, we got an interval between ϕ_S(θ) – e_4th(θ) and ϕ_S(θ) + e_4th(θ) that was divided into equal horizontal angular distances by 0.2° perpendicular to the meridian, where ϕ_S(θ) is the azimuth angle of point P_θ. Thus, the measured Sun position P_E can be found in this inaccuracy interval. (h) Distribution (frequency f) of the North uncertainty Δω_N with a maximum at angle ω_max and half bandwidth Δω being the full width at half maximum. (i) The two possibilities to project the estimated sun position E onto the forenoon and afternoon half of the gnomonic line. (j) The gnomonic lines for the spring equinox (21 March) and the summer solstice (21 June), onto which the real and estimated sun positions were projected. The angular deviation from the gnomonic lines gives the navigation uncertainty. G: gnomon. S_E: real Sun position projected onto the equinoctial line. S′_E: estimated erroneous Sun position projected onto the equinoctial line. P_E: point that we get after rotating S′_E to fit onto the equinoctial line (the main step of North uncertainty computation). S_S: real Sun position projected onto the solstice line. S′_S: estimated erroneous Sun position projected onto the solstice line. P_S: point that we get after rotating S′_S to fit onto the solstice line (the main step of North uncertainty computation). Grey: angles Δω_N with which S′_E and S′_S need to be rotated to fit onto the equinoctial and solstice line, respectively. More details can be read in the text.