Modeling Viking Voyages to Greenland

To determine the success of a voyage from Bergen to Greenland, we defined the borderline near the coastline of Greenland from where the navigator could see the mountains of Greenland. The distance of this border from the coast was calculated as follows (electronic supplementary material, figure S37): (i) the average height of the mountains of Greenland is m = 1000 m; the first mountains are on average at distance c = 1000 m from the coast inside the island. A Viking observer could climb up on the mast of a Viking ship up to a height h = 21 m (above sea level) in order to see the mountains of Greenland as soon as possible. (ii) The radius of the Earth is r = 6372.8 km, and the Earth's shape was approximated by a sphere. (iii) The distance from where the top of the mountains can already be seen is d = r(α + β) − c, where α = arc cos[r/(r  +  m)] is the angular distance where the tangential straight line from the top of the mountain reaches the Earth's surface, and β = arc cos[r/(r + h)] is the angular distance measured from α where this tangential line reaches the observer at a height h on the ship's mast (electronic supplementary material, figure S37).
A voyage was considered as successful if the simulated sailing route crossed the borderline of visibility of the mountains of Greenland, otherwise it was unsuccessful. For a given sunstone crystal (calcite, cordierite and tourmaline), a given date (spring equinox and summer solstice) and a given navigation periodicity Δt (=1, 2, 3, 4, 5, 6 h) we simulated N = 1000 voyages, from which N_s was successful and N_u was unsuccessful (N = N_s + N_u). Finally, we computed the navigation success s = N_s/N in all 36 cases = 3 (sunstones) × 2 (dates) × 6 (navigation periodicities).
We also simulated reversed voyages from Hvarf (Greenland) to Norway along the 60°21′55″ N latitude. However, in these cases the voyages were always successful, because the simulated sailing routes always reached somewhere on the coasts of Europe.