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Phil. Trans. R. Soc. B (2011) 366, 772–782

doi:10.1098/rstb.2010.0194

Research

* Autho

Electron10.1098

One cobiologic

On the trail of Vikings with polarizedskylight: experimental study of the

atmospheric optical prerequisites allowingpolarimetric navigation by Viking seafarersGabor Horvath1,*, Andras Barta1, Istvan Pomozi1, Bence Suhai1,

Ramon Hegedus2, Susanne Akesson3, Benno Meyer-Rochow4,5

and Rudiger Wehner6,7

1Environmental Optics Laboratory, Department of Biological Physics, Physical Institute, Eotvos University,Pazmany setany 1, Budapest 1117, Hungary

2Computer Vision and Robotics Group, University of Girona, Campus de Montilivi, Edifici P4,17071 Girona, Spain

3Department of Animal Ecology, Lund University, Ecology Building, SE-22362 Lund, Sweden4Faculty of Engineering and Science, Jacobs University of Bremen, PO Box 750561,

D-28725 Bremen, Germany5Department of Zoology, Biological Institute, University of Oulu, Oulu, Finland

6Brain Research Institute, University of Zurich, Winterthurerstrasse 190, CH-8057 Zurich, Switzerland7Biocenter, University of Wurzburg, Am Hubland, D-97074 Wurzburg, Germany

Between AD 900 and AD 1200 Vikings, being able to navigate skillfully across the open sea, werethe dominant seafarers of the North Atlantic. When the Sun was shining, geographical north couldbe determined with a special sundial. However, how the Vikings could have navigated in cloudy orfoggy situations, when the Sun’s disc was unusable, is still not fully known. A hypothesis was for-mulated in 1967, which suggested that under foggy or cloudy conditions, Vikings might havebeen able to determine the azimuth direction of the Sun with the help of skylight polarization,just like some insects. This hypothesis has been widely accepted and is regularly cited by research-ers, even though an experimental basis, so far, has not been forthcoming. According to this theory,the Vikings could have determined the direction of the skylight polarization with the help of an enig-matic birefringent crystal, functioning as a linearly polarizing filter. Such a crystal is referred to as‘sunstone’ in one of the Viking’s sagas, but its exact nature is unknown. Although accepted by many,the hypothesis of polarimetric navigation by Vikings also has numerous sceptics. In this paper, wesummarize the results of our own celestial polarization measurements and psychophysical labora-tory experiments, in which we studied the atmospheric optical prerequisites of possiblesky-polarimetric navigation in Tunisia, Finland, Hungary and the high Arctic.

Keywords: Viking navigation; sky polarization; imaging polarimetry; atmospheric optics

1. THE VIKING SUNDIAL AS A COMPASSOne of the most important Viking shipping routesbetween Norway and Greenland took them along lati-tude 618 N (figure 1). Archaeologists found a piece ofstone and a fragment of a wooden disc (figure 2a),both featuring straight and hyperbolic carvings(figure 2b) [1]. It turned out that the two items hadbeen parts of sundials used by the Vikings as a com-pass during their sea-crossings along latitude 618 N.

r for correspondence (gh@arago.elte.hu).

ic supplementary material is available at http://dx.doi.org//rstb.2010.0194 or via http://rstb.royalsocietypublishing.org.

ntribution of 20 to a Theme Issue ‘New directions inal research on polarized light’.

772

A stick (the so-called gnomon) was mounted into thewooden disc perpendicular to the disc. If the discwas held horizontally at latitude 618 N during the sail-ing season of the Vikings (May to August) and the Sunwas shining, then the tip of the shadow of the gnomoncast onto the disc followed the hyperbolic carving fromsunrise through noon to sunset. Straight and hyper-bolic carvings corresponded to the equinox and thesummer solstice, respectively.

After carving the trajectories of the tip of the gnomon’sshadow into the disc, the Vikings gained an instrument:the sundial. With it they could locate geographical northalong latitude 618N from May to August even on thehigh seas, provided the Sun was shining. All theyneeded to do was to hold the dial’s disc horizontally intothe Sun and rotate it around its vertical axis (coinciding

This journal is q 2011 The Royal Society

lattitude 61° N

Nyaland (New Foundland)

Jolduhlaub

Hvarf

Gunbjorns Skerry

or Blaserk

Oulu

Svalbard

StadHernam

Langeness

Horn (eastern)

Utbygder

Jan Mayen

Kolbeinsey

Krisev

igs K

lippe

fjel

Reykjan

essSneefjeldsness

7

4

3

2

1

56

5

Figure 1. The seven (1–7) main Viking sailing routes. Route 3 connected Hernam (nowadays the Norwegian Bergen) withHvarf in South Greenland along latitude 618 N [1].

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with the axis of the gnomon) until the tip of the gnomon’sshadow reached the corresponding carving on the dial. Anotch on the disc then showed the geographical northerndirection (figure 2b).

In a sailing competition across the Atlantic along lati-tude 618N in recent years, the captains having a numberof vessels were given replicas of the original Viking sun-dial and were asked to use these devices as compasses fora certain time during the competition. The result showedthat it was possible to accurately navigate on the openocean using only the Viking sundial, provided the Sunwas shining [1]. This method of navigation is knownas ‘solar Viking navigation’.

2. THE HYPOTHESIS OF SKY-POLARIMETRICNAVIGATION BY VIKINGSAs the Viking sundial can only be used when the Sunshines, the question arises as to how the Vikings couldhave navigated when the Sun was occluded by cloudsor fog, a situation that can last for days along majorparts of the North Atlantic sailing route of ancientViking seafaring folk. At the end of the 1960s, theDanish archaeologist Ramskou [2] suggested thatVikings made use of polarized skylight when the Sunwas behind clouds or fog:

— First, the Viking navigator had to determine thedirection of skylight polarization at two distinctpoints of the sky-dome by using a birefringent

Phil. Trans. R. Soc. B (2011)

crystal (the sunstone) as a linear polarizer(figure 2d). Although it is not clear what the sun-stone was made out of, it could have beencordierite or tourmaline, both common in Scandi-navia. By rotating such a crystal to and fro andlooking at the sky through it, the sky appears tobrighten and fade periodically, because, with theexception of the polarizational neutral points [3],the skylight is partially linearly polarized. Lookingthrough the birefringent crystal, the Viking naviga-tor could calibrate the sunstone by adjusting it insuch a way that a patch of the clear sky appearsbrightest. A line pointing to the true position ofthe Sun would then be scraped into the crystal.After such calibration, the direction of the Sunhidden by clouds can then be determined by look-ing at a clear patch of sky through the crystal androtating the latter until the sky appears the bright-est. The scratch on the sunstone shows thedirection of the invisible Sun, if the direction ofpolarization of skylight corresponds to Rayleigh’stheory of first-order light scattering, i.e. the direc-tion of polarization of light from an arbitrarypoint of the clear sky is perpendicular to theplane of scattering determined by the observer,the Sun and the celestial point observed.

— The Viking navigator could then estimate the inter-section of the two great circles running throughthe two investigated celestial points parallel to thescratches on the crystals. If the pattern of the

sky-dome

zenith

ground-basedobserver

Sun

(c)

(b)

(d)

(a)

Figure 2. (a) The wooden disc fragment (found in southern Greenland) onto which the Viking navigators scratched somehyperbolic curves [1]. (b) The reconstructed sundial used by the Vikings for navigation on the open sea. The left, grey partof the disc has not been found. (c) Three-dimensional drawing of the Viking sundial with a conical vertical gnomon and itsshadow, the endpoint of which touches the hyperbola scratched into the horizontal wooden disc. (d) Sky-polarimetric naviga-tion by Vikings can only function, if the direction of skylight polarization (symbolized by double-headed arrows) is

perpendicular to the plane of scattering (determined by the Sun, the observer and the celestial point observed). Thiscorresponds to Rayleigh’s theory of first-order light scattering.

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direction of polarization of skylight correspondedto Rayleigh’s theory, then the point of intersectionwould give the position of the invisible Sun.

— Finally, it would have been necessary to somehowimitate the rays of light from the invisible Sun tocast the imaginary shadow of the gnomon ontothe dial. For example, an assistant could haveheld a burning torch so that the navigator wouldsee it in the estimated direction of the invisibleSun. In this way, the Sun would be replaced bythe torch, and the gnomon’s shadow falling ontothe dial. Another method could have involved arotating thin tube (e.g. a piece of reed), attachedto the tip of the gnomon. The navigator couldhave directed this tube into the estimated positionof the invisible Sun, sliding a thin, straight stick(a straw, for example) into the tube; thus modellingthe sunbeam through the gnomon’s tip.

Ramskou [2] believed that by using this method,geographical north could be located even under

Phil. Trans. R. Soc. B (2011)

cloudy or foggy conditions. Since this method isbased on the pattern of the direction of skylight polar-ization, it is called ‘sky-polarimetric navigation byVikings’. This theory of polarimetric Viking navigationis accepted and frequently cited, in spite of a total lackof experimental evidence. In one of the Viking sagas,the Sigurd legend, a reference to sky-polarimetric navi-gation by Vikings appears to have been made [2]: ‘Theweather was very cloudy, it was snowing. Holy Olaf,the king sent out somebody to look around, butthere was no clear point in the sky. Then he askedSigurd, to tell him, where the Sun was. After Sigurdcomplied, he grabbed a sunstone, looked at the skyand saw from where the light came, from which heguessed the position of the invisible Sun. It turnedout, that Sigurd was right’.

Further to this obscure Sigurd saga, there is anotherargument supporting sky-polarimetric navigation byVikings: pilots of Scandinavian Airlines’ DC-8 air-planes were using a polarimetric instrument tonavigate with the help of the polarization of skylight

thin clouds (cirrus) thick cloudscumuli

Figure 3. Three full-sky photographs (taken in Oulu, 65800 N, 258260 E) from 25 pictures of cloudy skies presented 12 times on

a monitor to 18 test persons in the first psychophysical laboratory experiment. The task of the test persons was to guess theposition of the Sun, hidden by clouds, with the naked eye. The centre and the perimeter of the circular pictures are the zenithand the horizon, respectively. The Sun positions estimated by the test persons are shown by tiny white dots with blackperimeter, while the large black dots with white perimeter represent the average of these guessed solar positions.

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when flying close to the North Pole. This instrument isKollsman’s sky compass, which was developed for theUS Airforce in 1948 to navigate in twilight aftersunset. The instrument was equipped with lenses,adjustment screws and scales and contained a linearlypolarizing crystal. This sky compass was directed tothe zenith, while the navigator was rotating the crystalaround its vertical axis until the sky appeared brightestor darkest. The position of the Sun could then beestimated from the orientation of the crystal.

The third argument for sky-polarimetric navigationby Vikings is related with the discovery of Frisch [4],that honeybees (Apis mellifera) are sensitive to polar-ized light and use the direction of skylightpolarization for navigation, when the Sun is hiddenby clouds, but some clear regions of the sky are stillvisible. A similar capacity to detect polarization pat-terns and orient by means of sky polarization waslater found in many other arthropod species [5].

The reason for the wide acceptance of the unprovenhypothesis of sky-polarimetric navigation by Vikings isits elegance, appeal and lack of credible alternative.According to the widely accepted theory, the Vikingswere able to navigate on the open sea with the helpof sky polarization in all meteorological situations:under clear, partly cloudy, totally overcast and foggyskies. Since these claims were never tested, we decidedto investigate whether the atmospheric optical prere-quisites of sky-polarimetric navigation by Vikings aresatisfied. Our aim was to measure the polarization pat-tern of clear, partly cloudy, totally overcast and foggyskies to determine which parts of the sky permitsky-polarimetric navigation by Vikings. In the follow-ing sections, we summarize the main results of ourexperimental studies [6–10].

3. VISUAL ESTIMATION OF THE SUN POSITION INCLOUDY AND TWILIGHT SKIESAccording to Roslund & Beckman [11], Vikings did notneed to use sky-polarimetric navigation, because ‘Evenwhen the Sun is hidden behind clouds, its location canoften be found quite accurately for most navigationalneeds from the pattern of the Sun’s illumination ofclouds, from the bright lining of cloud tops and the

Phil. Trans. R. Soc. B (2011)

crepuscular rays emanating from the Sun. On overcastdays, careful observations of the sky may reveal thefaint disk of the Sun if the cloud cover is not toodense. [. . .] Nor does polarimetry give clues to theSun’s position when it is below the horizon that othermethods do not. The arcs of dawn and twilightappear distinct enough for the naked eye to make outin which direction the Sun is’ [11, p. 4754].

However, the theory of sky-polarimetric navigationby Vikings cannot be dismissed by such qualitativecounter-arguments. If the hypothesis that the positionof the Sun behind clouds or under the horizon canquite accurately be estimated by the naked eye iscorrect, then the Vikings would not have to use thesky-polarimetric navigation method to locate theposition of the invisible Sun.

Barta et al. [6] investigated quantitatively if Roslund &Beckman [11] were right with their claim. We tookphotographs of several cloudy skies on the seashoresof the island of Hailuoto and the city of Oulu in north-ern Finland through a 1808 field-of-view fisheye lens.Thus, we could map the whole sky into a circularcolour picture, in which the centre of the circle corre-sponded to the zenith and the perimeter to the horizon(figure 3). We took a second series of photographs oftwilight skies, when the Sun was close to but belowthe sea horizon (figure 4).

In our first psychophysical laboratory experiment,using a colour monitor, we showed 25 differentcloudy sky photographs (figure 3) 12 times to 18 testpersons in a dark room. In these photographs, theSun was occluded by clouds, and the test persons’task was to visually estimate the position of the invisibleSun by using the computer’s mouse. Our computerprogram stored the estimated Sun positions (c: zenithangle, w: azimuth angle from an arbitrary reference azi-muth direction) and calculated their averages (,c.

and ,w.) and standard deviations (sk, s? and sw)(table 1). In our second psychophysical laboratoryexperiment, we showed 15 different twilight sky photo-graphs (figure 4) six times in a dark room on a colourmonitor to 18 test persons. In these photographs, theSun was under the sea horizon, and the test persons’task was to visually estimate the azimuth direction ofthe invisible Sun by using the mouse. As before, our

clear strongly cloudyweakly cloudy

Figure 4. Three 1808 field-of-view photographs (taken in Oulu, 65800 N, 258260 E) from the 15 pictures of twilight skies pre-

sented six times on a monitor to 18 test persons in the second psychophysical laboratory experiment. The task of the testpersons was to guess the azimuth direction of the Sun below the sea horizon with the naked eye. The centre of the circularpictures points to the horizon, while the zenith and the nadir are the uppermost and lowermost points of the circle. Theupper/lower halves of the pictures depict the sky/sea. The solar azimuth directions estimated by the test persons are shown

by short vertical bars below the horizon. The long vertical bar above the horizon represents the average of these guessedazimuth directions, while their standard deviation is shown by the short vertical bars at the ends of the horizontal bar.

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computer program stored the estimated Sun azimuthdirections (w: azimuth angle from an arbitrary referenceazimuth direction) and calculated their averages (,w.)and standard deviations (sw). The test persons weregents from Bremen, Budapest and Roskilde, agedbetween 23 and 45 years. Details of these experimentscan be found in Barta et al. [6].

The standard deviations of the estimated solar pos-itions in cloudy skies were between smin

k ¼ 1.18 andsmin? ¼ 1.48 (the Sun was almost visible through thin

cirrus clouds) and smaxk ¼ 20.28 and s?

max ¼ 25.28(the Sun was hidden by a large, thick cloud) depend-ing on the degree of cloud cover. The maximumangular distance dmax between the estimated Sun pos-itions varied between 8.18 and 162.98 (electronicsupplementary material, table S1). The averages ofthe standard deviations sk, s? and sw and of the maxi-mum angular distances dmax for the 25 pictures were,sk. ¼7.48, ,s?. ¼ 11.98, ,sw. ¼22.38 and,dmax. ¼70.78. Barta et al. [6] has also obtaineddata on locating the Sun when it was visible behindthin cirrus clouds (sky 2 in fig. 1 and table 1 of [6]).According to these data, the average inherent errorsof the people tested in this psychophysical experimentwere the following: sk ¼ 1.18, s? ¼ 1.48, dmax ¼ 8.18and sw ¼ 2.18.

Depending on the solar elevation under the horizonand the degree of cloudiness, the standard deviationsof the estimated Sun positions in the twilight pictureswere between smin

w ¼ 0.68 (the Sun was still visible onthe horizon) and smin

w ¼ 28, while the maximum angu-lar distance between the estimated Sun azimuthdirections gmax varied between 2.18 (the Sun wasabove the horizon) and 998 (the Sun was belowthe horizon). The averages of sw and gmax for the15 twilight pictures were ,sw. ¼11.48 and,gmax. ¼37.38. The averages of ,sw. and,gmax. for the 18 test persons were �sw� ¼ 5.98and �gmax� ¼ 14.58. Data for locating the Sunwhen it was visible just on the horizon are also avail-able (twilight sky 10 in fig. 2 and table 3 of [6]).The average inherent errors that people tested in thisexperiment made were: sw ¼ 0.68 and gmax ¼ 2.18.

Phil. Trans. R. Soc. B (2011)

The averages of the standard deviations sk, s? andsw and maximal angular distances dmax and gmax of thecloudy and twilight skies characterizing the accuracy ofthe visual estimation of the Sun position for all cloudypictures (,sk. ¼ 78, ,s?. ¼128, ,sw. ¼ 228 and,dmax. ¼ 718), for all twilight pictures (,sw. ¼ 118and ,gmax. ¼ 378), and for all test persons (cloudyskies: �sk� ¼ 38, �s?� ¼ 88 and �dmax� ¼258; twilight skies: �sw� ¼ 68 and �gmax� ¼ 158)were quite high. The measured maximum valueswere smax

k ¼ 208, smax? ¼ 258, smax

w ¼ 808,max(dmax) ¼ 1638 for the cloudy skies, and smax

w ¼

428, max(gmax) ¼ 998 for the twilight skies. Theseobviously high errors do not support the assumptionthat under cloudy or twilight skies the position of theinvisible Sun can be estimated quite accurately byusing the colour and intensity patterns of the sky.

Although these results underestimate the accuracyof the visual estimation of the Sun position by anexperienced Viking navigator, the investigated counter-argument (that the position of the invisible Sun incloudy skies could be estimated quite accurately, evenby the naked eye, so that Viking navigators did notneed polarizing crystals to determine the position ofthe Sun hidden) cannot be taken seriously as a valid criti-cism of the theory of sky-polarimetric navigation byVikings. Our results disclaim only one counter-argumentof the polarimetric Viking navigation and imply thatViking navigators might have needed some aid to navi-gate on open seas during cloudy or foggy weatherconditions. Such an aid could, for example, have beenthe sky-polarimetric navigation method.

4. PROPORTION OF CLEAR AND PARTLYCLOUDY SKIES THAT CAN BE USED FORSKY-POLARIMETRIC NAVIGATION BY VIKINGSThe two atmospheric optical prerequisites of sky-polarimetric navigation by Vikings are the following:

— The plane of the oscillation of skylight is perpen-dicular to the plane of scattering, i.e. thedirection of polarization of skylight is the same as

clear skies(a)

1(sunriseqs = 0°)

1(sunriseqs = 0°)

2(qs = 9°)

3(qs = 26°)

non-Rayleigh(n)

Rayleigh(r)

overexposed(o)

4(qs = 34°)

5(qs = 40°)

6(qs = 56°)

7(noon

qs = 60°)

2(qs = 9°)

3(qs = 21°)

4(qs = 35°)

5(qs = 45°)

6(qs = 57°)

7(noon

qs = 65°)

(b) (c) (d)cloudy skies

Figure 5. (a,c) 1808 field-of-view colour photographs of Tunisian clear and partly cloudy skies as a function of the solarelevation angle uS from the horizon. The centre/perimeter of the circular pictures is the zenith/horizon, and the zenithangle c is proportional to the radius from the centre (czenith ¼ 08, chorizon ¼ 908). (b,d) Maps of the proportion r of the sky

that follows the pattern of the angle of polarization aRayleigh predicted by Rayleigh’s theory of first-order scattering for clearskies in the red (650 nm), spectral range versus solar elevation angle uS. ‘Rayleigh’ points with Da ¼ jameasured 2 aRayleighj �58 are shaded in grey, ‘non-Rayleigh’ points with Da . 58 are white, and overexposed points are black. Hence, the grey/white celestial regions are appropriate/inappropriate for sky-polarimetric navigation by Vikings, while nothing is knownabout the black areas. The radial bar in the circular pictures is the wire of the Sun occulter. The approximately hourly positions

of the Sun are represented by dots or the disc of the Sun occulter. The numerical values of r, n and o in rows 1–7 of tables S2and S3 in the electronic supplementary material were determined for these clear and partly cloudy skies.

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that predicted by Rayleigh’s theory. The accuracyof sky-polarimetric navigation by Vikings is deter-mined by that part of the sky, where the directionof polarization obeys first-order Rayleigh scattering.

— The degree of linear polarization p of skylight isso high that the periodic brightening and darken-ing of the sky seen through a rotating sunstone(functioning as a polarizer) can be observed, andthus the direction of skylight polarization can bedetermined with sufficient accuracy.

The fulfilment of these conditions could not be exam-ined earlier owing to the lack of wide field-of-viewimaging polarimeters. We began to study this topicduring an expedition to the Tunisian desert in 1999,when we investigated how similar polarization patternsof partly cloudy skies were to those of clear skies as afunction of solar elevation angle uS above the horizon.

Phil. Trans. R. Soc. B (2011)

This was important to understand the navigation ofthe desert ants, Cataglyphis bicolor based on the polariz-ation of skylight [9]. Although earlier there had beenseveral attempts to determine whether skylight polariz-ation obeyed Rayleigh’s theory, these studies werelimited to a few directions of the sky owing to theusage of point-source polarimeters.

We computed the difference Da ¼ jameasured 2

aRayleighj between the measured and theoretical(Rayleigh) angles of polarization ameasured and aRayleigh

of skylight for every celestial point by using the celes-tial pattern of the direction of polarization measuredby 1808 field-of-view imaging polarimetry at a givensolar position and for a given spectral range (red,green and blue). Then we counted the numberNRayleigh of celestial points for which Da , athreshold ¼58. From this we determined the ratio r ¼ NRayleigh/Nof the N ¼ 150 000 examined points of the sky for

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which the direction of polarization differed from thatof Rayleigh’s theory by less than 58. We also deter-mined the number Nnon-Rayleigh of points in the skyfor which Da . athreshold ¼ 58. There were Noverexposed

points in the sky near the Sun for which the detectorwas overexposed (here the polarization of skylightwas unknown). We also calculated the n ¼ Nnon-

Rayleigh/N and o ¼ Noverexposed/N ratios. The relationamong variables r, n and o are: r þ n þ o ¼ 1. Thesecalculations were performed for clear and partlycloudy skies measured in Tunisia as a function of thesolar elevation uS (figure 5). The clouds in the picturesof cloudy skies were detected by a custom algorithm,thus we could also determine which parts of the skyfollowed Rayleigh’s theory with an accuracy ofathreshold ¼ 58 for both the clear and cloudy celestialregions. From our results [9,10] we concluded the fol-lowing (e.g. figure 5 and electronic supplementarymaterial, tables S2 and S3):

— At a given solar elevation angle (uS) and in a givenspectral range (central wavelength l), the pro-portion r of the sky that is usable for sky-polarimetric navigation by Vikings is always higherfor a clear sky than for a partly cloudy sky. Depend-ing on uS and l, for clear skies r varies between 13and 70 per cent, while for partly cloudy skies r isbetween 4 and 69 per cent. If the Sun is near oron the horizon, then the r-values of partly cloudyskies are close to those of the clear skies.

— The lower the solar elevation uS, the higher theratio r of the sky that is appropriate for sky-polari-metric navigation by Vikings in connection withclear and partly cloudy skies, independent of thewavelength. For clear skies in the red spectralrange (where the proportion o of the overexposedregions is smallest, i.e. where the accuracy of themeasured r-values is highest) r increases from 19to 65 per cent, while uS decreases from 658(noon) to 08 (sunset/sunrise). For partly cloudyskies in the red spectral range r increases from 4per cent to 56–65% while uS decreases from thehighest solar elevation to zero.

— In the case of high solar elevations for clear andpartly cloudy skies r is highest in the blue andlowest in the red spectral range. For lower solarelevations rgreen . rred, but rblue , rgreen.

— Sometimes in the cloudy regions of the sky rela-tively large areas (12–34%) of the celestiala-pattern follow Rayleigh’s theory, and theseareas increase with the decrease of solar elevationuS, independent of the wavelength.

If the a-pattern of the overexposed sky regions nearthe Sun were known, these regions would increase theratio n of the sky inappropriate for sky-polarimetricnavigation by Vikings, because these regions overlapwith the surroundings of the neutral points [3], posses-sing a polarization pattern quite different from theRayleigh pattern. That is why the r-values in the elec-tronic supplementary material, tables S2 and S3 areonly slightly different from the reality, in spite of therelatively high o-values.

Phil. Trans. R. Soc. B (2011)

Multiple scattering of the light in clouds can causethe direction of polarization of cloudlight to be differ-ent from that of skylight described by Rayleigh’s theorybased on first-order scattering. However, if clouds andthe air layer beneath them are lit by direct sunlight,then it is highly probable that cloudlight (originatingeither directly from the cloud, or from the aircolumn beneath it) reaches a ground-based observerafter its first scattering by cloud particles or by theair column below the cloud. The lower the solarelevation uS, the greater the chance that the cloudsand the air beneath them are lit by direct sunlight;thus, the lower uS, the higher the proportion rcloudy

of cloudy skies usable for sky-polarimetric navigationby Vikings. The value of rcloudy can even reach the r-values of clear skies, if the Sun is on the horizon.

We conclude that for clear skies, the ratio r of thesky that follows Rayleigh’s theory with an accuracy ofathreshold ¼ 58 is high, depending on uS. This is par-ticularly true for lower solar elevations (uS � 138),when 40% , r , 70%. Depending on the cloudinessand how the clouds are lit by sunlight, r decreasesunder cloudy conditions, but r can be surprisinglyhigh, particularly for lower solar elevations (e.g.rmax ¼ 69% for uS ¼ 08). Usually, large parts of thea-patterns of clear and partly cloudy skies quite accu-rately follow Rayleigh’s theory, which is the base of theViking’s sky-polarimetric navigation.

5. THE POSSIBILITY OF SKY-POLARIMETRICNAVIGATION BY VIKINGS IN FOGVikings sailing the North Atlantic (figure 1), mustoften have encountered poor visibility with fog sodense that even the Sun’s disc would have been invis-ible to them, particularly with the Sun near thehorizon. Could the Vikings have navigated by skypolarization under foggy weather conditions? Aboardthe Swedish icebreaker ‘Oden’, Susanne Akesson &Gabor Horvath crossed the Arctic Ocean as membersof the Beringia 2005 six-week international expedition,organized by the Swedish Polar Research Secretariatin August and September 2005. They reached theNorth Pole on 12 September 2005. During thisexpedition they measured the polarization patterns offoggy or totally overcast Arctic skies, when the Sun’sdisc was not discernible [7] (figure 6 and electronicsupplementary material, table S4).

In certain Arctic meteorological situations, the foglayer was lit by direct sunlight, because the Sunabove the horizon was not occluded by clouds, butthe Sun was invisible owing to the dense, sunlit fog.The averages of the degree of linear polarization pand the noisiness 1 of the a-pattern of cloudy skies(pcloudy ¼ 10–25% and 1cloudy ¼ 4–15%) werebetween those of the clear (pclear ¼ 16–34% and1clear ¼ 3–6%) and foggy (pfoggy ¼ 4–15% and1foggy ¼ 5–45%) skies. For the similarity s of the a-pat-terns of clear, partly cloudy and foggy skies to thetheoretical a-patterns, we obtained sclear ¼ 65.8–70.7%, scloudy ¼ 49.0–61.8% and sfoggy ¼ 41.4–50.0%, while their minima and maxima were 45% �sclear � 81%, 36% � scloudy � 72% and 19% � sfoggy �71%, respectively. Thus, we see that if the fog layer is

Table 1. Definitions of the parameters used in this paper.

symbol of parameter meaning

p degree (%) of linear polarization of skylighta angle of polarization of skylight measured from the local meridian

c zenith angle (angular distance from the zenith)w azimuth angle from an arbitrary reference azimuth direction,x. mean of parameter x�x� average of the mean of parameter xsk,s? standard deviations (along two orthogonal great circles crossing each other at the average Sun

position) of the positions of the invisible Sun behind clouds located with the naked eyesw standard deviation of the azimuth angle w of the Sundmax maximum angular distance between the estimated Sun positionsgmax maximum angular distance between the estimated solar azimuth directions

max(dmax), max(gmax) maximum values of dmax and gmax

uS elevation angle of the Sun above the horizonDa ¼ jameasured 2 aRayleighj difference between the measured (ameasured) and the theoretical (aRayleigh) angles of polarization

of skylightNRayleigh number of celestial points for which Da , athreshold ¼ 58Nnon-Rayleigh number of celestial points for which Da . athreshold ¼ 58Noverexposed number of celestial points for which the detector was overexposedr ¼ NRayleigh/N proportion of the N examined points of the sky for which the direction of polarization differs

from that of Rayleigh’s theory by less than athreshold ¼ 58n ¼ Nnon-Rayleigh/N proportion of the N examined points of the sky for which the direction of polarization differs

from that of Rayleigh’s theory by more than athreshold ¼ 58o ¼ Noverexposed/N proportion of the overexposed celestial pointsrred, rgreen, rblue values of r in the red, green and blue parts of the spectrum1 the noisiness n of the celestial a-patterns similarity of the celestial a-pattern to the theoretical (Rayleigh) a-pattern

d dissimilarity of the celestial a-pattern in comparison with Rayleigh’s theory

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not too thick, then the a-pattern of the sunlit sky isvery similar to the theoretical a-pattern. On theother hand, the relationships between the averages arethe following: pfoggy , pcloudy , pclear, 1clear , 1cloudy ,

1foggy, sfoggy , scloudy , sclear.Our data from the Arctic and from Hungary ([7];

figure 6 and electronic supplementary material, tableS4) allow us to conclude that if the fog is lit by directsunlight, then the a-pattern of the foggy sky is quitesimilar to that of the clear sky. As a consequence, thefirst condition of the sky-polarimetric navigation by Vik-ings is met for foggy conditions nearly as well as for clearskies. On the other hand, the degrees of linear polariz-ation p of foggy skies are often so low, that the secondcondition of polarimetric Viking navigation is usuallynot satisfied. Thus, the limiting factor of thesky-polarimetric navigation method is the degree ofpolarization, rather than the direction of polarization.Under partly cloudy conditions, both conditions of theViking’s sky-polarimetric navigation are generallysatisfied.

6. SKY-POLARIMETRIC NAVIGATION BY VIKINGSUNDER CONDITIONS OF TOTAL OVERCAST?Vikings undoubtedly often had to sail under totallyovercast conditions, perhaps for days on end and inthe open water far away from land. For this reason,we also measured the polarization characteristics oflight under totally overcast skies in the Arctic regionand in Hungary [8] (figure 7 and electronic sup-plementary material, table S5), when the ground wascovered by high-albedo snow and ice, and it was

Phil. Trans. R. Soc. B (2011)

sometimes snowing or raining, allowing us to drawsome conclusions about the composition of theclouds (ice or water). To our great surprise, the pat-terns of the direction of polarization under totallyovercast skies were very similar to those of the clearskies (figures 5 and 7). We conclude that the first pre-requisite of sky-polarimetric navigation by Vikings ismet, even in totally overcast conditions, for (althoughvery noisy: electronic supplementary material, tableS5) large parts of the pattern of the direction of polar-ization of overcast skies exhibit the Rayleigh pattern.However, the degrees of linear polarization p of over-cast skies are so low (electronic supplementarymaterial, table S5) that it is very unlikely that Vikingnavigators were able to use the sky’s polarization intotally overcast conditions, for if p is low, it is uselessrotating the polarizing sunstone in front of the obser-ver’s eye. The periodic oscillation of the intensity oflight from the totally overcast sky is undetectable or atbest hardly visible. As a result, the direction of skylightpolarization could be determined only very inaccurately.

7. FURTHER RESEARCHOur investigations [6–10] determined the meteorolo-gical situations under which the atmospheric opticalprerequisites for sky-polarimetric navigation byVikings are or are not satisfied. What remains to bemeasured in psychophysical laboratory experimentswith a large number of probands are the:

— error in determinations of the oscillation directionof partially linearly polarized light with different

100% angle of polarization afrom the local meridian

angl

e of

pol

ariz

atio

n a

theo

retic

alm

easu

red

angl

e of

pol

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0°+45°

+90°

+135°

–45°

–90°

–135°180°

0%degree of linear

A

A A

B

B B

S

S S

foggy

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gree

of

linea

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n, p

clear partly cloudy

polarization, p

(a) (b) (c)

(d) (e) ( f )

(g) (h) (i)

( j) (k) (l)

Figure 6. (a–c) 1808 field-of-view photographs and (d–i ) polarization patterns of Arctic foggy, clear and partly cloudy skiesmeasured by full-sky imaging polarimetry in the blue (450 nm) part of the spectrum. ( j– l) Patterns of the angle of polarizationa (from the local meridian) calculated for the solar positions in patterns (a–i) on the basis of the Berry model, which candescribe the polarization of the clear sky more accurately than the Rayleigh model [7]. In the a-patterns, white dots show

the positions of the Sun (S), the Arago (A) and Babinet (B) neutral points.

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sunstones (e.g. birefringent cordierite, tourmalinor calcite crystals), functioning as linear polarizers,as a function of the degree of linear polarization(first step of the sky-polarimetric navigation byVikings);

— error of determining the position of the Sun hiddenby clouds/fog with estimating the intersection ofthe two great circles passing through two arbitrarycelestial points parallel to the local plane of oscil-lation of skylight (second step of the polarimetricViking navigation);

— error of determining the geographical northerndirection with a Viking sundial, if the position ofthe Sun occluded by clouds/fog is known (thirdstep of the sky-polarimetric navigation by Vikings).

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Once these psychophysical experiments have providedus with data on the above-mentioned error functions,computer modelling should allow us to estimate thelikelihood that in the chosen meteorological situationthe geographical north could be determined with acertain amount of deviation by using the polarimetricViking method. Finally, we ought to be able toanswer the question in which meteorological situationsthe Vikings could have satisfactorily navigated bymeans of sky polarization.

Since the psychophysical experiments, outlinedabove, cannot be performed with Viking navigators,we plan to measure the error functions by using maleGerman, Hungarian and Swedish students. Thesemeasurements are in progress.

photograph photograph photographangle ofpolarization a

S1

S2

S3

S4

S5

0°–10°a

10°–20° 20°–30° 30°–40° 40°–50° 50°–60° 60°–70° 70°–80° 80°–90°

180°–170° 170°–160° 160°–150° 150°–140° 140°–130° 130°–120° 120°–110° 110°–100° 100°–90°

S6

S7

S8

S9

S10

S11

S12

S13

S14

S15

angle ofpolarization a

angle ofpolarization a

Figure 7. 1808 field-of-view photographs and patterns of the angle of polarization a (clockwise from the local meridian) oftotally overcast skies measured by full-sky imaging polarimetry in the blue (450 nm) part of the spectrum on the ArcticOcean (S1–S8) and in Hungary (S9–S15). The ground was covered by high-albedo white snow and ice, both in the Arcticand Hungary.

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Our research was supported by an equipment donation fromthe Alexander von Humboldt Foundation to G.H. TheTunisian expedition was financed by the Swiss ScienceFoundation. The Finnish measuring campaign wassupported by the University of Oulu through a grant to B.Meyer-Rochow. The Beringia 2005 expedition wasorganized by the Swedish Polar Research Secretariat andLund University, and participation by S.A. and G.H. wasfinanced by a grant from the Swedish Research Council.Ramon Hegedus is a SCAR 6CI (Scientific Committee onAntarctic Research, International Polar Year’s 6th ContinentInitiative) fellow and holder of a Marie-Curie post-doctoralgrant. He is thankful to SCAR, the International PolarFoundation and the European Commission for theirsupport. Many thanks are due to Dr Jozsef Gal (JacobsUniversity, Bremen, Germany) and Dr Robert Horvath(University of Roskilde, Denmark) for organizing some ofthe psychophysical laboratory experiments in Bremen andRoskilde. We thank Dr David Selmeczi (University ofRoskilde, Denmark) for translating some publications ofThorkild Ramskou from Danish into Hungarian.

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