Abstract

It is widely accepted that Vikings used sun-compasses to derive true directions from the cast shadow of a gnomon. It has been hypothesized that when a cast shadow was not formed, Viking navigators relied on crude skylight polarimetry with the aid of dichroic or birefringent crystals, called “sunstones.” We demonstrate here that a simple tool, that we call “shadow-stick,” could have allowed orientation by a sun-compass with satisfying accuracy when shadows were not formed, but the sun position could have reliably been estimated. In field tests, we performed orientation trials with a set composed of a sun-compass, two calcite sunstones, and a shadow-stick. We show here that such a set could have been an effective orientation tool for Vikings only when clear, blue patches of the sky were visible.

© 2013 Optical Society of America

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References

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  1. E. G. Taylor, W. E. May, R. B. Motzo, and T. C. Letherbridge, “A Norse bearing-dial?,” J. Navig. 7, 78–84 (1954).
    [CrossRef]
  2. C. V. Sølver, “The discovery of an early bearing-dial,” J. Navig. 6, 294–296 (1953).
    [CrossRef]
  3. B. E. Schaefer, “Vikings and polarization sun-dials,” Sky Telesc. 93, 91–94 (1997).
  4. S. Thirslund, Viking Navigation: Sun-Compass Guided Norsemen First to America (Gullanders Bogtrykkeri a-s, Skjern, 2001).
  5. B. Bernáth, M. Blahó, A. Egri, A. Barta, and G. Horváth, “An alternative interpretation of the Viking sundial artefact: an instrument to determine latitude and local noon,” Proc. R. Soc. A 469, 20130021 (2013).
    [CrossRef]
  6. T. Ramskou, “Solstenen,” Skalk 2, 16–17 (1969).
  7. G. Horváth and D. Varjú, Polarized Light in Animal Vision—Polarization Patterns in Nature (Springer-Verlag, 2004).
  8. G. Horváth, A. Barta, I. Pomozi, B. Suhai, R. Hegedüs, S. Åkesson, B. Meyer-Rochow, and R. Wehner, “On the trail of Vikings with polarized skylight: experimental study of the atmospheric optical prerequisites allowing polarimetric navigation by Viking seafarers,” Phil. Trans. R. Soc. B 366, 772–782 (2011).
    [CrossRef]
  9. B. Suhai and G. Horváth, “How well does the Rayleigh model describe the E-vector distribution of skylight in clear and cloudy conditions? A full-sky polarimetric study,” J. Opt. Soc. Am. A 21, 1669–1676 (2004).
    [CrossRef]
  10. K. L. Coulson, Polarization and Intensity of Light in the Atmosphere (A. Deepak Publishing, 1988).
  11. I. Pomozi, G. Horváth, and R. Wehner, “How the clear-sky angle of polarization pattern continues underneath clouds: full-sky measurements and implications for animal orientation,” J. Exp. Biol. 204, 2933–2942 (2001).
  12. R. Hegedüs, S. Åkesson, R. Wehner, and G. Horváth, “Could Vikings have navigated under foggy and cloudy conditions by skylight polarization? On the atmospheric optical prerequisites of polarimetric Viking navigation under foggy and cloudy skies,” Proc. R. Soc. A 463, 1081–1095 (2007).
    [CrossRef]
  13. R. Hegedüs, S. Åkesson, and G. Horváth, “Polarization patterns of thick clouds: overcast skies have distribution of the angle of polarization similar to that of clear skies,” J. Opt. Soc. Am. A 24, 2347–2356 (2007).
    [CrossRef]
  14. C. Brosseau, Fundamentals of Polarized Light (Wiley, 1998).
  15. G. Ropars, G. Gorre, A. Le Floch, J. Enoch, and V. Lakshminarayanan, “A depolarizer as a possible precise sunstone for Viking navigation by polarized skylight,” Proc. R. Soc. A 468, 671–684 (2012).
    [CrossRef]
  16. A. Le Floch, G. Ropars, J. Lucas, S. Wright, T. Davenport, M. Corfield, and M. Harrisson, “The sixteenth century Alderney crystal: a calcite as an efficient reference optical compass?,” Proc. R. Soc. A 469, 20120651 (2013).
    [CrossRef]
  17. S. V. Berdyugina, A. V. Berdyugin, D. M. Fluri, and V. Piirola, “Polarized reflected light from the exoplanet HD189733b: first multicolor observations and confirmation of detection,” Astrophys. J. Lett. 728, L6 (2011)..
    [CrossRef]
  18. C. Roslund and C. Beckman, “Disputing Viking navigation by polarized skylight,” Appl. Opt. 33, 4754–4755 (1994).
    [CrossRef]
  19. A. Barta, G. Horváth, and V. B. Meyer-Rochow, “Psychophysical study of the visual sun location in pictures of cloudy and twilight skies inspired by Viking navigation,” J. Opt. Soc. Am. A 22, 1023–1034 (2005).
    [CrossRef]
  20. IGCRS (Institute of Geodesy, Cartography, and Remote Sensing) http://www.fomi.hu/portal_en/index.php/home .
  21. United States Naval Observatory (USNO), Naval Oceanography Portal: http://aa.usno.navy.mil/data/docs/AltAz.php .
  22. J. H. Zar, Biostatistical Analysis (Pearson Prentice Hall, 2010).
  23. L. K. Karlsen, Secrets of the Viking Navigators (One Earth, 2003).

2013 (2)

B. Bernáth, M. Blahó, A. Egri, A. Barta, and G. Horváth, “An alternative interpretation of the Viking sundial artefact: an instrument to determine latitude and local noon,” Proc. R. Soc. A 469, 20130021 (2013).
[CrossRef]

A. Le Floch, G. Ropars, J. Lucas, S. Wright, T. Davenport, M. Corfield, and M. Harrisson, “The sixteenth century Alderney crystal: a calcite as an efficient reference optical compass?,” Proc. R. Soc. A 469, 20120651 (2013).
[CrossRef]

2012 (1)

G. Ropars, G. Gorre, A. Le Floch, J. Enoch, and V. Lakshminarayanan, “A depolarizer as a possible precise sunstone for Viking navigation by polarized skylight,” Proc. R. Soc. A 468, 671–684 (2012).
[CrossRef]

2011 (2)

S. V. Berdyugina, A. V. Berdyugin, D. M. Fluri, and V. Piirola, “Polarized reflected light from the exoplanet HD189733b: first multicolor observations and confirmation of detection,” Astrophys. J. Lett. 728, L6 (2011)..
[CrossRef]

G. Horváth, A. Barta, I. Pomozi, B. Suhai, R. Hegedüs, S. Åkesson, B. Meyer-Rochow, and R. Wehner, “On the trail of Vikings with polarized skylight: experimental study of the atmospheric optical prerequisites allowing polarimetric navigation by Viking seafarers,” Phil. Trans. R. Soc. B 366, 772–782 (2011).
[CrossRef]

2007 (2)

R. Hegedüs, S. Åkesson, R. Wehner, and G. Horváth, “Could Vikings have navigated under foggy and cloudy conditions by skylight polarization? On the atmospheric optical prerequisites of polarimetric Viking navigation under foggy and cloudy skies,” Proc. R. Soc. A 463, 1081–1095 (2007).
[CrossRef]

R. Hegedüs, S. Åkesson, and G. Horváth, “Polarization patterns of thick clouds: overcast skies have distribution of the angle of polarization similar to that of clear skies,” J. Opt. Soc. Am. A 24, 2347–2356 (2007).
[CrossRef]

2005 (1)

2004 (1)

2001 (1)

I. Pomozi, G. Horváth, and R. Wehner, “How the clear-sky angle of polarization pattern continues underneath clouds: full-sky measurements and implications for animal orientation,” J. Exp. Biol. 204, 2933–2942 (2001).

1997 (1)

B. E. Schaefer, “Vikings and polarization sun-dials,” Sky Telesc. 93, 91–94 (1997).

1994 (1)

1969 (1)

T. Ramskou, “Solstenen,” Skalk 2, 16–17 (1969).

1954 (1)

E. G. Taylor, W. E. May, R. B. Motzo, and T. C. Letherbridge, “A Norse bearing-dial?,” J. Navig. 7, 78–84 (1954).
[CrossRef]

1953 (1)

C. V. Sølver, “The discovery of an early bearing-dial,” J. Navig. 6, 294–296 (1953).
[CrossRef]

Åkesson, S.

G. Horváth, A. Barta, I. Pomozi, B. Suhai, R. Hegedüs, S. Åkesson, B. Meyer-Rochow, and R. Wehner, “On the trail of Vikings with polarized skylight: experimental study of the atmospheric optical prerequisites allowing polarimetric navigation by Viking seafarers,” Phil. Trans. R. Soc. B 366, 772–782 (2011).
[CrossRef]

R. Hegedüs, S. Åkesson, R. Wehner, and G. Horváth, “Could Vikings have navigated under foggy and cloudy conditions by skylight polarization? On the atmospheric optical prerequisites of polarimetric Viking navigation under foggy and cloudy skies,” Proc. R. Soc. A 463, 1081–1095 (2007).
[CrossRef]

R. Hegedüs, S. Åkesson, and G. Horváth, “Polarization patterns of thick clouds: overcast skies have distribution of the angle of polarization similar to that of clear skies,” J. Opt. Soc. Am. A 24, 2347–2356 (2007).
[CrossRef]

Barta, A.

B. Bernáth, M. Blahó, A. Egri, A. Barta, and G. Horváth, “An alternative interpretation of the Viking sundial artefact: an instrument to determine latitude and local noon,” Proc. R. Soc. A 469, 20130021 (2013).
[CrossRef]

G. Horváth, A. Barta, I. Pomozi, B. Suhai, R. Hegedüs, S. Åkesson, B. Meyer-Rochow, and R. Wehner, “On the trail of Vikings with polarized skylight: experimental study of the atmospheric optical prerequisites allowing polarimetric navigation by Viking seafarers,” Phil. Trans. R. Soc. B 366, 772–782 (2011).
[CrossRef]

A. Barta, G. Horváth, and V. B. Meyer-Rochow, “Psychophysical study of the visual sun location in pictures of cloudy and twilight skies inspired by Viking navigation,” J. Opt. Soc. Am. A 22, 1023–1034 (2005).
[CrossRef]

Beckman, C.

Berdyugin, A. V.

S. V. Berdyugina, A. V. Berdyugin, D. M. Fluri, and V. Piirola, “Polarized reflected light from the exoplanet HD189733b: first multicolor observations and confirmation of detection,” Astrophys. J. Lett. 728, L6 (2011)..
[CrossRef]

Berdyugina, S. V.

S. V. Berdyugina, A. V. Berdyugin, D. M. Fluri, and V. Piirola, “Polarized reflected light from the exoplanet HD189733b: first multicolor observations and confirmation of detection,” Astrophys. J. Lett. 728, L6 (2011)..
[CrossRef]

Bernáth, B.

B. Bernáth, M. Blahó, A. Egri, A. Barta, and G. Horváth, “An alternative interpretation of the Viking sundial artefact: an instrument to determine latitude and local noon,” Proc. R. Soc. A 469, 20130021 (2013).
[CrossRef]

Blahó, M.

B. Bernáth, M. Blahó, A. Egri, A. Barta, and G. Horváth, “An alternative interpretation of the Viking sundial artefact: an instrument to determine latitude and local noon,” Proc. R. Soc. A 469, 20130021 (2013).
[CrossRef]

Brosseau, C.

C. Brosseau, Fundamentals of Polarized Light (Wiley, 1998).

Corfield, M.

A. Le Floch, G. Ropars, J. Lucas, S. Wright, T. Davenport, M. Corfield, and M. Harrisson, “The sixteenth century Alderney crystal: a calcite as an efficient reference optical compass?,” Proc. R. Soc. A 469, 20120651 (2013).
[CrossRef]

Coulson, K. L.

K. L. Coulson, Polarization and Intensity of Light in the Atmosphere (A. Deepak Publishing, 1988).

Davenport, T.

A. Le Floch, G. Ropars, J. Lucas, S. Wright, T. Davenport, M. Corfield, and M. Harrisson, “The sixteenth century Alderney crystal: a calcite as an efficient reference optical compass?,” Proc. R. Soc. A 469, 20120651 (2013).
[CrossRef]

Egri, A.

B. Bernáth, M. Blahó, A. Egri, A. Barta, and G. Horváth, “An alternative interpretation of the Viking sundial artefact: an instrument to determine latitude and local noon,” Proc. R. Soc. A 469, 20130021 (2013).
[CrossRef]

Enoch, J.

G. Ropars, G. Gorre, A. Le Floch, J. Enoch, and V. Lakshminarayanan, “A depolarizer as a possible precise sunstone for Viking navigation by polarized skylight,” Proc. R. Soc. A 468, 671–684 (2012).
[CrossRef]

Fluri, D. M.

S. V. Berdyugina, A. V. Berdyugin, D. M. Fluri, and V. Piirola, “Polarized reflected light from the exoplanet HD189733b: first multicolor observations and confirmation of detection,” Astrophys. J. Lett. 728, L6 (2011)..
[CrossRef]

Gorre, G.

G. Ropars, G. Gorre, A. Le Floch, J. Enoch, and V. Lakshminarayanan, “A depolarizer as a possible precise sunstone for Viking navigation by polarized skylight,” Proc. R. Soc. A 468, 671–684 (2012).
[CrossRef]

Harrisson, M.

A. Le Floch, G. Ropars, J. Lucas, S. Wright, T. Davenport, M. Corfield, and M. Harrisson, “The sixteenth century Alderney crystal: a calcite as an efficient reference optical compass?,” Proc. R. Soc. A 469, 20120651 (2013).
[CrossRef]

Hegedüs, R.

G. Horváth, A. Barta, I. Pomozi, B. Suhai, R. Hegedüs, S. Åkesson, B. Meyer-Rochow, and R. Wehner, “On the trail of Vikings with polarized skylight: experimental study of the atmospheric optical prerequisites allowing polarimetric navigation by Viking seafarers,” Phil. Trans. R. Soc. B 366, 772–782 (2011).
[CrossRef]

R. Hegedüs, S. Åkesson, R. Wehner, and G. Horváth, “Could Vikings have navigated under foggy and cloudy conditions by skylight polarization? On the atmospheric optical prerequisites of polarimetric Viking navigation under foggy and cloudy skies,” Proc. R. Soc. A 463, 1081–1095 (2007).
[CrossRef]

R. Hegedüs, S. Åkesson, and G. Horváth, “Polarization patterns of thick clouds: overcast skies have distribution of the angle of polarization similar to that of clear skies,” J. Opt. Soc. Am. A 24, 2347–2356 (2007).
[CrossRef]

Horváth, G.

B. Bernáth, M. Blahó, A. Egri, A. Barta, and G. Horváth, “An alternative interpretation of the Viking sundial artefact: an instrument to determine latitude and local noon,” Proc. R. Soc. A 469, 20130021 (2013).
[CrossRef]

G. Horváth, A. Barta, I. Pomozi, B. Suhai, R. Hegedüs, S. Åkesson, B. Meyer-Rochow, and R. Wehner, “On the trail of Vikings with polarized skylight: experimental study of the atmospheric optical prerequisites allowing polarimetric navigation by Viking seafarers,” Phil. Trans. R. Soc. B 366, 772–782 (2011).
[CrossRef]

R. Hegedüs, S. Åkesson, R. Wehner, and G. Horváth, “Could Vikings have navigated under foggy and cloudy conditions by skylight polarization? On the atmospheric optical prerequisites of polarimetric Viking navigation under foggy and cloudy skies,” Proc. R. Soc. A 463, 1081–1095 (2007).
[CrossRef]

R. Hegedüs, S. Åkesson, and G. Horváth, “Polarization patterns of thick clouds: overcast skies have distribution of the angle of polarization similar to that of clear skies,” J. Opt. Soc. Am. A 24, 2347–2356 (2007).
[CrossRef]

A. Barta, G. Horváth, and V. B. Meyer-Rochow, “Psychophysical study of the visual sun location in pictures of cloudy and twilight skies inspired by Viking navigation,” J. Opt. Soc. Am. A 22, 1023–1034 (2005).
[CrossRef]

B. Suhai and G. Horváth, “How well does the Rayleigh model describe the E-vector distribution of skylight in clear and cloudy conditions? A full-sky polarimetric study,” J. Opt. Soc. Am. A 21, 1669–1676 (2004).
[CrossRef]

I. Pomozi, G. Horváth, and R. Wehner, “How the clear-sky angle of polarization pattern continues underneath clouds: full-sky measurements and implications for animal orientation,” J. Exp. Biol. 204, 2933–2942 (2001).

G. Horváth and D. Varjú, Polarized Light in Animal Vision—Polarization Patterns in Nature (Springer-Verlag, 2004).

Karlsen, L. K.

L. K. Karlsen, Secrets of the Viking Navigators (One Earth, 2003).

Lakshminarayanan, V.

G. Ropars, G. Gorre, A. Le Floch, J. Enoch, and V. Lakshminarayanan, “A depolarizer as a possible precise sunstone for Viking navigation by polarized skylight,” Proc. R. Soc. A 468, 671–684 (2012).
[CrossRef]

Le Floch, A.

A. Le Floch, G. Ropars, J. Lucas, S. Wright, T. Davenport, M. Corfield, and M. Harrisson, “The sixteenth century Alderney crystal: a calcite as an efficient reference optical compass?,” Proc. R. Soc. A 469, 20120651 (2013).
[CrossRef]

G. Ropars, G. Gorre, A. Le Floch, J. Enoch, and V. Lakshminarayanan, “A depolarizer as a possible precise sunstone for Viking navigation by polarized skylight,” Proc. R. Soc. A 468, 671–684 (2012).
[CrossRef]

Letherbridge, T. C.

E. G. Taylor, W. E. May, R. B. Motzo, and T. C. Letherbridge, “A Norse bearing-dial?,” J. Navig. 7, 78–84 (1954).
[CrossRef]

Lucas, J.

A. Le Floch, G. Ropars, J. Lucas, S. Wright, T. Davenport, M. Corfield, and M. Harrisson, “The sixteenth century Alderney crystal: a calcite as an efficient reference optical compass?,” Proc. R. Soc. A 469, 20120651 (2013).
[CrossRef]

May, W. E.

E. G. Taylor, W. E. May, R. B. Motzo, and T. C. Letherbridge, “A Norse bearing-dial?,” J. Navig. 7, 78–84 (1954).
[CrossRef]

Meyer-Rochow, B.

G. Horváth, A. Barta, I. Pomozi, B. Suhai, R. Hegedüs, S. Åkesson, B. Meyer-Rochow, and R. Wehner, “On the trail of Vikings with polarized skylight: experimental study of the atmospheric optical prerequisites allowing polarimetric navigation by Viking seafarers,” Phil. Trans. R. Soc. B 366, 772–782 (2011).
[CrossRef]

Meyer-Rochow, V. B.

Motzo, R. B.

E. G. Taylor, W. E. May, R. B. Motzo, and T. C. Letherbridge, “A Norse bearing-dial?,” J. Navig. 7, 78–84 (1954).
[CrossRef]

Piirola, V.

S. V. Berdyugina, A. V. Berdyugin, D. M. Fluri, and V. Piirola, “Polarized reflected light from the exoplanet HD189733b: first multicolor observations and confirmation of detection,” Astrophys. J. Lett. 728, L6 (2011)..
[CrossRef]

Pomozi, I.

G. Horváth, A. Barta, I. Pomozi, B. Suhai, R. Hegedüs, S. Åkesson, B. Meyer-Rochow, and R. Wehner, “On the trail of Vikings with polarized skylight: experimental study of the atmospheric optical prerequisites allowing polarimetric navigation by Viking seafarers,” Phil. Trans. R. Soc. B 366, 772–782 (2011).
[CrossRef]

I. Pomozi, G. Horváth, and R. Wehner, “How the clear-sky angle of polarization pattern continues underneath clouds: full-sky measurements and implications for animal orientation,” J. Exp. Biol. 204, 2933–2942 (2001).

Ramskou, T.

T. Ramskou, “Solstenen,” Skalk 2, 16–17 (1969).

Ropars, G.

A. Le Floch, G. Ropars, J. Lucas, S. Wright, T. Davenport, M. Corfield, and M. Harrisson, “The sixteenth century Alderney crystal: a calcite as an efficient reference optical compass?,” Proc. R. Soc. A 469, 20120651 (2013).
[CrossRef]

G. Ropars, G. Gorre, A. Le Floch, J. Enoch, and V. Lakshminarayanan, “A depolarizer as a possible precise sunstone for Viking navigation by polarized skylight,” Proc. R. Soc. A 468, 671–684 (2012).
[CrossRef]

Roslund, C.

Schaefer, B. E.

B. E. Schaefer, “Vikings and polarization sun-dials,” Sky Telesc. 93, 91–94 (1997).

Sølver, C. V.

C. V. Sølver, “The discovery of an early bearing-dial,” J. Navig. 6, 294–296 (1953).
[CrossRef]

Suhai, B.

G. Horváth, A. Barta, I. Pomozi, B. Suhai, R. Hegedüs, S. Åkesson, B. Meyer-Rochow, and R. Wehner, “On the trail of Vikings with polarized skylight: experimental study of the atmospheric optical prerequisites allowing polarimetric navigation by Viking seafarers,” Phil. Trans. R. Soc. B 366, 772–782 (2011).
[CrossRef]

B. Suhai and G. Horváth, “How well does the Rayleigh model describe the E-vector distribution of skylight in clear and cloudy conditions? A full-sky polarimetric study,” J. Opt. Soc. Am. A 21, 1669–1676 (2004).
[CrossRef]

Taylor, E. G.

E. G. Taylor, W. E. May, R. B. Motzo, and T. C. Letherbridge, “A Norse bearing-dial?,” J. Navig. 7, 78–84 (1954).
[CrossRef]

Thirslund, S.

S. Thirslund, Viking Navigation: Sun-Compass Guided Norsemen First to America (Gullanders Bogtrykkeri a-s, Skjern, 2001).

Varjú, D.

G. Horváth and D. Varjú, Polarized Light in Animal Vision—Polarization Patterns in Nature (Springer-Verlag, 2004).

Wehner, R.

G. Horváth, A. Barta, I. Pomozi, B. Suhai, R. Hegedüs, S. Åkesson, B. Meyer-Rochow, and R. Wehner, “On the trail of Vikings with polarized skylight: experimental study of the atmospheric optical prerequisites allowing polarimetric navigation by Viking seafarers,” Phil. Trans. R. Soc. B 366, 772–782 (2011).
[CrossRef]

R. Hegedüs, S. Åkesson, R. Wehner, and G. Horváth, “Could Vikings have navigated under foggy and cloudy conditions by skylight polarization? On the atmospheric optical prerequisites of polarimetric Viking navigation under foggy and cloudy skies,” Proc. R. Soc. A 463, 1081–1095 (2007).
[CrossRef]

I. Pomozi, G. Horváth, and R. Wehner, “How the clear-sky angle of polarization pattern continues underneath clouds: full-sky measurements and implications for animal orientation,” J. Exp. Biol. 204, 2933–2942 (2001).

Wright, S.

A. Le Floch, G. Ropars, J. Lucas, S. Wright, T. Davenport, M. Corfield, and M. Harrisson, “The sixteenth century Alderney crystal: a calcite as an efficient reference optical compass?,” Proc. R. Soc. A 469, 20120651 (2013).
[CrossRef]

Zar, J. H.

J. H. Zar, Biostatistical Analysis (Pearson Prentice Hall, 2010).

Appl. Opt. (1)

Astrophys. J. Lett. (1)

S. V. Berdyugina, A. V. Berdyugin, D. M. Fluri, and V. Piirola, “Polarized reflected light from the exoplanet HD189733b: first multicolor observations and confirmation of detection,” Astrophys. J. Lett. 728, L6 (2011)..
[CrossRef]

J. Exp. Biol. (1)

I. Pomozi, G. Horváth, and R. Wehner, “How the clear-sky angle of polarization pattern continues underneath clouds: full-sky measurements and implications for animal orientation,” J. Exp. Biol. 204, 2933–2942 (2001).

J. Navig. (2)

E. G. Taylor, W. E. May, R. B. Motzo, and T. C. Letherbridge, “A Norse bearing-dial?,” J. Navig. 7, 78–84 (1954).
[CrossRef]

C. V. Sølver, “The discovery of an early bearing-dial,” J. Navig. 6, 294–296 (1953).
[CrossRef]

J. Opt. Soc. Am. A (3)

Phil. Trans. R. Soc. B (1)

G. Horváth, A. Barta, I. Pomozi, B. Suhai, R. Hegedüs, S. Åkesson, B. Meyer-Rochow, and R. Wehner, “On the trail of Vikings with polarized skylight: experimental study of the atmospheric optical prerequisites allowing polarimetric navigation by Viking seafarers,” Phil. Trans. R. Soc. B 366, 772–782 (2011).
[CrossRef]

Proc. R. Soc. A (4)

B. Bernáth, M. Blahó, A. Egri, A. Barta, and G. Horváth, “An alternative interpretation of the Viking sundial artefact: an instrument to determine latitude and local noon,” Proc. R. Soc. A 469, 20130021 (2013).
[CrossRef]

R. Hegedüs, S. Åkesson, R. Wehner, and G. Horváth, “Could Vikings have navigated under foggy and cloudy conditions by skylight polarization? On the atmospheric optical prerequisites of polarimetric Viking navigation under foggy and cloudy skies,” Proc. R. Soc. A 463, 1081–1095 (2007).
[CrossRef]

G. Ropars, G. Gorre, A. Le Floch, J. Enoch, and V. Lakshminarayanan, “A depolarizer as a possible precise sunstone for Viking navigation by polarized skylight,” Proc. R. Soc. A 468, 671–684 (2012).
[CrossRef]

A. Le Floch, G. Ropars, J. Lucas, S. Wright, T. Davenport, M. Corfield, and M. Harrisson, “The sixteenth century Alderney crystal: a calcite as an efficient reference optical compass?,” Proc. R. Soc. A 469, 20120651 (2013).
[CrossRef]

Skalk (1)

T. Ramskou, “Solstenen,” Skalk 2, 16–17 (1969).

Sky Telesc. (1)

B. E. Schaefer, “Vikings and polarization sun-dials,” Sky Telesc. 93, 91–94 (1997).

Other (8)

S. Thirslund, Viking Navigation: Sun-Compass Guided Norsemen First to America (Gullanders Bogtrykkeri a-s, Skjern, 2001).

G. Horváth and D. Varjú, Polarized Light in Animal Vision—Polarization Patterns in Nature (Springer-Verlag, 2004).

K. L. Coulson, Polarization and Intensity of Light in the Atmosphere (A. Deepak Publishing, 1988).

C. Brosseau, Fundamentals of Polarized Light (Wiley, 1998).

IGCRS (Institute of Geodesy, Cartography, and Remote Sensing) http://www.fomi.hu/portal_en/index.php/home .

United States Naval Observatory (USNO), Naval Oceanography Portal: http://aa.usno.navy.mil/data/docs/AltAz.php .

J. H. Zar, Biostatistical Analysis (Pearson Prentice Hall, 2010).

L. K. Karlsen, Secrets of the Viking Navigators (One Earth, 2003).

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Figures (6)

Fig. 1.
Fig. 1.

Deriving true compass directions with a Viking sun-compass, a shadow-stick and sunstones. A Under clear skies the gnomon casts a clear sharp shadow on the horizontal dial of a levelled sun-compass. Navigation is possible without auxiliary tools. B A cast shadow cannot be seen when the sun is occluded. The Viking navigator must estimate the elevation and azimuth angles of the sun. These data can be gained also by estimating the position of the antisolar point. C A shadow-stick is a small item provided with a series of sockets representing various solar elevation angles. Since the sockets must not overlap, the smallest resolution of elevation angles is determined by the dimensions of the shadow-stick and the diameter of the sockets. D To derive true compass directions with a shadow-stick and a sun-compass, the socket on the shadow-stick corresponding to a given solar elevation is applied on the gnomon tip, and then the end of the stick is turned to point toward the solar meridian. The shadow-stick now replaces the missing or poorly visible cast shadow. To find true compass directions, the navigator must rotate the sun-compass while keeping the shadow-stick still until the gnomonic line fits to the tip of the shadow-stick. E A marked replica Viking round-shield with a diameter of 80 cm was used as a crude sextant with satisfying precision to provide a secondary estimation of the solar elevation as suggested by Captain Jensen [4]. F The estimation of elevation angles of celestial bodies or celestial points with fists and extended arms is a practice frequently used by amateur astronomers. The observer counts the numbers of fists and fingers needed to subtend the arc in question.

Fig. 2.
Fig. 2.

Calcite rhombohedron used to measure the direction of polarization of transmitted skylight. All faces of the crystal are covered by a black adhesive carton paper, and only two narrow slits perpendicular to the crystallographic c-axis of the calcite remain open. Partially linearly polarized skylight entering the lower slit is separated into totally linearly polarized ordinary and extraordinary rays and form two images of the lower slit in the upper slit in the exit face.

Fig. 3.
Fig. 3.

Estimating the sun position with two birefringent calcite sunstones under totally overcast skies. The direction of polarization of skylight is symbolized by a dashed line, the degree of linear polarization of skylight is symbolized by the thickness of the dashed line. A Zone d of high degrees of polarization of skylight is located by scanning along zone b about 20°–30° above the horizon with a sunstone held with its slits enclosing about 45° with the local meridian a. This method provides a fair contrast of the slit images in the sunstone when the sun (S) is not close to the horizon or the zenith (Z). B The sunstone is rotated to reach equal intensities of the slit images to read the direction of polarization of skylight. The sun is located along the celestial great circle c that lies in the plane of scattering and is perpendicular to d. C The zone of highly polarized skylight d is verified. The sunstone is moved along d and then along c with its slit parallel to d. Along d the contrast of slit images is expected to remain perceivable. Along c the contrast of slit images is expected to quickly decrease. D A second patch appropriate for sky-polarimetry is chosen along d. The second sunstone is used to identify the direction of polarization of skylight here to mark out the great circle e. The intersections of c and e mark the positions of the sun and the antisolar point. Both celestial positions can be used to estimate the solar elevation angle ΘS and the direction of the solar meridian sm. The estimation is more reliable, if the intersecting angle κ of c and e is close to 90°.

Fig. 4.
Fig. 4.

Photographs of a calcite crystal prepared as a sunstone and transilluminated by totally linearly polarized light. A If the direction of polarization of the transilluminating light is parallel to the axis of the slits, only one of the slit images can be seen. B When the direction of polarization is rotated by 45°, the slit images will have equal light intensities. When analyzing the polarization properties of skylight, this orientation is appropriate to appoint the direction of the sun. C When the direction of polarization is rotated by further 45°, the other slit image will reach maximal intensity, while the first one darkens.

Fig. 5.
Fig. 5.

Histograms of directional angles considered to be true North (0°) under weather situations A 1; B, C 2; D, E 3; F, G 4. Dashed arrows mark the directions of the mean vectors. The number of individual orientations N, the direction of the mean vector a with 95% confidence limits, the length of the mean vector r and the significance level p of the Rayleigh test for uniformity of distribution of the measured directions are given in the lower half of the dials. A–E The distributions of direction considered as North are significantly directional under weather situations 1, 2, and 3; F and G while they are uniform under weather situation 4. E and G the test values r2 and significance levels p of circular rank correlation tests mark significant correlations between directional angles identified as true North by the observer using the sunstone and shadow-stick, and D and F between their unaided guesses.

Fig. 6.
Fig. 6.

Histograms of the deviation of solar elevation angles estimated in “fists” from real solar elevations under weather situations A 2, B 3, and C 4. A dashed line marks the deviation of 0°, and positive deviations mark overestimations of the solar elevation angle. The number of individual estimations N and the mean deviation ΔΘS with 95% confidence limits are given in the lower half of the dials.

Tables (1)

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Table 1. Discrete Solar Elevation Angle ΘS Measurable with Fist and Fingers, and Corresponding Lengths of the Shadow and Shadow-Stick for a 9.8 mm High Gnomon

Equations (1)

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f(α,n)=1/sin(n·α/4),

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