![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.](/figures/fig-1-deriving-true-compass-directions-with-a-viking-sun-2tei28tr.webp)
Orientation with a Viking sun-compass, a
shadow-stick, and two calcite sunstones
under various weather conditions
Balázs Bernáth,
1,
* Miklós Blahó,
1
Ádám Egri,
1
András Barta,
2
György Kriska,
3,4
and Gábor Horváth
1
1
Environmental Optics Laboratory, Department of Biological Physics, Physical Institute,
Eötvös University, H-1117 Budapest, Pázmány sétány 1, Hungary
2
Estrato Research and Development Ltd., H-1121 Budapest, Mártonlak utca 13, Hungary
3
Group for Methodology in Biology Teaching, Biological Institute, Eötvös University,
H-1117 Budapest, Pázmány sétány 1, Hungary
4
Danube Research Institute, Centre for Ecological Research, Hungarian Academy of Sciences,
A
lkotmány út 2-4, H-2163 Vácrátót, Hungary
*Corresponding author: bbernath@angel.elte.hu
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
p
olarimetry 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.
1. Introduction used primitive solar positioning instruments, or primitive sun-compasses to derive true directions
From the 8th century Vikings dominated the North
from the cast shadow of a gnomon at least from the
Atlantic Ocean and even founded settlements in
10th century [2
–5].
Greenland and Newfoundland without knowing about
A sun-compass is an inverted sun-dial: instead of
the magnetic compass. In 1948 in Uunartoq (Green
following the movement of the shadow tip along a
land) a fragment of a wooden disk—with a diameter
hyperbolic gnomonic line on a fixed dial, the com
of 70 mm carved in with a compass rose, the mark of
pass-dial is rotated until fitting a previously drawn
N
orth, and two gnomonic lines—was found [1]. Based
hyperbolic gnomonic line right under the shadow
on this artifact, it has been hypothesized that Vikings
tip. In this position, the major axis of the hyperbola
points toward the true North. Although gnomonic 1559-128X/13/256185-10$15.00/
0 lines a
inappropriate curve in the morning and in the afternoon
compensate each other [4
]. Sun-compasses with dimensions
identical with that of the Uunartoq artifact were tested by
modern sailors on short-haul trips (maximum about 50
nautical miles) and were found to be remarkably reliable in
clear weather [4
].
Sun-compasses are sensitive to the changing lighting
conditions. Thin cloud layers, dust, smoke, and
atmospheric haze scatter or absorb direct sunlight, shadows
are poorly contrasted when the intensity of direct sunlight
is comparable to or lower than that of scattered skylight.
Under such circumstances a Viking navigator could easily
underestimate the shadow length and derive compass
directions with significant errors. When the sun is
occluded, cast shadows do not appear at all, thus sun-
compasses cannot be used. Vikings are hypothesized to
overcome this problem by crude sky-polarimetry using
mysterious sunstones that are mentioned in the saga o
f
King Olaf the Holy [6].
N
avigation by means of the polarization pattern of the sky
is a natural ability of several animal species (e.g., insects
and birds), but it was also applied on board trans-arctic
flights of the 1960s [6
–8]. Skylight origins mainly from the
Rayleigh scattering of sunlight on atmospheric particles
and is predominantly partially linearly polarized.
Depending on the meteorological conditions and the sola
r
elevation angle, its direction of polarization is more or less
p
erpendicular to the plane of scattering determined by the
sun, the observed celestial point, and the observer [9], and
forms a celestial polarization pattern, whose axis of mirror
symmetry is the solar–antisolar meridian [7
,10]. This
p
attern exists even under thick clouds, although the degree
of polarization of skylight is very lowinsuchsituations[10–
13]. By measuring the direction of polarization of skylight
in two or more celestial points, one can mark out celestial
great circles (perpendicular to the local direction o
f
p
olarization), the intersection point of which provides a
good estimation of the position of the occluded sun [8
].
Sunstones are hypothesized to be dichroic crystals (e.g.,
tourmaline or cordierite) or birefringent crystals (e.g.,
calcite), that can be used to identify the direction of
polarization of skylight [6
,14–16]. Theoretically, Vikings
could use such a primitive skylight polarimetry to locate
the occluded sun and use this information for navigation
[6
,8,15,16]. In modern astrophysics, comparison of the
irradiances of ordinary and extraordinary beams in
birefringent calcite is used to detect extremely weak
polarized light produced in the atmospheres of exoplanets
[17
]. Theoretically, the contrast sensitivity of the human
eye should allow human observers to use calcite sunstones
to identify the direction of polarization of light with an
accuracy of 1°, even if its degree of polarization is very low
[15
]. Thus, sky-polariemtric navigation under an overcast
sky is theoretically possible. Roslund and Beckman [18
]
criticized this theory, suggesting that the sun can be located
in most situations without skylight polarimetry, but the
psychophysical survey by Barta et al. [19
] proved that such
estimations are unreliable. The actual threshold minimum
degree of polarization allowing reliable non-instrumental
measurement of the direction of skylight polarization in the
field is still not known. The atmospheric prerequisites and
the optical elements of such a sky-polarimetric navigational
method have been studied earlier [11
–13,15], but complete
procedures of using sunstones with sun-compasses have not
been tested in practice.
In this work, we suggest that a simple tool, called “shadow-
stick” could substitute the gnomon shadow at given solar
elevation angles and could extend the usability of sun-
compasses (Fig. 1
). This shadow-stick should be small and
elongated in general, but can take many forms, like a
carved fang or metal pendant, easy to be carried on the
body of the navigator. It must be provided with a series o
f
sockets corresponding to discreet solar elevation angles.
Positions of the sockets could be appointed eithe
r
empirically or by calculation. Theoretically, a set composed
of a sun-compass, a shadow-stick, and two sunstones [Figs.
1C
, 1D,and 2] could form an ideally compact all-weather
solar navigation toolkit with the potential of replacing a
magnetic compass.
To reveal the accuracy and reliability of such a navigation
toolkit under clear, partially cloudy, or totally overcast
skies, we performed an extensive series of field tests. We
demonstrate here that the solar elevation angle can be
assessed with satisfying accuracy even without dedicated
instruments, and the shadow-stick functions
p
erfectly in
situations when the sun position can be estimated by the
naked eye. However, locating the sun in overcast skies with
the aid of sunstones was found to be too inaccurate for
navigation purposes.
2. Materials and Methods
A series of orientation trials was performed with a
navigational toolkit consisting of a pendulum-levelled sun-
compass, two calcite sunstones, and a shadow-stick (Figs. 1
and 2) in the field in Northern Hungary (47° 28’ N, 19° 3’
E). The toolkit was operated by three male members of the
Environmental Optics Laboratory (Eötvös University,
Budapest, Hungary), aged between 26 and 37 years. All o
f
them were highly experienced in measuring and analyzing
skylight polarization patterns, thus they were eligible to
impersonate experienced Viking navigators.
The accuracy of orienting the sun-compass was measured
in the field under weather conditions of four categories,
classified on the basis of available information on the sun
position. Category 1: Under illumination dominated by
direct sunlight the cast shadow of the gnomon is
continuously clearly visible. Then, navigation with the sun-
compass was possible by fitting the tip of the gnomon
shadow to the gnomonic line. Category 2: The gnomon
shadow is not formed, but one can see either the faint sun
disk behind the clouds, or the sunbursts formed by the
atmospheric Tyndall effect unambiguously mark
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.
th
e sun position. Then, the sun position was esti-The exact position of the sun disk cannot be estimated by the naked eye,
and a shadow-stick was mated by the naked eye, but the intensity pattern of used to replace the cast gnomon shadow. To
measure the sky unambiguously marks the solar hemisphere. the accuracy of orienting the sun-compass with the Then, the
sun position or the antisolar point was aid of sunstones, the position of the antisolar point estimated with two calcite
sunstones, and a was estimated to ensure that the observers are not shadow-stick was used to replace the cast shadow.
influenced by the sight of the sun disk. Category 3: Category 4: When the sky is totally overcast, neither
the exact sun position, nor the solar hemisphere can be
identified by the naked eye. Then, the sun position or the
antisolar point was estimated with two calcite sunstones,
and a shadow-stick was used to replace the cast shadow.
Measurements in weather situations 1 and 2 were carried
out at a single location on an undisturbed platform of a
steel-frame building of the Eötvös University in Budapes
t
(47° 28.43’ N, 19° 3.69’ E). The local magnetic compass
deviation of 6.8° was calculated using the method o
f
Indian circles and was taken into consideration during the
evaluation. All measurements in weather categories 3 and 4
were carried out at different locations in four subur
b
an
areas of Budapest (47° 24.5’ N, 19° 5.5’ E; 47° 25.5’
N
, 19°
0.6’ E; 47° 35’ N, 19° 6.1’ E; 47° 23.8’ N, 18° 53.5’ E)
characterized by irregular street networks. The 2.8°
average magnetic compass deviation in Northern Hungary
[20
] was considered at these localities. Between
measurements, the observers were blindfolded and
transported by a car that followed a zig–zag path. After all
observers signaled that they had lost their sense o
f
direction, the driver chose a new locality, from which at
least about 80% of the sky was visible. All observers made
an independent educated guess on the direction of North;
the adequate directional angles were recorded by the driver.
Then, they independently estimated the sun position using
two calcite sunstones and oriented the sun-compass using
the shadow-stick. The real solar elevation angle was
obtained from the data service of the USNO Naval
Oceanography Portal [21].
Two polished rhombohedra of Icelandic spar (that is a
transparent variety of calcite) sized 5 cm × 5 cm × 2.5 cm
was used as sunstones (Figs. 2
–4). These crystals were
purchased in set from a specialized mineral bourse trader
(Kristálycentrum Kft., Budapest, Hungary;
www.kristalycentrum.hu), thus their exact geographical
origin was untraceable. A calcite crystal with similar size
was found between navigational instruments in a 16th
century shipwreck at Alderney [16
]. All faces of ou
r
rhombohedra were covered by a black adhesive carton
p
aper only leaving clear a 3 mm wide incoming slit and a 6
mm wide exit slit, both being perpendicular to the
crystallographic c-axis of the calcite (Figs. 2
and 4). The
greatest faces of the rhombohedra were used as incoming
and exit faces. Partially linearly polarized light entering the
b
irefringent calcite through the incoming slit is separated
into totally linearly polarized ordinary and extraordinary
rays with a walk-off distance of 3 mm, and these rays form
two parallel images in the entrance slit on the exit face
(Figs. 2
and 4). The irradiances of the two slit images are
proportional to the square of sine or cosine of the angle
enclosed by the slit axis and the direction of polarization o
f
light entering the crystal. The slit images are equally bright
when the axis of the entrance slit encloses 45° with the
direction of polarization of incoming light. This direction
was marked on the exit face [Fig. 4B
]. Maximal and
minimal irradiances of
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
p
olarized skylight entering the lower slit is separated into totally linearly
p
olarized ordinary and extraordinary rays and form two images of the
lower slit in the upper slit in the exit face.
slit images are transposed when the rhombohedron is
rotated by 90° (Fig. 4
). Observers measured the direction o
f
polarization of skylight by rotating the crystals—further on
“sunstones”—until reaching equally bright slit images.
In weather situations 2, 3, and 4, the position of either the
sun or the antisolar point was estimated by performing
skylight polarimetry using the two sunstones (Figs. 4
). The
sun position can be estimated by appointing the celestial
great circles perpendicular to the local direction o
f
p
olarization of skylight in sky patches characterized with
high degree of polarization. The intersection of such two
great circles or the error triangle defined by three great
circles appoints the sun position. The accuracy of such an
estimation is influenced chiefly by the angular distance
b
etween the sun and the clear sky patches, the intersecting
angle κ of the appointed great circles and the accuracy o
f
determining the direction of polarization.
Totally overcast skies are characterized by low and
fluctuating degrees of polarization, and the observers must
systematically look up points with relatively high degrees
of polarization (Fig. 3
). According to the single-scattering
Rayleigh model, the degree of polarization of skylight is
the highest in a zone d, at 90° from the sun. To locate zone
d, a horizontal zone b of the sky about 20°–30° above the
horizon is scanned with a sunstone held with its slits
enclosing about 45° with the local meridian a until finding
the greatest contrast of the two slit images [Fig. 3A
]. This
method provides a fair contrast of the slit images in most
situations when the sun is not close to the horizon or the
zenith. Held toward the point of b characterized by the
highest contrast of slit images, the sunstone is rotated to
reach equal intensities of the slit images to read the
direction of polarization of skylight [Fig. 3B
]. Then, the
sun is located along
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 fai
r
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 o
f
skylight. The sun is located along the celestial great circle c that lies in the
p
lane of scattering and is perpendicular to d. C The zone of highly
p
olarized 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-
p
olarimetry is chosen along d. The second sun-stone 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
p
oint. Both celestial positions can be used to estimate the solar elevation
angle Θ
S 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. Photographs of a calcite crystal prepared as a sunstone and
transilluminated by totally linearly polarized light. A If the direction o
f
p
olarization 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 o
f
p
olarization is rotated by 45°, the slit i
m
ages 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.
the celestial great circle c perpendicular to zone d, lying in
the plane of scattering. The zone d of highly polarized
skylight should be verified [Fig. 3C
]. The sunstone is
turned until its slit is parallel to the assumed zone d, then i
t
is moved along d and also along
c. Along d the contrast of slit images is expected to
fluctuate due to the changing degree of skylight
polarization, but to remain perceivable. Along c the
contrast of slit images is expected to decrease while the
sunstone deviates from d. After verifying zone d, a second
patch appropriate for sky-polarimetry is chosen along d
[Fig. 3D
]. The second sunstone is used to identify the
direction of polarization in this point to mark out the great
circle e. The observer sweeps c and e along with the
sunstones until pointing toward their intersections, which is
the estimated position of the sun or the antisolar point.
Either point can be used to estimate the solar elevation
angle Θ
S and the position of the solar meridian [Fig. 1B].
The estimation is more reliable if the intersecting angle κ o
f
c and e is close to 90°.
In weather situations 2, 3, and 4 the elevation angle of the
sun or the antisolar point was estimated