On the trail of Vikings with polarized skylight: experimental study of the atmospheric optical prerequisites allowing polarimetric navigation by Viking seafarers.

Between AD 900 and AD 1200 Vikings, being able to navigate skillfully across the open sea, were the dominant seafarers of the North Atlantic. When the Sun was shining, geographical north could be determined with a special sundial. However, how the Vikings could have navigated in cloudy or foggy situations, when the Sun's disc was unusable, is still not fully known. A hypothesis was formulated in 1967, which suggested that under foggy or cloudy conditions, Vikings might have been 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 researchers, 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 enigmatic 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, we summarize the results of our own celestial polarization measurements and psychophysical laboratory experiments, in which we studied the atmospheric optical prerequisites of possible sky-polarimetric navigation in Tunisia, Finland, Hungary and the high Arctic.

Wall thickness of gas- and marrow-filled avian long bones: measurements on humeri, femora and tibiotarsi in crows (Corvus corone cornix) and magpies (Pica pica).

We studied how the ratio K of the internal to external diameter of gas- and marrow-filled avian long bones follows the biomechanical optima derived for tubular bones with minimum mass designed to fulfil various mechanical requirements. We evaluated radiographs of numerous humeri, femora and tibiotarsi in Corvus corone cornix and Pica pica. The K-values of the gas-filled humerus (K=0.78+/-0.03) and the marrow-filled femur (K=0.79+/-0.02) in Corvus are practically the same, while K of the marrow-filled tibiotarsus (K=0.71+/-0.04) is significantly smaller. The same is true for the gas-filled humerus (K=0.78+/-0.02) and the marrow-filled femur (K=0.77+/-0.02) and tibiotarsus (K=0.67+/-0.05) in Pica. K in Corvus is slightly larger than K in Pica, but the differences are statistically not significant. The standard deviation DeltaK of the tibiotarsi (DeltaK=0.04-0.05) is approximately two times as large as that of the humeri (DeltaK=0.02-0.03) and femora (DeltaK=0.02) in both species. Accepting the assumption of earlier authors that the ratio Q of the marrow to bone density is 0.5, our data show that the marrow-filled tibiotarsi of Corvus and Pica are optimized for stiffness, while the marrow-filled femora are far from any optimum. The relative wall thickness W=1-K of the gas-filled avian humeri studied is much larger than the theoretical optimum W*=1-K*=0.07, and thus these bones are thicker-walled than the optimal gas-filled tubular bone with minimum mass.

How does the relative wall thickness of human femora follow the biomechanical optima? An experimental study on mummies.

We studied how the ratio (K) of the internal:external diameter of human femora follows the biomechanical optima derived earlier by other researchers for marrow-filled tubular bones with circular cross section and minimum mass designed to withstand yield and fatigue, or stiffness, or bending fracture, or impact strengths. With evaluation of radiographs of 107 femora from 57 human mummies the values of K were measured. We found that K(posterior)=0.498+/-0.085 for the posterior radiographic view, and K(medial)=0.589+/-0.070 for the medial view with K(min)=0.345 and K(max)=0.783. The theoretical optima for K depend on the ratio (Q) of the marrow:bone density. Accepting the assumption of earlier authors that Q=0.50, our data show that human femora are optimised to withstand bending fracture, or yield and fatigue strengths. There were no sex-, age- and length-specific differences in K, and the means of K of the right and left femora of individuals were statistically not significantly different. The biomechanical optimization for K of human femora is not finely tuned. Compared with fox femora, K of human femora follows the biomechanical optimum to a much lesser extent. Although the relative wall thickness W=1-K of human femora are optimised, the very low relative mass increment due to deviation of K from the optimum and the considerable intraspecific variance of K make it probable that an accurate optimization of the relative wall thickness is irrelevant in humans.

How well does the Rayleigh model describe the E-vector distribution of skylight in clear and cloudy conditions? A full-sky polarimetric study.

We present the first high-resolution maps of Rayleigh behavior in clear and cloudy sky conditions measured by full-sky imaging polarimetry at the wavelengths of 650 nm (red), 550 nm (green), and 450 nm (blue) versus the solar elevation angle thetas. Our maps display those celestial areas at which the deviation deltaalpha = /alphameas - alphaRyleigh/ is below the threshold alphathres = 5 degrees, where alphameas is the angle of polarization of skylight measured by full-sky imaging polarimetry, and alphaRayleigh is the celestial angle of polarization calculated on the basis of the single-scattering Rayleigh model. From these maps we derived the proportion r of the full sky for which the single-scattering Rayleigh model describes well (with an accuracy of deltaalpha = 5 degrees) the E-vector alignment of skylight. Depending on thetas, r is high for clear skies, especially for low solar elevations (40% < r < 70% for thetas < or = 13 degrees). Depending on the cloud cover and the solar illumination, r decreases more or less under cloudy conditions, but sometimes its value remains remarkably high, especially at low solar elevations (rmax = 69% for thetas = 0 degrees). The proportion r of the sky that follows the Rayleigh model is usually higher for shorter wavelengths under clear as well as cloudy sky conditions. This partly explains why the shorter wavelengths are generally preferred by animals navigating by means of the celestial polarization. We found that the celestial E-vector pattern generally follows the Rayleigh pattern well, which is a fundamental hypothesis in the studies of animal orientation and human navigation (e.g., in aircraft flying near the geomagnetic poles and using a polarization sky compass) with the use of the celestial alpha pattern.

Testing the biomechanical optimality of the wall thickness of limb bones in the red fox (Vulpes vulpes).

The optimum for the ratio K of the internal to external diameter of a marrow-filled tubular bone with minimum mass designed to withstand a given type of strength (yield/fatigue, stiffness, fracture or impact) depends on Q = rhom/rhob only, where rhom and rhob are the densities of marrow and bone. With computer-assisted evaluation of radiographs of 62 femurs in the red fox (Vulpes vulpes) we measured the values of K. The mean and standard deviation of K are 0.68 and 0.036, and K changes in the rather wide range from 0.59 to 0.74. Accepting the assumption of earlier authors that Q = 0.50 or 0.44, our data would support the hypothesis that the fox femurs are optimized to withstand yield, fatigue or stiffness strengths. However, since the Q-values are unknown, the possibility cannot be excluded that any studied fox bone with an appropriately selected Q-value is optimized for any strength type. Assuming Q = 0.50 or 0.44, the relative mass increments mu of the investigated fox bones are smaller than 5% under all four mechanical conditions. The evolutionary relevance of such tiny mu-values is questionable.

First observation of the fourth neutral polarization point in the atmosphere.

In the clear sky there are three commonly known loci, the Arago, Babinet, and Brewster neutral points, where the skylight is unpolarized. These peculiar celestial points, bearing the names of their discoverers, have been the subject of many ground-based investigations, because their positions are sensitive indicators of the amount and type of atmospheric turbidity. According to theoretical considerations and computer simulations, there should exist an additional neutral point approximately opposite to the Babinet point, which can be observed only at higher altitudes in the air or space. Until now, this anonymous "fourth" neutral point has not been observed during air- or space-borne polarimetric experiments and has been forgotten, in spite of the fact that the neutral points were a basic tool in atmospheric research for a century. Here, we report on the first observation of this fourth neutral point from a hot air balloon. Using 180 degrees-field-of-view imaging polarimetry, we could observe the fourth neutral point at 450, 550, and 650 nm from different altitudes between 900 and 3500 m during and after sunrise at approximately 22 degrees - 40 degrees below the anti-solar point along the anti-solar meridian, depending on the wavelength and solar elevation. We show that the fourth neutral point exists at the expected location and has characteristics similar to those of the Arago, Babinet, and Brewster points. We discuss why the fourth neutral point has not been observed in previous air- or space-borne polarimetric experiments.

Ground-based full-sky imaging polarimetry of rapidly changing skies and its use for polarimetric cloud detection.

For elimination of the shortcomings of imaging polarimeters that take the necessary three pictures sequentially through linear-polarization filters, a three-lens, three-camera, full-sky imaging polarimeter was designed that takes the required pictures simultaneously. With this polarimeter, celestial polarization patterns can be measured even if rapid temporal changes occur in the sky: under cloudy sky conditions, or immediately after sunrise or prior to sunset. One of the possible applications of our polarimeter is the ground-based detection of clouds. With use of the additional information of the degree and the angle of polarization patterns of cloudy skies measured in the red (650 nm), green (550 nm), and blue (450 nm) spectral ranges, improved algorithms of radiometric cloud detection can be offered. We present a combined radiometric and polarimetric algorithm that performs the detection of clouds more efficiently and reliably as compared with an exclusively radiometric cloud-detection algorithm. The advantages and the limits of three-lens, three-camera, full-sky imaging polarimeters as well as the possibilities of improving our polarimetric cloud detection method are discussed briefly.