The Rayleigh sky model describes the observed polarization pattern of the daytime sky. Within the atmosphere, Rayleigh scattering of light by air molecules, water, dust, and aerosols causes the sky's light to have a defined polarization pattern. The same elastic scattering processes cause the sky to be blue. The polarization is characterized at each wavelength by its degree of polarization, and orientation (the e-vector angle, or scattering angle). The polarization pattern of the sky is dependent on the celestial position of the Sun. While all scattered light is polarized to some extent, light is highly polarized at a scattering angle of 90° from the light source. In most cases the light source is the Sun, but the Moon creates the same pattern as well. The degree of polarization first increases with increasing distance from the Sun, and then decreases away from the Sun. Thus, the maximum degree of polarization occurs in a circular band 90° from the Sun. In this band, degrees of polarization near 80% are typically reached.
When the Sun is located at the zenith, the band of maximal polarization wraps around the horizon. Light from the sky is polarized horizontally along the horizon. During twilight at either the vernal or autumnal equinox, the band of maximal polarization is defined by the north-zenith-south plane, or meridian. In particular, the polarization is vertical at the horizon in the north and south, where the meridian meets the horizon. The polarization at twilight at an equinox is represented by the figure to the right. The red band represents the circle in the north-zenith-south plane where the sky is highly polarized. The cardinal directions (N, E, S, W) are shown at 12-o'clock, 9 o'clock, 6 o'clock, and 3 o'clock (counter-clockwise around the celestial sphere, since the observer is looking up at the sky). Note that because the polarization pattern is dependent on the Sun, it changes not only throughout the day but throughout the year. When the sun sets toward the South, in the northern hemisphere's winter, the North-Zenith-South plane is offset, with "effective" North actually located somewhat toward the West. Thus if the sun sets at an azimuth of 255° (15° South of West) the polarization pattern will be at its maximum along the horizon at an azimuth of 345° (15° West of North) and 165° (15° East of South). During a single day, the pattern rotates with the changing position of the sun. At twilight, it typically appears about 45 minutes before local sunrise and disappears 45 minutes after local sunset. Once established it is very stable, showing change only in its rotation. It can easily be seen on any given day using polarized sunglasses. Many animals use the polarization patterns of the sky at twilight and throughout the day as a navigation tool. Because it is determined purely by the position of the Sun, it is easily used as a compass for animal orientation. By orienting themselves with respect to the polarization patterns, animals can locate the Sun and thus determine the cardinal directions.
Theory
Geometry
The geometry for the sky polarization can be represented by a celestial triangle based on the Sun, zenith, and observed pointing (or the point of scattering). In the model, γ is the angular distance between the observed pointing and the Sun, Θs is the solar zenith distance (90° – solar altitude), Θ is the angular distance between the observed pointing and the zenith (90° – observed altitude), Φ is the angle between the zenith direction and the solar direction at the observed pointing, and ψ is the angle between the solar direction and the observed pointing at the zenith. Thus, the spherical triangle is defined not only by the three points located at the Sun, zenith, and observed point but by both the three interior angles as well as the three angular distances. In an altitude-azimuth grid the angular distance between the observed pointing and the Sun and the angular distance between the observed pointing and the zenith change while the angular distance between the Sun and the zenith remains constant at one point in time.
The figure to the left shows the two changing angular distances as mapped onto an altitude-azimuth grid (with altitude located on the x-axis and azimuth located on the y-axis). The top plot represents the changing angular distance between the observed pointing and the Sun, which is opposite to the interior angle located at the zenith (or the scattering angle). When the Sun is located at the zenith this distance is greatest along the horizon at every cardinal direction. It then decreases with rising altitude moving closer toward the zenith. At twilight the sun is setting in the west. Hence the distance is greatest when looking directly away from the Sun along the horizon in the east, and lowest along the horizon in the west. The bottom plot in the figure to the left represents the angular distance from the observed pointing to the zenith, which is opposite to the interior angle located at the Sun. Unlike the distance between the observed pointing and the Sun, this is independent of azimuth, i.e. cardinal direction. It is simply greatest along the horizon at low altitudes and decreases linearly with rising altitude.
The figure to the right represents the three angular distances. The left one represents the angle at the observed pointing between the zenith direction and the solar direction. This is thus heavily dependent on the changing solar direction as the Sun is perceived as moving across the sky. The middle one represents the angle at the Sun between the zenith direction and the pointing. Again this is heavily dependent on the changing pointing. This is symmetrical between the North and South hemispheres. The right one represents the angle at the zenith between the solar direction and the pointing. It thus rotates around the celestial sphere.
Degree of polarization The Rayleigh sky model predicts the degree of sky polarization as:
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