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:
As a simple example one can map the degree of polarization on the horizon. As seen in the figure to the right it is high in the North (0° and 360°) and the South (180°). It then resembles a cosine function and decreases toward the East and West reaching zero at these cardinal directions.
The degree of polarization is easily understood when mapped onto an altitude-azimuth grid as shown below. As the sun sets due West, the maximum degree of polarization can be seen in the North-Zenith-South plane. Along the horizon, at an altitude of 0° it is highest in the North and South, and lowest in the East and West. Then as altitude increases approaching the zenith (or the plane of maximum polarization) the polarization remains high in the North and South and increases until it is again maximum at 90° in the East and West, where it is then at the zenith and within the plane of polarization.
Click on the adjacent image to view an animation that represents the degree of polarization as shown on the celestial sphere. Black represents areas where the degree of polarization is zero, whereas red represents areas where the degree of polarization is much larger. It is approximately 80%, which is a realistic maximum for the clear Rayleigh sky during day time. The video thus begins when the sun is slightly above the horizon and at an azimuth of 120°. The sky is highly polarized in the effective North-Zenith-South plane. This is slightly offset because the sun's azimuth is not due East. The sun moves across the sky with clear circular polarization patterns surrounding it. When the Sun is located at the zenith the polarization is independent of azimuth and decreases with rising altitude (as it approaches the sun). The pattern then continues as the sun approaches the horizon once again for sunset. The video ends with the sun below the horizon.
Polarization angle
The scattering plane is the plane through the Sun, the observer, and the point observed (or the scattering point). The scattering angle, γ, is the angular distance between the Sun and the observed point. The equation for the scattering angle is derived from the law of cosines to the spherical triangle (refer to the figure above in the geometry section). It is given by:
In the above equation, ψs and θs are respectively the azimuth and zenith angle of the Sun, and ψ and θ are respectively the azimuth and zenith angle of the observed point.
This equation breaks down at the zenith where the angular distance between the observed pointing and the zenith, θs is 0. Here the orientation of polarization is defined as the difference in azimuth between the observed pointing and the solar azimuth.
The angle of polarization (or polarization angle) is defined as the relative angle between a vector tangent to the meridian of the observed point, and an angle perpendicular to the scattering plane.
The polarization angles show a regular shift in polarization angle with azimuth. For example, when the sun is setting in the West the polarization angles proceed around the horizon. At this time the degree of polarization is constant in circular bands centered around the Sun. Thus the degree of polarization as well as its corresponding angle clearly shifts around the horizon. When the Sun is located at the zenith the horizon represents a constant degree of polarization. The corresponding polarization angle still shifts with different directions toward the zenith from different points.
The video to the right represents the polarization angle mapped onto the celestial sphere. It begins with the Sun located in a similar fashion. The angle is zero along the line from the Sun to the zenith and increases clockwise toward the East as the observed point moves clockwise toward the East. Once the sun rises in the East the angle acts in a similar fashion until the sun begins to move across the sky. As the sun moves across the sky the angle is both zero and high along the line defined by the sun, the zenith, and the anti-sun. It is lower South of this line and higher North of this line. When the Sun is at the zenith, the angle is either fully positive or 0. These two values rotate toward the west. The video then repeats a similar fashion when the sun sets in the West.
Q and U Stokes parameters
The angle of polarization can be unwrapped into the Q and U Stokes parameters. Q and U are defined as the linearly polarized intensities along the position angles 0° and 45° respectively; -Q and -U are along the position angles 90° and −45°.
If the sun is located on the horizon due west, the degree of polarization is then along the North-Zenith-South plane. If the observer faces West and looks at the zenith, the polarization is horizontal with the observer. At this direction Q is 1 and U is 0. If the observer is still facing West but looking North instead then the polarization is vertical with him. Thus Q is −1 and U remains 0. Along the horizon U is always 0. Q is always −1 except in the East and West.
The scattering angle (the angle at the zenith between the solar direction and the observer direction) along the horizon is a circle. From the East through the West it is 180° and from the West through the East it is 90° at twilight. When the sun is setting in the West, the angle is then 180° East through West, and only 90° West through East. The scattering angle at an altitude of 45° is consistent.
The input stokes parameters q and u are then with respect to North but in the altitude-azimuth frame. We can easily unwrap q assuming it is in the +altitude direction. From the basic definition we know that +Q is an angle of 0° and -Q is an angle of 90°. Therefore, Q is calculated from a sine function. Similarly U is calculated from a cosine function. The angle of polarization is always perpendicular to the scattering plane. Therefore, 90° is added to both scattering angles in order to find the polarization angles. From this the Q and U Stokes parameters are determined:
and