Rayleigh sky model

During twilight at either the vernal or autumnal equinox, the band of maximal polarization is defined by the north-zenith-south plane, or meridian.

The red band represents the circle in the north-zenith-south plane where the sky is highly polarized.

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.

By orienting themselves with respect to the polarization patterns, animals can locate the Sun and thus determine the cardinal directions.

When the Sun is located at the zenith this distance is greatest along the horizon at every cardinal direction.

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.

This is thus heavily dependent on the changing solar direction as the Sun is perceived as moving across the sky.

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.

Click on the adjacent image to view an animation that represents the degree of polarization as shown on the celestial sphere.

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).

At this time the degree of polarization is constant in circular bands centered around the Sun.

If the sun is located on the horizon due west, the degree of polarization is then along the North-Zenith-South plane.

From this the Q and U Stokes parameters are determined: and The scattering angle, derived from the law of cosines is with respect to the Sun.

These neutral points can depart from their regular positions due to interference from dust and other aerosols.

The skylight polarization switches from negative to positive while passing a neutral point parallel to the solar or antisolar meridian.

The Rayleigh sky causes a clearly defined polarization pattern under many different circumstances.

The Rayleigh sky may undergo depolarization due to nearby objects such as clouds and large reflecting surfaces such as the ocean.

The polarization pattern caused by twilight remains fairly consistent throughout this time period.

This is because the sun is moving below the horizon nearly perpendicular to it, and its azimuth therefore changes very slowly throughout this time period.

Initially it is greatest in the ultraviolet, but as light moves to the Earth's surface and interacts via multiple-path scattering it becomes high at middle to long wavelengths.

From the polarization patterns, these species can orient themselves by determining the exact position of the Sun without the use of direct sunlight.

These species have specialized photoreceptors in their eyes that respond to the orientation and the degree of polarization near the zenith.

The best example is the halicitid bee Megalopta genalis, which inhabits the rainforests in Central America and scavenges before sunrise and after sunset.

[3] Not only does this case exemplify the fact that polarization patterns are present during twilight, but it remains as a perfect example that when light conditions are challenging the bee orients itself based on the polarization patterns of the twilight sky.

It has been suggested that Vikings were able to navigate on the open sea in a similar fashion, using the birefringent crystal Iceland spar, which they called "sunstone", to determine the orientation of the sky's polarization.

However, the polarization of these objects due to resonant scattering, emission, reflection, or other phenomena can differ from that of the background illumination.

There is a wide range of remote sensing applications in which polarization is useful for detecting objects that are otherwise difficult to see.