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The Rayleigh sky at sunset or sunrise

Rayleigh sky modelFrom Wikipedia, the free encyclopedia

The Rayleigh sky model describes the observed polarization pattern of the daytime sky. Within the atmosphereRayleigh scattering of light from air molecules, water, dust, and aerosols causes the sky's light to have a definedpolarization pattern. The same elastic scattering processes cause the sky to be blue. The polarization ischaracterized at each wavelength by its degree of polarization, and orientation (the e-vector angle, or scatteringangle).

The polarization pattern of the sky is dependent on the celestial position of the sun. While all scattered light ispolarized to some extent, light is highly polarized at a scattering angle of 90° from the light source. In mostcases the light source is the sun, but the moon creates the same pattern as well. The degree of polarization firstincreases with increasing distance from the sun, and then decreases toward the anti-sun. Thus, the maximumdegree 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 maximalpolarization wraps around the horizon. Light from the sky ispolarized horizontally along the horizon. During twilight ateither the Vernal or Autumnal equinox, the band of maximalpolarization is defined by the North-Zenith-South plane, ormeridian. In particular, the polarization is vertical at thehorizon in the North and South, where the meridian meetsthe horizon. The polarization at twilight at an equinox isrepresented by the figure to the right. The red bandrepresents the circle in the North-Zenith-South plane wherethe sky is highly polarized. The cardinal directions N, E, S,W are shown at 12-o'clock, 9 o'clock, 6 o'clock and 3o'clock (counter-clockwise around the celestial sphere sincethe 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 butthroughout the year. When the sun sets toward the South, in the 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 appearsabout 45 minutes before local sunrise and disappears 45 minutes after local sunset. Once established it is verystable, 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

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the cardinal directions.

Contents1 Theory

1.1 Geometry

1.2 Degree of polarization

1.3 Polarization angle

1.4 Q and U Stokes parameters

1.5 Neutral points and lines

1.6 Depolarization2 Uses

2.1 Navigation

2.2 Non-polarized objects

3 Notes and references

4 External links

Theory

Geometry

The geometry for the sky polarization can be represented by a celestial triangle based on the sun, zenith, andobserved pointing (or the point of scattering). In the model, γ is the angular distance between the observedpointing and the sun, Θs is the solar zenith distance (90° – solar altitude), Θ is the angular distance between theobserved pointing and the zenith (90° – observed altitude), Φ is the angle between the zenith direction and thesolar direction at the observed pointing, and ψ is the angle between the solar direction and the observedpointing at the zenith.

Thus, the spherical triangle is defined not only by the three points located at the sun, zenith, and observed pointbut by both the three interior angles as well as the three angular distances. In an altitude-azimuth grid theangular distance between the observed pointing and the sun and the angular distance between the observed

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The geometry representing the Rayleigh sky

pointing and the zenith change while the angular distance between the sun and the zenith remains constant atone point in time.

The figure to the left shows the two changing angular distances as mapped onto an altitude-azimuth grid (withaltitude located on the x-axis and azimuth located on the y-axis). The top plot represents the changing angulardistance 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 everycardinal direction. It then decreases with rising altitude moving closer toward the zenith. At twilight the sun issetting in the west. Hence the distance is greatest when looking directly away from the sun along the horizon inthe east, and lowest along the horizon in the west.

The bottom plot in the figure to the left represents theangular 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 thesun, this is independent of azimuth, i.e. cardinal direction. Itis simply greatest along the horizon at low altitudes anddecreases linearly with rising altitude.

The figure to the right represents the three angular distances.The left one represents the angle at the observed pointingbetween the zenith direction and the solar direction. This isthus heavily dependent on the changing solar direction asthe sun moves across the sky. The middle one represents theangle at the sun between the zenith direction and thepointing. Again this is heavily dependent on the changingpointing. This is symmetrical between the North and Southhemispheres. The right one represents the angle at the zenithbetween the solar direction and the pointing. It thus rotatesaround the celestial sphere.

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The angular distances between the observed pointing and the sun (topplot) and between the observed pointing and the zenith (bottom plot)

The three interior angles of the celestial triangle.

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 itis high in the North (0° and 360°) and the South (180°). It then resembles a cosine function and decreasestoward the East and West reaching zero at these cardinal directions.

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The polarization along the horizon.

The degree of sky polarization as mapped onto the celestial sphere.

The degree of polarization is easily understood when mapped onto an altitude-azimuth grid as shown below. Asthe sun sets due West, the maximum degree of polarization can be seen in the North-Zenith-South plane. Alongthe horizon, at an altitude of 0° it is highest in the North and South, and lowest in the East and West. Then asaltitude increases approaching the zenith (or the plane ofmaximum polarization) the polarization remains high in theNorth and South and increases until it is again maximum at90° in the East and West, where it is then at the zenith andwithin the plane of polarization.

Click on the image to the right to view an animation that represents the degree of polarization as shown on thecelestial sphere. Black represents areas where the degree of polarization is zero, whereas red represents areaswhere the degree of polarization is much larger. It is approximately 80%, which is a realistic maximum for theclear Rayleigh sky during day time. The video thus begins when the sun is slightly above the horizon and at anazimuth of 120°. The sky is highly polarized in the effective North-Zenith-South plane. This is slightly offsetbecause the sun's azimuth is not due East. The sun moves across the sky with clear circular polarization patternssurrounding it. When the sun is located at the zenith the polarization is independent of azimuth and decreaseswith rising altitude (as it approaches the sun). The pattern then continues as the sun approaches the horizon onceagain for sunset. The video ends with the sun below the horizon.

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The degree of polarization. Red ishigh (approximately 80%) and blackis low (0%).

The polarization angle. Red is high(approximately 80%) and black is low(0%).

Polarization angle

The relative azimuth between the observed pointing and the sun, ψ, isthe scattering angle and can be found by applying the law of cosines tothe spherical triangle. It gives:

This equation breaks down at the zenith where the angular distancebetween the observed pointing and the zenith, θs is 0. Here theorientation of polarization is defined as the difference in azimuthbetween the observed pointing and the solar azimuth.

The scattering plane is the plane through the sun, the observer, and thepoint observed (or the scattering point). The angle, ψ, located at thezenith between the solar direction and the observed pointing is thescattering angle. This angle of polarization is always perpendicular tothe scattering plane.

The polarization angles show a regular shift in polarization angle with azimuth. For example, when the sun issetting in the West the polarization angles proceed around the horizon. At this time the degree of polarization isconstant in circular bands centered around the sun. Thus the degree of polarization as well as its correspondingangle clearly shifts around the horizon. When the sun is located at the zenith the horizon represents a constantdegree of polarization. The corresponding polarization angle still shifts with different directions toward thezenith from different points.

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The q and u input.

The video to the right represents the polarization angle mapped onto the celestial sphere. It begins with the sunlocated in a similar fashion. The angle is zero along the line from the sun to the zenith and increases clockwisetoward the East as the observed point moves clockwise toward the East. Once the sun rises in the East the angleacts in a similar fashion until the sun begins to move across the sky. As the sun moves across the sky the angleis both zero and high along the line defined by the sun, the zenith, and the anti-sun. It is lower South of this lineand higher North of this line. When the sun is at the zenith, the angle is either fully positive or 0. These twovalues 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 unwrappedinto the Q and U Stokes parameters. Q andU are defined as the linearly polarizedintensities along the position angles 0° and45° respectively; -Q and -U are along theposition angles 90° and −45°.

If the sun is located on the horizon due west,the degree of polarization is then along theNorth-Zenith-South plane. If the observerfaces West and looks at the zenith, thepolarization is horizontal with the observer.At this direction Q is 1 and U is 0. If theobserver is still facing West but lookingNorth instead then the polarization isvertical with him. Thus Q is −1 and Uremains 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 thehorizon is a circle. From the East through the West it is 180° and from the West through the East it is 90° attwilight. When the sun is setting in the West, the angle is then 180° East through West, and only 90° Westthrough 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 caneasily unwrap q assuming it is in the +altitude direction. From the basic definition we know that +Q is an angleof 0° and -Q is an angle of 90°. Therefore, Q is calculated from a sine function. Similarly U is calculated from acosine function. The angle of polarization is always perpendicular to the scattering plane. Therefore, 90° isadded to both scattering angles in order to find the polarization angles. From this the Q and U Stokes parametersare determined:

and

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The q input. Red is high(approximately 80%) and black is low(0%). (Click for animation)

The u input. Red is high(approximately 80%) and black is low(0%). (Click for animation)

The scattering angle, derived from the law of cosines is with respect to the sun. The polarization angle is theangle with respect to the zenith, or positive altitude. There is a line of symmetry defined by the sun and thezenith. It is drawn from the sun through the zenith to the other side of the celestial sphere where the "anti-sun"would be. This is also the effective East-Zenith-West plane.

The first image to the right represents the q input mapped onto thecelestial sphere. It is symmetric about the line defined by the sun-zenith-anti-sun. At twilight, in the North-Zenith-South plane it is negativebecause it is vertical with the degree of polarization. It is horizontal, orpositive in the East-Zenith-West plane. In other words, it is positive inthe ±altitude direction and negative in the ±azimuth direction. As the sunmoves across the sky the q input remains high along the sun-zenith-anti-sun line. It remains zero around a circle based on the sun and the zenith.As it passes the zenith it rotates toward the south and repeats the samepattern until sunset.

The second image to the right represents the u input mapped onto thecelestial sphere. The u stokes parameter changes signs depending onwhich quadrant it is in. The four quadrants are defined by the line ofsymmetry, the effective East-Zenith-West plane and the North-Zenith-South plane. It is not symmetric because it is defined by the angles ±45°.In a sense it makes two circles around the line of symmetry as opposedto only one.

It is easily understood when compared with the q input. Where the qinput is halfway between 0° and 90°, the u input is either positive at+45° or negative at −45°. Similarly if the q input is positive at 90° ornegative at 0° the u input is halfway between +45° and −45°. This canbe seen at the non symmetric circles about the line of symmetry. It thenfollows the same pattern across the sky as the q input.

Neutral points and lines

Areas where the degree of polarization is zero (the skylight isunpolarized), are known as neutral points. Here the Stokes parameters Qand U also equal zero by definition. The degree of polarization thereforeincreases with increasing distance from the neutral points.

These conditions are met at a few defined locations on the sky. TheArago point is located above the antisolar point, while the Babinet and Brewster points are located above andbelow the sun respectively. The zenith distance of the Babinet or Arago point increases with increasing solarzenith distance. These neutral points can depart from their regular positions due to interference from dust andother aerosols.

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Light pollution is mostly unpolarized,and its addition to moonlight resultsin a decreased polarization signal.

The skylight polarization switches from negative to positive while passing a neutral point parallel to the solar orantisolar meridian. The lines that separate the regions of positive Q and negative Q are called neutral lines.

Depolarization

The Rayleigh sky causes a clearly defined polarization pattern under many different circumstances. The degreeof polarization however, does not always remain consistent and may in fact decrease in different situations. TheRayleigh sky may undergo depolarization due to nearby objects such as clouds and large reflecting surfacessuch as the ocean. It may also change depending on the time of the day (for instance at twilight or night).

In the night, the polarization of the moonlit sky is very strongly reduced in the presence of urban light pollution,because scattered urban light is not strongly polarized.[1]

Extensive research shows that the angle of polarization in a clear skycontinues underneath clouds if the air beneath the cloud is directly lit bythe sun. The scattering of direct sunlight on those clouds results in thesame polarization pattern. In other words, the proportion of the sky thatfollows the Rayleigh Sky Model is high for both clear skies and cloudyskies. The pattern is also clearly visible in small visible patches of sky.The celestial angle of polarization is unaffected by clouds.

Polarization patterns remain consistent even when the sun is not presentin the sky. Twilight patterns are produced during the time periodbetween the beginning of astronomical twilight (when the sun is 18°below the horizon) and sunrise, or sunset and the end of astronomicaltwilight. The duration of astronomical twilight depends on the length ofthe path taken by the sun below the horizon. Thus it depends on the time of year as well as the location, but itcan last for as long as 1.5 hours.

The polarization pattern caused by twilight remains fairly consistent throughout this time period. This isbecause the sun is moving below the horizon nearly perpendicular to it, and its azimuth therefore changes veryslowly throughout this time period.

At twilight, scattered polarized light originates in the upper atmosphere and then traverses the entire loweratmosphere before reaching the observer. This provides multiple scattering opportunities and causesdepolarization. It has been seen that polarization increases by about 10% from the onset of twilight to dawn.Therefore, the pattern remains consistent while the degree changes slightly.

Not only do polarization patterns remain consistent as the sun moves across the sky, but also as the moon movesacross the sky at night. The moon creates the same polarization pattern. Thus it is possible to use thepolarization patterns as a tool for navigation at night. The only difference is that the degree of polarization is notquite as strong.

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Underlying surface properties can affect the degree of polarization of the daytime sky. The degree ofpolarization has a strong dependence on surface properties. As the surface reflectance or optical thicknessincrease, the degree of polarization decreases. The Rayleigh sky near the ocean can therefore be highlydepolarized.

Lastly, there is a clear wavelength dependence in Rayleigh scattering. It is greatest at short wavelengths,whereas skylight polarization is greatest at middle to long wavelengths. 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 tolong wavelengths. The angle of polarization shows no variation with wavelength.

Uses

Navigation

Many animals, typically insects, are sensitive to the polarization of light and can therefore use the polarizationpatterns of the daytime sky as a tool for navigation. This theory was first proposed by Karl von Frisch whenlooking at the celestial orientation of honeybees. The natural sky polarization pattern serves as an easilydetected compass. From the polarization patterns, these species can orient themselves by determining the exactposition of the sun without the use of direct sunlight. Thus under cloudy skies, or even at night, animals can findtheir way.

Using polarized light as a compass however is no easy task. The animal must be capable of detecting andanalyzing polarized light. These species have specialized photoreceptors in their eyes that respond to theorientation and the degree of polarization near the zenith. They can extract information on the intensity andorientation of the degree of polarization. They can then incorporate this visually to orient themselves andrecognize different properties of surfaces.

There is clear evidence that animals can even orient themselves when the sun is below the horizon at twilight.How well insects might orient themselves using nocturnal polarization patterns is still a topic of study. So far, itis known that nocturnal crickets have wide-field polarization sensors and should be able to use the night-timepolarization patterns to orient themselves. It has also been seen that nocturnally migrating birds becomedisoriented when the polarization pattern at twilight is unclear.

The best example is the halicitid bee Megalopta genalis, which inhabits the rainforests in Central America andscavenges before sunrise and after sunset. This bee leaves its nest approximately 1 hour before sunrise, foragesfor up to 30 minutes, and accurately returns to its nest before sunrise. It acts similarly just after sunset.

Thus, this bee is an example of an insect that can perceive polarization patterns throughout astronomicaltwilight.[2] Not only does this case exemplify the fact that polarization patterns are present during twilight, but itremains as a perfect example that when light conditions are challenging the bee orients itself based on thepolarization patterns of the twilight sky.

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It has been suggested that Vikings were able to navigate on the open sea in a similar fashion, using thebirefringent crystal Iceland spar, which they called "sunstone", to determine the orientation of the sky'spolarization.[3][4][5][6][7] This would allow the navigator to locate the sun, even when it was obscured by cloudcover. An actual example of such a "sunstone" was found on a sunken (Tudor) ship dated 1592, in proximity tothe ship's navigational equipment.[8]

Non-polarized objects

Both artificial and natural objects in the sky can be very difficult to detect using only the intensity of light.These objects include clouds, satellites, and aircraft. However, the polarization of these objects due to resonantscattering, emission, reflection, or other phenomena can differ from that of the background illumination. Thusthey can be more easily detected by using polarization imaging. There is a wide range of remote sensingapplications in which polarization is useful for detecting objects that are otherwise difficult to see.

Notes and referencesPolarization Patterns of the Twilight Sky. Cronin T.W. et al., 2005, SPIE, 5888, 389Polarization patterns of the summer sky and its neutral points measured by full-sky imaging polarimetry in FinnishLapland north of the Arctic Circle. Gál J. et al. 2001, Proc. R. Soc. Lond. 457, 1385Polarized radiance distribution measurement of skylight. Liu Y. & Voss K., 1997, ApOpt, 36, 8753How the clear-sky angle of polarization pattern continues underneath clouds: full-sky measurements and implicationsfor animal orientation. Pomozi, I. et al., 2001, J. Exp. Biology, 204, 2933

1. Kyba, C. C. M.; Ruhtz, T.; Fischer, J.; Hölker, F. (17 December 2011). "Lunar skylight polarization signal polluted byurban lighting". Journal of Geophysical Research 116 (D24). Bibcode:2011JGRD..11624106K.doi:10.1029/2011JD016698.

2. Cronin, T.W.; Warrant, E.J.; Greiner, B. (2006). "Celestial polarization patterns during twilight". Appl. Opt. 45: 5582.Bibcode:2006ApOpt..45.5582C. doi:10.1364/ao.45.005582.

3. Suhai, B.; Horváth, G. (2004). "How well does the Rayleigh model describe the E-vector distribution of skylight in clearand cloudy conditions? A full-sky polarimetric study". JOSA A 21: 1669. Bibcode:2004JOSAA..21.1669S.doi:10.1364/josaa.21.001669.

4. The Viking Sunstone (http://www.polarization.com/viking/viking.html), from Polarization.net. Retrieved February 8,2007.

5. Secrets of the Viking Navigators, by Leif K. Karlsen. One Earth Press, 2003. ISBN 978-0-9721515-0-46. Could Vikings have navigated under foggy and cloudy conditions by skylight polarization? On the atmospheric optical

prerequisites of polarimetric Viking navigation under foggy and cloudy skies, by Ramón Hegedüs et al.[1](http://rspa.royalsocietypublishing.org/content/463/2080/1081.full)

7. Horvath, G. Et al. (2011). 'On the trail of Vikings with polarized skylight: experimental study of the atmospheric opticalprerequisites allowing polarimetric navigation by Viking seafarers' Phil. Trans. R. Soc. B (2011) 366, 772–782doi:10.1098/rstb.2010.0194

8. Wade, Lizzie (March 5, 2013). "Sunstone Unearthed From Shipwreck". Science (American Association for theAdvancement of Science). Retrieved March 11, 2013.

External linksPolarization of the sky (http://www.polarization.com/sky/sky.html)

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Blue Sky and Rayleigh Scattering (http://hyperphysics.phy-astr.gsu.edu/hbase/atmos/blusky.html)Polarized Light in Nature and Technology (http://polarization.com/)

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