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# Saturn's Seasonal Sundial

This Demonstration simulates the seasonal shift of the shadows cast by Saturn's rings on the planet's surface.
As a result of its axial tilt of 26.75°, the shadows of Saturn's rings move up and down its surface as the planet makes its way around the sun in approximately 29.5 Earth years. The position of these shadows marks Saturn's seasons in its orbit around the sun, just like the shadow of the style on a sundial indicates the time of day on Earth.
These shadows are not easily visible from Earth because, from our vantage point near the Sun, the rings block their shadows. However, the Cassini robotic spacecraft, currently orbiting Saturn, made some remarkable images of these shadows.
In this Demonstration, the viewpoint is taken to be tangential to the orbit of Saturn (similar to Cassini), in order to get a good view of the ring shadows.

### DETAILS

If we suppose Saturn is a perfect sphere, the shadows are the intersections of a sphere and a set of concentric elliptical cylinders formed by parallel lines passing through the rings. (The lines converge to the Sun.)
These can be found by solving the equations , resulting in space curves with parametric form , with being the declination of Saturn.
Since we take the orbit of Saturn around the Sun to be circular and 26.75° as its axial tilt, the declination at time is equal to .
The ring radii are approximated by a middle third Cantor set with three iterations between the radii 1.12 and 2.3. These correspond to the radii of the D and F rings (approx. 67,000 km and 140,000 km) if we set Saturn's equatorial radius (approx. 60,000 km) equal to 1.
See the images of the shadows of Saturn's rings taken by the Cassini spacecraft: "Shadows of a Seasonal Sundial".
See also the article and images on "Saturn shadows shift with the seasons" from "The Planetary Society".
Other interesting images and information can be found in ''The Seasons of Saturn" at "Astronomy Picture of the Day" from July 2, 2001.
Snapshot 1: the ring shadows at northern vernal equinox (last occurrence August 2009)
Snapshot 2: the ring shadows at northern summer solstice (next occurrence May 2017)
Snapshot 3: the ring shadows at northern autumnal equinox (next occurrence in 2024)
Snapshot 4: the ring shadows at northern winter solstice (next occurrence in 2039)
As of the end of 2011, the "today" button will show only a slight shift of the shadows toward the south. It is only some 820 days past vernal equinox (August 10, 2009) within a Saturn year of 10,759 days.

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#### Related Curriculum Standards

US Common Core State Standards, Mathematics

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