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
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.
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.