A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: David Iwancio
Date: 2020 Feb 12, 14:27 -0800
In the image, the equator will be a single line, crossing the axis of the figure-8 at a right angle. The equator is parallel to the path of your daily trip around the earth's axis, so its projection onto the sky will be as constant as your latitude.
The ring of the ecliptic is not parallel to your perpetual spinning around the earth's axis. The ecliptic and the equator intersect at two points (two great circles always intersect each other at two points 180° apart). For lack of imagination, I shall call these points "Aries" and "Libra." As you stand on the rotating earth, Aries and Libra slide along the portion of the equator you can see in your sky. Aries crosses your meridian, then Libra crosses your meridian 11 hours and 58 minutes later (assuming you're still at the same longitude), and then Aries crosses again after another 11 hours and 58 minutes.
The key to building these multiple-exposure photographs of an analemma is that, between exposures, Aries has moved 361° (a full circle plus an extra 1°) or some multiple thereof (722°, 1083°, etc.) If Aries were shown in the photographs, the distance between the images of Aries would be 1° apart (about 2 sun diameters) for every day that passed between exposures.
The analemma itself is about 47° tall (double the angle that the equator and ecliptic intersect at). The equator crosses the analemma's axis at a right angle at its midpoint. If our picture of the analemma is going to be a 50° "square" and if exposures are taken weekly, images of Aries would be 7° (about 14 sun diameters) apart and only appear in the photograph itself 6 or 7 times (Libra would also show up 6 or 7 times, appearing about 20 weeks after Aries moves out of the picture's frame).
If you were to then draw arrows from the sun along the shortest distance towards either Aries or Libra (whichever was closer at the time), those arrows would be along the ecliptic, sloping towards the equator at a different angle for every exposure.* But for most of those exposures, neither Aries nor Libra are in the picture so those arrows would be pointing off-screen. For those exposures when either Aries or Libra is in the picture, the sun is necessarily near the equator.
*The math of the changing direction of these arrows is exactly the same as the math for determining azimuth at different points along a great circle sailing.