# From 3D length to 3D duration

We observe the sun and the moon traversing the sky. We know that the moon is objectively orbiting the earth but the sun is not. Where then is the sun that is observed traversing the sky in daily and annual cycles? It is not in 3D length. It is in 3D duration.

Binary stars orbit their common barycenter. If the sun and earth were the only celestial bodies, it might not be clear as to which was orbiting which. But since there are other planets orbiting the sun, the only objective view is that all the planets are orbiting the sun (more precisely, the barycenter of the solar system, which is in or near the sun).

Compare sun-centered (heliocentric) and earth-centered (geocentric) frames of reference (click to enlarge):

The sun (A) is at a focus of the ellipse in the heliocentric frame, and the earth (B) is at a focus of an identical ellipse rotated 180°. The orbital path p(t) in the heliocentric frame transforms to the path –p(t) in the geocentric frame.

But the path in the earth-centered frame is not a spatial path; it is a virtual path called the ecliptic. Although the ecliptic is considered to be a great circle of the celestial sphere, the above figure shows that it is really an ellipse as observed from the frame of the earth.

(Ecliptic is also defined in a different sense as the plane of the earth’s orbit. Here the ecliptic is “an imaginary line that marks the path of the sun.”)

Is the ecliptic unreal, an imaginary picture that of something does not exist? It doesn’t exist in 3D length. Could the ecliptic exist in 3D duration? Yes.

The ecliptic shows the sun as it is observed from the earth. This is the path of the sun that gives us not only sunrise, noon, and sunset but also equinox, solstice, and the calendar. These are all terms for duration, not length. The sun is a giant clock and calendar.

We commonly think of time as a scalar quantity, but the sun shows more than a scalar. The sun rises in the east and sets in the west. In the northern hemisphere the sun is further north in the summer and further south in the winter. The angle of the sun varies by day and by hour. The full time shown by the sun includes the direction; it is a 3D duration.

The object-centered frame is the 3D space frame. The 3D duration frame is subject-centered, meaning it varies with the subject, which in this case is the view from the earth. Switching from an object-centered frame to a subject-centered frame switches from 3D length to 3D duration. The orbital elements switch, too: measures of space become measures of time and vice versa.

Note: the terms “heliocentric” and “geocentric” here refer to reference frames, not the historic conflict between theories of a static earth vs. a static sun.