Geometrical basics or why 1+1 equals 23 ½

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To simplify the problems of astronomy and geometry, let us first agree on a viewing model for our planet and its star. We want to look at the earth as a sphere, which appears to an observer at any geographical point as a disk bounded by the horizon (the horizon in the mathematical sense is a great circle that divides the celestial sphere into two equal halves and whose pole is the zenith. The natural or landscape horizon is the boundary line between the sky and the earth, which depends on local conditions). The celestial sphere or vault, which we perceive as a hemisphere, rises above it. Its highest point is the zenith, which is 90° away from the horizon and exactly opposite to the nadir (base point).

In this simple horizon system, from a position far away in space, we would first recognize that the earth orbits the sun counterclockwise on a slightly elliptical path and takes 365 days to complete one orbit. For the sake of simplicity, let us first imagine that the earth’s axis is not inclined, contrary to reality, but is perpendicular to the orbit around the sun. If we were to stand together on the equator of this hypothetical earth, with the sun directly overhead, and if we were able to see the stars in addition to the sun, what would we discover? While the earth moves on its orbit around the sun, the sun moves against the background of the stars every day, seemingly slightly to the east, and returns to its initial position after a year. We call this apparent solar orbit the ecliptic, and if we were to artificially straighten our earth, it would also be straight and coincide with the celestial equator, which extends to the celestial sphere.

However, the axis of our real earth is tilted by a good 23.5° and the key to understanding what is happening is that the direction of this tilt always remains almost the same. So at one point in the orbit, the earth tilts more of its northern hemisphere toward the sun, and at another, it tilts more of its southern hemisphere toward our lightspending star. Let’s go back to our common position on the earth’s equator and see what is really happening. As the earth moves on its orbit around the sun, the sun seems to move a little to the east every day against the background of the stars. Since the movement of the earth around the sun now alters the relative inclination of the earth to the sun, the apparent sun course no longer runs straight, as previously described, but instead wanders along the celestial equator from north to south over the course of a year, creating the rising and falling line known as the ecliptic. How far does the sun travel north and south? Simple. It describes its northernmost arc in the northern summer (on June 21) at 23.5° north of the equator, and its southernmost arc in the northern winter (on December 21) at 23.5° south of the equator. Halfway between these two turning points, it passes the celestial equator in each case, which we know as the equinox (March 21 and September 23), because day and night relate to each other in exactly the same way. So the appearance of the ecliptic depends completely on the orbit of the earth around the sun and the inclination of the earth axis.

And we can even depict this movement photographically by taking pictures of the sun in a multiple exposure over a whole year at regular intervals of seven or nine days, each at the same time and from the same place. The finished image then presents this phenomenon, known as analemna, as a shape reminiscent of the infinity symbol, with its horizontal dimensions determined by the sun’s seasonal north/south motion and its inclination dependent on the location’s latitude. At the equator it lies flat, and at the poles it is vertical.

Thus, we have established the most crucial earth-sun connections in this paragraph. Everything that follows refers to this.

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