We are all familiar with the daily movement of the sun in the sky. It rises eastward, rises in the sky until noon, then begins a descent for several hours to sink to the western horizon.
You may also know our planet’s more imposing annual journey. For Northern Hemisphere residents, as summer approaches, they move a little higher in the sky every day until noon until June’s bliss, and begin to go low every day until December’s bliss.
These movements are empty clockwork. They repeat with sufficient accuracy and we become the basis for their passing time measurements. However, this universe’s mechanism has more than three gears. Additionally, subtle and elegant, somewhat oddly, there is a cog that repeatedly moves the position of the sun in the sky during the year, swaying up and down.
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If you stand in the same place every day and take a photo of the sun at the same time point (ignoring the shift when saving daylight savings), after the whole year, the sun’s position will form a skewed figure in the sky. This phenomenon is called ananma and is a reflection of the axial tilt of the Earth and the ellipsoidity of the planet’s orbit. They combine figures in subtle ways to create eight patterns, and the best way to understand all of this is to examine these effects individually.
First, if the Earth has no axial tilt and its orbit is a perfect circle, what is the position of the sun in the sky? In that case, the sun follows the exact same path in the sky every day. If you take photos every day in the same place, the sun is always in the same place in the sky. The analogues of Figure Class 8 collapse into simple points. We actually have the name of this ideal movement: the average sun (meaning average average). The movement of the average sun across the sky is constant and unshakable. This virtual sun can literally set the clock. In fact, we do! We can think of the average sun as a clock. It always reaches the best point of the day at noon.
So let’s mess with it and add the tilt of the earth. Our planets tend to be about 23.5 degrees compared to the plane of orbit around the Sun. This changes the actual location of the sun in the sky (what we call the “true sun”) and deviates from the average sun’s path. The real sun is what you use to measure time if you had a sundial, but perhaps sadly, over the past centuries, They collapsed somewhat.
The tilt of the earth is what causes the seasons. In summer, when the poles of our planet tilt towards the sun, the days become longer and the daily path of the true sun across the sky is higher. The opposite is true in winter.
This will quickly explain the north-south range of analogues! The true sun is higher in summer and lower in winter. Therefore, at noon every day, the height changes slightly from the horizon. Over the course of a year, the true sun travels twice the Earth’s tilt, or about 47 degrees in the sky (it’s directly above the head compared to the 90 degree angle from the horizon to the zenith).
However, the analogue has an east-west range, and its appearance gives it eight shapes. This is also mainly due to the tilt of the earth..
Imagine you are in the Northern Hemisphere when the true sun is at the lowest point in the hollow year at noon. Every day for the next six months, it moves slowly north at first, then accelerates, reaching maximum speeds in equinox. Then it will be late again until the summer solstice.
It is moving northward, so it is not moving at the same speed as the average sun. It starts to fall behind, a little later to the east, and takes a little more time to reach the peak of each day. (After the clock appears at noon, it reaches its highest point.) Every day, it falls a little behind and takes a long time to peak the daily path.
But this is where things become strange: as the true sun’s delay increases, rate Where it’s late small every day. It’s still lagging behind the average sun, but every day the extra amount gets smaller. The eastward movement of the true sun then ceases, reverses, and begins to catch up with the average sun. It eventually pulls forward west of the average sun and reaches its peak in front Average sun is noon. This lasts several more weeks until the westward movement slows down, stops and reverses, then begins to move east again. Eventually, the average sun catches up to it and is slow enough to pass through it, and the whole process begins again.
This back and forth movement, combined with the up and down movement, creates an overall figure 8 shape of Ananma.
The actual dependence of this movement is a complex feature of algebraic trigonometry. (and Spherical geometry), but in the end very Roughly, over a year, the sun moves ahead of the average sun for three months, then three more, then first, and then back again. Bubble, rinse, repeat.
The true sun departure from the average sun is caused by astronomers saying, “The equation of time.“The Sundial time is used to figure out how far away the average Sun’s 24-hour clock, increasing the approximately eight-degree excursion to the east and west. This is much smaller than the 47-degree north-south movement, making the anus narrow and taller.
This is complicated by the elliptical orbit of the Earth. The Earth orbits at the fastest speed when closest to the Sun, and moves more slowly as it gets further away. If the Earth is closest to the Sun (called an approximation), its orbital velocity is faster than average, so the true Sun is behind the average Sun. The opposite is true at the farthest orbital point from the Sun (Apherion), and the true sun moves before the average Sun. If the Earth does not have an axial tilt, its orbital ellipticity means that the true sun lined up east-west throughout the year to create short lines.
There will continue to be a perspective effect. At perihelion, the true sun’s movement in the sky is faster than that of Apherion, which also affects the shape of the analogue. Combined with the tilt of the earth, this excludes the north-south symmetry in Figure 8. The Southern Loop (made when the sun appears to move faster nearer) is bigger and wider than the northern loop. This will make the overall shape look like a bowling pin.
This becomes even more complicated. The dates in recent years (around January 4th every year) are nearby, but they are not the bliss of December (around December 21st). It knocks on the Katawan Pass and distorts its appearance a littledistorted from perfect symmetry.
If this all applies to Earth, what about other planets? They also get an annanma, as they have an oval trajectory and similarly have an axial tilt, but they are shaped differently. For example, Mars tilts at about 25 degrees, similar to Earth. However, its orbit is much more elliptical, which governs the analogue, It becomes more tear-like. Jupiter has a tilt of only three degrees, but its orbit is more elliptical than Earth, making its anamma a simpler ellipse due to its sky. Neptune may have an annamma of all of them in the most pleasant form: its inclination is 28 degrees, but the eccentricity of the orbit is so slight that its analogue is almost symmetrical Figure 8.
All of this is a bit complicated and can be hard to imagine in your head. I struggled with some of the more subtle moves myself (this was not aided by many false explanations of the phenomenon I found online). It’s probably surprising that the clockwise movement of heaven isn’t as easy as you would like! But I actually find beauty in all this: repetitive cycles, subtle movements, brilliant symmetry against them. And it’s all predictable by combining physics, geometry and mathematics. There are some truths to the idea of Musica universalis“Sphere Music,” and the music of the great composer, there is a dignity of complexity.
