THIS IS A WORK IN PROGRESS.
Prelude: Why did only people 1000 year ago get to discover basic geometrical facts about the universe? Now it's "obvious" that Earth is round?
The goal of this blog is to find the curve that the sun takes in the sky. One interesting thought is how high the sun is during the day, Intuitively, we know that when the sun rises in the east, let's call sunrise $\alpha(t)$, we know $\alpha(0)=\alpha(T)=0$, little less obvious, but nontheless true is that the sun's maximum is achieved at $arg \max \alpha(t)=T/2$. These are some basic properties that we'll keep in mind for or full formula later.
To calculate the sun's height we need the day of year, time of day, and location on earth. But really the day of the year is not a fundamental quanty, rather we can see time of year as a point in $[0, 2\pi)$, but really this does not matter either cuz really what matters is Earth's tilt towards the sun, the declination $\delta\in[-23^\circ, 23^\circ]$. Then we have time of day, one day is one revolution around earth's rotational axis. Hence, an angle hthink in terms of hours and minutes, but more natural is just a number in $[0, 2\pi)$ that says how much of.
Then for the position, on earth every position can be given by a pair $(\phi, \lambda)$ but since the longitude $\lambda$ simply shifts the start of the day, we don't care about it and we can assume $\lambda=0$ it's i.e on the prime meridian. In the animation below I've set the latitude to $\phi\approx 60^\circ$, the same latitude as Sandviken, Sweden.
Now we have the goal, to find $$\alpha(\delta, t, \phi).$$ To start thinking about some properties we can assume time of year is equanox, I.e between the winter and summer solstice, so $\delta=0$. Note that having $\delta=0$ is not saying that Earth's tilt has changed, it's always fixed at $23^\circ$, but it just means that axis is orthogonal to the vector connecting sun's center and earth's center.
Sun Earth
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Facts:
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sun always shines 50% of earths surface always.
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even if earth was a point source we can assume all light rays hit parallel
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Sun is 1 austronimal unit or 8 lightminutes away from earth. When you look up it looks like 0.5 degrees in the sky. Formula for is radial size is. Sun is 400x times size of moon, but 400x times further away, so moon is also 0.5 degrees. Meaning there can be perfect solar eclipses. Since
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Since the sun is not a point source, when it shines on moon it creates a large penumbra, while the eclipse on jupiter has almost no penumbra, since the sun in that case can almost be seen as point source.
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All planets except venus in solar system spins prograde.
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euler angles. Precession so our north star changes a bit.
The sun's diameter is ~100x of the earth, so this illustration is not just the wrong scale, it's proportions too.
Reading the diagram
The bold red point sits at 35° north and follows the rotating surface. It becomes translucent on the far side of Earth, so you can follow its full path around the tilted latitude circle.
Enable Show axes (23°) to draw the vertical orbital-plane normal and the tilted spin axis through Earth. The globe’s grid turns about that tilted axis. The north pole tilts away from the Sun in winter and toward it in summer. At the equinoxes, the tilt points into or out of the screen, so the projected lines overlap even though the 3D tilt remains 23°.
Use Earth rotation to turn the globe from 0° to 360° without changing its orbital position. Use Orbital phase to move continuously around the Sun without changing the spin angle. Either slider pauses the animation for inspection; play resumes Earth’s spin, its orbit, and the background stars together.
A point on a spinning Earth
This view holds the orbital position fixed, sets Earth’s tilt to 0°, and shows only its spin. The red point is still at 35° north. Enable Surface normal to see the outward unit vector and a small tangent plane centered on the point. The plane represents the local ground beneath an observer; the vector points straight up from it.
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Standing on the ground
Now keep the observer’s ground plane fixed and let the Sun direction move instead. Earth is removed from this view, but the location is still 35° north, with 0° axial tilt. The upright normal v stays fixed; s traces the Sun’s direction as Earth turns.
Here, “up” is v, not a direction fixed to the page in the earlier globe view. The Sun vector is expressed relative to the observer’s local east, north, and up directions. Its component along v tells us whether the Sun is above the horizon; comparing the two vectors is the starting point for describing sunlight on the ground.
From picture to equations
This is the starting diagram. The next step is to choose a reference frame and define the quantities on it: time, orbital angle, the Sun–Earth separation, and Earth's spin angle. Keeping rotation and revolution separate here should make those definitions easier to connect to the picture.
Gnomon
well, it's hard to have a bird's eye view of the solar system. So thinking locally, from our locally flat looking plane?