I conducted a long series of observations for nearly 4 months. No scientific discoveries were expected. I was just curious.
We have a window that faces south. In the summer months, the roof overhang blocked the sunlight from shining through the glass. Starting about September 1st, the lower elevation angle of the Sun at noon allowed it to shine onto the floor. As the weeks passed, the shadow of the window sill moved farther from the wall. I found a long strip of paper to record the edge of the shadow of the sill at various dates. This image illustrates the sunlight paths for the 1st of Sep, Oct, Nov, and Dec.
Closer inspection shows the shadow lines for the first of each month. There is a faint line in the bright light at the far right of the strip. Today is the winter solstice December 21. It marks the farthest shadow from the window sill because the Sun was at its lowest elevation in the sky.
The lines marked in the image below are not in an orderly pattern. Some days were cloudy, I skipped a few days, or I forgot. I tried to mark the shadow lines at the same time when the Sun was perpendicular to the window. The vertical portion of the window cast a shadow parallel to the boards on the floor. I used that as my time indicator.
Some analysis can be applied to the situation. This diagram illustrates my location in the northern hemisphere. It is where the top yellow ray meets the Earth at my horizon. If I subtract the measure of the angle marked in blue from 90˚, it gives me the elevation of the Sun above the horizon. Sliding down the dotted line to the center of the Earth, that angle marked in blue is the same measure as the one at the surface. It is the sum of my latitude plus the declination of the Sun from the plane of the Equator. On this date, the Sun has a declination of 23.5˚. My latitude is 41.7˚. The angle marked in blue is 65.2˚. Today, the elevation of the Sun at noon is 90˚- 65.2˚ = 24.8˚.
How does that calculation compare with the geometry in my top photograph? Measuring from the base of the wall out to the shadow line on December 21 gives 131 cm. Measuring up the wall to the edge of the window sill gives 59 cm. Drawn in black is a right triangle. The ratio of 59 divided by 131 is the tangent of the elevation angle of the Sun. Tan (Elev) = 59/131 = 0.450 which means the elevation angle = 24.2˚ of the Sun. That is a favorable value compared with the 24.8˚ calculated above.
Hurray for science and math. Happy winter solstice to you.