Angle of Sunlight Over Time

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.

Light rays over the window sill at the four selected dates.

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.


17 thoughts on “Angle of Sunlight Over Time

  1. I knew you wouldn’t let the hibernal solstice pass without offering something interesting and educational. Let’s make your trail name “professor.”

    • I will proudly wear the trail moniker ‘professor’. 🙂
      Back in 1987, I ventured to Mexico City with two other physics teachers to represent the education office of Fermilab at a physics teaching conference. It was usual and customary to use professor when addressing instructors. I liked the sound of it. It never happened in the US.

        • We wish you peaceful and loving holidays in the coming couple of weeks. We will be home and hunkered most of the time. With a daughter on Saturday.

    • Our sunsets are coming a little later now since Dec. 4. Still a few days of later sunrises until Jan. 4. I am ready.

  2. Why is this phenomenon more satisfying in autumn? I scarcely notice it in the spring, but for some reason delight in the changing light and shadows all fall. Fascinating stuff!

  3. Think of all the science and discoveries that were revealed by those same observations and calculations. You are a good teacher. I bet there wasn’t many who said, ‘when will we ever use this?’. I was never much for formulas but if they had a told me you could use it along with your camp, the mountain and clouds to figure out how far the river was below you, I’d have been interested. You are able to make math apply.
    Take care, say hello to Melanie.

    • Thank you for those kind words. I will tell her hello for you. Is your shopping done for Lisa. Time is getting short. 🙂

      • Everything is ready for Christmas. Lisa and I bought snowshoes for each other, it is not a surprise but they should be fun just the same! Merry Christmas to you and Melanie and your family.🎄

  4. A very logical and clear explanation of the seasonal effect on the angle of sunbeams, something which many people would not even cross people’s minds.

    I’m glad the maths backed up your results and I’m looking forward to your next experiment!

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