- Fig. 1. An illustration showing the solar angle in Chicago on the Winter and Summer Solstices. Adapted from the original at WGNTV.com.
- ChicagoWinterSolstice.jpg (35.24 KiB) Viewed 1249 times
It occurred to me the information contained in the drawing would make possible the calculation of the Latitude of Chicago and the tilt of the Earth's axis of rotation with respect to the plane of its orbit around the Sun. The mathematics are quite simple. And a bit of basic astronomy will be required.
To solve these two problems requires understanding that the angle of the Sun's ray relative to the ground at the equator at local noon on equinox (when the sun will be directly overhead) will be 90-degrees, and the angle decreases as latitude increases to the North or South. Mathematically:
- Solar angle = 90 - Latitude
At the Solstices, the sun angle will be increased (Summer) or decreased (Winter) by the amount of axis tilt of the Earth. This change in the apparent Solar angle is called the Declination of the Sun.
- Solar angle = 90 - Latitude ± Declination
This means that the difference between the two angles, 71.6-degrees and 24.6-degrees, represent twice the angle of the Earth's rotation axis tilt. In one case it is being added, and the other it is being subtracted. Mathematically:
- Axis tilt = (71.6 - 24.6) / 2
Axis tilt = 47 / 2
Axis tilt = 23.5 degrees
A check of the accepted value for the tilt of the Earth's axis of rotation revealed it was 23.436-degrees, so the error in my calculation was only 0.064-degrees. Now that we know the amount of influence from the Earth axis tilt on the solstices, we can solve easily for Latitude. Using the Winter Solstice example with Solar angle of 24.6-degrees gives:
- Solar angle = 90 - Latitude - 23.5
Substituting the sun angle of 24.6 gives
24.6 = 90 - Latitude -23.5
Rearranging
Latitude = 90 - 24.6 - 23.5
Latitude = 41.9 degrees North
A check on the location of downtown Chicago showed its location was roughly 41.9-degrees North Latitude.
An implication of this method is that a very patient observer could deduce his latitude by observing the angles on the two Solstices. How would he know the when the Solstice occurred? That is simple, the solar angle would be at its highest or lowest value. That is all the information contained in the newspaper graphic above.
Of particular importance for getting an accurate answer is the angles of the sun need to be accurate. Fortunately in the article the angle values were given to a tenth of a degree, and this allowed the Latitude of Chicago and the axis tilt of the Earth to be calculated to a similar precision. Out patient observer would need to have similar precision in his angle measurements.
On a related topic, see my experiment to measure the Amplitude of the Sun at Sunset:
Navigation Calculation
https://continuouswave.com/ubb/Forum6/HTML/003307.html
The above thread also includes another experiment in how to measure the solar angle at local noon using a step ladder, a pole, and a ruler. And from that how to deduce the circumference of the Earth.
[Fixed several typo’s in the original version. Finally found the source URL for the article that inspired this post, and gave credit to the author. ]