Noon Shadows and the Seasons
At a school in Massachusetts, students measure the shadow of a 1.0 m vertical stick at noon on four dates. They use the same stick, same location, and same measuring method each time. The shadow lengths are: Sept. 22, 1.0 m; Dec. 21, 2.0 m; Mar. 21, 1.0 m; June 21, 0.4 m. They also record the Sun's height above the horizon at noon: about 45 degrees in September and March, about 24 degrees in December, and about 69 degrees in June. A student says, 'Summer is warmer because Earth is closer to the Sun, and the short June shadow proves the Sun is closer.' Another student says the shadow data show the Sun's angle, not Earth's distance. The class needs to explain the seasonal pattern using the evidence and identify what additional data would test the distance claim.
1. Explain how the shadow length is related to the Sun's angle above the horizon. Use the September and December data.
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For a vertical stick, a lower Sun angle makes a longer shadow, and a higher Sun angle makes a shorter shadow. The relationship is approximately tan(angle) = stick height / shadow length. In September, a 1.0 m stick casts a 1.0 m shadow, giving tan(angle) = 1.0, or about 45 degrees, which matches the recorded angle. In December, the same stick casts a 2.0 m shadow, giving tan(angle) = 1.0 / 2.0 = 0.5, or about 27 degrees, close to the recorded 24 degrees. The June shadow of 0.4 m gives tan(angle) = 2.5, or about 68 degrees, close to the recorded 69 degrees. The shadow length is therefore evidence about the Sun's angle, not a direct ruler measurement of the Sun's distance.
2. Evaluate the claim 'Summer is warmer because Earth is closer to the Sun, and the short June shadow proves the Sun is closer.'
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The claim does not follow from the evidence. The short June shadow shows that the Sun is higher in the sky at noon, not that the Sun is physically closer. A change in distance would not by itself make the noon shadow shorter in the same way; the shadow depends mainly on the Sun's angle above the horizon. Also, Earth's orbit is only slightly elliptical, and in the Northern Hemisphere summer Earth is actually near its farthest point from the Sun, not its closest. The seasonal warmth is better explained by Earth's tilted axis: in summer, sunlight strikes the Northern Hemisphere more directly and days are longer, so more solar energy is received. The shadow data support the angle explanation.
3. What additional evidence would help test whether distance to the Sun causes the seasons? Describe a comparison.
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One strong test is to compare the two hemispheres at the same time of year. If distance to the Sun caused summer, then both hemispheres would have summer at the same time, because they are the same distance from the Sun. In reality, when the Northern Hemisphere has summer, the Southern Hemisphere has winter. Another test is to measure the Sun's apparent size or the solar energy reaching the top of Earth's atmosphere over a year. The distance change is small, only a few percent, while the seasonal change in sunlight angle and day length is much larger. These comparisons would show that distance alone cannot explain the seasonal pattern.