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Seasons: Sun angle, day length and energy per square metre
The geometry of the seasons: Earth's rotation and orbit, the noon Sun formula α = 90° − φ + δ, and how the Sun's elevation sets the energy on each square metre. With a shadow-stick measurement.
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Earth rotates once every 23.9 hours and orbits the Sun once every 365.25 days. Its rotation axis is tilted by about 23.4° to the perpendicular of its orbit. In June the Northern Hemisphere is tilted towards the Sun and in December the Southern Hemisphere is.[2][3]
Distance is not the cause. Earth's distance from the Sun varies by only about 3%: about 147 million km in January (perihelion) and 152 million km in July (aphelion). The Northern Hemisphere is in winter when Earth is closest, and the two hemispheres have opposite seasons at the same distance.[4][3][1]
The Sun's height at noon depends on latitude φ and the Sun's declination δ, which changes through the year from about +23.4° (June solstice) through 0° (equinoxes) to about −23.4° (December solstice). In the Northern Hemisphere the noon elevation is α = 90° − φ + δ. Example: on the equator (φ = 0°) on 21 June the noon Sun is about 67° high.[6][4]
The elevation sets how concentrated the sunlight is. The power on a horizontal surface is Shorizontal = Sincident · sin α. At α = 30°, sin α = 0.5, so the same beam is spread over twice the area it would cover with the Sun overhead.[7][4]
The tilt also changes day length. Near the equator days are about 12 hours all year. At the June solstice the Sun is overhead at the Tropic of Cancer (latitude equal to the tilt) and everywhere north of the Arctic Circle (latitude 90° minus the tilt) has 24 hours of daylight; at the poles there are six months of daylight and six of darkness. At the equinoxes every place has about 12 hours of day and 12 of night.[4]
The hottest and coldest weather comes a month or more after the solstices, because the air and especially the oceans take time to heat up and cool down.[4]
Try it: Measure the Sun's height with a shadow stick
You need
- A straight stick or metre rule, at least 50 cm long
- A spirit level or plumb line
- A measuring tape
- Chalk
- A watch
Safety
NEVER look directly at the Sun, not even with sunglasses; look only at the shadow. Work away from traffic and keep the stick from falling on anyone.
What to do
- On flat, level ground, stand the stick exactly vertical (check with the level or plumb line). Measure its height h above the ground.
- From about 20 minutes before local solar noon until 20 minutes after, mark the tip of the shadow with chalk every 5 minutes.
- Find the shortest shadow: this is local solar noon. Measure its length L.
- Calculate the Sun's elevation: tan α = h / L, so α = arctan(h / L).
- Look up your latitude φ and today's solar declination δ (for example from an almanac or a solar-position tool) and calculate α = 90° − φ + δ.
- Compare the measured and calculated angles. Then calculate sin α: what fraction of the full overhead sunlight is reaching each square metre of ground?
What happens, and why?
The shadow is shortest when the Sun is highest, at local solar noon. The stick, its shadow and the sunray form a right-angled triangle, so tan α = h / L. The measured angle should agree with α = 90° − φ + δ to within a degree or two; repeating on other dates shows α rising towards June and falling towards December.[6][7]
Play: Noon Sun calculator
Use α = 90° − φ + δ (Northern Hemisphere) with δ = +23.4° in June, 0° at the equinoxes and −23.4° in December. Put each case in the right box.
Your best score is kept only on this device. Nothing is sent anywhere.
Quiz
Other ages: Ages 6–7 · Ages 10–11