Seasons: Sun angle, day length and energy per square metre

Lesson page: echoesof.science/school/day-night-seasons/ages-15-16/ · Pupil worksheet: echoesof.science/school/day-night-seasons/ages-15-16/print/

Learning aims

Describe Earth's rotation, orbit and axial tilt with values.
Use α = 90° − φ + δ for the noon Sun elevation.
Use S_horizontal = S_incident · sin α to compare energy per square metre.
Explain day length and the polar day and night from the tilt.
Measure an angle with a shadow stick and evaluate uncertainty.

Lesson plan

Time	Activity
5 min	Starter: the 'closer to the Sun' misconception. Give the January and July distances and ask the class to judge.
10 min	Read the text; work through the 51° N June and December examples.
15 min	Shadow-stick measurement (or demonstrate with a lamp and stick indoors if it is cloudy).
10 min	Calculate sin α for June and December and the ratio.
5 min	Quiz, or the game as homework.

Materials for the experiment

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.

Common misconceptions

Pupils may think…	Actually…
Seasons are caused by Earth's distance from the Sun.	The distance changes by only about 3% and Earth is closest in January.
The Sun is directly overhead at noon everywhere.	Only between the tropics, and only on certain dates; elsewhere α = 90° − φ + δ is less than 90°.
The longest day is also the hottest day.	The hottest weather lags a month or more behind the solstice.

Worksheet answers

June: 90 − 51 + 23.4 = 62.4°. December: 90 − 51 − 23.4 = 15.6°. Equinox: 90 − 51 = 39°.
sin 62.4° / sin 15.6° ≈ 0.886 / 0.269 ≈ 3.3 times.
Distance varies only about 3% (147 to 152 million km); Earth is closest in January during northern winter; north and south have opposite seasons at the same distance.
Answers will vary. Uncertainties: stick not vertical, ground not level, blurry shadow edge, finding the exact shortest shadow, measuring L.

Quiz answers

Using α = 90° − φ + δ, what is the noon Sun elevation at latitude 51° N on the June solstice (δ = +23.4°)? 62.4°. 90 − 51 + 23.4 = 62.4°.[6]
And at the same place on the December solstice (δ = −23.4°)? 15.6°. 90 − 51 − 23.4 = 15.6°.[6]
With S_horizontal = S_incident · sin α, what fraction of overhead sunlight reaches a horizontal surface when α = 30°? 0.5. sin 30° = 0.5.[7]
By roughly how much does Earth's distance from the Sun vary over a year? About 3%. From about 147 to 152 million km.[4][3]
If Earth's tilt were 16°, at what latitude would the Arctic Circle be? 74° N. Arctic Circle latitude = 90° − tilt.[4]
Why are the hottest months usually July and August rather than June in the north? Air and oceans take time to heat up. The seasons lag behind the Sun because of the slow heating of air and water.[4]

Going further

At the North Pole (φ = 90°) the formula gives α = δ. What does this mean for the Sun's height on the June solstice and at the equinoxes, and how does it explain six months of daylight?

Sources used in this lesson

[1] What Causes the Seasons? (NASA Space Place): reliability High
[2] Earth: Facts (NASA Science): reliability High
[3] Why Does Earth Have Seasons? (NOAA NESDIS (U.S. satellite and information service)): reliability High
[4] Astronomy, 4.2 The Seasons (OpenStax, Rice University): reliability High
[6] Elevation Angle (PVEducation (C. Honsberg and S. Bowden)): reliability Medium-high
[7] Solar Radiation on a Tilted Surface (PVEducation (C. Honsberg and S. Bowden)): reliability Medium-high

Full list with links and a note on each source's reliability: echoesof.science/school/day-night-seasons/sources/

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