EXPLORATION 24·Astronomy · Tilt and sunlight

How does a tilted axis create the seasons?

Connect axial tilt, equinoxes, solstices and polar day through Sun height, daylight, shadows and a zero-tilt comparison.

3:01 filmSuggested ages 10+

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Preparing the experiment…

0:00One orbit, opposite seasons
Follow the gold latitude circle, then compare the two daylight ribbons. The northern day is long while the southern day is short. Half an orbit later, they trade places. Both locations are almost equally far from the Sun. We even keep this orbit circular. Look instead at the direction of the incoming light.
0:22Keep the axis fixed
Look at the four small axes. They remain parallel as Earth travels around the Sun. The axis does not turn to keep pointing toward the light. That leaves the northern end leaning sunward during one part of the orbit, and the southern end during another. The tilt itself has not changed.
0:44Spread the same beam
Watch the gold beam. Measured across the rays, its width stays at one. As the Sun lowers, the bright footprint on the ground stretches. The same incoming beam is shared by more horizontal area, so each square metre intercepts less power. This is a change in geometry before we consider air or weather.
1:06Give sunlight time
Angle is only part of the story. Let the post experience a winter day and then a summer day. The Sun crosses the sky and the shadow turns around the base. Notice the width of the input curve. Longer daylight extends the receiving time. The area underneath combines Sun height with time.
1:28At an equinox
Now approach an equinox. The day–night boundary passes through both poles, and the Sun’s direct overhead point returns to the equator. The two daylight ribbons match. We define sunrise and sunset using the Sun’s centre on a geometric horizon. That gives twelve hours here; real observations also include refraction and the Sun’s disk.
1:51Sunlight at midnight
Move the observing field to seventy-five degrees north. Near the June solstice, follow the Sun right through midnight: it still stays above the horizon. The entire latitude circle receives daylight. Half a year later, the Sun remains below the horizon all day. Polar day and night extend the same geometry to high latitudes.
2:15Try a world without tilt
Return to fifty degrees north for a controlled comparison. Set the axis upright and continue around the orbit. The noon shadow stops changing, daylight remains twelve hours, and the annual input curves become flat. The place and the incoming beam are unchanged. Removing tilt removes this yearly variation in our circular-orbit model.
2:37What the geometry tells us
Restore Earth’s tilt and watch a full year return. The two latitude curves run half an orbit apart while the shadow lengthens and shortens. We have measured accumulated geometric input, not temperature. Clouds, the atmosphere, oceans and heat stored in the ground still affect the result. The hottest day need not be the solstice.

Why do the hemispheres have opposite seasons?

Place identical posts in the northern and southern hemispheres. In June, the northern post has a shorter noon shadow and a longer day. Half a year later, the two places exchange roles.

Earth’s axis is tilted. As Earth travels around the Sun, that axis keeps nearly the same direction in space. One hemisphere then gets more direct sunlight and longer days, while the other gets the opposite.

Follow the shadow and daylight ribbon. Ask separately how directly sunlight arrives and how long it lasts. Both affect incoming sunlight; oceans and air also affect temperature.

Test a guess with a shadow

Advance half a year. Keep the same place and solar noon. Compare the shadow and length of daylight.
Follow an equal-width beam. When light arrives at a shallow angle, does it cover more or less ground?
Set the axis upright. Travel around the Sun again. Does sunlight at the same place still change through the year as before?

Inside the observing model

More: additional comparisons and controls

Enter exploration, keep the observer at 50° north and solar noon, then advance half a year. Compare the shadow, Sun height and daylight ribbons.
Select “Equal-width beam.” Its cross-section stays the same width while its footprint changes. A lower Sun spreads the same beam over more ground.
Choose “Midnight at 75° N.” Near the June solstice the Sun is still above the horizon at midnight. Advance half a year to see polar night.
Set the axis upright and move around the orbit. Except at the poles’ special horizon boundary, daylight lasts twelve hours. At a fixed latitude, noon shadow length and daily geometric input no longer change through the year.
Set solar longitude to 0° or 180° and change latitude. At an equinox the Sun’s declination is zero. Most latitudes divide the day in half; at either exact pole, our ideal point Sun remains on the horizon all day.

From a fixed axis to the Sun in a local sky

We use a circular orbit, a fixed rotation axis and parallel sunlight. Obliquity ε is measured from the normal to the orbital plane and defaults to 23.44°. Solar longitude λ starts at the March equinox: 0°, 90°, 180° and 270° mark the March equinox, June solstice, September equinox and December solstice. This is an orbital angle, not a calendar date.

The dot product of the fixed axis and the direction toward the Sun determines declination δ. Latitude φ and local solar time locate the observer’s surface normal. That same geometry determines solar altitude α:

sin δ = sin ε · sin λ

sin α = sin φ sin δ + cos φ cos δ cos H

Here H = 15° × (solar hour − 12). Local solar noon is when the Sun crosses the observer’s meridian. It need not match noon on a civil clock. We omit longitude, time-zone and daylight-saving conversions and the equation of time. The altitude relationship is documented in NOAA’s solar-position equations. Declination here comes directly from our circular orbit rather than NOAA’s calendar-based fit.

A shadow is the continuation of a light ray

Write the unit vector toward the Sun in local east, north and up components E, N and U. When the Sun is above the horizon, the shadow tip of our one-metre post lies at (−E/U, −N/U). Its distance from the base is 1 / tan α metres. The line from the post’s tip to the shadow tip is parallel to the incoming light. No finite direct-sun shadow is drawn at or below the horizon.

A fixed-width beam spreads across the horizontal ground by a factor of 1 / sin α. The incoming power per horizontal square metre, relative to a surface facing the beam directly, is sin α. Shadows longer than the observing plate stop at its edge; the numerical reading retains their full length. At very low Sun heights, the beam diagram also clips footprints outside its drawing area and labels that truncation.

For an ordinary sunrise and sunset, cos H₀ = −tan φ tan δ, and geometric daylight lasts 24H₀/π hours. The model handles polar day and night by checking the day’s maximum and minimum Sun heights before evaluating an inverse cosine. Its geometric horizon is the centre of a point Sun at altitude zero. We deliberately omit atmospheric refraction and the Sun’s angular radius, including NOAA’s usual 90.833° zenith correction for observed sunrise and sunset.

Incoming geometry is not a temperature forecast

Let S₀ be the incoming power per square metre on a plane facing the Sun. The horizontal-plane input in this model is S₀ × max(sin α, 0). Integrate over one day and divide by S₀ to obtain the displayed Q / S₀ in hours. This is the equivalent duration of perpendicular illumination from the same beam.

Q / S₀ = ∫ max(sin α, 0) dt

The gold and blue annual curves give this daily integral for opposite latitudes. The pale dashed line is the zero-tilt comparison at the same latitude. We do not calculate absorption, scattering, clouds, reflectivity, heat storage or energy transport. The curve is neither surface temperature nor weather. A solstice’s geometric maximum does not require the hottest day to fall on that date.

Earth’s real orbit is slightly elliptical, and changing distance modulates incoming sunlight. We hold the radius at 1 AU to isolate the role of tilt. This teaching choice does not claim that the real orbit is a perfect circle. NASA’s explanation of the seasons connects the tilted axis to the opposing seasonal cycles of the two hemispheres.

Why do the hemispheres have opposite seasons?

Test a guess with a shadow

Inside the observing model

Further reading & sources

Where will your curiosity go next?