Seasons

Axial tilt

Goal: connect the 23.4° tilt to the Sun's declination, and to everything you can observe.

The one number

Earth's rotation axis is tilted 23.44° from the perpendicular to its orbit. The axis keeps pointing the same way in space all year; it does not wobble to follow the Sun.

That fixed tilt, plus the orbit, produces everything.

Declination is the tilt made visible

The Sun's declination is its angle north or south of the celestial equator. It is, equivalently, the latitude where the Sun is directly overhead today.

import datetime as dt
from moonfield import sun, time as mtime

for month in range(1, 13):
    when = dt.datetime(2026, month, 21, 12, tzinfo=mtime.UTC)
    print(f"{when:%b}  {sun.declination(when):+7.2f} deg")

It swings between +23.44° and −23.44° and back, once a year, roughly sinusoidally.

The tropics are defined by the tilt. They are the latitudes the Sun can reach overhead. The polar circles at 66.56° (= 90 − 23.44) are where the Sun can stay up or down for a full day.

Change the tilt and you move all four lines.

The formula that ties it together

From module 05:

noon altitude = 90° − |latitude − declination|

Latitude is fixed. Declination is the season. That single expression contains your whole year of noon Suns.

Try it yourself

  1. Plot declination against date for a year. How close to a sine is it?
  2. Compute the Arctic Circle latitude from the tilt alone
  3. What would seasons be like with a 0° tilt? With 90°? (Uranus is at 98°.)
  4. Track your own noon shadow length weekly for a term and plot it

Checkpoint

Next: Solstices and equinoxes.