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Seasons — it's not the distance

In early January, Earth is closer to the Sun than at any other time of year — and in Poland it's winter. Seasons come from the 23.44° tilt of Earth's axis. Rotate the scene, move the date, and see where long summer days, white nights and the midnight sun come from, why winter is the shortest season in the north — and why, some twelve thousand years from now, Vega will shine at the celestial pole.

“We're closer to the Sun in summer”→perihelion ≈ January 3 (147.1 million km), aphelion in early July (152.1 million km)

Drag to rotate the scene; scroll or pinch to zoom. Earth and the Sun are not to scale.
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←→ day Shift+←→ month Space play / pause T today 1–5 modes +− zoom

Where do the seasons come from?

Earth's rotation axis is tilted by ε = 23.44° from the perpendicular to the orbital plane, and all year round it points to the same place in the sky — near the North Star. So in June the Northern Hemisphere is turned toward the Sun, and in December away from it. When the Sun is higher, the same beam of light falls on a smaller area, and the day is longer — and that's what does the heating.

The distance from the Sun changes by only about 3.4% (147.1 → 152.1 million km), so the flux of sunlight differs by about 6.9% between perihelion and aphelion. In Warsaw, the summer–winter difference is almost sevenfold.

Solar declination

The declination δ is the latitude over which the Sun stands at the zenith at noon. A simple approximate formula (N — day number of the year, January 1 is N = 1):

δ ≈ −23.44° · cos(360°365· (N + 10))

The simulation is more precise: from Kepler's orbit we find the Sun's ecliptic longitude λ, and then sin δ = sin ε · sin λ. That way the equinoxes, solstices, and perihelion fall on the right days.

Height of the Sun at noon

h = 90° − φ + δfor the Northern Hemisphere, when the Sun culminates in the south; in general h = 90° − |φ − δ|

In Warsaw (φ = 52.2°) this gives about 61.2° in June and only 14.3° in December.

Day length

cos H0 = sin h0 − sin φ sin δcos φ cos δ,   day = 2H015°/h

H0 is the hour angle of sunrise. We take h0 = −0.833°: the atmosphere lifts the Sun's image by about 0.57° (refraction), and sunrise is counted from the upper limb of the disk (radius 0.27°). When the right-hand side falls outside the range −1 to 1, the Sun doesn't set (polar day) or doesn't rise (polar night).

Insolation

Q = S0π · (d̄d)2 · (H0 sin φ sin δ + cos φ cos δ sin H0)

This is the average power of sunlight per 1 m² of horizontal surface over a day, above the atmosphere. S0 ≈ 1361 W/m² is the solar constant, d̄/d is the ratio of the mean to the current distance from the Sun, and H0 (in radians) is computed here for the center of the disk on the horizon (h0 = 0). The factor (d̄/d)² varies over the year only from about 0.967 to 1.034.

Twilights

At midnight the Sun is at its lowest: hmin = φ + δ − 90° (Northern Hemisphere). On the solstice (δ = 23.44°), white nights — civil twilight all night long (hmin ≥ −6°) — begin at 60.56° N, and the Sun stops setting from about 65.7° N — a little south of the Arctic Circle (66.56°), thanks to refraction and the size of the disk.

Kepler's orbit and the unequal seasons

M = E − e sin E,   tan(ν2) = √1 + e1 − e tan(E2),   r = a(1 − e²)1 + e cos ν

The mean anomaly M grows steadily with time, and Kepler's equation gives the position on the ellipse (e = 0.0167). The seasons are quarters of the orbit (every 90° of the Sun's longitude), but Earth runs through them at different speeds: today northern spring lasts about 92.7 d, summer 93.7 d, autumn 89.9 d, and winter 89.0 d.

The Great Year

The gravity of the Sun and Moon acts on the flattened Earth as on a spinning top: the axis traces a cone once every 25,770 years or so (precession, 50.29″ per year, i.e. 1° in about 72 years). At the same time, the orbital ellipse itself rotates. Together, this makes perihelion shift through the calendar by about 1 day every 58 years, and a full cycle relative to the seasons takes about 21,000 years.

Model simplifications
  • Earth's orbit is a Keplerian ellipse with a constant eccentricity of 0.0167; we ignore the influence of the planets. The perihelion and aphelion dates include a correction for the Moon (Earth's center orbits the barycenter of the Earth–Moon system), so they match the tables to within a few hours.
  • The axial tilt is fixed at 23.44°; in reality it changes slowly (about 23.44° today and decreasing).
  • Sunrise and sunset times are given in solar time; clock time for cities is computed from longitude, time zone, and the equation of time. Refraction is constant (standard); in reality it depends on the weather and can shift sunrise by a minute or two.
  • We take the declination as constant during the day (computed at noon).
  • Insolation is computed above the atmosphere, without clouds, absorption, or reflection.
  • Precession is a uniform rotation around a circle; perihelion shifts linearly, and eccentricity and tilt are constant in mode 5. We ignore nutation and the stars' proper motions.
  • The Milankovitch curves are a sketch with the right periods and ranges, not an orbital solution (such as Laskar's).
  • For Jupiter, Uranus, and Venus, the direction of perihelion relative to the seasons is approximate or arbitrary.

Did you know…?

  1. We're closest to the Sun in January. Perihelion falls around January 2–5 (about 147.1 million km), aphelion around July 3–6 (about 152.1 million km). The exact day varies from year to year, partly because of the Moon.
  2. In January, Earth moves about 1 km/s faster than in July: about 30.3 km/s versus 29.3 km/s. That's why winter is the shortest season in the north — it lasts about 89 days, and summer about 93.7.
  3. During the southern summer, about 7% more sunlight reaches the top of the atmosphere than during the northern summer, because Earth is closer to the Sun then. That doesn't make heat waves much worse, though: about 81% of the Southern Hemisphere is ocean, which moderates the climate.
  4. The North Star wasn't always — and won't always be — the pole star. Around 2800 BCE, when the pyramids were being built, the celestial pole lay near Thuban in Draco, and in about 12,000 years it will be about 5° from Vega.

Challenges 0/6

  • Set the date of perihelion

    Find the day when Earth is closest to the Sun (modes 1–4, Earth, without playback).

    not completed
  • Set the tilt so the seasons disappear

    At what axial tilt does the Sun follow the same path every day?

    not completed
  • Find the latitude where the Sun doesn't set on June 21

    Set the date to around June 21 and a latitude with polar day (including refraction).

    not completed
  • Create true white nights

    The Sun must set, but not sink more than 6° below the horizon all night.

    not completed
  • Find the day when the noon Sun stands highest in Warsaw

    Choose Warsaw and look for the day with the highest noon Sun.

    not completed
  • Shift time so that perihelion falls in summer

    In Great Year mode, find an epoch when Earth is closest to the Sun during the northern summer.

    not completed

Event dates are given in Polish time. The model computes positions to within a few hours, so an event just before midnight may fall a day later or earlier in the tables.