Earth Seasons - 2D

What drives the seasons

Seasons come from the tilt of the rotation axis (obliquity eps = 23.44 deg), not from the changing distance to the Sun. The axis keeps pointing at the same place in space while Earth goes round the Sun, so each hemisphere alternately leans toward and away from it.

Equations used

• Kepler's equation M = E - e sin E, true anomaly nu = 2 atan2(sqrt(1+e) sin(E/2), sqrt(1-e) cos(E/2)), distance r = a (1 - e cos E); perihelion on 3 Jan, e = 0.0167.
• Solar ecliptic longitude lambda = nu + 282.94 deg; declination sin(delta) = sin(eps) sin(lambda); right ascension tan(alpha) = cos(eps) tan(lambda).
• Equation of time EoT = L0 - alpha (mean longitude minus true right ascension); sub-solar longitude = -(15 deg (UTC - 12 h) + EoT).
• Sun height sin(h) = sin(phi) sin(delta) + cos(phi) cos(delta) cos(H), noon height 90 deg - |phi - delta|.
• Day length cos(H0) = (sin(h0) - sin(phi) sin(delta)) / (cos(phi) cos(delta)) with h0 = -0.833 deg (refraction plus solar radius); day length = 2 H0 / 15 deg per hour.
• Daily mean top-of-atmosphere insolation Q = (S0 / pi) (a/r)^2 (H0 sin phi sin delta + cos phi cos delta sin H0), S0 = 1361 W/m2.

Checks shown in the page

(1) a one-minute scan of the 3D Sun vector reproduces the closed-form day length; (2) Kepler's second law holds for the solver; (3) the area-weighted yearly mean of Q equals S0/4/sqrt(1-e^2); (the 3D companion adds a fourth check against its live scene).

Honest simplifications

Orbit size, Sun and Earth sizes are not to scale (the Earth would be a speck). The orbit is a pure Kepler ellipse (no planetary perturbations, precession or nutation), the atmosphere only enters through the fixed -0.833 deg horizon correction, insolation is at the top of the atmosphere (no clouds, no albedo), and the continents are a hand-simplified outline textured in Blender. Try tilt = 0 (no seasons), 90 deg (a Uranus-like world with polar days) and e = 0.2 (distance matters only slightly).

What to try

Press Observer in Camera and run the year: the gnomon shadow is shortest at noon and shortest overall at the summer solstice. Move the latitude beyond the polar circle and watch the day length saturate at 24 h. Compare the dashed mirror-latitude curve: the two hemispheres are six months out of phase.