Planet Rotation (2D)

Sidereal day, solar day and the year

A planet turns once relative to the stars in the sidereal period T_sid, but the Sun returns to the same position in the sky only after the solar day T_sol, because the planet also moves along its orbit (period T_year). The spin and the orbit combine as angular velocities:

For Earth 1/23.934 h − 1/8766.1 h gives 24.00 h. Venus spins backwards (negative T_sid = −5832.5 h) so both terms add and a Venus solar day lasts only about 116.8 Earth days. The speed of the equator is v = 2πR/T_sid (about 465 m/s for Earth).

Axial tilt and the Sun's declination

The spin axis keeps a fixed direction in space, tilted by ε from the orbit normal. With λ the orbital angle measured from the spring equinox, the latitude of the subsolar point (the solar declination) is

The Sun is above the horizon at latitude φ when sin(elevation) = sin φ sin δ + cos φ cos δ cos H > 0, where H is the hour angle. Setting the elevation to zero gives the half-day arc cos H₀ = −tan φ tan δ (polar day when the right side is below −1, polar night above +1). The daily mean sunlight on a flat surface at the top of the atmosphere is

Averaged over the whole sphere (weights cos φ) this gives exactly S/4, whatever the tilt: the page checks this numerically.

Honest simplifications

• Circular orbits: no eccentricity, so no equation of time and no perihelion effect on the seasons (important for Mars, small for Earth).
• The Sun is a point; no refraction or atmosphere, so sunrise is when the centre crosses the geometric horizon. Gas giants have no solid surface, so the pin sits at a cloud-top latitude.
• Rotation rates of the giant planets are system-III values; real cloud layers rotate at different speeds. Surface patterns are procedural art, not maps.
• Uranus's 97.8 degree tilt and Venus's 177 degree tilt are both listed as measured (the sense of spin differs by convention).
• The orbit is sped up by the fast-forward factor so that seasons can be seen; the day-to-night cycle is drawn at four seconds per solar day at speed 1.

• Press Uranus and move the orbit position: at the solstice a whole hemisphere sits in daylight for decades, and the daylight graph at 60 N goes flat at 100 %.
• On Earth set the latitude to 66 and sweep the orbit; check where daylight hits 0 % and 100 %.
• Drag the tilt slider to 0: every graph collapses to the same line, the Sun stays on the equator and there are no seasons.
• Press Venus: watch the Sun rise in the west and see how long its solar day is compared to its sidereal day.
• Compare the sunlight at the observer on Mars and Earth: the same tilt gives different seasons only because the year and the distance differ.