Kepler's 2nd law — equal areas in equal times

Kepler's second law: the line from the Sun to a planet sweeps out equal areas in equal times. Because the ellipse is narrow near the Sun, the planet must move faster there (perihelion) and slower when far away (aphelion) — the purple sector keeps the same area everywhere on the orbit.

What the animation shows

The planet is integrated by conservation of angular momentum, so its speed is not constant. The purple sector is the area swept in a fixed time window — watch it stay equal while its shape changes. Live r and v follow the vis-viva equation; the orbit period obeys Kepler's third law. (Eccentricity is exaggerated for clarity.)

Quantities

• a = 1.00 AU — semi-major axis (1 AU = 1.496×10¹¹ m).
• e — eccentricity (0 = circle, →1 = thin ellipse).
• T = 1.00 yr — period, T² = 4π²a³/GM.
• r — Sun–planet distance, r = a(1−e²)/(1+e·cosθ).
• v — orbital speed, v = √(GM(2/r − 1/a)); ≈ 29.8 km/s at 1 AU.
• dA/dt = L/2 — areal velocity, constant (this IS the 2nd law).

Sources

Kepler, Astronomia Nova 1609 & Harmonices Mundi 1619; Newton, Principia 1687; standard solar GM.