Law of Equal Areas

KEPLER'S 2ND LAW

THE LAW OF EQUAL AREAS

ORBITAL VELOCITY:

Variable Speed

Planets do not travel at a constant speed. They move faster when near the Sun and slower when further away.

AREA SWEEP

dA/dt = C

Constant area over time.

ORBITAL SPEED

DYNAMIC

Inversely proportional to distance.

Kepler Law II

The Rhythmic Swing. Kepler’s Second Law is Momentum Balance. It proves that planets don't just "float"—they accelerate and decelerate in a perfect, gravitational dance.

🏎️ Velocity Delta: Speed is highest when distance is lowest.
🍰 Equal Slices: The "Area" of the orbit is the universal constant.
⏳ Dwell Time: Spacecraft spend most of their life at the "slow" end of the orbit.

MOMENTUM SYNC

dA/dt = Const.

V-MAX: ✅ AT PERIAPSIS

AREA-LOCK ACTIVE ✅

Vector L

The Slingshot Effect. This corrected loop shows the Angular Velocity spike. When the rocket gets close to the Sun, it must speed up to conserve momentum.

🎯 Focus Locking: The orbit is now centered on the primary mass.
🏎️ Sprint Phase: Watch the rocket accelerate at the "turn."
🌌 Deep Space: The slow drift at the furthest point.

☀️

🚀

MOMENTUM SYNC

v 1/r

ALIGNMENT: ✅ OPTIMIZED

SLINGSHOT ACTIVE ✅

Vector Δt

The Universal Pacing. Time Interval is the Standard Unit of Area. In Kepler’s world, time and space are traded like currency to keep the angular momentum in perfect check.

⏳ Fixed Duration: Δt is the independent variable in the sweep.
🗓️ Mission Planning: Measured in Earth Days for logistical sync.

TIME INTERVAL (Δt)

14.5 DAYS

AREA SWEEP: ✅ CALCULATED

TIME-LOCK ACTIVE ✅

Vector v(r)

The Living Force. Instantaneous Velocity is Energy in Motion. It tells the navigator exactly how much kinetic energy is fighting against the gravity well at this precise second.

📍 r-Dependency: Your speed is a direct function of your distance.
🔥 The Delta-v: Changes in instantaneous velocity redefine the entire orbit.
⚡ Kinetic Peak: Velocity is always highest at the Periapsis.

INSTANTANEOUS V

7.6 km/s

r-OFFSET: ✅ 42,164 km

VELOCITY-LOCK ACTIVE ✅

Kepler's 2nd Law

A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.

Perihelion = Fastest speed

Aphelion = Slowest speed

Area Rate = dA / dt = Constant