Law of Equal Areas

KEPLER'S SECOND LAW

LAW OF EQUAL AREAS DYNAMICS

PHYSICS CORE:

The Law of Equal Areas

Kepler's Second Law states that a line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. This implies that a planet moves faster when it is closer to the Sun (perihelion) and slower when it is farther away (aphelion).

ORBITAL SWEEP

A₁ = A₂

The area swept out by the radius vector is constant for any equivalent time segment along the orbit.

VELOCITY VARIATION

CONSTANT AREA RATE

Planets accelerate near perihelion and decelerate toward aphelion to maintain constant area sweep rates.

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 ✅

ORBITAL MECHANICS / DYNAMICS

THE LAW OF EQUAL AREAS

Kepler’s Second Law states that a line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. This is not just a geometric observation; it is a direct consequence of the conservation of angular momentum. It explains why a planet's orbital speed is not constant.

Core Principle dA / dt = constant

Fastest Point Perihelion (Closest)

Slowest Point Aphelion (Farthest)