§ 17. Elliptical Orbit & Kepler's Second Law (Grade 10–11)
Elliptical planetary orbit, equal-area sweep rate & orbital energy exchange. A planet's radius vector sweeps out equal areas in equal times: orbital speed peaks at perihelion and drops to a minimum at aphelion.
Theory Summary & Scientific Mechanism
Planetary and satellite orbits in the central gravitational field of a body of mass M are governed by Newton’s law of gravitation F = -G(Mm)/(r³)r. Because a central force exerts zero torque about the force center (τ = [r × F] = 0), the planet’s orbital angular momentum L = m[r × v] = const is strictly conserved.
Angular momentum conservation directly implies Kepler’s Second Law (Law of Equal Areas): the areal velocity swept out by the radius vector is constant throughout the orbit, (dS)/(dt) = (L)/(2m) = const. Every bound orbit (E < 0) traces an ellipse r(θ) = (a(1 - e²))/(1 + ecos θ) of semi-major axis a and eccentricity e ∈ [0, 1) with the Sun at one focus.
At perihelion (rₘin = a(1 - e)), gravitational potential energy Eₚ = -(GMm)/(r) is lowest and orbital speed peaks at vₚ = √((GM)/(a)(1+e)/(1-e)). At aphelion (rₘax = a(1 + e)), speed drops to vₐ = √((GM)/(a)(1-e)/(1+e)), satisfying the exact ratio vₚ / vₐ = (1 + e)/(1 - e).
Everyday Analogy
Just like an ice skater spinning faster when pulling their arms in, a comet or planet speeds up dramatically when swinging closest to the Sun (perihelion) and slows down far away (aphelion)!
Interactive Question & Derivation
A planet orbits a star along an elongated ellipse with eccentricity e > 0. How does the planet's orbital speed v change as it moves from its closest approach (perihelion rₚ) to its farthest point (aphelion rₐ)?
- A: The orbital speed remains constant at every point along the ellipse
- B: Speed is maximum at perihelion vₚ = vₐ (1+e)/(1-e) and decreases toward aphelion due to conservation of angular momentum L = mvr_⊥
- C: Speed is highest at aphelion because the orbital radius is largest
Study Hint: By Kepler’s second law (angular momentum conservation r · v_⊥ = const): is orbital speed highest at perihelion (closest to the star) or aphelion (farthest)?
Proof & Explanation: Because gravity F_g is a central force (zero torque about the star), the planet's angular momentum L = m v r sin α is conserved. This yields Kepler's Second Law: the areal velocity dS/dt = L/(2m) is constant. At perihelion rₚ = a(1-e) is smallest so speed is greatest; at aphelion rₐ = a(1+e) speed is lowest.
Key Equations & 3D Simulation Parameters
rₚ = a(1 - e), rₐ = a(1 + e), (vₚ)/(vₐ) = (1 + e)/(1 - e)
- Orbital eccentricity e: 0 … 0.75 (default: 0.45)
- Semi-major axis a: 3 … 5.6 AU (default: 4.4 AU)
- Central star mass M: 0.8 … 2.5 M☉ (default: 1.5 M☉)
- Planet mass m: 0.5 … 2.5 M⊕ (default: 1 M⊕)