力学 Classical Mechanics · Simulation

Kepler's Laws of Planetary Motion

Aim: to simulate planetary and cometary motion under Newton's inverse-square law of universal gravitation, verify the elliptic orbit equation (Kepler I), prove constancy of areal velocity (Kepler II), and establish the harmonic relation \(T^2 \propto a^3\) across the solar system (Kepler III).

Orbital Plane & Swept Areal Velocity

Orbit: Stable Ellipse
Watch the swept area sectors. When "Equal Time Sectors" is checked, each shaded wedge has identical surface area \(\Delta A\).

Kepler's Third Law: \(T^2\) against \(a^3\)

Orbital Energy & Conservation of \(L\)

Solar System Celestial Catalog

Newton's Derivation of Kepler's Laws

In an inverse-square central gravitational field \(\mathbf{F} = -\frac{G M m}{r^2}\hat{\mathbf{r}}\), angular momentum \(\mathbf{L} = \mathbf{r} \times m\mathbf{v}\) is strictly conserved because the torque \(\boldsymbol{\tau} = \mathbf{r} \times \mathbf{F} = 0\).

  1. Kepler's First Law (Law of Ellipses): The orbit of a planet is an ellipse with the Sun at one focus: $$ r(\theta) = \frac{a(1 - e^2)}{1 + e\cos\theta} $$ The perihelion distance is \(r_p = a(1 - e)\) and aphelion is \(r_a = a(1 + e)\).
  2. Kepler's Second Law (Equal Areas in Equal Times): The area swept out by the position vector in differential time \(dt\) is: $$ dA = \frac{1}{2} |\mathbf{r} \times d\mathbf{r}| = \frac{1}{2} r^2 \dot{\theta} dt \implies \frac{dA}{dt} = \frac{L}{2m} = \text{constant} $$ A planet moves fastest at perihelion and slowest at aphelion, sweeping out equal area sectors \(\Delta A\) in equal time intervals \(\Delta t\).
  3. Kepler's Third Law (Harmonic Law): Integrating the areal velocity over an entire orbital period \(T\): $$ \text{Area} = \pi a b = \pi a^2 \sqrt{1 - e^2} = \left(\frac{L}{2m}\right) T $$ Combining with the focal parameter \(L^2 / (G M m^2) = a(1 - e^2)\) yields the exact harmonic relationship: $$ T^2 = \left(\frac{4\pi^2}{G M}\right) a^3 $$

The Vis-Viva Equation & Numerical Stability

The velocity \(v\) at any radial distance \(r\) from the central star is governed by the vis-viva energy equation:

$$ v^2 = G M \left(\frac{2}{r} - \frac{1}{a}\right) $$

The total specific orbital energy \(\varepsilon = \frac{E}{m}\) remains invariant:

$$ \varepsilon = \frac{v^2}{2} - \frac{G M}{r} = -\frac{G M}{2a} $$

To prevent numerical energy drift and artificial perihelion precession over thousands of orbits, PrincipleSim integrates the equations of motion using a second-order symplectic Verlet scheme that exactly preserves the Laplace–Runge–Lenz vector.

Viva Questions

Why does Kepler's second law hold for ANY central force?

Because \(\mathbf{F} \parallel \mathbf{r}\), meaning the cross product \(\mathbf{r} \times \mathbf{F} = 0\). Torque vanishes for any central force (even \(1/r^3\) or a harmonic spring), guaranteeing conservation of angular momentum and constant areal velocity.

What distinguishes an inverse-square force (\(1/r^2\)) from other central forces?

Bertrand's theorem states that only \(1/r^2\) (gravitational/Coulomb) and \(r\) (harmonic oscillator) produce closed, non-precessing orbits for all bound initial conditions. The Laplace–Runge–Lenz vector is conserved exclusively in \(1/r^2\) potentials.