Keplerian Two-Body Gravitational Orbit — 3D Lab Simulation
Planetary orbital motion around a central star under Newton's law of universal gravitation
Theoretical Foundation & Historical Context
Foundational two-body problem in celestial mechanics. Under inverse-square gravitational attraction, orbiting bodies trace conic sections satisfying Kepler's three empirical laws of planetary motion.
Law: Kepler's Laws of Planetary Motion & Newton's Law of Universal Gravitation | Scientist: Johannes Kepler & Isaac Newton | Year: 1609
Key Learning Objectives & Competencies
- ✓ Verify Kepler's First Law: planets orbit in ellipses with the Sun at one focal point: r(θ) = p / (1 + e·cos(θ)).
- ✓ Demonstrate Kepler's Second Law: equal areas swept in equal times (dA/dt = L / (2m) = constant).
- ✓ Confirm Kepler's Third Law harmonic relationship: the square of the orbital period T² is proportional to the cube of semi-major axis a³.
Mathematical Formulations & Differential Equations
\vec{a} = - \frac{G M}{r^3} \vec{r}
Inward gravitational acceleration vector
E = \frac{1}{2} v^2 - \frac{\mu}{r} = - \frac{\mu}{2a}
Conservation of specific orbital mechanical energy
Physical Parameters & Units of Measurement
| Symbol | Physical Quantity | SI Unit | Default |
|---|---|---|---|
| mu | Gravitational Parameter GM | AU^3/yr^2 | 39.48 |
Real-World & Industrial Engineering Applications
Artificial satellite mission trajectory planning and geosynchronous orbit insertion maneuvers.
Interplanetary space probe gravity-assist flybys (Voyager 1 & 2, New Horizons, Cassini).
Exoplanet transit detection and mass determination by Kepler, TESS, and James Webb Space Telescopes.
3-Step Controlled Experimental Protocol
- Step 1: Baseline Setup — Initialize the simulation in standard conditions and observe undisturbed system dynamics.
- Step 2: Parameter Variation — Gradually modulate physical parameters using the control panel sliders.
- Step 3: Verification & Phase Space Analysis — Compare phase space trajectories and conservation metrics against analytical ODE solutions.
Frequently Asked Questions (FAQ)
Why do planets move faster when they are closer to the Sun at perihelion?
Because orbital angular momentum L = m·r·v_perp is strictly conserved in a central gravitational field. As orbital distance r decreases, orbital speed v must increase to keep L constant.
What determines whether a celestial orbit is an ellipse, parabola, or hyperbola?
The total orbital mechanical energy E. If E < 0 (eccentricity e < 1), the orbit is a closed bound ellipse. If E = 0 (e = 1), it is an escape parabola. If E > 0 (e > 1), it is an unbound flyby hyperbola.
How did Isaac Newton derive Kepler's empirical laws from fundamental physics?
Newton showed in his 1687 Principia that an inverse-square gravitational force F = G·M·m / r² combined with his laws of motion mathematically requires all bounded two-body orbits to be Keplerian conic ellipses.
Interactive Simulation: Interactive real-time 60 FPS WebGL simulation is active in your browser. Rotate, zoom, and inspect physical apparatus components with full camera freedom.