Adjust orbital elements and watch planets trace ellipses—explore eccentricity, inclination, and orbital period
Johannes Kepler discovered the three laws of planetary motion in the early 1600s by analyzing Tycho Brahe's precise observations of Mars. These empirical laws—that orbits are ellipses, that planets sweep equal areas in equal times, and that orbital period depends on semi-major axis—were later derived by Newton from his universal law of gravitation. Kepler's laws apply to any two-body system: planets around stars, moons around planets, and binary stars.
The white ellipse is the orbital path. The yellow star sits at one focus (Kepler's First Law). The other focus is empty—a purely geometric point. Eccentricity e measures how elongated the orbit is: e=0 is a circle, e→1 is highly elongated. The perihelion (closest approach) and aphelion (farthest point) lie along the major axis. Real planets have low eccentricity (Earth e≈0.017, Mars e≈0.09), but comets can have e>0.9.
The blue shaded sector shows the area swept by the radius vector in a fixed time interval. As the planet orbits, this sector rotates. According to Kepler's Second Law, the planet moves faster near perihelion and slower near aphelion such that the swept area per unit time is constant. This is a consequence of angular momentum conservation: L = mr²(dθ/dt) is constant, so dA/dt = L/(2m) is constant.
Kepler's Third Law states P² = (4π²/GM)a³, where P is the orbital period, a is the semi-major axis, G is Newton's constant, and M is the central mass. For the solar system with a in AU and P in years, this simplifies to P² = a³. This law is how we measure masses: observing a and P lets us infer M. It's the foundation of exoplanet characterization and dynamical mass measurements.
Any Keplerian orbit is specified by six orbital elements: (1) semi-major axis a (size), (2) eccentricity e (shape), (3) inclination i (tilt relative to reference plane), (4) longitude of ascending node Ω (where orbit crosses plane), (5) argument of perihelion ω (orientation of ellipse in orbital plane), and (6) mean anomaly M₀ at epoch (where the planet is at t=0). This toy lets you explore a, e, and i. The remaining elements control 3D orientation and phasing.
Circular Orbit: Set e=0. The orbit becomes a perfect circle with the star at the center. Speed is constant (no equal-area law visible since the planet moves uniformly). This is the limiting case—real orbits always have some eccentricity due to perturbations, but many planets have nearly circular orbits.
Watch the Equal-Area Law: Set e=0.7 and watch the blue sector. Near perihelion (closest point), the planet zips along quickly and the sector sweeps rapidly. Near aphelion, the planet crawls and the sector barely moves. Yet the area swept in any fixed time interval is identical. This is conservation of angular momentum in action.
Kepler's Third Law: Set a=1 AU (Earth's orbit). The period is 1 year. Now set a=4 AU. The period becomes 8 years (since 4³ = 64 and √64 = 8). Double the semi-major axis and the period increases by 23/2 ≈ 2.83. This power-law relation holds across the solar system and beyond.
Earth's Orbit: Click "Earth" preset. You get e≈0.017 (nearly circular) and a=1 AU. Earth's orbit is so circular it's hard to see the eccentricity. This low eccentricity is good for climate stability—if Earth had Mercury's eccentricity (e=0.21), seasonal temperature variations would be extreme.
Mars: Click "Mars" preset. With e=0.09 and a=1.52 AU, Mars has a more eccentric orbit than Earth. Martian seasons vary in intensity depending on whether Mars is near perihelion or aphelion during that hemisphere's summer. This asymmetry affects dust storm activity.
Mercury: Click "Mercury". With e=0.21 (highest among planets), Mercury's orbit is noticeably elongated. Its perihelion precesses due to general relativity—an effect not present in Newtonian gravity. This precession (43 arcsec/century) was a key early test of Einstein's theory.
Halley's Comet: Click "Halley's Comet". With e=0.967, this is a highly eccentric orbit. Halley spends most of its 76-year period far from the Sun in the cold outer solar system, then dives through the inner solar system briefly. Comets on such orbits sublimate vigorously near perihelion, producing their iconic tails.
Inclination Effects: Increase inclination to 60°. The orbit now tilts steeply relative to the viewing plane. From our perspective, the planet appears to move in and out of the plane. Most solar system planets have low inclinations (~0-7°) because they formed in a flattened protoplanetary disk. High inclination suggests capture or scattering events.
Energy and Orbit Shape: The total orbital energy E = -GMm/(2a) depends only on the semi-major axis, not eccentricity. Circular and highly eccentric orbits with the same a have identical energies. Eccentricity determines how that energy is partitioned between kinetic and potential—high e means large speed variations.
Kepler's laws are the foundation of celestial mechanics. They describe the motion of planets, asteroids, comets, satellites, and binary stars. Every spacecraft trajectory is a Keplerian orbit (or a sequence of them connected by burns). The laws enable:
Newton derived Kepler's laws from F=GMm/r² and his laws of motion. This unification of terrestrial and celestial mechanics was a watershed moment—the same force that makes an apple fall governs the Moon's orbit. Kepler's empirical laws became consequences of a deeper, universal principle. Today, we use these laws to navigate spacecraft across billions of kilometers with meter-level precision.