Kepler's Laws

The three laws

Kepler's First Law (1609). The orbit of every planet is an ellipse with the Sun at one focus.

Kepler's Second Law (1609). A line connecting a planet to the Sun sweeps out equal areas in equal times.

Kepler's Third Law (1619). The square of a planet's orbital period is proportional to the cube of its semi-major axis:

T² = (4π² / GM) · a³

First law — ellipses

An ellipse is defined by two focal points and a constant — for any point P on the ellipse, |PF₁| + |PF₂| = constant (= 2a, where a is the semi-major axis).

The Sun sits at ONE focus (not at the center). The planet's distance from the Sun varies between perihelion (closest, a(1-e)) and aphelion (farthest, a(1+e)), where e is eccentricity.

Object	Semi-major axis	Eccentricity	Perihelion	Aphelion
Mercury	0.387 AU	0.206	0.307 AU	0.467 AU
Earth	1.000 AU	0.0167	0.983 AU	1.017 AU
Mars	1.524 AU	0.0934	1.381 AU	1.666 AU
Pluto	39.48 AU	0.249	29.66 AU	49.31 AU
Halley's Comet	17.83 AU	0.967	0.587 AU	35.08 AU

Second law — equal areas in equal times

This is conservation of angular momentum. Mathematically: dA/dt = L/(2m), where L is angular momentum (constant for a Keplerian orbit because gravity exerts no torque about the focus).

Consequence — planets move FASTER when closer to the Sun (perihelion) and SLOWER when farther (aphelion):

• Earth at perihelion (early January): v = 30.29 km/s.
• Earth at aphelion (early July): v = 29.29 km/s.
• Difference: 1 km/s, or about 3% — small because Earth's orbit is nearly circular.

For Halley's comet (e = 0.97), the speed difference is enormous — 54 km/s at perihelion vs 0.9 km/s at aphelion.

Third law — period vs orbital size

For any orbit around the same central mass:

T² = (4π² / GM) · a³

Or, for solar system orbits with T in years and a in AU (using 4π²/GM_sun ≈ 1):

T² = a³ (in solar system units)

Planet	a (AU)	T (years)	a³	T²
Mercury	0.387	0.241	0.058	0.058
Earth	1.000	1.000	1.000	1.000
Mars	1.524	1.881	3.54	3.54
Jupiter	5.203	11.86	140.85	140.66
Saturn	9.539	29.46	867.97	867.89
Pluto	39.48	247.94	61,548	61,475

The match is essentially exact (small deviations from interplanetary perturbations and relativity).

Newton's derivation

Starting from F = GMm/r² (his universal law), Newton derived all three Kepler laws:

1. Bound orbits are conics — ellipses for stable bound orbits, circles in the special case of e=0, parabolas at exactly escape, hyperbolas for unbound trajectories.
2. Angular momentum conserved — gravity is a central force (along the radius), so it exerts no torque about the focus. Equal areas in equal times follows.
3. T² ∝ a³ — direct algebra from F = ma applied to centripetal motion (or more carefully, applying the orbital integral).

This unification — three empirical laws derived from one universal force — is one of the great achievements of physics.

JavaScript — Kepler's laws calculations

const G = 6.674e-11;
const M_SUN = 1.989e30;
const M_EARTH = 5.972e24;
const AU = 1.496e11;
const YEAR = 365.25 * 24 * 3600;

// Kepler's 3rd law: orbital period from semi-major axis
function orbitalPeriod(a_meters, M_central) {
  return 2 * Math.PI * Math.sqrt(a_meters * a_meters * a_meters / (G * M_central));
}

// In solar system units (a in AU, T in years)
function keplerSimplified(a_AU) {
  return Math.pow(a_AU, 1.5);  // T = a^1.5 (since T² = a³)
}

console.log(`Mars (a = 1.524 AU): T = ${keplerSimplified(1.524).toFixed(3)} yr`); // 1.881
console.log(`Jupiter (a = 5.203 AU): T = ${keplerSimplified(5.203).toFixed(2)} yr`); // 11.86

// Earth around Sun
const T_earth = orbitalPeriod(AU, M_SUN);
console.log(`Earth period: ${(T_earth / YEAR).toFixed(3)} years`); // 1.000

// Speed at perihelion and aphelion (vis-viva equation)
function vAtRadius(a, r, M_central) {
  // v² = GM(2/r - 1/a)
  return Math.sqrt(G * M_central * (2/r - 1/a));
}

const a_earth = AU;
const e_earth = 0.0167;
const r_perihelion = a_earth * (1 - e_earth);
const r_aphelion = a_earth * (1 + e_earth);
const v_perihelion = vAtRadius(a_earth, r_perihelion, M_SUN);
const v_aphelion = vAtRadius(a_earth, r_aphelion, M_SUN);
console.log(`Earth at perihelion: ${(v_perihelion/1000).toFixed(2)} km/s`); // 30.29
console.log(`Earth at aphelion: ${(v_aphelion/1000).toFixed(2)} km/s`);    // 29.29

// Halley's Comet: extreme eccentricity
const a_halley = 17.83 * AU;
const e_halley = 0.967;
const r_p_halley = a_halley * (1 - e_halley);
const v_p_halley = vAtRadius(a_halley, r_p_halley, M_SUN);
console.log(`Halley at perihelion: ${(v_p_halley/1000).toFixed(2)} km/s`); // ~54.6

// Find central mass from orbital period and semi-major axis
function centralMass(T, a) {
  return 4 * Math.PI * Math.PI * a * a * a / (G * T * T);
}

// Moon orbits Earth: a = 384,400 km, T = 27.3 days
const M_e_from_moon = centralMass(27.3 * 86400, 384400e3);
console.log(`Earth's mass from Moon: ${M_e_from_moon.toExponential(3)} kg (actual: ${M_EARTH.toExponential(3)})`);
// Slight difference because Moon contributes too (M_e + M_moon ~ 6.05e24)

Where Kepler's laws show up

• Astronomy. Solar system mapping, exoplanet detection (Kepler's 3rd determines stellar mass from orbital period and radius), binary star systems.
• Spacecraft trajectory design. Hohmann transfer orbits, gravitational slingshots, orbital insertions.
• Satellite operations. GPS satellites, communications satellites, ISS — all in elliptical (often nearly circular) orbits following Kepler.
• Cosmology. Galactic rotation, dark matter inferred from rotation deviations from Keplerian predictions.
• Astrophysics. Black hole mass measurements via orbiting objects (Sagittarius A* mass measured this way).
• History of science. Kepler's laws + Newton's derivation = paradigm of mathematical physics.
• Education. Standard introductory astronomy and physics topic; demonstrates physics's predictive power.

Common mistakes

• Confusing semi-major axis with average distance. Semi-major axis is a, the longest "radius" of the ellipse. Average distance varies depending on what averaging is done (over time, over orbit, etc.).
• Putting the central body at the center of the ellipse. Sun is at a FOCUS, not the geometric center. Geometric center is offset toward the other (empty) focus.
• Treating planets as moving at constant speed. 2nd law says equal areas in equal times — meaning faster at perihelion, slower at aphelion. Speed varies along the orbit.
• Misapplying 3rd law to non-Keplerian orbits. Works only for inverse-square gravity. Galaxies don't follow Kepler (extra dark matter), and tightly-orbiting binaries near black holes need GR.
• Forgetting the central mass dependence in 3rd law. T²/a³ = 4π²/GM. Different M (Sun vs Earth vs Jupiter) gives different proportionality constants. Compare orbits around the SAME central body.
• Treating eccentric orbits like circles. Earth's e = 0.0167 — small but not zero. Aphelion-perihelion difference is ~5 million km. Other planets are more eccentric.