Orbital mechanics
Kepler spent eight years wrestling Mars into an ellipse. Newton needed one equation to explain all of it: every orbit is a conic section, every sweep covers equal areas in equal times, and the square of every period scales with the cube of the semi-major axis. Click to place orbiting bodies, drag to set their velocity, and watch Keplerian mechanics unfold — the geometry that holds the solar system together.
The first law — Ellipses
Every planet moves in an ellipse with the Sun at one focus. This was Kepler’s great departure from two millennia of circular astronomy. Circles are a special case — eccentricity zero. Most orbits are ellipses: slightly squashed circles for Earth (e = 0.017), dramatically elongated ovals for comets. When the eccentricity reaches one, the ellipse opens into a parabola; beyond one, a hyperbola. All four curves are conic sections, and all four arise naturally from Newtonian gravity. The simulation marks both foci of each orbit and draws the semi-major and semi-minor axes so you can see the geometry directly.
The second law — Equal areas
A line from the planet to the Sun sweeps out equal areas in equal intervals of time. This means a planet moves faster near perihelion (closest approach) and slower near aphelion (farthest point). The effect is dramatic for eccentric orbits: a comet screams past the Sun in weeks and drifts through the outer solar system for years, but every equal time interval covers the same area. This law is equivalent to conservation of angular momentum — there is no tangential force in a central force field, so L = r × mv is constant. Toggle the Kepler overlays to see the swept areas shaded in equal time slices.
The third law — Periods and distances
The square of the orbital period is proportional to the cube of the semi-major axis: T² ∝ a³. This is why the outer planets take so much longer to orbit. Mars, with a semi-major axis 1.52 times Earth’s, takes 1.88 Earth years. Neptune, at 30 AU, takes 165 years. The proportionality constant depends only on the central mass, which is what makes this law so powerful: measure T and a for any orbiting body, and you can weigh the object it orbits. This is how we know the masses of stars, black holes, and galaxies.
Bound and unbound orbits
Every orbiting body has kinetic energy (KE = ½mv²) and gravitational potential energy (PE = −GMm/r). The total orbital energy E = KE + PE determines the orbit’s shape. When E < 0, the body is bound — trapped in an ellipse, endlessly falling toward the central body and endlessly missing it. When E = 0, the body follows a parabola: it escapes to infinity but barely, arriving with zero velocity. When E > 0, the orbit is a hyperbola: the body passes through, deflected, and escapes forever. In this simulation, bound orbits are colored in blues and greens; escape trajectories glow red.
The vis-viva equation
The speed at any point in an orbit is given by v² = GM(2/r − 1/a), where r is the current distance and a is the semi-major axis. For a circular orbit (r = a always), this simplifies to v = √(GM/r). For a parabolic escape (a → ∞), v = √(2GM/r) — exactly √2 times the circular velocity at the same distance. This √2 factor is the difference between orbiting and escaping: just 41% more speed, and a bound orbit becomes an unbound one.
Circles, ellipses, parabolas, hyperbolas
All orbits under an inverse-square force are conic sections — curves formed by slicing a cone with a plane. Tilt the plane perpendicular to the axis and you get a circle. Tilt slightly and you get an ellipse. Tilt until the plane is parallel to the cone’s surface and you get a parabola. Tilt further and you get a hyperbola. Apollonius of Perga classified these curves in the third century BCE; two thousand years later, Newton showed that gravity produces exactly these shapes and no others. The eccentricity e parameterizes the whole family: e = 0 is a circle, 0 < e < 1 is an ellipse, e = 1 is a parabola, and e > 1 is a hyperbola.
Why conics?
The deep reason is that an inverse-square central force conserves not just energy and angular momentum, but a third vector: the Laplace-Runge-Lenz vector, which points from the focus toward the perihelion and has magnitude proportional to the eccentricity. Its conservation is equivalent to saying the orbit closes on itself — a property unique to the 1/r² and r² force laws. Any other power law produces orbits that precess, never quite closing. This is why Kepler’s first law is exact for Newtonian gravity and only approximate in general relativity, where Mercury’s orbit precesses 43 arcseconds per century.
Energy conservation
In an isolated gravitational system, the total energy E = KE + PE is constant along every orbit. A body falling inward gains kinetic energy and loses potential energy in exact balance. At perihelion, KE is maximum and PE is minimum; at aphelion, the reverse. The total never changes. For bound orbits, E = −GMm/(2a), depending only on the semi-major axis, not on the eccentricity. Two orbits with the same semi-major axis have the same energy regardless of shape.
Angular momentum conservation
Because gravity is a central force — always directed toward the center — there is no torque about the center of mass. Therefore angular momentum L = mr²ω is constant. This single fact implies Kepler’s second law: dA/dt = L/(2m), a constant. Angular momentum also constrains the orbit to a plane: an orbiting body never spontaneously tilts its orbital plane because that would change the direction of L. In three dimensions, L is a vector perpendicular to the orbit plane, and its constancy means the plane is fixed in space.
Kepler and the death of circles
Johannes Kepler published his first two laws in 1609 in Astronomia Nova, after eight years of trying to fit Mars’s orbit to various curves. He tried circles, ovals, egg shapes — everything but the correct answer. The breakthrough came when he abandoned the ancient conviction that celestial motion must be circular. An ellipse with the Sun at one focus fit Tycho Brahe’s observations to within two arcminutes. The third law followed in 1619 in Harmonices Mundi. Kepler had the patterns but not the cause.
Newton and the cause
Isaac Newton, in the Principia (1687), showed that all three of Kepler’s laws follow from a single assumption: a central force that falls off as the square of the distance. F = GMm/r² explains not just planetary orbits but tides, comets, the shape of the Earth, and the precession of the equinoxes. Newton proved that an inverse-square force produces exactly conic section orbits — and conversely, that elliptical orbits imply an inverse-square force. The Principia unified terrestrial and celestial mechanics for the first time: the same force that makes an apple fall keeps the Moon in orbit.
From Newton to Einstein
Newtonian gravity reigned for over two centuries. Its first serious challenge came from Mercury: Le Verrier showed in 1859 that Mercury’s perihelion precesses 43 arcseconds per century more than Newtonian theory predicts. Einstein’s general relativity (1915) explained the discrepancy exactly — spacetime curvature adds a tiny correction to the 1/r² force, breaking the conservation of the Laplace-Runge-Lenz vector and causing orbits to precess. But for every other planet in the solar system, and for the simulation above, Newton’s law is effectively exact.