What is Kepler's third law?

The square of a planet's orbital period is proportional to the cube of its semi-major axis: T² = 4π²a³/GM. Measured in years and astronomical units round the Sun, it becomes simply T² = a³, so Jupiter at 5.2 AU takes 11.9 years.

Why does a planet move faster when it is closer to the Sun?

Gravity does no net work over an orbit but it does pull the planet inwards and outwards along the way. Angular momentum is conserved because the force points at the Sun, so r × v stays constant and a smaller r needs a bigger v. That is Kepler's second law, equal areas in equal times.

What is the difference between orbital speed and escape speed?

Circular orbital speed is √(GM/r). Escape speed is √(2GM/r), exactly √2 times bigger. At escape speed the total energy is zero, so the object can coast to infinity. For Earth's orbit they are 29.8 km/s and 42.1 km/s.

Does the mass of the planet change its orbit?

Hardly at all, as long as the planet is much lighter than the star. The planet's mass cancels from its acceleration, GM/r², so a pebble and a planet launched the same way follow the same path.

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Orbits and Kepler's laws

Give a planet a sideways push and gravity bends its path into an ellipse. Its speed, its distance and the star's mass decide the shape and the year.

Energy per kilogram vs time

Readouts

What's happening

Gravity pulls the planet towards the star with a force GMm/r². Launched sideways at just the right speed, √(GM/r), it falls round in a circle; a little faster or slower and the path becomes an ellipse with the star at one focus. Kepler found three rules in the planets' motion. The orbits are ellipses (first law). The line from star to planet sweeps out equal areas in equal times, so the planet speeds up near the star and crawls far away (second law, which is conservation of angular momentum). And the square of the period is proportional to the cube of the semi-major axis, T = 2π√(a³/GM) (third law). The total energy, kinetic plus gravitational potential, stays constant; if it reaches zero the planet has escape speed, √2 times the circular speed, and never comes back. The simulation works in astronomical units, years and solar masses and integrates the motion with a symplectic method that keeps the energy steady over many orbits.

F = GMm / r²v_circ = √(GM / r)T = 2π √(a³ / GM)v_esc = √(2GM / r)E = ½v² − GM/r = −GM / 2a

A-Level Physics (AQA, OCR, Edexcel): gravitational fields, orbits of planets and satellites, Kepler's third law, escape velocity. First-year university mechanics: central forces and the Kepler problem.

Work through the numbers with Physics Formulas and Scientific Calculator.

Challenge

Predict first: Mars orbits the Sun with a semi-major axis of 1.524 AU. How long is a Martian year, in Earth years? Choose Mars, type your answer in, check, then run it and read the measured period.

FAQ

What is Kepler's third law?: The square of a planet's orbital period is proportional to the cube of its semi-major axis: T² = 4π²a³/GM. Measured in years and astronomical units round the Sun, it becomes simply T² = a³, so Jupiter at 5.2 AU takes 11.9 years.
Why does a planet move faster when it is closer to the Sun?: Gravity does no net work over an orbit but it does pull the planet inwards and outwards along the way. Angular momentum is conserved because the force points at the Sun, so r × v stays constant and a smaller r needs a bigger v. That is Kepler's second law, equal areas in equal times.
What is the difference between orbital speed and escape speed?: Circular orbital speed is √(GM/r). Escape speed is √(2GM/r), exactly √2 times bigger. At escape speed the total energy is zero, so the object can coast to infinity. For Earth's orbit they are 29.8 km/s and 42.1 km/s.
Does the mass of the planet change its orbit?: Hardly at all, as long as the planet is much lighter than the star. The planet's mass cancels from its acceleration, GM/r², so a pebble and a planet launched the same way follow the same path.