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Kepler orbits

Set a planet's launch speed and watch its orbit become a circle, an ellipse or an escape; see equal areas swept in equal times, check T² = a³ against the planets, and find how g changes above and below the Earth's surface.

Read the full guide: Kepler's laws of planetary motion, with a launch you can change

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  1. Launch
  2. Ellipse
  3. T² = a³
  4. g and depth
Nothing is uploadedWorks offline once open
Level
Drag the arrow tip to change the launch speed.

The orbit

Try this

  • Set the launch speed to 29.8 km/s: a circle, the Earth's orbit, one year round.
  • Go faster: the orbit stretches into an ellipse with the star at one focus, and the year gets longer.
  • Watch the shaded slices: each is swept in the same time. Near the star the planet moves faster, so its slices are short and wide.
  • Push past 42.1 km/s, the escape speed: the planet never comes back.
  • In g above and below, go up one Earth radius: g falls to a quarter. Go half-way to the centre: g is half.

Check yourself

What it does

A planet is launched sideways at 1 AU (the Earth’s distance) from a Sun-like star. Change the launch speed with the slider or by dragging the arrow tip, and its orbit changes in front of you: at 29.8 km/s it is a circle (the Earth’s orbit), faster or slower it is an ellipse with the star at one focus, and at 42.1 km/s or more it escapes. The orbit is shaded in twelve slices, each swept in the same time, so you can see the equal areas law: short, wide slices near the star, long thin ones far away. Press Play and the planet speeds up and slows down as it should.

The T² = a³ view puts the eight planets on log scales, where Kepler’s third law is a straight line, with your launched orbit added as an extra dot. The g above and below view shows how gravity weakens with height (inverse square) and falls to zero at the centre of a model Earth.

It follows the school topics on gravitation: the universal law of gravitation, free fall, g and the difference between mass and weight (Class 9 Science, gravitation), and Kepler’s laws, the variation of g with height and depth, escape speed and orbital energy (Class 11 Physics, gravitation). For two bodies orbiting each other see two-body orbit sandbox, for real orbital elements orbit elements, and for energy on Earth energy skate track.

How to use it

  1. Pick a Level. Class 9 keeps to the laws in words, the inverse square law and weight. Class 11 adds the line to the star, the area of each slice, the orbit’s energy and the formulas for g inside and outside.
  2. In Launch a planet, drag the arrow tip up or down, or use the slider. Tick or untick the shading. Press Play to watch one orbit take about six seconds.
  3. In T² = a³, read the table: T² ÷ a³ is close to 1 for every planet.
  4. In g above and below, slide from 6,371 km deep (the centre) to three Earth radii up.

Worked example

Open the page: launched at 36 km/s. The orbit is an ellipse with e = 0.46, closest to the star at 1 AU (where it was launched) and farthest at 2.709 AU. The semi-major axis is a = 1.855 AU, so T = a^1.5 = 2.526 years (923 days); the page checks T² = 6.38 = a³. Each shaded slice has the same area, 0.7993 AU², swept in T/12 = 76.9 days.

Slow it to 25 km/s: it is now launched at its farthest point and falls inwards to 0.544 AU; its year is 248 days.

In g above and below, one Earth radius up (6,371 km) the page shows g = 2.45 m/s², a quarter of 9.8, and a 50 kg student weighs 122.5 N instead of 490 N. The escape speed from the surface is 11.2 km/s.

Tips

  • A planet in an ellipse is fastest at its closest point and slowest at its farthest, because the line to the star sweeps equal areas in equal times.
  • The period depends only on the semi-major axis, not on how stretched the ellipse is.
  • Mass is the amount of matter and stays the same everywhere; weight is the pull of gravity, m × g, and changes.

FAQ

Why does the escape speed equal √2 times the circular speed? Escape needs the kinetic energy ½mv² to match the gravitational energy GMm/r, which gives v = √(2GM/r); a circular orbit needs v = √(GM/r).

Is the inside of the Earth really like this? No. The model treats the Earth as a uniform ball. The real Earth has a dense core, so g actually rises a little as you go down at first.

Is anything stored? No. Everything is worked out in your browser.

Where the numbers come from

  • Newton's law of gravitation, Kepler's three laws, escape speed √(2GM/r) and g = g₀R²/r² outside and g₀r/R inside a uniform sphere: standard school physics found in any textbook. Planet distances and periods are rounded general values; 1 AU/yr = 4.7405 km/s from the IAU astronomical unit and the Julian year.
  • Positions from Kepler's equation were checked against an independent velocity-Verlet integration of Newton's gravity; the swept areas by summing small triangles; g and escape speed against hand-worked values.

Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.

Read more on the Blog

Last updated 2026-11-09.

Change log for Kepler orbits

  1. Keep labels inside the pictures on phones (#1023, #1026-#1028) Details on GitHub
  2. Add #1027 Kepler's laws orbit lab Details on GitHub

Every change to fibub

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