Home / Learn / Physics / Gravity variationTool #L201 (formerly #1339)

Gravity variation

Slide a point from the centre of the Earth, Mars or the Moon out into space and watch g change: in a straight line inside, as 1/r² outside. Read your weight there, the speed and period of a satellite at that height, the escape speed and the height of the geostationary orbit.

Keep it one tap away for everyday use.

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Planet

Inside, the planet is taken as a ball of even density, so g falls in a straight line to zero at the centre (the real Earth's dense core makes g rise a little first). Planet data: NASA fact sheets, rounded.

g here, and a satellite here

Try this

  • Go up by one Earth radius (6371 km). g is a quarter of its surface value, not half: it falls as 1/r².
  • Go halfway down to the centre. g is half: inside, it falls in a straight line.
  • Tap "Stationary orbit". The satellite takes as long to go round as the Earth takes to turn, so it hangs over one place.
  • Compare the escape speed with the orbit speed at any height: it is always √2 ≈ 1.41 times bigger.

Check yourself

What it does

Gravity gets weaker as you go up, but it also gets weaker as you go down a deep mine. This tool shows both on one graph. Slide a point from the centre of the Earth (or Mars, or the Moon) out to seven times its radius and watch g, the pull on each kilogram, change: it rises in a straight line from zero at the centre to its surface value, then falls as 1/r² out in space.

At any height above the surface it also treats the point as a satellite going round in a circle and gives its speed, the time for one orbit, the escape speed from there and its energy. The “Stationary orbit” button finds the height at which a satellite goes round once per turn of the planet, so it hangs over one place: the geostationary orbit used for TV and weather satellites.

It follows the school topics on gravitation: g and how it changes with height and depth, escape speed and Earth satellites (Class 11 Physics, gravitation), and the mass and weight ideas from Class 9. For the shape of orbits and Kepler’s laws see kepler orbits, for two bodies pulling on each other two-body orbit sandbox, and for moving from one orbit to another orbit transfer calculator.

How to use it

  1. Pick the planet: Earth, Mars or the Moon.
  2. Drag the slider, or drag along the graph, to move the point. Minus heights are depths below the surface. You can also type an exact height in kilometres.
  3. Type your mass to see your weight there. The mass never changes; the weight does.
  4. Tap a preset: the surface, halfway down to the centre, the Space Station’s height (400 km) or the stationary orbit.

The dashed line near the surface is the school short cut g(1 − 2h/R). It is good for heights much smaller than the radius and goes wrong further up.

Worked example

Open the page: the Earth, 400 km up, a 50 kg student. The Earth’s GM is 3.986 × 10¹⁴ m³/s² and its mean radius 6371 km, so at the surface g = 3.986 × 10¹⁴ ÷ 6 371 000² = 9.820 m/s² (school books round it to 9.8). At 400 km, r = 6771 km and g = 3.986 × 10¹⁴ ÷ 6 771 000² = 8.694 m/s², which is 88.5 % of the surface value. The student weighs 434.7 N instead of 491.0 N. Astronauts float not because gravity is gone but because they are falling round the Earth with their station.

A satellite at that height moves at √(GM/r) = 7.673 km/s and goes round in 92.4 min. The escape speed from there is 10.851 km/s, √2 times as much. Each kilogram has kinetic energy 29.43 MJ and potential energy −58.87 MJ, so its total energy is −29.43 MJ.

Tap Halfway down: at a depth of 3185 km, about half the radius, g = 9.820 × (1 − 3185 ÷ 6371) = 4.911 m/s², half the surface value. Tap Stationary orbit: 35 786 km above the equator.

Tips

  • Going up one radius quarters g; going halfway down only halves it.
  • The orbit speed does not depend on the satellite’s mass.
  • A higher orbit is slower and takes longer, but has more total energy.

FAQ

Is g really zero at the centre? Yes: the planet pulls equally in every direction there.

Why does the real Earth differ inside? The Earth’s core is much denser than its rocks, so g actually rises a little going down before falling. The even-density model is what the school books use.

Why no stationary orbit for the Moon? The Moon turns once in 27.3 days, so the orbit would be about 88 000 km up, where the Earth’s pull is stronger than the Moon’s.

Is anything stored? No. It all runs in your browser.

Where the numbers come from

  • Newton's law of gravitation, g = GM/r² outside a planet and g₀(1 − d/R) inside a planet of even density (shell theorem), circular orbit speed √(GM/r), period 2πr/v, escape speed √(2GM/r) and orbital energy −GMm/2r: standard school physics found in any textbook. G = 6.6743 × 10⁻¹¹ N m²/kg² (CODATA 2018).
  • GM, mean and equatorial radii and sidereal rotation periods of the Earth, Mars and the Moon: NASA planetary fact sheets (public domain), rounded.
  • Checked against known values: the geostationary height of 35 786 km, the Space Station's 7.67 km/s at 400 km, the Earth's escape speed 11.19 km/s and Mars' stationary orbit near 17 030 km.

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.

Last updated 2026-11-09.

Change log for Gravity variation

  1. Add #1339 g with height, depth and satellites Details on GitHub

Every change to fibub

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