Gravitational Fields and Field Strength

A gravitational field is the region around a mass in which another mass experiences a force; the gravitational field strength g at a point is the force per unit mass, g = F ÷ m, and for a spherical mass M at distance r from its center, g = GM ÷ r2, directed toward the center. In brief: g is measured in N/kg, which is the same as m/s2. Near the Earth’s surface g ≈ 9.8 N/kg and the field is approximately uniform, with parallel, equally spaced field lines. Far from the Earth, the field is radial, with lines converging on the center, and its strength falls with the square of the distance.

The idea of a field explains how the Earth acts on the Moon without touching it. This guide defines the gravitational field and field strength, shows the field of a planet and how it varies with distance, discusses uniform and radial fields, compares the surface gravity of bodies in a table, works through numerical examples and corrects common mistakes.

What is a gravitational field?

Any mass M modifies the space around it, so that another mass m placed there experiences a force. This is called a field. The concept answers the question of how gravity acts across empty space: the mass M creates a field, and the field acts on m. A gravitational field acts on mass only and is always attractive. It provides the “weight” of objects, and the force that keeps planets in orbit.

What is field strength?

The gravitational field strength at a point is the force acting on a small test mass at that point divided by the test mass: g = F ÷ m, in N/kg. It is a vector pointing toward the mass creating the field. By Newton’s law of gravitation, F = GMm ÷ r2, so for a spherical (or point) mass M, outside it:

g = GM ÷ r2.

The field strength depends only on M and r, not on the test mass. The value of g falls by a factor of four if the distance from the center doubles. At the surface of the Earth, g = GM ÷ R2 = 9.82 N/kg; it varies from 9.78 N/kg at the equator to 9.83 N/kg at the poles, because of the rotation of the Earth and its shape, and with altitude and local geology. Inside a uniform sphere, the field strength is proportional to the distance from the center, g = GMr ÷ R3, falling to zero at the center. For several masses, fields add as vectors, so there are points between two bodies where the net field is zero (such as the point between the Earth and the Moon, about 90 percent of the way to the Moon, where the pulls are equal).

Gravitational field strength against distance from the center of the Earth 0 R 2R 3R 4R 0 2 4 6 8 10 Distance r from the center of the Earth g (N/kg) 9.82 2.46 1.09 g ∝ r outside: g ∝ 1/r²
Field strength g against distance from the center, treating the Earth as a uniform sphere. Inside, g rises in proportion to r to 9.82 N/kg at the surface (r = R). Outside, it falls as 1/r²: 2.46 N/kg at 2R and 1.09 N/kg at 3R, as in the worked example.

How are fields drawn?

Field lines show the direction of the force on a mass. For a spherical mass, the lines are radial, pointing inward, and closer together near the surface where the field is stronger. Near the surface of a planet, over distances small compared with its radius, the field lines are parallel and evenly spaced, which represents a uniform field of constant g. The arrows of the lines show the direction of the force, and the density of lines shows the strength. Lines do not cross. Unlike electric field lines, gravitational field lines always point toward the mass, since there is no negative mass.

g = const.g = GM ÷ r²

Surface gravity of bodies compared

Approximate gravitational field strength at the surface (equatorial radius)
Body Mass (kg) Radius (km) g (N/kg)
Moon7.35 × 10221,7371.62
Mars6.42 × 10233,3903.73
Earth5.97 × 10246,3719.82
Jupiter (1 bar level)1.90 × 102769,911about 25

Worked example: field strength calculations

  1. Moon’s surface. g = GM ÷ r2 = 6.674 × 10−11 × 7.35 × 1022 ÷ (1.737 × 106)2 = 4.905 × 1012 ÷ 3.017 × 1012 = 1.63 N/kg.
  2. Force on an astronaut. An astronaut of 100 kg (with equipment) has a weight on the Moon of F = mg = 100 × 1.63 = 163 N, compared with 982 N on the Earth.
  3. Altitude. At a height of one Earth radius above the surface (r = 2R), g = 9.82 ÷ 4 = 2.46 N/kg. At r = 3R, it is 9.82 ÷ 9 = 1.09 N/kg.
  4. Geostationary orbit. At r = 4.22 × 107 m: g = 3.986 × 1014 ÷ (4.22 × 107)2 = 0.224 N/kg, which supplies the centripetal acceleration of a satellite.
  5. Point of zero field. Between the Earth and Moon (separation d = 3.84 × 108 m), the point where the fields cancel is where GME ÷ x2 = GMM ÷ (d − x)2. The ratio √(ME ÷ MM) = √81.2 = 9.01, so x = 9.01(d − x), x = 0.90d = 3.46 × 108 m from the Earth (90 percent of the way, or 3.8 × 107 m from the Moon).

How did the field concept develop?

Newton himself was uneasy about “action at a distance”. Michael Faraday introduced the notion of a field in the 1840s for electric and magnetic forces, and Pierre-Simon Laplace and Siméon Poisson developed the concept of a gravitational potential and field. Albert Einstein’s general theory of relativity (1915) replaced the field of force with the curvature of spacetime. The NASA GRACE satellites mapped small variations in Earth’s gravitational field from 2002 to 2017, used to track groundwater and ice-sheet changes.

Which mistakes are common?

Mistake 1: Using the height above the surface as r

r is the distance from the center of the body.

Mistake 2: Saying that the field strength depends on the test mass

It depends only on the source mass and the distance. The force depends on the test mass.

Mistake 3: Treating g as constant at all altitudes

g decreases with height, but is nearly constant over small heights.

Mistake 4: Confusing the field strength with the force

g is force per unit mass (N/kg), while the force on a mass m is mg.