13.1–13.3 - Gravitational forces and fields

A field maps force per unit mass

Does a planet's gravitational field disappear when no satellite happens to be there? No. The field describes what force a small test mass would experience at each position; the test mass helps define or measure it, rather than switching it on.

A mass is the source of a gravitational field. The field is a model mapping the gravitational force per unit mass through space. It is not material emitted by the source. At a particular point, a larger test mass experiences a proportionally larger force, while the force per unit mass is unchanged.

The field is a vector: its direction is the direction of force on a test mass. Gravitational forces are attractive, so the field around an isolated spherical planet points towards its centre.

The unit $\mathrm{N\,kg^{-1}}$ is equivalent to $\mathrm{m\,s^{-2}}$, since a newton is $\mathrm{kg\,m\,s^{-2}}$. Thus field strength is also the acceleration of a freely falling test mass when gravity is its only force. A supported object can experience a gravitational force without accelerating because another force balances it.

[DIAGRAM: asset_name: Gravitational field lines are a map; asset_slug: gravitational-field-map; caption: Inward radial lines map the exterior field; a small surface region can be approximated by parallel lines; description: The left panel shows a spherical mass with twelve non-crossing radial lines outside it and arrows pointing inward. The right panel enlarges a small near-surface patch and shows approximately parallel evenly spaced downward arrows. Labels state that lines represent field direction and relative strength, not emitted material.]

Field-line arrows show direction. Closer line spacing represents greater strength in the chosen drawing; lines do not cross because the field at one point has one resultant direction. The number of lines an artist chooses is not a count of physical objects. A small surface patch is often drawn with parallel lines as a local approximation, not as a claim that the whole planet's field is parallel.

Newton's law and spherical masses

Every pair of masses attracts gravitationally. Increasing either mass increases the force; increasing their separation weakens it. Newton's law gives the quantitative relation for point masses.

The inverse square is decisive: doubling separation reduces the force to one quarter. Doubling one mass doubles the force; doubling both masses multiplies it by four. The force on each body has the same magnitude, even when their masses are very different, by Newton's third law. Their accelerations can differ because acceleration is force divided by the body's own mass.

For a point outside a uniform sphere, its gravitational effect is the same as if the sphere's mass were concentrated at its centre. This justifies using a centre-to-centre distance for spherical planets. A height above the surface is not that distance: $r=R+h$, where $R$ is planet radius and $h$ altitude.