D.1 Gravitational fieldsIB Physics SL: Revision notes

Kepler described planetary motion from observations before Newton explained it:

1. First law: each planet moves in an ellipse with the Sun at one focus.
2. Second law: the line joining the Sun and a planet sweeps out equal areas in equal times. So a planet moves fastest at perihelion (closest) and slowest at aphelion (furthest).
3. Third law: T² ∝ r³, where T is the orbital period and r the mean orbital radius (semi-major axis). For bodies orbiting the same central mass, T²/r³ is the same.

For a circular orbit, equating GMm/r² to mv²/r with v = 2πr/T gives T²/r³ = 4π²/GM.

Every pair of masses attracts with a force F = Gm₁m₂/r², where r is the separation of their centres and G = 6.67 × 10⁻¹¹ N m² kg⁻². The force is always attractive, acts along the line joining the centres, and the two bodies feel equal and opposite forces. It is an inverse-square law: doubling r reduces F to one quarter.

Newton's law is written for point masses. A real body may be treated as a point mass at its centre when:

• it is spherically symmetric (a uniform sphere, or a sphere made of uniform shells); outside it, its field is exactly that of a point mass at its centre, or
• its size is very small compared with the separation of the bodies.

Irregular bodies close to another object (a probe near a small asteroid) cannot be treated as point masses.

The gravitational field strength g at a point is the force per unit mass on a small test mass placed there: g = F/m. For a point (or spherical) mass M, g = GM/r². Units: N kg⁻¹, which are equivalent to m s⁻². g is a vector directed towards the mass. Near Earth's surface g ≈ 9.8 N kg⁻¹; at the height of the ISS it is still about 8.6 N kg⁻¹.

Astronauts in orbit feel weightless not because g = 0, but because they and their spacecraft are in free fall together.

Field lines show the direction of the force on a small test mass. For a spherical mass they are radial and point inwards to the centre. The density of the lines shows the field strength: lines spread out with distance, so the field weakens. Close to the surface of a large planet, over a small region, the lines are almost parallel and equally spaced, so the field is nearly uniform. Field lines never cross.