D.1 Gravitational fieldsIB Physics HL: Revision notes

Kepler's laws: (1) planets move in ellipses with the Sun at one focus; (2) the line joining the Sun and planet sweeps out equal areas in equal times, so a planet is fastest at perihelion; (3) T² ∝ r³ for bodies orbiting the same mass.

Newton's law: F = Gm₁m₂/r², always attractive, r between centres. For a circular orbit GMm/r² = mv²/r with v = 2πr/T gives T²/r³ = 4π²/GM, which explains Kepler's third law.

A body can be treated as a point mass if it is spherically symmetric, or if it is very small compared with the separation.

Gravitational field strength g = F/m, the force per unit mass on a small test mass; for a spherical mass g = GM/r² (N kg⁻¹). Field lines around a spherical mass are radial and point inwards; their density shows the strength. Near the surface, over a small region, the field is almost uniform.

The gravitational potential energy Ep of a system is the work done to assemble the system from infinite separation. For two bodies, Ep = −Gm₁m₂/r, with r the separation of their centres of mass.

Ep is zero at infinity and negative everywhere else, because the attractive force does work as the bodies come together: you would have to do work to pull them apart again. Ep becomes more negative as the bodies get closer.

The gravitational potential Vg at a point is the work done per unit mass in bringing a small mass from infinity to that point: Vg = −GM/r (J kg⁻¹). It is a scalar, so potentials from several masses simply add.

The work done moving a mass m between two points is W = mΔVg. It depends only on the start and end potentials, not on the path taken.

The field strength is the negative potential gradient: g = −ΔVg/Δr. The minus sign shows that the field points towards decreasing potential (towards the mass). On a graph of V against r, the gradient at a point gives g there.

Equipotential surfaces join points of equal potential. No work is done moving along one. Around a spherical mass they are concentric spheres; near the surface over a small region they are almost flat, parallel planes. Field lines are always perpendicular to equipotentials. For equal steps of potential, equipotentials are closer together where the field is stronger.

For a circular orbit, GMm/r² = mv²/r gives the orbital speed v = √(GM/r), independent of the satellite's mass.

The escape speed is the minimum speed a body needs to reach infinity with no drive: ½mv² − GMm/r = 0, so v_esc = √(2GM/r). It does not depend on the body's mass or on the direction of launch (ignoring air resistance).

In a circular orbit: Ek = GMm/(2r), Ep = −GMm/r and total energy E = −GMm/(2r). A lower orbit has more kinetic energy but less total energy.

A satellite in low orbit feels a small viscous drag from the thin upper atmosphere. Drag removes energy, so the total energy E = −GMm/(2r) decreases and the orbit's height falls. Surprisingly, the satellite's speed increases: as r falls, Ep decreases by twice as much as Ek increases, so the satellite gains kinetic energy while losing total energy. The satellite gradually spirals inwards, faster and faster, and eventually burns up in the denser lower atmosphere unless it is boosted.