Kepler's laws of planetary motion
Johannes Kepler, using careful observations by Tycho Brahe, found three laws:
Law of orbits: every planet moves in an ellipse with the Sun at one of its foci.
Law of areas: the line joining a planet to the Sun sweeps out equal areas in equal intervals of time. So a planet moves faster when it is closer to the Sun (at perihelion) and slower when farther (aphelion). This law follows from the conservation of angular momentum, because gravity is a central force.
Law of periods: the square of the time period of revolution is proportional to the cube of the semi-major axis of the orbit: T² ∝ a³.
Example. A planet's orbit has a semi-major axis 4 times the Earth's. Its period is √(4³) = 8 years.
Newton's universal law
The force between point masses m₁ and m₂ a distance r apart is
F = G m₁m₂ ⁄ r², attractive, along the line joining them, with G = 6.67 × 10⁻¹¹ N m² kg⁻², measured first by Henry Cavendish.
Gravitational forces obey the principle of superposition: the net force on a mass is the vector sum of the forces from each other mass. A uniform spherical shell attracts outside masses as if all its mass were at its centre, and exerts no net force on a mass inside it.
Acceleration due to gravity
On the Earth's surface, g = GM_E ⁄ R_E² ≈ 9.8 m/s².
Variation with height h above the surface:
g(h) = GM_E ⁄ (R_E + h)², which for h ≪ R_E is approximately g(h) ≈ g(1 − 2h/R_E).
Variation with depth d below the surface: only the mass of the inner sphere of radius (R_E − d) attracts, so
g(d) = g(1 − d/R_E).
g therefore decreases both above and below the surface, and is zero at the Earth's centre.
Example. At what height does g fall by 1%? Using the approximation, 2h/R_E = 0.01, so h = 0.005 R_E ≈ 32 km. At a depth of 0.01 R_E ≈ 64 km, g also falls by 1%.
g also varies with latitude: it is slightly larger at the poles than at the equator, because the Earth is flattened and because of its rotation.
Gravitational potential energy
For masses far apart, the gravitational potential energy is
V = −G M m ⁄ r, taking V = 0 at infinite separation.
The negative sign shows the system is bound: work must be done to separate the masses. Near the surface, the difference in V between two heights reduces to mgh.
Gravitational potential at a point is the potential energy per unit mass: −GM/r.
Escape speed
The escape speed is the minimum speed with which a body must be launched from the surface to escape the Earth's gravity completely. Setting the total energy to zero:
½mv_e² − GMm/R = 0, so v_e = √(2GM/R) = √(2gR).
For the Earth, v_e ≈ 11.2 km/s. It doesn't depend on the mass of the body or the direction of launch.
The Moon's escape speed is only about 2.3 km/s. Gas molecules at the Moon's surface move fast enough to escape, which is why the Moon has no atmosphere.
Satellites
A satellite of mass m orbiting the Earth at height h, radius r = R_E + h, needs gravity to supply the centripetal force:
mv²/r = GM_E m ⁄ r², so the orbital speed v = √(GM_E/r).
Close to the surface (r ≈ R_E), v = √(gR_E) ≈ 7.9 km/s. Note that v_e = √2 × v_orbital.
Time period T = 2π √(r³ ⁄ GM_E), which is Kepler's third law for satellites.
Energy of an orbiting satellite: K = GMm/2r, V = −GMm/r, so the total energy E = −GMm/2r. The negative total energy means the satellite is bound to the Earth.
Special satellites
Geostationary satellites orbit in the equatorial plane with a period of 24 hours, at a height of about 35,800 km. They appear fixed in the sky, so they are used for communication and TV broadcasting.
Polar satellites orbit at low heights (about 500–800 km) passing over the poles. As the Earth rotates beneath them they scan the whole surface, which is useful for weather forecasting and remote sensing.
Weightlessness. Astronauts in an orbiting satellite feel weightless because they and the satellite are both in free fall towards the Earth with the same acceleration. Gravity still acts on them.
Summary
Kepler: elliptical orbits, equal areas in equal times, T² ∝ a³.
g = GM/R² decreases with height (≈ 1 − 2h/R) and with depth (1 − d/R).
Gravitational PE = −GMm/r; escape speed √(2gR) ≈ 11.2 km/s.
Orbital speed √(GM/r); geostationary satellites have T = 24 h at about 35,800 km.