Orbits and satellites

Gravity takes the centripetal job

A satellite in a circular orbit is doing circular motion, so something must supply the centripetal force. Gravity does, entirely. No engine and no thrust come into it. The satellite is permanently falling towards the planet while its tangential speed carries it forever past the edge. Setting the two faces of one force equal is the master move of the whole topic,

from which the orbital speed at any radius, v = √(GM/r), falls straight out. Lower orbits are faster orbits.

Kepler's three laws

Most of a century before Newton wrote down a force law, Johannes Kepler was handed Tycho Brahe's naked-eye records of the planets and spent twenty years fitting curves to them. What came out were three laws, and they still frame this whole topic. Kepler described; Newton, later, explained. All three follow from the inverse-square law, and the third one you are about to derive yourself.

The first law is the law of ellipses. Every planet moves in an ellipse with the Sun at one focus. An ellipse has two foci, and the point examiners press on is that the second one is empty: nothing sits there, and the Sun is certainly not at the centre. A circle is the special case in which the two foci coincide, which is exactly why the circular treatment above is worth so much. Planetary orbits are ellipses of small eccentricity, and the Earth's distance from the Sun stays within two per cent of its average all year round.

The second law is the law of equal areas. The line joining a planet to the Sun sweeps out equal areas in equal times. Near the Sun that line is short, so it must swing through a wide angle to cover the area; far out it is long, and a narrow wedge is enough. The planet therefore moves fastest at perihelion, its closest approach, and slowest at aphelion, its furthest. Comets make the point violently, tearing round the Sun in weeks and then crawling through the outer part of the orbit for decades. Underneath it is conservation of angular momentum, which a purely radial pull has no way to change.

The third law is the law of periods. The square of the orbital period is proportional to the cube of the mean radius, with the mean radius being the semi-major axis of the ellipse, and for the near-circular orbits you will be given it is simply the radius. It is the only one of the three that puts one orbit into arithmetic with another, and it is the one the specifications ask you to derive.

The third law, derived

The derivation of the period-radius law is expected, and it runs to three lines. Replace v in the centripetal equation with the circumference over the period, v = 2πr/T, then rearrange.

so T² ∝ r³, and distant orbits are disproportionately slow. Plot T² against r³ for any family of satellites and every one sits on a single straight line through the origin, whose gradient measures the central mass. Astronomers weigh planets with exactly this graph.

Orbital energy and escape

An orbiting satellite holds kinetic energy and negative gravitational potential energy, and its total energy stays constant while it stays in one stable orbit. For a circular orbit each piece has a clean expression. The potential energy is U = -GMmr$U = -\frac{GMm}{r}$U = -GMm/r from the potential lesson; the orbit condition GMm/r² = mv²/r rearranges to E_k = 12mv² = GMm2r$E_k = \frac{1}{2}mv^{2} = \frac{GMm}{2r}$E_k = (1/2)mv^2 = GMm/2r, exactly half the well's depth; and adding them gives the total,

negative, as any bound orbit's total must be, and rising towards zero as r grows. Moving between orbits is where the energy balance has its famous quirk. Drop to a lower orbit and the satellite speeds up, its kinetic energy rising, yet the total falls, because the potential energy fell twice as far. Atmospheric drag on a low satellite plays that story out in slow motion: the drag drains a little total energy, the orbit sinks, and the fall converts enough potential energy to raise the kinetic energy despite the loss, so the satellite ends lower and faster, the opposite of every intuition about friction.

Leaving entirely means paying the whole depth of the well from radius r. Set the kinetic energy equal to that climb, ½mv² = GMm/r, and out comes the escape velocity,

about 11 km s⁻¹ from the Earth's surface, independent of the escaping mass.

The special orbits

A synchronous orbit has a period equal to its planet's rotation period. The geostationary orbit adds three more conditions. It is circular, it lies in the equatorial plane, and it travels the same way as the Earth's spin, so the satellite hangs over one fixed point on the equator. Kepler's law then fixes its radius at about 4.2 × 10⁷ m from the Earth's centre, roughly 36,000 km up. Every satellite dish that never moves is aimed at that one ring in the sky.

Low orbits, a few hundred kilometres up, trade coverage for closeness. Periods run near 90 minutes, imaging detail is fine and signal delays are small, but each satellite sees any given place only briefly, so low-orbit systems fly in constellations.