Orbits and satellites

Gravity as the centripetal force

Describe the drawing

The dot goes round at constant speed, and the cyan velocity arrow never changes length; but it never stops changing direction, so the velocity is changing and there must be an acceleration. The coral arrow shows it: aimed at the centre at every instant. Restyled, the same motion is a satellite held by gravity or a charge held by a magnetic field.

For an ideal circular orbit with negligible drag, gravity provides the centripetal force, so the satellite needs no continuous thrust to remain in orbit. It is accelerating towards the planet at all times, while its tangential speed keeps its distance from the centre constant. Equating the two expressions for the same force gives

from which the orbital speed at any radius, v = √(GM/r), follows directly. A lower circular orbit has a greater orbital speed.

Kepler's three laws

Nearly a century before Newton stated a force law, Johannes Kepler analysed Tycho Brahe's naked-eye records of the planets and spent twenty years fitting curves to them. The result was three laws. Kepler described the motion; Newton later explained it. All three laws follow from the inverse-square law, and the third is derived below.

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 second is empty: nothing occupies it, and the Sun is not at the centre. A circle is the special case in which the two foci coincide. Planetary orbits are ellipses of small eccentricity, which is why the circular treatment above applies well to them: the Earth's distance from the Sun stays within two per cent of its mean value throughout the year.

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 turn through a larger angle to sweep the same area; further out it is longer, and a narrower sector suffices. The planet therefore moves fastest at perihelion, its closest approach, and slowest at aphelion, its furthest. A comet passes close to the Sun in a few weeks and spends decades in the outer part of its orbit. The underlying reason is conservation of angular momentum, which a purely radial force cannot change.

Describe the drawing

An elliptical orbit drawn in cyan with the Sun marked at the left focus and the second focus left empty. Two shaded wedges of equal area run from the Sun: a short fat one at perihelion and a long thin one at aphelion. A long velocity arrow at perihelion and a short one at aphelion show the planet running three times faster at the near end than at the far end.

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

The third law, derived

The derivation of the period-radius law is required, and takes three lines. Replace v in the centripetal equation with the circumference divided by 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 is 4π²/GM and therefore gives the mass of the central body.

Orbital energy and escape

An orbiting satellite has kinetic energy and negative gravitational potential energy, and its total energy is constant while it stays in one stable orbit. For a circular orbit each term has a simple form. 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, half the magnitude of U; and adding them gives the total,

Its total energy is therefore −GMm/(2r): negative, as any bound orbit's total energy must be, and rising towards zero as r increases. A lower circular orbit has a more negative total energy but a greater orbital speed. When a low satellite loses a little total energy to atmospheric drag, its orbital radius decreases, and the decrease in potential energy is twice the loss in total energy, so the kinetic energy and the orbital speed both increase. The satellite therefore ends up lower and faster.

Escaping from the field means supplying the full change in potential energy from radius r to infinity. Setting the kinetic energy equal to that change, ½mv² = GMm/r, gives 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 conditions: it is circular, it lies in the equatorial plane, and it is in the same direction as the Earth's rotation, so the satellite stays above one fixed point on the equator. Kepler's third law then fixes its radius at about 4.2 × 10⁷ m from the Earth's centre, roughly 36 000 km above the surface. Fixed satellite dishes are aimed at satellites in this orbit.

Low orbits, a few hundred kilometres above the surface, have periods near 90 minutes. Imaging resolution is high and signal delays are short, but each satellite is above any given place only briefly, so low-orbit systems use constellations of satellites.