Skip to content

Lagrange points

Gravity and orbits · level 8, lesson 3 of 6

A question to start

The James Webb telescope orbits the Sun 1.5 million km beyond the Earth, yet it keeps pace with the Earth all year, although Kepler's third law says something farther out should go round more slowly. Is there a point like that near Mars, and what could we put there?

The idea

In the frame that turns with a planet round the Sun, there are five points where the pulls of the Sun and planet and the "centrifugal" effect of the turning frame all balance: the Lagrange points, found by Euler (L1 to L3) and Lagrange (L4, L5) in the 1760s and 1770s.

L1L2L3L4 (ahead)L5 (behind)Mars

L1 and L2 lie on the Sun–planet line, at a distance from the planet of about

\[r \approx a\left(\frac{m}{3M}\right)^{1/3}.\]

At L1 the planet's pull partly cancels the Sun's, so a slower orbit than Kepler's law would predict balances; at L2 the planet's pull adds to the Sun's, so a faster one does. Both are unstable: like a ball on a hilltop, a craft there must make small burns to stay.

L4 and L5, 60° ahead of and behind the planet, form equilateral triangles with the Sun and planet. When the planet is less than about 1/25 of the Sun's mass, these are stable: things that drift away circle back. Asteroids collect there and are called trojans.

Sun–Mars Sun–Earth
\(m/M\) \(3.23 \times 10^{-7}\) \(3.04 \times 10^{-6}\) (Earth with the Moon)
L1 and L2: distance from the planet about 1.08 million km about 1.5 million km
Trojans known about a dozen, most at L5, including 5261 Eureka two small ones at L4

The Hill sphere. The same distance, \(a(m/3M)^{1/3}\), is roughly the edge of a planet's gravitational "territory": inside it, a moon can orbit the planet stably; outside, the Sun takes over. For Mars, about 1.08 million km. Phobos (9,376 km) and Deimos (23,463 km) are deep inside.

Uses at Mars. A craft at Sun–Mars L1 would see solar storms coming towards Mars and could give the city from tens of minutes to about an hour of warning; one at L2 could relay signals for the far side. Small asteroids at the trojan points could one day be studied, or mined.

Worked examples

1. Sun–Mars L1. \(m/3M = 3.23\times10^{-7}/3 = 1.076\times10^{-7}\); cube root \(= 4.76\times10^{-3}\); times \(2.279\times10^8\) km \(= 1.08\times10^6\) km.

2. Sun–Earth L2. \(m/3M = 1.01\times10^{-6}\); cube root \(= 1.00\times10^{-2}\); times \(1.496\times10^8\) km \(= 1.50\times10^6\) km. ✓ (Webb.)

3. Mars–Phobos L1. \(m/M = 1.65\times10^{-8}\); \((m/3M)^{1/3} = 1.77\times10^{-3}\); times 9,376 km \(= 16.6\) km from Phobos's centre: only about 3 km above its surface at the ends of its long axis. Phobos almost fills its own Hill sphere, a sign of how close it is to being pulled apart.

Try it

  1. Find the distance of Sun–Jupiter L1 from Jupiter (\(m/M = 9.55 \times 10^{-4}\), \(a = 778\) million km).
  2. Why are L4 and L5 stable for Mars but L1 and L2 not?
  3. Estimate the Hill sphere of the Earth in the Sun's field. Is the Moon (384,400 km) inside?
  4. Why is Phobos's small Hill sphere a hint that it is in danger?
Answers
  1. \((9.55\times10^{-4}/3)^{1/3} = 0.0683\); times 778 = 53 million km.
  2. Near L4 and L5, the Coriolis effect in the turning frame turns a drifting craft back round the point; near L1 and L2, small displacements along the Sun–planet line grow.
  3. About 1.5 million km. Yes, the Moon is well inside.
  4. Loose material on its surface is barely bound to Phobos; as it spirals inwards its Hill sphere shrinks below its own size, and it will be pulled apart.

Common mistake

Thinking the Lagrange points are places of zero gravity. Gravity is strong there. They are places where a craft can keep the same position relative to the planet, because the pulls and the turning of the frame balance.

Where it leads