Lagrange points

How can a space telescope hold station relative to the Earth and Sun for years without an engine running?

The James Webb Space Telescope sits about 1.5 million kilometres from Earth, directly away from the Sun, and has held that arrangement since 2022 with no engine running. The tempting reading is that it found a place where gravity cancels. But look at what it is actually doing: it circles the Sun once a year, in step with Earth, from a radius about one percent larger than Earth's.

That should be impossible. Kepler's third law is unambiguous — the further out you are, the longer your orbit takes. A free body at 1.01 astronomical units should fall behind Earth and drift away within months. So what lets it keep pace?

Start with what a circular orbit actually is. A body goes round at a steady rate because the gravity acting on it supplies exactly the centripetal force that rate and radius demand — no more, no less. Too little and it spirals outward; too much and it falls inward. Further out, the Sun's pull is weaker, so the sustainable rate is slower. That is Kepler's law restated as a force budget.

Now put Earth in the picture. A spacecraft directly outside Earth's orbit feels the Sun pulling it inward and Earth pulling it inward too, since Earth lies between it and the Sun. The two add. So the centripetal budget at that radius is larger than the Sun alone provides — and a larger budget supports a faster rate. Is there a distance where the sum is exactly right for one revolution per year? There is, and only one: about 1.5 million kilometres out. That is L2.

Run the same argument on the sunward side and Earth's pull now subtracts from the Sun's. A body closer in would normally orbit faster than a year; weakening the inward force slows it, and again one distance gives exactly a year. That is L1, where SOHO and DSCOVR watch the Sun. A third, L3, sits far round on the opposite side of the Sun.

Two more exist, and they are stranger. Place a body 60 degrees ahead of Earth in its orbit, at the same radius, and it forms an equilateral triangle with the Sun and Earth. Add the two pulls as vectors and the resultant points at the common centre of mass with precisely the magnitude co-rotation requires — and remarkably, this works for any ratio of the two masses. Those are L4 and L5.

So five balance points. But balanced is not the same as staying put. Ask what happens to a body nudged off each one. Drift from L2 toward Earth and Earth's pull strengthens, pulling you further in — the error feeds itself. The three collinear points are unstable, with a wandering spacecraft doubling its displacement in a matter of weeks. Nudge a body off L4, however, and something else happens: as it starts to slide away it picks up speed, and in the rotating frame a moving body is deflected sideways. The deflection curves it back around the point in a long looping path. L4 and L5 are stable, provided the larger body outweighs the smaller by roughly 25 to 1 — which the Sun and Jupiter comfortably do, which is why many thousands of Trojan asteroids are catalogued there.

One more detail, usually skipped: Webb does not sit at L2. It flies a halo orbit around it, hundreds of thousands of kilometres wide, taking about six months to go round. Sitting exactly at L2 would put it in Earth's shadow, cutting its solar power and thermally shocking an instrument built to stay near 40 kelvin, and would point its antenna straight past the Sun. The halo keeps it permanently in sunlight and off the line.

Three refinements worth holding onto.

First, nothing is cancelled. The Sun still pulls Webb with almost the force it pulls Earth, and the telescope is travelling around it at some 30 kilometres per second. What is special is the combination, not the absence.

Second, instability is not the same as expense. Because the collinear points are only just unstable, correcting the drift costs a couple of metres per second per year. Webb's thrusters can only push it away from the Sun — they sit on the sunlit side of the sunshield and cannot be turned around without exposing the optics — so its path is deliberately biased sunward and every correction adds to that push. An overshoot at launch could not have been undone; as it was, the accuracy left propellant for past twenty years.

Third, this is an idealisation. The classical result assumes two bodies on a circular orbit and a third of negligible mass; the real system is elliptical and includes the Moon and other planets, so the points are approximate regions that shift slightly rather than mathematical dots.

• Why the stability condition works out to a mass ratio near 25 to 1, and which pairs in the solar system fail it.
• How a spacecraft reaches a halo orbit at all, given that the point it circles exerts no force of its own.

Every term the collection defines is gathered in the glossary.