Earth, planets and orbits
1772
Balance points in space
Joseph-Louis Lagrange
Are there places in space where you could park and never drift away?
Lagrange found five places where a small object can keep step with two big ones, such as the Sun and Jupiter. In 1906 the first asteroid was found at one of them, sharing Jupiter’s orbit. Today the James Webb Space Telescope works near another, beyond the Earth.
First, make a guess
Which balance points keep a gently nudged probe nearby?
- Mass ratio μ
- 0.01215
- L4 and L5 stable?
- Yes (μ is below 0.0385)
- What happened
- Probes still nearby
This view turns with the two bodies, so they stand still. Bright regions are hilltops of the combined gravity-and-spin landscape. Release a probe at a balance point with a tiny nudge, or tap anywhere to drop one at rest.
How they figured it out
In 1772 Lagrange, born in Turin and working in Berlin, entered a Paris Academy prize problem about three bodies pulling on one another, a problem with no general solution. He found five special arrangements that keep their shape as they turn. Euler had already found the three on the line between the bodies; Lagrange added the two at the corners of equilateral triangles.
For over a century they were mathematical curiosities. Then in 1906 Max Wolf found an asteroid, now called 588 Achilles, sitting near one of Jupiter’s triangle points. Thousands of these “Trojans” are now known.
The equation
Where gravity and the turning of the system balance, the push in every direction is zero.
See it in real life
The saddle and the merry-go-round
A marble balanced on a horse’s saddle stays only if you place it perfectly: that’s L1, L2 and L3. The James Webb Space Telescope works near L2 and fires its thrusters about every three weeks to stay on the saddle. L4 and L5 are different: the spin of the system turns a drift into a gentle loop.
Where engineers use it
Space telescopes
JWST and Gaia work near Sun–Earth L2 and fire small thrusters to stay there.
Solar observatories
SOHO watches the Sun from L1, giving early warning of solar storms.
Where it came from
- Universal gravitation 1687
Led to
Lagrange solved a special case of Newton’s three-body problem.
- Eigenvalues 1829
Same technique
Whether a probe at a Lagrange point drifts or holds is decided by eigenvalues.
Where the comparison stops: Lagrange found the points in 1772; the stability analysis in eigenvalue language came later.