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Libratrix

Lagrange Points / Three-Body Equilibrium Physics

\[r_{L2} \approx R\left(\frac{M_2}{3M_1}\right)^{1/3}\]
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Libratrix is a Pixelated Physics law character, one of our forces designs. At the L2 Lagrange point, Sun and Earth gravity balance an orbit, but the balance is unstable, so spacecraft need nudges. This is a 3" × 3" pixel-art sticker. The physics panel below gives the equation, what each symbol means and the sources.

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Size3″ × 3″

3″ × 3″ kiss-cut sticker.

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The physics

Two bodies orbiting each other have five points where a small third body can keep pace with them. For the Sun and Earth, L2 lies a little beyond Earth, and the triangular points L4 and L5 are stable only when one mass greatly outweighs the other.

Libratrix slings her hammock where the pulls balance, and she knows how thin that balance is.

In 1772 Joseph-Louis Lagrange found that, for two large bodies circling each other, there are five places where a small third body can keep pace with them: three on the line through the pair and two at the corners of equilateral triangles. In the frame turning with the pair, gravity and the outward pull of the rotation cancel there.

Her favourite is L2 of the Sun and Earth, about 1.5 million km beyond Earth, directly away from the Sun, the first two lines on this page. Telescopes love it: the Sun, Earth and Moon all stay on one side, so one shield can block them all. The James Webb Space Telescope works there. Strictly, it does not sit at L2; it follows a large halo orbit around it.

The catch is that L2 is a saddle, not a bowl. Drift a little and the drift grows. So Webb fires small thrusters to stay on track, with station-keeping burns scheduled on average every 42 days; 21 days is only the minimum spacing. Libratrix gives a small nudge every six weeks or so.

Her cousins at L4 and L5 have it easier. Those points sit on hilltops of the effective potential, yet the Coriolis effect holds a body there in a stable loop, provided the larger mass outweighs the smaller by more than about 24.96 times, the last line. Edward Routh published that condition in 1874–75; Gascheau had found it in 1843. Sun and Jupiter qualify, and Jupiter’s Trojan asteroids gather there. In 2011 the first Earth Trojan was identified.

Balanced. Barely. Nudge me often.

Equations

\[r_{L2} \approx R\left(\frac{M_2}{3M_1}\right)^{1/3}\]
\[r_{L2} \approx 1.5\times10^{6}\ \text{km}\]
\[\frac{M_1}{M_2} > \frac{25+\sqrt{621}}{2} \approx 24.96\]
SymbolMeaningUnit
\(r_{L2}\) distance from the smaller body to L2 (second line: Sun–Earth L2, beyond Earth) \(\mathrm{m}\)
\(R\) distance between the two bodies \(\mathrm{m}\)
\(M_1, M_2\) masses of the larger and the smaller body \(\mathrm{kg}\)
\(M_1/M_2\) mass ratio; above about 24.96, L4 and L5 are stable (Routh) \(\mathrm{1}\)

Sources

  1. Encyclopaedia Britannica — Lagrange point (Lagrange 1772; L4/L5 stability above 24.96; Trojan asteroids)
  2. Routh (1874–75) On Laplace’s Three Particles, with a Supplement on the Stability of Steady Motion, Proc. London Math. Soc. s1-6, 86–97
  3. NASA Science — What is a Lagrange Point?
  4. STScI JWST User Documentation — JWST Orbit (orbits around L2; station-keeping on average every 42 days)
  5. NASA NTRS — Flight Dynamics Planning and Operations Support for the JWST Mission (AAS 22-611)
  6. Connors, Wiegert & Veillet (2011) Earth’s Trojan asteroid, Nature 475, 481
  7. Sicardy (2010) Stability of the triangular Lagrange points beyond Gascheau’s value, Celest. Mech. Dyn. Astron. 107, 145–155 (Gascheau 1843 precedes Routh)

Source Lagrange, J.-L. (1772). Essai sur le problème des trois corps. Prix de l’Académie Royale des Sciences de Paris, tome IX.

Field notes

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