Editing Lagrange point (section)

===Stability===
The triangular points ({{L4|nolink=yes}} and {{L5|nolink=yes}}) are stable equilibria, provided that the ratio of {{sfrac|''M''<sub>1</sub>|''M''<sub>2</sub>}} is greater than 24.96.<ref group="note" name="exact_stability_threshold">Actually {{sfrac|25 + 3{{sqrt|69}}|2}} ≈ {{val|24.9599357944}} {{OEIS|A230242}}</ref> This is the case for the Sun–Earth system, the Sun–Jupiter system, and, by a smaller margin, the Earth–Moon system. When a body at these points is perturbed, it moves away from the point, but the factor opposite of that which is increased or decreased by the perturbation (either gravity or angular momentum-induced speed) will also increase or decrease, bending the object's path into a stable, [[kidney bean]]-shaped orbit around the point (as seen in the corotating frame of reference).<ref name="cornish">{{cite web|url= https://wmap.gsfc.nasa.gov/media/ContentMedia/lagrange.pdf |title=The Lagrange Points |date=1998|publisher=NASA}}, Neil J. Cornish, with input from Jeremy Goodman</ref>

The points {{L1|nolink=yes}}, {{L2|nolink=yes}}, and {{L3|nolink=yes}} are positions of [[Mechanical equilibrium|unstable equilibrium]]. Any object orbiting at {{L1|nolink=yes}}, {{L2|nolink=yes}}, or {{L3|nolink=yes}} will tend to fall out of orbit; it is therefore rare to find natural objects there, and spacecraft inhabiting these areas must employ a small but critical amount of [[Orbital station-keeping|station keeping]] in order to maintain their position.