Editing Lagrange point (section)

==Lagrange points==
{{Also|List of objects at Lagrange points}}

The five Lagrange points are labeled and defined as follows:

=== {{L1|nolink=yes}} point===
The {{L1|nolink=yes}} point lies on the line defined between the two large masses ''M''<sub>1</sub> and ''M''<sub>2</sub>. It is the point where the gravitational attraction of ''M''<sub>2</sub> and that of ''M''<sub>1</sub> combine to produce an equilibrium. An object that [[orbit]]s the [[Sun]] more closely than [[Earth]] would typically have a shorter orbital period than Earth, but that ignores the effect of Earth's gravitational pull. If the object is directly between Earth and the Sun, then [[Gravity of Earth|Earth's gravity]] counteracts some of the Sun's pull on the object, increasing the object's orbital period. The closer to Earth the object is, the greater this effect is. At the {{L1|nolink=yes}} point, the object's orbital period becomes exactly equal to Earth's orbital period. {{L1|nolink=yes}} is about 1.5&nbsp;million kilometers, or 0.01 [[Astronomical unit|au]], from Earth in the direction of the Sun.<ref name="Lagrange Cornish">{{cite web|last1=Cornish|first1=Neil J. |date=1998 |title=The Lagrange Points |publisher=WMAP Education and Outreach |url=http://www.physics.montana.edu/faculty/cornish/lagrange.pdf| archive-url     =https://web.archive.org/web/20150907090116/http://www.physics.montana.edu/faculty/cornish/lagrange.pdf|archive-date=7 September 2015|url-status=dead|access-date=15 December 2015}}</ref>

==={{L2|nolink=yes}} point===
The {{L2|nolink=yes}} point lies on the line through the two large masses beyond the smaller of the two. Here, the combined gravitational forces of the two large masses balance the centrifugal force on a body at {{L2|nolink=yes}}. On the opposite side of Earth from the Sun, the orbital period of an object would normally be greater than Earth's. The extra pull of Earth's gravity decreases the object's orbital period, and at the {{L2|nolink=yes}} point, that orbital period becomes equal to Earth's. Like L<sub>1</sub>, L<sub>2</sub> is about 1.5&nbsp;million kilometers or 0.01 [[Astronomical unit|au]] from Earth (away from the sun). An example of a spacecraft designed to operate near the Earth–Sun L<sub>2</sub> is the [[James Webb Space Telescope]].<ref name="stsci.edu">{{cite web |url=http://www.stsci.edu/jwst/overview/design/orbit|archive-url=https://wayback.archive-it.org/all/20140203174537/http://www.stsci.edu/jwst/overview/design/orbit|url-status=dead|archive-date=3 February 2014 |title=L2 Orbit|publisher=Space Telescope Science Institute|access-date=28 August 2016}}</ref> Earlier examples include the [[Wilkinson Microwave Anisotropy Probe]] and its successor, ''[[Planck (spacecraft)|Planck]]''.

==={{L3|nolink=yes}} point===
The {{L3|nolink=yes}} point lies on the line defined by the two large masses, beyond the larger of the two. Within the Sun–Earth system, the {{L3|nolink=yes}} point exists on the opposite side of the Sun, a little outside Earth's orbit and slightly farther from the center of the Sun than Earth is. This placement occurs because the Sun is also affected by Earth's gravity and so orbits around the two bodies' [[Barycentric coordinates (astronomy)|barycenter]], which is well inside the body of the Sun. An object at Earth's distance from the Sun would have an orbital period of one year if only the Sun's gravity is considered. But an object on the opposite side of the Sun from Earth and directly in line with both "feels" Earth's gravity adding slightly to the Sun's and therefore must orbit a little farther from the barycenter of Earth and Sun in order to have the same 1-year period. It is at the {{L3|nolink=yes}} point that the combined pull of Earth and Sun causes the object to orbit with the same period as Earth, in effect orbiting an Earth+Sun mass with the Earth-Sun barycenter at one focus of its orbit.

==={{L4|nolink=yes}} and {{L5|nolink=yes}} points===
[[File:L4 diagram.svg|thumb|Gravitational accelerations at {{L4|nolink=yes}}]]

The {{L4|nolink=yes}} and {{L5|nolink=yes}} points lie at the third vertices of the two [[equilateral triangle]]s in the plane of orbit whose common base is the line between the centers of the two masses, such that the point lies 60° ahead of ({{L4|nolink=yes}}) or behind ({{L5|nolink=yes}}) the smaller mass with regard to its orbit around the larger mass.

===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.