Figure: The Lagrange points illustrated and explicated

Image 1 Caption: An image showing the five Lagrange points (L-points) for the Sun-Earth system (idealized as a 2-body system) with the Earth's orbit being idealized as a perfectly circular orbit with the Sun center taken as the center of mass for the 2-body system. The Lagrange points corotate with the 2 relatively large astro-bodies which determine the five Lagrange points.

Further explication:

1. The (five) Lagrange points exist for any 2-body system with orbits which are approximately circular orbits.

   The exact arrangement is simplest for the idealized case which is what Image 1 depicts.

2. The (five) Lagrange points are points of mechanical equilibrium (i.e., zero force) in the corotating reference frame determined by the two astro-bodies.

   The mechanical equilibria are due to a combination of the gravitational forces of the two astro-bodies, the centrifugal force, and, for the L4 and L5 points, the Coriolis force (see Wikipedia: Lagrange point: Stability).

   Test particles (i.e., astro-bodies which are negligible in mass relative to the masses of the two astro-bodies) put at rest exactly at the Lagrange points will stay at the those points in the corotating reference frame.

   To be clear, the Lagrange points corotate with the 2 large astro-bodies.

3. The L1, L2, and L3 points are collinear with the two astro-bodies

   The L4 and L5 points complete equilateral triangles (i.e., triangles where all angles are 60°) with the two astro-bodies as seen in Image 1. This means L4 and L5 points are 60° away from the line joining the two astro-bodies.

4. The 3 collinear Lagrange points were mathematically discovered by Leonhard Euler (1707--1783) and the other 2 by Joseph-Louis Lagrange (1736--1813) (see Wikipedia: Lagrange points: History). Both persons are famous mathematicians and are the eponyms of many things in mathematics and physics: e.g., Euler's formula, Euler's number (much more commonly known as the exponential number e = 2.71828 ... ), the Euler-Lagrange equation, the Euler-Mascheroni constant γ = 0.5772156649 ... , Lagrangian mechanics, and Lagrange multipliers.

5. The L1, L2, and L3 points are unstable equilibria (see Wikipedia: Lagrangian point: Stability). In principle, any perturbation, NO matter how small, will cause test particles to escape to infinity eventually if there is NO effects to keep them close to those Lagrange points.

6. The L4 and L5 points are stable equilibria due to the Coriolis force which acts as restoring force (see Wikipedia: Lagrange point: Stability).

   A stable equilibrium is one where motions due sufficiently small perturbation on a test particle placed at the stable equilibrium damp out and the test particle average position remains the stable equilibrium. The test particle NEVER escapes to infinity unless there is some other effect causing them to escape.

7. In fact, all physically real stable equilibria are metastable (i.e., a large enough perturbation will cause a test particle to escape to infinity) and most (maybe all) physically real unstable equilibria are slightly metastable (i.e., a small enough perturbation will NOT cause a test particle to escape to infinity) for any test particle that stays at them for a finite time. However, we customarily do NOT use the term metastable state unless the state is obviously NOT near the limits of absolute stability or instability.

8. In planetary systems, each planet and its star establish Lagrange points.

   However, the gravitational perturbations of an n-body system with more than 2 astro-bodies make the Lagrange points less than ideal. In general, the larger planet, the larger its gravitational force, and the more ideal its Lagrange points.

   For example, in the Solar System, Jupiter's L4 and L5 points are particularly close to ideal, and therefore very stable equilibria. For that reason, Jupiter's trojan asteroids accumulated near its L4 and L5 points and tended to stay there over the Solar System's evolution.

   The word trojan has come to mean any small astro-body (e.g., an asteroid) that orbits near L4 and L5 points.

9. Now test particles can be kept at the L1 point, L2 point, and L3 point by orbital station-keeping using orbital maneuvers (AKA burns) using thruster burns (see also Wikipedia: Thrust, Wikipedia: Rocket, Wikipedia: Rocket engine, and Wikipedia: Spacecraft propulsion).

10. In fact, the Earth's Lagrange points have become favorite locations for spacecraft for special space missions.

11. In practice, it turns out that for some space missions, it is better to put spacecraft into Lissajous orbits (or their special case halo orbits) around L-points rather than at L-points. Lissajous orbits are quasi-periodic orbits around L-points that are NOT in the orbital plane of the 2-body system. In quasi-periodic orbits, the orbiting astro-body is bound, but the orbital trajectory never exactly repeats (see also Wikipedia: Orbit dynamics: asymptotically periodic orbit).

    Lissajous orbits are sort of like quasi-periodic epicycles come to life.

    Lissajous orbits can used for spacecraft at the unstable L1, L2, and L3 points.

12. The Earth's Lagrange point the L2 point is currently a particular favorite for important space missions.

    This L2 point is on Earth-Sun line at 0.010 AU (∼ 1.5*10**6 km) from the Earth.

    An astro-body at that distance from the Sun, NOT at an L2 point, would have an orbital period about the Sun that is slightly longer than a sidereal year = 365.256363004 days (J2000).

    An astro-body in Lissajous orbit about L2 has, of course, an orbital period about the Sun of exactly a sidereal year = 365.256363004 days (J2000).

13. Note that Earth's umbra extends out to ∼ 0.009 AU (∼ 1.4*10**6 km) (see Wikipedia: Umbra, penumbra and antumbra: Umbra; Richard Pogge: Lecture 9: Eclipses of the Sun & Moon: scroll down ∼ 25 %).

    So spacecraft at the L2 point are NEVER in the umbra. In fact, their Lissajous orbits probably NEVER put them in the penumbra either.

    Going into penumbra would decrease the solar power that powers most spacecraft.

14. Examples of spacecraft at the Earth's Lagrange points:
    1. At the L2 point, Gaia spacecraft (mission 2013--2025) (see also gaia_2013_2025.html), the James Webb Space Telescope (JWST; 2021--2041?), and Euclid spacecraft (2023--2029?) (see also videos Euclid in a nutshell (2023) | 1:14, The Euclid Space Telescope: tackling dark matter and dark energy mysteries (2023) | 14:54, etc.).

    2. The European Space Agency (ESA) is planning the ESA Vigil spacecraft (est. 2031--2036?) which will study solar weather (and space weather generally) and in particular give the early warning of CMEs, particularly dangerous super CMEs. ESA Vigil will be at Lagrange point L5.

       How early will be the warning for CMEs? Hopefully days, but maybe sometime only hours??? (see Wikipedia: Coronal mass ejection; Wikipedia: Coronal mass ejection: First traces). Early warning is particularly necessary for super coronal mass ejection (CMEs) (e.g., the Solar storm of 1859 (AKA the Carrington event) and the May 1921 geomagnetic storm (AKA New York Railroad Storm of 1921)) which could burn out the electrical power grids worldwide which would be a major disaster. Simply breaking the electrical power grid connections for a few hours while a super CME passes over could prevent said major disaster.

    
15. Image 2 Caption: Top and side views of an example trajectory from the Earth to a Lissajous orbit around the Sun-Earth system Lagrange L2 point.