*Analysis and insight from an 18th-century mathematician were used to place the James Webb Space Telescope in a “parking spot” optimally suited for its imaging, communication, and solar-power needs.*

Lagrange’s analysis was an analysis of the three-body problem and provided insight into the conditions under which it could be solved analytically. The five Lagrange points, dubbed L1 through L5, offer different space-observation vantage points. Three of the five Lagrange points are unstable, and two are stable **(Figure 1)**. The unstable Lagrange points, also called libration points, are labeled L1, L2 and L3, and lie along the line connecting the two large masses. Thus, when people say that the JWST is “parked” at L2, that’s mostly but not wholly accurate. (Note that five Lagrange points also exist with respect to the Earth and moon, but that’s a story for another time.)

The L1 point of the Earth-Sun system provides an uninterrupted view of the Sun and is currently home to the Solar and Heliospheric Observatory Satellite SOHO.

The L2 point of the Earth-Sun system was the home to Wilkinson Microwave Anisotropy Probe (WMAP) spacecraft; a NASA mission launched in 2001 to make fundamental measurements of cosmology, as well as the Planck observatory operated by the ESA from 2009 to 2013, mapping the anisotropies of the cosmic microwave background (CMB) at microwave and infrared frequencies.

Now, it has gained additional attention as the home to the JWST. L2 is ideal for astronomy because a spacecraft located there is close enough to communicate with Earth readily; it can keep Sun, Earth, and Moon behind the spacecraft for solar power; and (with appropriate shielding) provides a clear view of deep space for our telescopes **(Figure 2)**. The L1 and L2 points are unstable on a scale of approximately 23 days, which requires satellites orbiting these positions to undergo regular course and attitude corrections.

Scientists are unlikely to find any use for the L3 point since it always remains hidden behind the Sun. It has also spawned the idea of a hidden “Planet-X” at the L3 point as a popular topic in science fiction, including the 1951 cult classic The Man from Planet X.

The stable Lagrange points, L4 and L5, form the apex of two equilateral triangles with large masses at their vertices. L4 leads the orbit of Earth and L5 follows the orbit. The L4 and L5 points are home to stable orbits so long as the mass ratio between the two large masses exceeds 24.96.

This condition is satisfied for both the Earth-Sun and Earth-Moon systems and many other pairs of bodies in the solar system. Objects found orbiting at the L4 and L5 points are often called Trojans after the three large asteroids Agamemnon, Achilles, and Hector that orbit in the L4 and L5 points of the Jupiter-Sun system.

The JWST has rocket propellant onboard for midcourse correction and insertion into orbit around L2, of course, but also for necessary functions during the life of the mission, such as “station keeping” maneuvers—small thruster burns to adjust Webb’s orbit—as well as momentum management to maintains Webb’s orientation in space.

The project goal was to have enough fuel remaining on arrival at L2 to maintain them for five years. Still, the initial launch and mid-course connections were so accurate that project engineers are now confident there is enough left for ten or perhaps even 15 years. Unlike the Hubble Space Telescope, there is at no way to refuel the JWST due to distance and access (but perhaps there will be ten years from now—you never know).

The third part of this article looks at other implications of the n-body problem on which Lagrange worked, as well as issues associated with astrodynamics.

**EE World Related Content**

Reaction wheels ensure satellites maintain the right attitude: Part 2

Now *that’s* a test instrument: Inside the Webb space telescope

Testing On the James Webb Space Telescope

The Hubble Space Telescope: Part 4

**External References**

__JWST and Lagrange Points__

- European Space Agency, “James Webb Space Telescope”
- NASA, “L2 Will be the James Webb Space Telescope’s Home in Space”
- NASA, “What is a Lagrange Point?”
- NASA, “The Lagrange Points” (Detailed mathematical analysis)
- European Space Agency, “What are Lagrange points?”

__Three-body problem__

- Joseph-Louis Lagrange
**, “**Essai sur le Probleme des Trois Corps**”**(scan of his original published paper) - Ellis Strömgren, Translation of the paper “Connaissance actuelle des orbites dans le problème des trois corps” or “Current knowledge of orbits in the three body problem” (an early 20
^{th}-century paper which is a modern update on Lagrange’s work) - Sylvio R. Bistafa, University of Sao Paulo, “Euler’s three-body problem”
- Scholarpedia, “Three body problem”

__Mercury perihelion and precession anomaly__

- NASA, “Newtonian gravity and Einstein’s Theory of General Relativity”
- Wikipedia, “Test of General Relativity”

__Lagrange Multipliers__

- Machine Language Mastery, “A Gentle Introduction To Method Of Lagrange Multipliers”
- Khan Academy, “Lagrange multipliers, introduction”
- Wikipedia, “Lagrange Multiplier”

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