About the Lagrange Points Simulation
This simulation maps the five Lagrange points (L1–L5) of the circular restricted three-body problem for a chosen pair such as Sun–Earth or Sun–Jupiter. It computes L1, L2 and L3 by Newton's method as saddle points along the line joining the bodies, while L4 and L5 sit 60° ahead of and behind the secondary. The effective Jacobi potential in the co-rotating frame is drawn as a colored height field, and you can launch test satellites that are integrated with a fourth-order Runge–Kutta scheme, including Coriolis and centrifugal terms, to watch them librate or drift away.
Lagrange points are positions where the gravity of two massive bodies and the centrifugal force of the rotating frame cancel, letting a small object hover almost for free. First described by Joseph-Louis Lagrange in 1772, they are prime real estate for spacecraft: SOHO and DSCOVR sit at Sun–Earth L1, while the James Webb Space Telescope, Gaia and Planck occupy L2. The stable L4 and L5 points trap Trojan asteroids — more than 12,000 share Jupiter's orbit.
Frequently Asked Questions
What are Lagrange points?
They are five positions in a two-body system where the combined gravitational pull of the bodies and the centrifugal force in the rotating frame balance exactly. A small object placed there can stay nearly fixed relative to the two larger bodies.
Why are L4 and L5 stable but L1, L2 and L3 are not?
L1–L3 are saddle points of the effective potential, so a small nudge grows and the object drifts away. L4 and L5 are local maxima but the Coriolis force curves escaping motion into a stable libration — provided the mass ratio M₁/M₂ exceeds about 24.96.
What is the effective potential shown as the colour map?
It is the Jacobi potential in the co-rotating frame: the sum of both bodies' gravitational potentials plus the centrifugal potential. Lagrange points appear as saddles (L1–L3) or peaks (L4–L5) on this surface, marking where the net force vanishes.
Which spacecraft use Lagrange points?
SOHO and DSCOVR occupy Sun–Earth L1 for continuous solar viewing, while the James Webb Space Telescope, Gaia and Planck sit at Sun–Earth L2, where they can shield instruments from the Sun, Earth and Moon at once.
What are Trojan asteroids?
Trojans are bodies trapped at a planet's stable L4 and L5 points, 60° ahead of and behind it in orbit. Jupiter hosts the largest population, with more than 12,000 catalogued Trojans co-orbiting the Sun alongside the planet.
What is the mass ratio μ and why does it matter?
μ is the secondary body's fraction of the total mass, m₂/(m₁+m₂). It sets the positions of the Lagrange points and decides whether L4 and L5 are stable, which requires the primary-to-secondary mass ratio to exceed 24.96.
How are L1, L2 and L3 computed in this simulation?
They lie on the line through the two bodies and are found by solving the balance-of-forces equation numerically with Newton's method, iterating from good initial guesses until the position converges.
What does it mean for a particle to "librate"?
Libration is a slow, looping oscillation around a stable Lagrange point rather than a fixed sit. Trojan asteroids and well-placed satellites trace such tadpole or horseshoe paths instead of resting exactly at the point.
How is the test satellite's motion calculated?
Its trajectory is integrated in the rotating frame using a fourth-order Runge–Kutta method that includes gravity from both bodies plus the centrifugal and Coriolis accelerations, giving accurate, stable orbits.
Who discovered the Lagrange points?
Joseph-Louis Lagrange described the triangular points L4 and L5 in 1772 while studying the three-body problem; the collinear points L1–L3 had been found earlier by Leonhard Euler.