Calculating Collinear Lagrange Point Positions: L1, L2, L3 in Restricted 3-Body Problem | Topic 8

The unstable Lagrange points L1, L2, and L3 are along the line of the two primary masses, forming a syzygy. Computation of the x values of the collinear equilibrium points requires finding the maxima of the function Ū(x,0,0), i.e., the solutions of ∂Ū(x,0,0)/∂x = 0 which is a quintic equation after simplification, that is, a fifth-order polynomial. The distance of L_i , i = 1, 2 from the smaller primary mass, m2, is given by the unique positive solution γ_i.
An approximation of the distance is γ_i = r_h, where r_h = ( μ/3 )^(1/3) , the Hill radius, is the radius of the Hill sphere in the spatial problem, where μ = m2 / (m1 + m2) is the mass parameter. The Hill sphere is the ‘bubble’ in 3-D position space surrounding m2 inside of which the gravitational field of m2 has a greater effect on the particle’s motion than m1.
A Note on Terminology. Throughout the literature covering the equilibrium points in the circular restricted 3-body problem (CR3BP), the points are given various names, such as libration points, Lagrange points, and Lagrangian points.
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► Reference: Section 2.5, "Location of the Equilibrium Points" of my FREE PDF book:
Dynamical Systems, the Three-Body Problem and Space Mission Design.
Koon, Lo, Marsden, Ross (2011)
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► PDF Lecture Notes (Lecture 4 for this video)
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► Euler's 1767 paper on the collinear equilibrium points
De motu rectilineo trium corporum se mutuo attrahentium
Novi Commentarii academiae scientiarum Petropolitanae 11, 1767, pp. 144-151
Original (Latin) and English translation:
eulerarchive.ma...
The effective potential energy (also called the augmented potential) is a way to include both the effects of gravity and the centrifugal force of the rotating frame. The critical points of the effective potential energy function, Ū(x,y,z), are the Lagrange points, equilibrium points in the rotating frame (a.k.a., relative equilibria).
The circular restricted 3-body problem (CR3BP) describes the motion of a body moving in the gravitational field of two primaries that are orbiting in a circle about their common center of mass, with trajectories such as Lagrange points, halo orbits, Lyapunov planar orbits, quasi-periodic orbits, quasi-halos, low-energy trajectories, etc.
• The two primaries could be the Earth and Moon, the Sun and Earth, the Sun and Jupiter, etc.
• The equations have been non-dimensionalized
• The mass parameter μ = m2 / (m1 + m2) is main factor determining the type of motion possible for the spacecraft. It is analogous to the Reynold's number Re in fluid mechanics, as it determines the onset of new types of behavior.
► Dr. Shane Ross is an Aerospace Engineering Professor at Virginia Tech. He has a Ph.D. from Caltech (California Institute of Technology) and worked at NASA/JPL and Boeing.
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