Restricted Three-Body Problem
4. Restricted Three-Body Problem (Lagrange Points)
Governing Equations (Rotating Coordinate System)
\[ \begin{align} x - \left[ \frac{(1-\mu)(x+\mu)}{r_1^3} + \frac{\mu(x - (1-\mu))}{r_2^3} \right] &= 0 \\ y - \left[ \frac{(1-\mu)y}{r_1^3} + \frac{\mu y}{r_2^3} \right] &= 0 \end{align} \]
Parameter: $\mu = 0.01215$ (Earth-Moon Mass Ratio)
$r_1, r_2$: Distance from primary and secondary bodies
Characteristics
In the restricted three-body problem, the equilibrium points where the centrifugal force and the gravitational force exerted by two celestial bodies balance out are called "Lagrange Points".
Since these equations do not have analytical exact solutions (closed-form expressions), solutions are usually found via numerical analysis. The system has a total of 5 equilibrium points (L1 - L5), and each is known as a "special seat" for space exploration.
Effective Potential and Gravity Wells
To intuitively understand the positional relationship of Lagrange points, the "Effective Potential (Roche Potential)" in the rotating coordinate system is visualized in the figure below.
The saddle-like points where contour lines cross correspond to L1, L2, and L3, while the peaks (or basins) of the potential correspond to L4 and L5. Spacecraft utilize these "gravity contour lines" to travel efficiently.
Fig 1: Conceptual diagram of the positions of two celestial bodies and the 5 Lagrange points
Source: Wikimedia Commons
Results
5 Solutions (Theoretical Values)
Assuming a mass ratio $\mu=0.01215$ (Earth-Moon system), this equation has the following 5 coordinates $(x, y)$ as solutions. L1, L2, and L3 are called Euler's collinear solutions, and L4 and L5 are called Lagrange's triangular solutions.
| Point | Name / Location | X Coordinate (Theoretical) | Y Coordinate (Theoretical) |
|---|---|---|---|
| L1 | Collinear (Between bodies) | 0.8369 | 0.0000 |
| L2 | Collinear (Outer) | 1.1557 | 0.0000 |
| L3 | Collinear (Opposite side) | -1.0051 | 0.0000 |
| L4 | Triangular (Leading 60°) | 0.4879 | 0.8660 |
| L5 | Triangular (Trailing 60°) | 0.4879 | -0.8660 |
Simulation Results
Here are the results of analyzing the restricted three-body problem using Newton's method.
Instead of direct programming, we performed the analysis using the "SPICE-Oriented Analysis Method", a methodology based on the reverse idea of describing equations as circuits and solving them with the circuit simulator SPICE. By changing the initial values, we were able to find all 5 solutions.
Fig 2: Analysis results of the restricted three-body problem using the SPICE-oriented analysis method
Analysis Environment
- SPICE: Mac SPICE3
- PC: MacBook
- OS: macOS Monterey 12.7.6
- CPU: 1.2GHz Dual-Core Intel Core m5
- Memory: 8GB
Since the calculation of this scale converges instantly in the current PC environment, detailed measurement of calculation time is omitted.