Introduction

The Three-Body Problem is nothing more than the Two-Body Problem with one more body added to the system.

Consider a system with three finite masses and some origin in the inertial frame of reference:

../../_images/three-body_image.png

Figure 1.1 Three-Body Problem Definition

Such that the body masses are non-zero: P1 has mass m1, P2 has mass m2, and P3 has mass m3. Define the general relative vector notation as.

(1)rij=rjri

For example

r12=r2r1

and

r12=r21

We are interested in how vectors r1, r2, and r3 change as a function of time. Assuming they only interact gravitationally and each particle has a uniform spherical gravitational field.

From Newton’s Law of Gravitation, the mutual gravitational interaction on one finite point mass due to the other masses is:

(2)mir¨i=Fij+Fik

For the Three-Body System the three force interactions are:

(3)m1r¨1=G(m1m2)r213r21G(m1m3)r313r31
(4)m2r¨2=G(m2m1)r123r12G(m2m3)r323r32
(5)m3r¨3=G(m3m1)r133r13G(m3m2)r233r23

These are equations of motion for the Three-Body Problem.

Note

The Solar System Dynamics group at NASA JPL keeps track of parameters commonly used in Astrodynamics. Among the parameters listed on the Astrodynamics Parameters page is the Newtonian Constant of Gravitation (G).

G=6.67430(±0.00015)×1011kg1m3s2

Another reference of interest for the Three-Body Problem is the Physical Parameters of Planets page. There you will find a useful reference for the mass of a planetary body used in astrodynamic computations.

Each particle has a State Vector made up of its position and velocity:

(6)x=[xyzx˙y˙z˙]

To solve the for the Three-Body Problem we require 18 equations of motion due to 3 particle state vectors. Unfortunately, there are only 10 Classical Integrals available. The Three-Body Problem is a non-trivial problem, and generally, a non-integrable problem in dynamics.

The next sections will introduce a special case Three-Body Problem that is better formulated for numerical analysis. First let us introduce the Jacobian Coordinate Frame.

Jacobian Coordinate Frame

To help derive the equations of motion for the restricted Three-Body Problem, consider the Jacobi Coordinate formulation. Which is defined such that P3 is with respect to the barycenter of the other two particles by attaching a non-inertial coordinate system to the barycenter of P1 and P2 as shown in the figure below:

../../_images/jacobi_frame.png

Figure 1.2 Jacobi Coordinate Frame

Where the center of mass of P1 and P2 is defined as vector Rcm

(7)Rcm=m1r1+m2r2m1+m2

R is the positional vector from P1 to P2

(8)R=r12=r2r1

and the positional vector of P3 relative to the barycenter is defined as

(9)r=r3Rcm

Note

The Earth-Moon system is a classic example of using the Jacobian Coordinate Frame. Where P1 and P2 represent the Earth and Moon, and P3 can be considered a spacecraft relative to the Earth-Moon system.

By taking the derivative of Equation (8), Equation (9) and utilizing the Equations (3) - (5) we end up with the following two equations of motion:

(10)R¨=G(m1+m2)RR3+Gm3[rm1m1+m2R|rm1m1+m2R|3r+m2m1+m2R|r+m2m1+m2R|3]
(11)r¨=G(m1+m2+m3)m1+m2[m1(r+m2m1+m2R)|r+m2m1+m2R|3+m2(rm1m1+m2R)|rm1m1+m2R|3]

This is a very complicated solution to the equations of motion for the Jacobian Coordinate frame. Generally, it is useful to assume the third particle has zero or negligible mass.

m30

Which is certainly a valid assumption when analyzing satellite dynamics in the Earth-Moon system.