Non-Dimensional Circular Restricted Three-Body Problem¶

Non-dimensionalization of the Circular Restricted Three-Body Problem provides an advantage of solving one problem and applying the results to a general number of problems. The system is generalized for any Three-Body Problem by removing the dependence of the rotating reference rate.

Normalize the equations for the Three-Body Problem along 3 physical properties of the system

1. Mass¶

As shown by Figure 1.4 the primary masses are normalized such that:

(1)¶\[\begin{split}\begin{align*} \mu &= \frac{m_2}{m_1+m_2} \\ 1 - \mu &= \frac{m_1}{m_1+m_2} \end{align*}\end{split}\]

Allowing the equations of motion to be independent of the primary masses and relying more on their relative size ratio.

2. Length¶

As described by the Jacobian Coordinate Frame section, the characteristic length of the Three-Body system is the vector \(\mathbf{R}\). Define the non-dimensional length scale as \(\mathbf{R}\) thus the length scale is:

\[r_s = R\]

and yields the non-dimensional length term

(2)¶\[\mathbf{r}^* = \frac{\mathbf{r}}{r_s} = \frac{\mathbf{r}}{R}\]

3. Time¶

The time scale is simply normalized against the period of the circular orbit:

\[t_s = \frac{1}{n}\]

this yields the non-dimensional time term:

(3)¶\[\tau = nt\]

Non-Dimensional Equations Of Motion¶

Now apply the non-dimensional terms to Equation (16) to get non-dimensional equations of motion:

(4)¶\[\begin{split}\begin{align*} n^2R\ddot{x}^* - 2n^2R\dot{y}^* &= \frac{1}{R}\frac{\partial{\mathbf{V}}}{\partial{x^*}} \\ n^2R\ddot{y}^* - 2n^2R\dot{x}^* &= \frac{1}{R}\frac{\partial{\mathbf{V}}}{\partial{x^*}} \\ n^2R\ddot{z}^* &= \frac{1}{R}\frac{\partial{\mathbf{V}}}{\partial{z^*}} \end{align*}\end{split}\]

Normalize the force potential Equation (14) as:

(5)¶\[\mathbf{\tilde{V}} = \frac{\mathbf{V}}{n^2R^2}\]

The equations of motion for the Non-Dimensional Circular Restricted Three-Body Problem become:

(6)¶\[\begin{split}\begin{align*} \ddot{x}^* - 2\dot{y}^* &= \frac{\partial{\tilde{\mathbf{V}}}}{\partial{x^*}} \\ \ddot{y}^* + 2\dot{x}^* &= \frac{\partial{\tilde{\mathbf{V}}}}{\partial{y^*}} \\ \ddot{z}^* &= \frac{\partial{\tilde{\mathbf{V}}}}{\partial{z^*}} \end{align*}\end{split}\]

Note that this looks similar to the equations of motion for the Circular Restricted Three-Body Problem(16) it’s just that the definition of \(\mathbf{\tilde{V}}\) and \(\mathbf{\tilde{U}}\) are different. Different in the sense that we scaled the force potential.

(7)¶\[\mathbf{\tilde{V}}\left(x^*,y^*,z^*\right) = \frac{1}{2}\left( {x^*}^2 + {y^*}^2\right) + \mathbf{\tilde{U}}\]

(8)¶\[\mathbf{\tilde{U}} = \frac{U}{n^2R^2}\]

We assume for the Non-Dimensional Restricted Three-Body Problem that \(\mu < \frac{1}{2}\). This is not a bad assumptions for most problems we are interested in solving. If not one can always swap \(m_1\) and \(m_2\) to get towards \(\mu < \frac{1}{2}\).

The gravity potential can be written as

(9)¶\[\mathbf{\tilde{U}} = \frac{1-\mu}{r^*_1} + \frac{\mu}{r^*_2}\]

where \(r^*_1\) and \(r^*_2\) are:

\[\begin{split}\begin{align*} r^*_1 &= \sqrt{\left( x^* + \mu \right)^2+{y^*}^2+{z^*}^2} \\ r^*_2 &= \sqrt{\left( x^* - 1 + \mu \right)^2+{y^*}^2+{z^*}^2} \end{align*}\end{split}\]

Figure 1.6 below is an example of solving the Non-Dimensional Force Potential (7). Since Figure 1.5 gave an example for the Earth-Moon system (\(\mu = 0.012156\)) this example exaggerates the potential wells by looking at a large mass ratio of \(\mu=0.09\). Again, the reader is encouraged to interact with Figure 1.6 using their mouse to zoom into the region of space around the bodies and rotate the surface to inspect the potential wells.

Figure 1.6. Example of Non-Dimensional CR3BP Potential

Dimensional Transformation¶

Given a solution for the the Non-Dimensional Circular Restricted Three-Body Problem (\(\mathbf{r^*}\), \(\mathbf{\dot{r}^*}\)) we can transform back to dimensional system by introducing \(\mathbf{R}\), \(m_1\) and \(m_2\) and solving for the mean motion of the two masses:

\[n = \sqrt{\frac{G\left( m_1 + m_2 \right)}{R^3}}\]

\[\begin{split}\begin{matrix} x = Rx^* & \dot{x} = nR\dot{x}^* \\ y = Ry^* & \dot{y} = nR\dot{y}^* \\ z = Rz^* & \dot{z} = nR\dot{z}^* \\ \end{matrix}\end{split}\]

The advantage of this transformation is that by solving one non-dimensional problem, is actually solves infinite number of problems in a dimensional set!