# Circular Restricted Three-Body Problem By letting $m_3 \rightarrow 0$ then Equation {eq}`3bp_eom_1`-{eq}`3bp_eom_2` as derived from the [Jacobian Coordinate Frame](./three_body_problem.md/#jacobian-coordinate-frame) section simplifies to take on a similar form as the Two-Body Problem. :::{math} :label: \ddot{\mathbf{R}} = - G\frac{\left(m_1+m_2\right)}{R^3}\mathbf{R} ::: Many import results can be derived from this general form such as the analysis of the relative motion of the International Space Station, satellites, asteroids etc. among the Earth-Moon system. ## Force Potential Next, we can setup the total force potential as: :::{math} :label: cr3bp_U U = \frac{Gm_1}{|\mathbf{r}+\frac{m_2}{m_1+m_2}\mathbf{R}|} + \frac{Gm_2}{|\mathbf{r}-\frac{m_1}{m_1+m_2}\mathbf{R}|} ::: so that :::{math} :label: \ddot{\mathbf{r}} = \frac{\partial{U}}{\partial{\mathbf{r}}} ::: In many scenarios, assume $|\mathbf{R}|$ is constant such that $P_1$ and $P_2$ are in mutual circular orbit. This is certainly a valid assumption for systems such as Earth-Moon, or the Earth-Sun. ## The Rotating Frame In a rotating coordinate frame, the position vector for the Three-Body system can be represented as: :::{math} :label: \mathbf{R} = \left[ \cos\left(nt\right)\mathbf{\hat{x}}+ \sin\left(nt\right) \mathbf{\hat{y}} \right] ::: where $n$ is the mean motion of the primary body system. :::{math} :label: n = \sqrt{\frac{G\left(m_1+m_2\right)}{R^3}} = n_{12} ::: Shift the coordinate system to a rotating frame so that $\mathbf{R}$ is stationary along the local x-coordinate axis ($\hat{\mathbf{x}}_R$). For example, consider the Earth-Moon system, the x-coordinate axis is always pointing from the Earth to the Moon and is rotating at a rate of $\mathbf{n}$: ```{figure} ./figures/jacobi_frame_shift.png :name: fig:jacobi_frame_shift :width: 90% **Figure 1.3** Jacobi Coordinate Frame Shifted to The Rotating Frame ``` The angular velocity ($\Omega$) for this system is defined as: :::{math} :label: \mathbf{\Omega} = n\mathbf{\hat{z}} ::: It is useful to relate the inerital position vector ($\mathbf{r}_I$) to the rotational vector ($\mathbf{r}_R$): :::{math} :label: \mathbf{r}_I = \mathbf{r}_R ::: The velocity in the inertial frame is: :::{math} :label: \dot{\mathbf{r}}_I = \dot{\mathbf{r}}_R + \mathbf{\Omega}\times\mathbf{r}_R ::: The acceleration in the inertial frame is: :::{math} :label: r_ddot_i_to_rot \ddot{\mathbf{r}}_I = \ddot{\mathbf{r}}_R + \dot{\mathbf{\Omega}}\times\mathbf{r}_R + 2\mathbf{\Omega}\times\dot{\mathbf{r}}_R+\mathbf{\Omega}\times\left(\mathbf{\Omega}\times\mathbf{r}_R \right) ::: ```{important} Note that $\dot{\Omega} = 0$ since we assumed the orbit is circular. In other words $n$ is constant. ``` Let $\mathbf{r}_R$ be the position in the rotating coordinate frame. $\mathbf{R}_R = \mathbf{r}$ We can rewrite Equation {eq}`r_ddot_i_to_rot` as: :::{math} \ddot{\mathbf{r}}_I = \ddot{\mathbf{r}} + 2\mathbf{\Omega}\times\dot{\mathbf{r}}+\mathbf{\Omega}\times\left(\mathbf{\Omega}\times\mathbf{r} \right) = \frac{\partial{U}}{\partial{\mathbf{r}}} ::: :::{math} :label: r_ddot_i_to_rot_simp \ddot{\mathbf{r}}_I = \ddot{\mathbf{r}} + 2n\mathbf{\hat{z}}\times\dot{\mathbf{r}} + n^2\mathbf{\hat{z}} \times \left( \mathbf{\hat{z}} \times \mathbf{r}\right)= \frac{\partial{U}}{\partial{\mathbf{r}}} ::: ### Equations of Motion As part of the rotating coordinate frame, it is convenient to write Equation {eq}`r_ddot_i_to_rot_simp` in vector form by breaking [$2n\mathbf{\hat{z}}\times\dot{\mathbf{r}}$] and [$n^2\mathbf{\hat{z}} \times \left( \mathbf{\hat{z}} \times \mathbf{r}\right)$] into their components. Thus, the set of equations of the Three-Body Problem for the circular restricted problem in the rotating coordinate frame are: :::{math} :label: cr3bp_eom \ddot{x} - 2n\dot{y} = n^2x + \frac{\partial{U}}{\partial{x}} \ddot{y} + 2n\dot{x} = n^2y + \frac{\partial{U}}{\partial{y}} \ddot{z} = \frac{\partial{U}}{\partial{z}} ::: ```{note} The components of [$2n\mathbf{\hat{z}}\times\dot{\mathbf{r}}$] are usually referred to as the **tidal terms** in Astrodynamics. Think of this as $P_2$ orbits $P_1$ every orbital period there is a force experience directly as a function of $n$ :::{math} \begin{matrix} - 2n\dot{y} \\ + 2n\dot{x} \\ \end{matrix} = \text{Tidal Terms} ::: ``` ## Simplifying The Equations of Motion Equations {eq}`cr3bp_eom` represent where the spacecraft is relative to the Jacobian Coordinate frame. [Figure 1.4](jacobi_frame_shift_updated) below shows an updated frame of reference: ```{figure} ./figures/jacobi_frame_shift_updated.png :name: jacobi_frame_shift_updated :width: 90% **Figure 1.4** Circular Restricted Three-Body Problem in the Rotating Frame ``` ```{note} In Astrodynamics the mass ratios are generally defined as: :::{math} :label: \begin{align*} \mu &= \frac{m_2}{m_1+m_2} \\ 1 - \mu &= \frac{m_1}{m_1+m_2} \end{align*} ::: ``` The total force potential {eq}`cr3bp_U` for this system can be rewritten as: :::{math} :label: U\left(\mathbf{r} \right) = \frac{Gm_1}{\sqrt{\left( x + \frac{m_2}{m_1+m_2}R \right)^2+y^2+z^2}} + \frac{Gm_2}{\sqrt{\left( x - \frac{m_1}{m_1+m_2}R \right)^2+y^2+z^2}} ::: Let us write this a bit more concisely by defining a new force potential as $\mathbf{V}$ in order to simplify our equations of motion Equation {eq}`cr3bp_eom`: :::{math} :label: cr3bp_V \mathbf{V}\left(x,y,z\right) = \frac{1}{2}n^2\left( x^2 + y^2 \right) + \mathbf{U} ::: :::{math} :label: \mathbf{U} = \frac{Gm_1}{r_1} + \frac{Gm_2}{r_2} ::: where $r_1$ and $r_2$ are: :::{math} r_1 = \sqrt{\left( x + \frac{m_2}{m_1+m_2}R \right)^2+y^2+z^2} ::: :::{math} r_ 2 = \sqrt{\left( x - \frac{m_1}{m_1+m_2}R \right)^2+y^2+z^2} ::: ```{note} The potential energy is made up of three components 1. The potential energy due to the force induced by the rotating reference frame 2. The gravitational potential energy from $P_1$ 3. The gravitational potential energy from $P_2$ ``` [Figure 1.5](cr3bp_potent) below is an example of solving Force Potential {eq}`cr3bp_V` for the Earth-Moon system. The figure is interactive and generated using [Plotly](https://plotly.com/python/), an open-source graphing library for Python. The reader is encouraged to interact with [Figure 1.5](cr3bp_potent) using their mouse to zoom into the region of space around the bodies and rotate the surface to inspect the potential wells.
Figure 1.5. Example of CR3BP Potential
### Equations of Motion Finally, the generalized equations of motion for the **Circular Restricted Three-Body Problem** are: :::{math} :label: cr3bp_eom_simp \begin{align*} \ddot{x} - 2n\dot{y} &= \frac{\partial{\mathbf{V}}}{\partial{x}} \\ \ddot{y} + 2n\dot{x} &= \frac{\partial{\mathbf{V}}}{\partial{y}} \\ \ddot{z} &= \frac{\partial{\mathbf{V}}}{\partial{z}} \end{align*} :::