Co-Rotating Frame

Let us transform to a non-inertial frame of reference rotating with angular velocity $\omega$ about an axis normal to the orbital plane of masses $m_1$ and $m_2$, and passing through their center of mass. It follows that masses $m_1$ and $m_2$ appear stationary in this new reference frame. Let us define a Cartesian coordinate system $(x,\,y,\,z)$ in the rotating frame of reference which is such that masses $m_1$ and $m_2$ always lie on the $x$-axis, and the $z$-axis is parallel to the previously defined $\zeta$-axis. It follows that masses $m_1$ and $m_2$ have the fixed position vectors ${\bf r}_1=\mu_2\,(-1,\,0,\,0)$ and ${\bf r}_2=\mu_1\,(1,\,0,\,0)$ in our new coordinate system. Finally, let the position vector of mass $m_3$ be ${\bf r}=(x,\,y,\,z)$--see Figure 63.

Figure 63: The co-rotating frame.

According to Chapter 8, the equation of motion of mass $m_3$ in the rotating reference frame takes the form

$$\ddot{\bf r} + 2\,\mbox{\boldmath$\omega$}\times \dot{\bf r}... ...ldmath$\omega$}\times(\mbox{\boldmath$\omega$}\times {\bf r}),$$ (1113)

where $\mbox{\boldmath$\omega$}=(0,\,0,\,\omega)$, and

$$\rho_1^{\,2} = (x+\mu_2)^2+y^2 + z^2,$$ (1114)

$$\rho_2^{\,2} = (x-\mu_1)^2+y^2 + z^2.$$ (1115)

Here, the second term on the left-hand side of Equation (1113) is the Coriolis acceleration, whereas the final term on the right-hand side is the centrifugal acceleration. The components of Equation (1113) reduce to

$$\ddot{x} - 2\,\omega\,\dot{y} = - \frac{\mu_1\,(x+\mu_2)}{\rho_1^{\,3}}- \frac{\mu_2\,(x-\mu_1)}{\rho_2^{\,3}} + \omega^2\,x,$$ (1116)

$$\ddot{y} + 2\,\omega\,\dot{x} = - \frac{\mu_1\,y}{\rho_1^{\,3}}- \frac{\mu_2\,y}{\rho_2^{\,3}} + \omega^2\,y,$$ (1117)

$$\ddot{z} = - \frac{\mu_1\,z}{\rho_1^{\,3}}- \frac{\mu_2\,z}{\rho_2^{\,3}},$$ (1118)

which yield

$$\ddot{x} - 2\,\omega\,\dot{y} = -\frac{\partial U}{\partial x},$$ (1119)

$$\ddot{y} + 2\,\omega\,\dot{x} = -\frac{\partial U}{\partial y},$$ (1120)

$$\ddot{z} = -\frac{\partial U}{\partial z},$$ (1121)

where

$$U = - \frac{\mu_1}{\rho_1} - \frac{\mu_2}{\rho_2} - \frac{\omega^2}{2}\,(x^2+y^2)$$ (1122)

is the sum of the gravitational and centrifugal potentials.

Now, it follows from Equations (1119)-(1121) that

$$\ddot{x}\,\dot{x} - 2\,\omega\,\dot{x}\,\dot{y} = -\dot{x}\,\frac{\partial U}{\partial x},$$ (1123)

$$\ddot{y}\,\dot{y} + 2\,\omega\,\dot{x}\,\dot{y} = -\dot{y}\,\frac{\partial U}{\partial y},$$ (1124)

$$\ddot{z}\,\dot{z} = -\dot{z}\,\frac{\partial U}{\partial z}.$$ (1125)

Summing the above three equations, we obtain

$$\frac{d}{dt}\left[\frac{1}{2}\left(\dot{x}^2+\dot{y}^2+\dot{z}^2\right) + U\right] = 0.$$ (1126)

In other words,

$$C = - 2\,U - v^2$$ (1127)

is a constant of the motion, where $v^2=\dot{x}^2+\dot{y}^2+\dot{z}^2$. In fact, $C$ is the Jacobi integral introduced in Section 14.3 [it is easily demonstrated that Equations (1102) and (1127) are identical]. Note, finally, that the mass $m_3$ is restricted to regions in which

$$-2\,U \geq C,$$ (1128)

since $v^2$ is a positive definite quantity.