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Next: Jacobi integral Up: Three-body problem Previous: Introduction


Circular restricted three-body problem

Consider an isolated dynamical system consisting of three gravitationally interacting point masses, $ m_1$ , $ m_2$ , and $ m_3$ . Suppose, however, that the third mass, $ m_3$ , is so much smaller than the other two that it has a negligible effect on their motion. Suppose, further, that the first two masses, $ m_1$ and $ m_2$ , execute circular orbits about their common center of mass. In the following, we shall examine this simplified problem, which is usually referred to as the circular restricted three-body problem. The problem under investigation has obvious applications to the solar system. For instance, the first two masses might represent the Sun and a planet (recall that a given planet and the Sun do indeed execute almost circular orbits about their common center of mass), whereas the third mass might represent an asteroid or a comet (asteroids and comets do indeed have much smaller masses than the Sun or any of the planets).

Figure 9.1: Circular restricted three-body problem.
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Let us define a Cartesian coordinate system $ \xi,\,\eta,\,\zeta$ in an inertial reference frame whose origin coincides with the center of mass, $ C$ , of the two orbiting masses, $ m_1$ and $ m_2$ . Furthermore, let the orbital plane of these masses coincide with the $ \xi$ -$ \eta$ plane, and let them both lie on the $ \xi$ -axis at time $ t=0$ . (See Figure 9.1.) Suppose that $ a$ is the constant distance between the two orbiting masses, $ r_1$ the constant distance between mass $ m_1$ and the origin, and $ r_2$ the constant distance between mass $ m_2$ and the origin. Moreover, let $ \omega$ be the constant orbital angular velocity. It follows, from Section 4.16, that

    $\displaystyle \omega^{\,2}$ $\displaystyle = \frac{G\,M}{a^{\,3}},$ (9.1)
and   $\displaystyle \frac{r_1}{r_2}$ $\displaystyle = \frac{m_2}{m_1},$     (9.2)

where $ M=m_1+m_2$ .

It is convenient to choose our unit of length such that $ a=1$ , and our unit of mass such that $ G\,M=1$ . It follows, from Equation (9.1), that $ \omega =1$ . However, we shall continue to retain $ \omega$ in our equations, for the sake of clarity. Let $ \mu_1=G\,m_1$ and $ \mu_2=G\,m_2=1-\mu_1$ . It is easily demonstrated that $ r_1 = \mu_2$ and $ r_2=1-r_1=\mu_1$ . Hence, the two orbiting masses, $ m_1$ and $ m_2$ , have position vectors

    $\displaystyle {\bf r}_1$ $\displaystyle = (\xi_1,\,\eta_1,\,0)=\left(-\mu_2\,\cos (\omega\, t),\,-\mu_2\,\sin(\omega \,t),\,0\right),$ (9.3)
and   $\displaystyle {\bf r}_2$ $\displaystyle =(\xi_2,\,\eta_2,\,0)= \left(\mu_1\,\cos( \omega\, t),\,\mu_1\,(\sin\omega \,t),\,0\right),$ (9.4)

respectively. (See Figure 9.1.) Let the third mass have position vector $ {\bf r} = (\xi,\,\eta,\,\zeta)$ . The Cartesian components of the equation of motion of this mass are thus

    $\displaystyle \skew{3}\ddot{\xi}$ $\displaystyle = -\mu_1\,\frac{(\xi-\xi_1)}{\rho_1^{\,3}} - \mu_2\,\frac{(\xi-\xi_2)}{\rho_2^{\,3}},$ (9.5)
    $\displaystyle \skew{3}\ddot{\eta}$ $\displaystyle = -\mu_1\,\frac{(\eta-\eta_1)}{\rho_1^{\,3}} - \mu_2\,\frac{(\eta-\eta_2)}{\rho_2^{\,3}},$ (9.6)
and   $\displaystyle \skew{3}\ddot{\zeta}$ $\displaystyle = -\mu_1\,\frac{\zeta}{\rho_1^{\,3}} - \mu_2\,\frac{\zeta}{\rho_2^{\,3}},$ (9.7)

where

    $\displaystyle \rho_1^{\,2}$ $\displaystyle = (\xi-\xi_1)^2+(\eta-\eta_1)^2 + \zeta^{\,2},$ (9.8)
and   $\displaystyle \rho_2^{\,2}$ $\displaystyle = (\xi-\xi_2)^2+(\eta-\eta_2)^2 + \zeta^{\,2}.$ (9.9)


next up previous
Next: Jacobi integral Up: Three-body problem Previous: Introduction
Richard Fitzpatrick 2016-03-31