Exercises

1. Demonstrate directly from Equations (9.5)–(9.7) and (9.11)–(9.13) that the Jacobi integral $C$, which is defined in Equation (9.10), is a constant of the motion in the circular restricted three-body problem.

2. A comet approaching the Sun in a parabolic orbit of perihelion distance $r_p$ and inclination $I$ (with respect to Jupiter's orbital plane) is disturbed by a close encounter with Jupiter such that its orbit is converted into an ellipse of major radius $a'$, eccentricity $e'$, and inclination $I'$. Demonstrate that
   $$\sqrt{2\,r_p}\,\cos I \simeq \frac{1}{2\,a'} + \sqrt{(1-e'^{\,2})\,a'}\,\cos I',$$
   where all lengths are normalized to the major radius of Jupiter.

3. A comet approaching the Sun in a hyperbolic orbit of perihelion distance $r_p$ and inclination $I$ (with respect to Jupiter's orbital plane), whose asymptotes subtend an acute angle $\phi$ with respect to one another, is disturbed by a close encounter with Jupiter such that its orbit is converted into an ellipse of major radius $a'$, eccentricity $e'$, and inclination $I'$. Demonstrate that
   $$\frac{1}{2\,r_p} \,(1-e)+\sqrt{(1+e)\,r_p}\,\cos I \simeq \frac{1}{2\,a'} + \sqrt{(1-e'^{\,2})\,a'}\,\cos I',$$
   where $e=\sec(\phi/2)$, and all lengths are normalized to the major radius of Jupiter.

4. Let $(\xi,\,\eta,\,\zeta)$ be the coordinates of mass $m_3$ in the inertial frame, and let $(x,\,y,\,z)$ be the corresponding coordinates in the co-rotating frame. It follows that
   $${\bf x} = {\bf A}\,$$$\xi$$$,$$
   where ${\bf x}$ is the column vector of the co-rotating coordinates, $\xi$ is the column vector of the inertial coordinates, and
   $${\bf A} = \left(\begin{array}{ccc} \cos \omega t, & \sin \omega t,&0\\ [0.5ex] -\sin\omega t,&\cos\omega t,&0\\ [0.5ex] 0,&0,&1\end{array}\right).$$
   Demonstrate that ${\bf A}^T\,{\bf A} = {\bf 1}$, where $^T$ denotes a transpose, and
   $${\bf 1} = \left(\begin{array}{ccc} 1, & 0, &0\\ [0.5ex] 0,&1,&0\\ [0.5ex] 0,&0,&1\end{array}\right).$$
   Hence, deduce that ${\bf x}^T\,{\bf x}=$   $\xi$$^T\,$$\xi$, or $x^{\,2}+y^{\,2}=\xi^{\,2}+\eta^{\,2}$.

   Show that

   $$\skew{3}\dot{\bf x} = {\bf A}\,\skew{3}\dot{\mbox{\boldmath $\xi$}} - \omega\,{\bf B}\,\mbox{\boldmath $\xi$},$$
   where $\dot{\bf x}$ is the column vector of the time derivatives of the co-rotating coordinates, $\skew{3}\dot{\mbox{\boldmath $\xi$}}$ is the column vector of the time derivatives of the inertial coordinates, and
   $${\bf B} = \left(\begin{array}{ccc} \sin \omega t, & -\cos \omega t&0\\ [0.5ex] \cos\omega t,&\sin\omega t&0\\ [0.5ex] 0,&0,&0\end{array}\right).$$

   Demonstrate that

   $$\dot{{\bf x}}^T\,\dot{{\bf x}} = \skew{3}\dot{\mbox{\boldmath $\x... ...xi$}+\omega^{\,2}\,\mbox{\boldmath $\xi$}^T\,{\bf 1}'\,\mbox{\boldmath $\xi$},$$
   where
   $${\bf 1}' = \left(\begin{array}{ccc} 1, & 0, &0\\ [0.5ex] 0,&1,&0\\ [0.5ex] 0,&0,&0\end{array}\right),$$
   and
   $${\bf C} = \left(\begin{array}{ccc} 0,& 1, &0\\ [0.5ex] -1,&0,&0\\ [0.5ex] 0,&0,&0\end{array}\right).$$
   Hence, deduce that
   $$\skew{3}\dot{x}^{\,2}+\skew{3}\dot{y}^{\,2}+\skew{3}\dot{z}^{\,2}... ...w{3}\dot{\eta} -\eta\,\skew{3}\dot{\xi})+\omega^{\,2}\,(\xi^{\,2}+\eta^{\,2}).$$

   Finally, show that the Jacobi constant in the co-rotating frame,

   $$C = 2\left(\frac{\mu_1}{\rho_1}+\frac{\mu_2}{\rho_2}\right) + \om... ...2}+y^{\,2})-\skew{3}\dot{x}^{\,2}-\skew{3}\dot{y}^{\,2}-\skew{3}\dot{z}^{\,2},$$
   transforms to
   $$C = 2\left(\frac{\mu_1}{\rho_1}+\frac{\mu_2}{\rho_2}\right) +2\,\... ...i})-\skew{3}\dot{\xi}^{\,2}-\skew{3}\dot{\eta}^{\,2}-\skew{3}\dot{\zeta}^{\,2}$$
   in the inertial frame.

5. Derive the first three terms (on the right-hand side) of Equation (9.46) from Equation (9.45), and the first three terms of Equation (9.47) from Equation (9.46).

6. Derive the first three terms of Equation (9.50) from Equation (9.49), and the first three terms of Equation (9.51) from Equation (9.50).

7. Derive the first two terms of Equation (9.53) from Equation (9.52), and the first two terms of Equation (9.54) from Equation (9.53).

8. Derive Equations (9.62)–(9.66).

9. Employing the standard system of units for the circular restricted three-body problem, the equation defining the location of a zero-velocity curve in the $x$-$y$ plane of the co-rotating frame is
   $$C = x^{\,2}+y^{\,2} + 2\left(\frac{\mu_1}{\rho_1}+\frac{\mu_2}{\rho_2}\right),$$
   where $C$ is the value of the Jacobi constant, and $\rho_1=[(x+\mu_2)^2+y^{\,2}]^{1/2}$ and $\rho_2=[(x-\mu_1)^2+y^{\,2}]^{1/2}$ are the distances to the primary and secondary masses, respectively. The critical zero-velocity curve that passes through the $L_3$ point, when $C\simeq 3+\mu_2$, has two branches. Defining polar coordinates such that $x=r\,\cos \theta$ and $y=r\,\sin \theta$, show that when $\mu_2\ll 1$ the branches intersect the unit circle $r=1$ at $\theta=\pi$ and $\theta=\pm 23.9^\circ$. (Modified from Murray and Dermott 1999.)

10. In the circular restricted three-body problem (employing the standard system of units) the condition for the three co-linear Lagrange points to be linearly unstable is $A>1$, where $A=\mu_1/\rho_1^{\,3}+\mu_2/\rho_2^{\,3}$. Here, $\rho_1$ and $\rho_2$ are the distances to the masses $\mu_1$ and $\mu _2$, respectively. Let $\alpha = (\mu_2/3\,\mu_1)^{1/3}$ and $\beta=(7/12)\,(\mu_2/\mu_1)$. Consider the limit $\mu_2\rightarrow 0$. Show that close to $L_1$, where $\rho_2\simeq \alpha-\alpha^{\,2}/3$ and $\rho_1=1-\rho_2$, the parameter $A$ takes the value $A\simeq 4+6\,\alpha+{\cal O}(\alpha^{\,2})$. Likewise, show that close to $L_2$, where $\rho_2\simeq \alpha+\alpha^{\,2}/3$ and $\rho_1=1+\rho_2$, the parameter $A$ takes the value $4-6\,\alpha+ {\cal O}(\alpha^{\,2})$. Finally, show that close to $L_3$, where $\rho_1\simeq1-\beta$ and $\rho_2=1+\rho_1$, the parameter $A$ takes the value $1+(3/2)\,\beta+{\cal O}(\beta^{\,2})$. Hence, deduce that the three co-linear Lagrange points are all linearly unstable. Demonstrate that, in the case of the $L_3$ point, the growth-rate of the fastest growing instability is $\gamma\simeq \left[\sqrt{(21/8)\,\mu_2} + {\cal O}(\mu_2)\right]\omega$. (Modified from Murray and Dermott 1999.)

11. Consider the circular restricted three-body problem. Demonstrate that if $[x(t)$, $y(t)$, $z(t)]$ is a valid trajectory for $m_3$ in the co-rotating frame then $[x(t),\,y(t),\,-z(t)]$ and $[x(-t)$, $-y(-t)$, $z(-t)]$ are also valid trajectories. Show that if $[x(t),\,y(t),\,z(t)]$ is a valid trajectory when $\mu_2=\zeta$ (where $0\leq \zeta\leq 1$) then $[-x(t),\,-y(t),\,z(t)]$ is a valid trajectory when $\mu_2=1-\zeta$.

12. Consider the circular restricted three-body problem (adopting the standard system of units). Suppose that $\mu_2\ll 0.0385$, so that the $L_4$ and $L_5$ points are stable equilibrium points (in the co-rotating frame) for the tertiary mass. Consider motion (in the co-rotating frame) of the tertiary mass in the vicinity of $L_4$ that is confined to the $x$-$y$ plane. Let

    $$x = \frac{1}{2}-\mu_2 + \delta x,$$

    $$y = \frac{\sqrt{3}}{2}+ \delta y,$$

    where $\vert\delta x\vert$, $\vert\delta y\vert\ll 1$. It is helpful to rotate the Cartesian axes through $30^\circ$, so that
    $$\left(\begin{array}{c}\delta x\\ [0.5ex]\delta y\end{array}\right... ...}\right) \left(\begin{array}{c}\delta x'\\ [0.5ex]\delta y'\end{array}\right).$$
    Thus, $\delta x'$ parameterizes displacements from $L_4$ that are tangential to the unit circle on which the mass $m_2$, and the $L_3$, $L_4$, and $L_5$ points, lie, whereas $\delta y'$ parameterizes radial displacements. Writing $\delta x'(t)=\delta x_0'\,\exp(\gamma\,t)$ and $\delta y'(t)= \delta y_0' \,\exp(\gamma\,t)$, where $\delta x_0'$, $\delta y_0'$, $\gamma$ are constants, demonstrate that
    $$\left(\begin{array}{cc}\sqrt{3}\,\gamma^{\,2}/2+\gamma - 3\sqrt{3... ...a y_0'\end{array}\right)=\left(\begin{array}{c}0\\ [0.5ex]0\end{array}\right),$$
    and, hence, that
    $$\gamma^{\,4} + \gamma^{\,2} + \frac{27}{4}\,\mu_2\,(1-\mu_2)= 0.$$

    Show that the general solution to the preceding dispersion relation is a linear combination of two normal modes of oscillation, and that the higher frequency mode takes the form

    $$\delta x'(t) \simeq -2\,e\,\sin(\omega_+\,t-\phi_+),$$

    $$\delta y'(t) \simeq -e\,\cos(\omega_+\,t-\phi_+),$$

    where
    $$\omega_+ \simeq \left(1-\frac{27}{8}\,\mu_2\right)\omega,$$
    and $e$, $\phi_+$ are arbitrary constants. Demonstrate that, in the original inertial reference frame, the addition of the preceding normal mode to the unperturbed orbit of the tertiary mass (in the limit $0<e\ll 1$) converts a circular orbit into a Keplerian ellipse of eccentricity $e$. In addition, show that the perihelion point of the new orbit precesses (in the direction of the orbital motion) at the rate
    $$\skew{5}\dot{\varpi} = \frac{27}{8}\,\mu_2\, \omega.$$

    Demonstrate that (in the co-rotating reference frame) the second normal mode takes the form

    $$\delta x'(t) \simeq d\,\sin(\omega_-\,t-\phi_-),$$

    $$\delta y'(t) \simeq -\sqrt{3\,\mu_2}\,d\,\sin(\omega_-\,t-\phi_-),$$

    where
    $$\omega_- \simeq \frac{3}{2}\sqrt{3\,\mu_2}\,\omega,$$
    and $d$, $\phi_-$ are arbitrary constants. This type of motion, which entails relatively small amplitude radial oscillations, combined with much larger amplitude tangential oscillations, is known as libration.

    Finally, consider a Trojan asteroid trapped in the vicinity of the $L_4$ point of the Sun-Jupiter system. Demonstrate that the libration period of the asteroid (in the co-rotating frame) is approximately $148$ years, whereas its perihelion precession period (in the inertial frame) is approximately $3,690$ years. Show that, in the co-rotating frame, the libration orbit is an ellipse that is elongated in the direction of the tangent to the Jovian orbit in the ratio $18.7:1$.