Exercises

- Demonstrate directly from Equations (9.5)–(9.7) and (9.11)–(9.13) that the
Jacobi integral , which is defined in Equation (9.10), is a constant of the motion in the circular
restricted three-body problem.
- A comet approaching the Sun in a parabolic orbit of perihelion distance and inclination (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 , eccentricity , and inclination . Demonstrate that
- A comet approaching the Sun in a hyperbolic orbit of perihelion distance and inclination (with respect to Jupiter's orbital plane), whose
asymptotes subtend an acute angle 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 , eccentricity , and inclination . Demonstrate that
- Let
be the coordinates of mass in the inertial frame, and
let
be the corresponding coordinates in the co-rotating frame. It
follows that
**is the column vector of the inertial coordinates, and**Show that

Demonstrate that

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

- 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).
- 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).
- 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).
- Derive Equations (9.62)–(9.66).
- 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 - plane of the co-rotating frame is
- 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 , where
. Here, and are the distances to
the masses and , respectively. Let
and
. Consider the limit
.
Show that
close to , where
and
, the parameter takes the value
. Likewise, show that close to , where
and
,
the parameter takes the value
. Finally, show that close to , where
and
, the parameter takes the value
. Hence, deduce that the three co-linear Lagrange
points are all linearly unstable. Demonstrate that, in the case of the point,
the growth-rate of the fastest growing instability is
. (Modified from Murray and Dermott 1999.)
- Consider the circular restricted three-body problem. Demonstrate that if , , is a valid trajectory for in the co-rotating frame then
and , , are also valid trajectories. Show that
if
is a valid trajectory when
(where
) then
is a valid trajectory
when
.
- Consider the circular restricted three-body problem (adopting the standard system of units). Suppose that
, so that the and 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 that
is confined to the - plane.
Let
where , . It is helpful to rotate the Cartesian axes through , so that
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

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

where*libration*.Finally, consider a Trojan asteroid trapped in the vicinity of the point of the Sun-Jupiter system. Demonstrate that the libration period of the asteroid (in the co-rotating frame) is approximately years, whereas its perihelion precession period (in the inertial frame) is approximately 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 .