Stability of Lagrange points

The equations of motion of mass in the co-rotating frame are specified in Equations (9.27)-(9.29). Note that the motion in the - plane is complicated by presence of the Coriolis acceleration. However, the motion parallel to the -axis simply corresponds to motion in the potential . Hence, the condition for the stability of the Lagrange points (which all lie at ) to small displacements parallel to the -axis is simply (see Section 2.7)

(9.67) |

This condition is satisfied everywhere in the - plane. Hence, the Lagrange points are all stable to small displacements parallel to the -axis. It thus remains to investigate their stability to small displacements lying within the - plane.

Suppose that a Lagrange point is situated in the - plane at coordinates . Let us consider small amplitude - motion in the vicinity of this point by writing

where and are infinitesimal. Expanding about the Lagrange point as a Taylor series, and retaining terms up to second order in small quantities, we obtain

(9.71) |

where , , , and so on. However, by definition, at a Lagrange point, so the expansion simplifies to

Finally, substituting Equations (9.68)-(9.70), and (9.72) into the equations of - motion, (9.27) and (9.28), and only retaining terms up to first order in small quantities, we get

(9.73) | ||||||

and | (9.74) |

as .

Let us search for a solution of the preceding pair of equations of the form and . We obtain

(9.75) |

This equation only has a nontrivial solution if the determinant of the matrix is zero. Hence, we get

It is convenient to define

(9.77) | ||||||

(9.78) | ||||||

(9.79) | ||||||

and | (9.80) |

where all terms are evaluated at the point . It thus follows that

(9.81) | ||||||

(9.82) | ||||||

and | (9.83) |

Consider the co-linear Lagrange points, , , and . These all lie on the -axis, and are thus characterized by , , and . It follows, from the preceding equations, that and . Hence, , , and . Equation (9.76) thus yields

(9.84) |

where . For a Lagrange point to be stable to small displacements, all four of the roots, , of Equation (9.76) must be purely imaginary. This, in turn, implies that the two roots of the preceding equation,

(9.85) |

must both be real and negative. Thus, the stability criterion is

(9.86) |

Figure 9.13 shows calculated at the three co-linear Lagrange points as a function of , for all allowed values of this parameter (i.e., ). It can be seen that is always greater than unity for all three points. Hence, we conclude that the co-linear Lagrange points, , , and , are intrinsically unstable equilibrium points in the co-rotating frame.

Let us now consider the triangular Lagrange points, and . These points are characterized by . It follows that , , , and . Hence, , , and , where the upper and lower signs corresponds to and , respectively. Equation (9.76) thus yields

(9.87) |

for both points, where . As before, the stability criterion is that the two roots of the preceding equation must both be real and negative. This is the case provided that , which yields the stability criterion

(9.88) |

In unnormalized units, this criterion becomes

We thus conclude that the and Lagrange points are stable equilibrium points, in the co-rotating frame, provided that mass is less than about percent of mass . If this is the case then mass can orbit around these points indefinitely. In the inertial frame, the mass will share the orbit of mass about mass , but it will stay approximately ahead of mass if it is orbiting the point, or behind if it is orbiting the point. (See Figure 9.12.) This type of behavior has been observed in the solar system. For instance, there is a subclass of asteroids, known as the Trojan asteroids, that are trapped in the vicinity of the and points of the Sun-Jupiter system [which easily satisfies the stability criterion in Equation (9.89)], and consequently share Jupiter's orbit around the Sun, staying approximately ahead of and behind, Jupiter, respectively. These asteroids are shown in Figures 9.14 and 9.15. The Sun-Jupiter system is not the only dynamical system in the solar system that possess Trojan asteroids trapped in the vicinity of its and points. In fact, the Sun-Neptune system has eight known Trojan asteroids, the Sun-Mars system has four, and the Sun-Earth system has one (designated 2010 TK7) trapped at the point. The and points of the Sun-Earth system are also observed to trap clouds of interplanetary dust.