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Stability of Lagrange Points

We have seen that the five Lagrange points, to , are the equilibrium points of mass in the co-rotating frame. Let us now determine whether or not these equilibrium points are stable to small displacements.

Now, the equations of motion of mass in the co-rotating frame are specified in Equations (1056)-(1058). 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 3.2)

 (1096)

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

 (1097) (1098) (1099)

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
 (1100)

where , , , etc. However, by definition, at a Lagrange point, so the expansion simplifies to
 (1101)

Finally, substitution of Equations (1097)-(1099), and (1101) into the equations of - motion, (1056) and (1057), yields
 (1102) (1103)

since .

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

 (1104)

This equation only has a non-trivial solution if the determinant of the matrix is zero. Hence, we get
 (1105)

Now, it is convenient to define
 (1106) (1107) (1108) (1109)

where all terms are evaluated at the point . It thus follows that
 (1110) (1111) (1112)

Consider the co-linear Lagrange points, , , and . These all lie on the -axis, and are thus characterized by , , and . It follows, from the above equations, that and . Hence, , , and . Equation (1105) thus yields

 (1113)

where . Now, in order for a Lagrange point to be stable to small displacements, all four of the roots, , of Equation (1105) must be purely imaginary. This, in turn, implies that the two roots of the above equation,
 (1114)

must both be real and negative. Thus, the stability criterion is
 (1115)

Figure 56 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/lower signs corresponds to and , respectively. Equation (1105) thus yields

 (1116)

for both points, where . As before, the stability criterion is that the two roots of the above equation must both be real and negative. This is the case provided that , which yields the stability criterion
 (1117)

In unnormalized units, this criterion becomes
 (1118)

We thus conclude that the and Lagrange points are stable equilibrium points, in the co-rotating frame, provided that mass is less than about 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 will stay approximately ahead of mass , if it is orbiting the point, or behind, if it is orbiting the point--see Figure 55. This type of behavior has been observed in the Solar System. For instance, there is a sub-class of asteroids, known as the Trojan asteroids, which are trapped in the vicinity of the and points of the Sun-Jupiter system (which easily satisfies the above stability criterion), and consequently share Jupiter's orbit around the Sun, staying approximately ahead of, and behind, Jupiter, respectively. Furthermore, the and points of the Sun-Earth system are occupied by clouds of dust.

Next: Lunar Motion Up: The Three-Body Problem Previous: Zero-Velocity Surfaces
Richard Fitzpatrick 2011-03-31