Lagrange points

The Lagrange points satisfy in the rotating frame. It thus follows, from Equations (9.27)-(9.29), that the Lagrange points are the solutions of

(9.37) |

It is easily seen that

(9.38) |

Because the term in brackets is positive definite, we conclude that the only solution to the preceding equation is . Hence, all of the Lagrange points lie in the - plane.

If , it is readily demonstrated that

(9.39) |

where use has been made of the fact that . Hence, Equation (9.30) can also be written

The Lagrange points thus satisfy

which reduce to

One obvious solution of Equation (9.44) is , corresponding to a Lagrange point that lies on the -axis. It turns out that there are three such points: lies between masses and , lies to the right of mass , and lies to the left of mass . (See Figure 9.6.) At the point, we have and . Hence, from Equation (9.43),

Assuming that , we can find an approximate solution of Equation (9.45) by expanding in powers of :

This equation can be inverted to give

where

(9.48) |

is assumed to be a small parameter.

At the point, we have and . Hence, from Equation (9.43),

Again, expanding in powers of , we obtain

Finally, at the point, we have and . Hence, from Equation (9.43),

Let . Expanding in powers of , we obtain

where is assumed to be a small parameter.

Let us now search for Lagrange points that do not lie on the -axis. One obvious solution of Equations (9.41) and (9.42) is

(9.55) |

giving, from Equation (9.40),

(9.56) |

or

(9.57) |

because . The two solutions of this equation are

(9.58) | ||||||

and | (9.59) |

and specify the positions of the Lagrange points designated and . Note that the point and the masses and lie at the apexes of an equilateral triangle. The same is true for the point . We have now found all of the possible Lagrange points.

Figure 9.6 shows the positions of the two masses, and , and the five Lagrange points, to , calculated for the case where .