Chapter 4.8, Problem 42E

The figure shows the sun located at the origin and the earth at the point (1, 0). (The unit here is the distance between the centers of the earth and the sun, called an astronomical unit: 1 AU ≈ 1.496 × 10⁸ km.) There are five locations L₁, L₂, L₃, L₄, and L₅ in this plane of rotation of the earth about the sun where a satellite remains motionless with respect to the earth because the forces acting on the satellite (including the gravitational attractions of the earth and the sun) balance each other. These locations are called libration points. (A solar research satellite has been placed at one of these libration points.) If m₁ is the mass of the sun, m₂ is the mass of the earth, and r = m₂/(m₁ + m₂), it turns out that the x-coordinate of L₁ is the unique root of the fifth-degree equation

$\begin{array}{c}p(x)=x^5-(2+r)x^4+(1+2r)x^3-(1-r)x^2\\ +2(1-r)x+r-1=0\end{array}$

and the x-coordinate of L₂ is the root of the equation

$p(x)-2r{x}^2=0$

Using the value r ≈ 3.04042 × 10⁻⁶, find the locations of the libration points (a) L₁ and (b) L₂.