Chapter 3, Section 3.13, Figure 3.28

The zero velocity curves for Hill's equations are given by Eq.(3.205) with \(TraditionalForm\`\(\(x\& . \)\(\ \)\)\)= \(TraditionalForm\`\(\(y\& . \)\(\ \)\)\)= 0.  We define a function Chill that we can then plot.

Chill[x_,y_,μ_] := 3x^2 + 2 μ /Sqrt[x^2+y^2];

We want to make sure that the critical curve gets included.  This is given by Eq.(3.208) with ζ = (3^(4/3)).  We are going to use a mass ratio μ = 0.1.

μval=0.1;

crit=3^(4/3)μval^(2/3)

Now proceed with the contour plot.

ContourPlot[Chill[x,y,0.1],{y,-1.5,1.5},{x,-0.8,0.8},
 AspectRatio->Automatic, ContourShading->False,
 PlotPoints->200, ContourShading->Automatic,
 Contours->{1.8,1.6,1.4,1.2,1.0,crit,
            0.8,0.6,0.4,0.2,0.0}]