Lagrange Points of a Binary Star System

First we define some initialization constants

q =  .4;
μ = M1 q/(M1  + q M1);
a = 1;
M1 = 1;
G = 1;
Porb = Sqrt[4 (π^2) (a^3)/(G (1 + q))];
Ωorb = 2 π/Porb;
M2 = q M1;

Next, we define the potential

Φ[x_, y_, 
        z_] := -((G M1)/(Sqrt[(((x + q/(1 + q)))^2) + (y^2) + (z^2)])) - ((G M1 q)/(Sqrt[(((x - a + q/(1 + q)))^2) + (y^2) + (z^2)])) - 
        (1/2) Ω(orb^2) ((((x - μ  + q/(1 + q) a))^2) + (y^2))

Here, we determine the location of several critical points, namely the Lagrange points

deriv = D[Φ[x, 0, 0], x];
posAtL1 = FindRoot[deriv == 0, {x,  .3}];
ΦAtL1 = Φ[x /. posAtL1, 0, 0];
posAtL2 = FindRoot[deriv == 0, {x, 1.5}];
ΦAtL2 = Φ[x /. posAtL2, 0, 0];
posAtL3 = FindRoot[deriv == 0, {x, -1}];
ΦAtL3 = Φ[x /. posAtL3, 0, 0];
posAtL4 = 
        FindRoot[
          D[Φ[x /. posAtL1, y, 0], y] == 0, {y, 1}];
ΦAtL4 = Φ[x /. posAtL1, y /. posAtL4, 
          0];

Here, we define a color function

ccolor = (_RGBColor | _Hue | _GrayLevel | _CMYKColor);

ablendsequence[c : {ccolor, ccolor .. }, u_?NumericQ] /; 
        And[SameQ[Head[#] & /@ c], 
          SameQ[Length[#] & /@ c]] := Head[
              First[c]] @@ (({1 - #, #} &[
                    FractionalPart[#]]) . (List @@@ 
                    Take[RotateLeft[c, Floor[#]], 2])) &[
        u*(-1 + Length[c])]

Finally, we construct the graphic

fs = 14;

\(gr = 
      ContourPlot[Φ[x, y, 0], {x, \(-1.5\), 1.5}, {y, \(-1.5\), 
          1.5}, Contours -> {ΦAtL1 - 
              1, ΦAtL1 -  .5, ΦAtL1, ΦAtL2, ΦAtL3, ΦAtL4, ΦAtL4 -  .1}, ImageSize -> {500, 500}, Frame -> False, 
        PlotPoints -> 200, ContourStyle -> {Thickness[ .003]}, 
        ColorFunction -> \((ablendsequence[{RGBColor[0, 0,  .5], 
                  RGBColor[ .3,  .5,  .3], Brown, 
                  RGBColor[1, 1, 1]}, #] &)\), 
        Epilog -> {White, 
            Text[StyleForm["L1", FontSize -> fs, 
                FontFamily -> "Arial", 
                FontWeight -> "Bold"], {x /. posAtL1, \(- .1\)}], 
            Text[StyleForm["L2", FontSize -> fs, 
                FontFamily -> "Arial", 
                FontWeight -> "Bold"], {\((x /. posAtL2)\) +  .1, 0}], 
            Text[StyleForm["L3", FontSize -> fs, 
                FontFamily -> "Arial", 
                FontWeight -> "Bold"], {\((x /. posAtL3)\) -  .1, 0}], 
            Black, Text[
              StyleForm["L4", FontSize -> fs, 
                FontFamily -> "Arial", 
                FontWeight -> "Bold"], {x /. 
                  posAtL1, \((y /. posAtL4)\) +  .1}, {0, \(-1\)}], 
            Text[StyleForm["L5", FontSize -> fs, 
                FontFamily -> "Arial", 
                FontWeight -> "Bold"], {x /. 
                  posAtL1, \(-\((y /. posAtL4)\)\) -  .1}, {0, 
                1}]}]\)