Chapter 4, Section 4.14, Figure 4.19

Define the function Lorb that returns the orbital angular momentum of the Moon as a function of its semi-major axis a in units of Earth radii.  This is the first term on the right-hand side of Eq.(4.234).

Lorb[a_] :=  0.832*Sqrt[a/60.336]

Define a function Lspin that returns the spin angular momentum of the Earth as a function of the Moon's semi-major axis a.  This is effectively the second term on the right-hand side of Eq.(4.234).  Note how we calculate the spin period, T, and rearrange the conservation relation given in Eq.(4.234).

\(\(Lspin[a_] := 
        Module[{}, 
		t = 
            1/\((\((1 - Lorb[a])\)/\((24*0.168)\))\); 
		0.168*24/
              t];\)\)

Generate the plot for the evolution of the orbital angular momentum as a function of a.

p1 = 
      ListPlot[Table[{a, Lorb[a]}, {a, 0, 100, 1}], 
        PlotJoined -> True]

Generate the plot for the evolution of the spin angular momentum as a function of a.

p2 = 
      ListPlot[Table[{a, Lspin[a]}, {a, 0, 100, 1}], 
        PlotJoined -> True]

Combine the two together.

p3 = 
Show[p1, p2, Frame -> True, 
        PlotRange -> {{0, 90}, {0, 1}}]

For the points we first decide on the values of a to use.

avalue = {9.4, 31, 60, 86.54};

Now we define a function Tvalue to determine the period for a given semi-major axis.

Tvalue[a_]:=1/((1-Lorb[a])/(24*0.168))

Calculate the values of Lspin for each of our chosen values of a.

Lsvalue = Map[Lspin,avalue]

Calculate the values of Lorb for each of our chosen values of a.

Lovalue = Map[Lorb,avalue]

The spin periods are given by ...

period=
Transpose[
  avalue /.{x_ -> {x,Tvalue[x]}}];

.. or, in table form :

TableForm[period]

Express the final period in days.

period[[4,2]]/24

Create a graphics object for the spin points ...

spoints = 
Graphics[{PointSize[0.02],
 Map[Point,Transpose[{avalue,Lsvalue}]]}]

and the orbit points.

opoints = 
Graphics[{PointSize[0.02],
 Map[Point,Transpose[{avalue,Lovalue}]]}]

Display the final plot.

Show[p3,spoints,opoints]