Chapter 3, Section 3.12, Figure 3.27

We want to express Eq.(3.199) in a form where H((Subscript[θ, min])) is a function of ((Subscript[m, 1])+ (Subscript[m, 2]))/(Subscript[m, c]).  To do this we first take a fixed value for (Subscript[s, 180])/a  (see the text) and define the function const.

const[mass_] := ((3.32*10^(-4))^2 * (3/4) / mass)+1

Now define the function H(θ) as in Eq.(3.179) making sure we express θ in degrees.

H[θ_]:=1/Sin[(θ °)/2]-2*Cos[θ °]-2

Set up a table of values of the log of the mass ratio ( =(total mass of satellites)/(mass of planet)) and the value of (Subscript[θ, min]).  Here we are solving Eq.(3.199) to find (Subscript[θ, min])for a given value of mass.

θmin =
N[Table[{Log[10,mass],
 θ /. FindRoot[H[θ] == const[mass], 
      {θ, -Log[10,mass]}]},
{mass, 10^(-10), 10^(-8), 10^(-10)}]];

Plot the resulting curve.

p=
ListPlot[θmin, PlotJoined->True, Frame->True,
 PlotRange->{{-10,-8},{0,10}}]

The observed value of (Subscript[θ, min])is 5.64°.

θminobs=5.64;

Hence, from Eq.(3.199), the corresponding value of the log of the mass ratio is:

logmassobs = 
Log[10,mass /. FindRoot[
  H[θminobs] == const[mass],{mass,10^(-8)}]]

Create a graphics object for the observed data point ...

obs=
Graphics[
 {PointSize[0.03],Point[{logmassobs,θminobs}]}]

... and add it to the plot.

Show[p,obs]