Chapter 10, Section 10.3.2, Figure 10.8

We start by using Eq.(10.19) to define an approximation for the radius, r, as a function of the longitude, θc, in the corotating frame.

r[a_, e_,  m_,  θc_]  :=  
      a (1 - e*Cos[m*θc])

Now define a function, StreamLine, that will plot the streamline path of mode m.

\(StreamLine[m_] := 

      ParametricPlot[
{r[1, 0.2, m, θc]*
            Cos[θc], 
          r[1, 0.2, m, θc]*Sin[θc]}, {θc, 0, 
          2*Pi}, 
AspectRatio -> 1, 
        PlotRange -> {{\(-1.5\), 1.5}, {\(-1.5\), 1.5}}]\)

Plot the streamline for m = 0.  The result is a circle centred on the origin.  (See Fig.10.8a.)

StreamLine[0]

Plot the streamline for m = 1.  The result is an ellipse (or displaced circle).  (See Fig.10.8b.)

StreamLine[1]

Plot the streamline for m = 2.  The result is a centred ellipse or "two-lobed figure".  (See Fig.10.8c.)

StreamLine[2]

Plot the streamline for m = 7.  The result is seven-lobed figure.  (See Fig.10.8d.)

StreamLine[7]