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By copying  alphanumerical data of  Table [ f [t], { t,tmin,tmax}] // Table Form   output from CDF Player 9 how to  paste into Notepad  to get text in same Table Format ?  I get full Mathematica code characters. TIA.
[size=3][font=georgia, serif]May be you ran the Newton's eqilibrium equation formulation  including inverse square law already to verify angular momentum conservation, the ellipse orbit etc.. Suggest taking mass of earth = 1, radius =1 , or better in...
Easier and fun if you try the cycloid tautochrone at first. Radius of curvature is bisected by x-axis, it is not too complicated.  The property of same time period  for any boundary condition ( found by Christian  Huygens) can be verified. hth.
I like to use "Manipulate" command. Write coordinates using symbols and vary their values on a slider.
[font=georgia, serif]Try giving boundary condition at  a slightly higher e value,  say  e = 11.72, and trap all " the poles" , where denominator may be going  to zero. Or may be your solution itself there has a vertical tangent...or returning...
Did you try FindRoot ? Narasimham