Hi Stephan,
you simply can let 'r' run from 0.00.. to 'a':
Manipulate[
ParametricPlot[{v*
t (1 + 1/(1 - r/a*Cos[w*t])), (r*Sin[w*t])/(1 -
r/a*Cos[w*t])}, {t, 0, 10},
PlotRange -> {{-.5, 200}, {-100, 100}}], {{r, 1}, 0.000000001, a,
Appearance -> "Labeled"}, {{a, 2}, 0.000000001, 100,
Appearance -> "Labeled"}, {{v, 0.1}, 0.000000001, 20,
Appearance -> "Labeled"}, {{w, 2 Pi}, 0.000000001, 20,
Appearance -> "Labeled"}]
Also please note the way of initialization of the parameters which was probably inteded by you originally.
Yours Henrik