c = 1; \[Epsilon]0 = 1; e = 1; m = 1;
re = 1/(4 \[Pi] \[Epsilon]0) e^2/(m c^2);
r0 = 20 re;
tmax = r0^3/(4 c re^2);
r[t_] := (r0^3 - 4 c re^2 t)^(1/3)
\[Omega][t_] := 1/(4 \[Pi] \[Epsilon]0) e^2/(m r[t]^3)
tmp = Integrate[\[Omega][t], t]
\[Theta][t_] := \[Pi] Log[500] - \[Pi] Log[500 - \[Pi] t]
p0 = Table[
   Show[
    ParametricPlot[ r[t] {Cos[\[Theta][t]], Sin[\[Theta][t]]}, {t, 0, \[Tau]}, PlotStyle -> {Orange}, Axes -> False, PlotPoints -> 10^4, PlotRange -> r0 {{-1.1, 1.1}, {-1.1, 1.1}}]
    ,
    Graphics[{
      Disk[{0, 0}, 0.05],
      Disk[r[\[Tau]] {Cos[\[Theta][\[Tau]]], Sin[\[Theta][\[Tau]]]}, 0.02 ]
      }]
    ]
   , {\[Tau], tmax/500, tmax, tmax/500}];
ListAnimate[p0]