comet[eccentricity_]:=Module[{e = eccentricity},
velocity = NDSolve[{v'[t]*Power[1+e, 2]/Power[1+Cos[v[t]]*e, 2] == 1, v[0]==0},
v, {1, -1000, 1000}][[1, 1, 2]];

y[tt_, ec_]:=(1+ec)*Cos[velocity[tt]]/(1+ec*Cos[velocity[tt]]);

y[tt_, ec_]:=(1+ec)*Sin[velocity[tt]]/(1+ec*Cos[velocity[tt]]);

Animate[Graphics[{{Yellow, Disk[{0,0}, 0.4]}, PointSize[0.005],
Point@Table[{x[tt, e],y[tt,e]},{tt, -t, t, 1}]}, Axes->True,
PlotRange->{{-20, 1.5},{-10,10}}, ImageSize->{500, 350}], {t, 1, 100}]]
comet[0.5]