ClearAll["Global`*"]
Off[Solve::ifun];Off[Plot::plln];Off[Solve::ratnz];
a=1;
b=(1/2)*a;
p=((b^2)/a);
e=(Sqrt[1-((((b/a)))^2)]);
ω = -1;
Ω =1;
(Subscript[ϕ, 0])=(Pi/2);
ee=.05;
R=(a/(2*ee));
x0[t_]=R*Cos[Ω*t];
y0[t_]=R*Sin[Ω*t];
(*r[t_]=(p/(1+e*Cos[ω*t]));*)
r[t_]=(1/(Sqrt[(((((Cos[ω*t+(Subscript[ϕ, 0])])/a)))^2)+(((((Sin[ω*t+(Subscript[ϕ, 0])])/b)))^2)]));
orbs[i_] := If[Mod[ω*i,1]==0,i,orbs[i+1]]
T=orbs[1]*(2*Pi);
xp[t_] = r[t]*(-Sin[ω*t+(Subscript[ϕ, 0])]);
yp[t_] = r[t]*Cos[ω*t+(Subscript[ϕ, 0])];
ifx[t_] = xp[t]*Cos[Ω*t]-yp[t]*Sin[Ω*t];
ify[t_] = xp[t]*Sin[Ω*t]+yp[t]*Cos[Ω*t];
planet[t_] = {ifx[t]+x0[t],ify[t]+y0[t]};
dt=(Pi/100);
orbit = Table[planet[t],{t,0,T,dt}];
ϵ=.05;
dAe[t_] = Plot[{},{x,Min[-a,-a+x0[t]]-ϵ,Max[a+x0[t],a]+ϵ},AspectRatio->Automatic,PlotRange->{{-b-R-ϵ,b+R+ϵ},{-b-R-ϵ,b+R+ϵ}},Axes->False,Background->Black];
Animate[Show[{dAe[t]},Graphics[{Thickness[.008],Dashed,Blue,Circle[{0,0},{R,R}]}],Graphics[{Thickness[.007],Dashed,Purple,Line[{{x0[t],y0[t]},planet[t]}]}],
Graphics[{Thickness[.007],Dashed,Blue,Line[{{0,0},{x0[t],y0[t]}}]}],
Graphics[{Thickness[.01],Opacity[.9],Darker[Green],Circle[{-R*ee,0},{R,R*(Sqrt[1-(ee^2)])}]}],
Graphics[{Thickness[.01],Opacity[.3],White,Line[orbit[[1;;Length[orbit]]]]}],Graphics[{Thickness[.009],Opacity[.9],Pink,Line[orbit[[1;;Floor[(t/dt)]+1]]]}],Graphics[Rotate[{Thickness[.009],Dashed,Purple,Circle[{x0[t],y0[t]},{b,a}]},Ω*t]],Graphics[{PointSize[.04],Darker[Red],Point[planet[t]]}],Graphics[{PointSize[.03],Opacity[.7],Lighter[Blue],Point[{x0[t],y0[t]}]}],Graphics[{PointSize[.07],Yellow,Point[{0,0}]}]],{t,0,T},DefaultDuration->orbs[1]*5, RefreshRate->40]

Export["Guided_Center_005.gif",Table[Panel[Show[{dAe[t]},Graphics[{Thickness[.008],Dashed,Blue,Circle[{0,0},{R,R}]}],Graphics[{Thickness[.007],Dashed,Purple,Line[{{x0[t],y0[t]},planet[t]}]}],
Graphics[{Thickness[.007],Dashed,Blue,Line[{{0,0},{x0[t],y0[t]}}]}],
Graphics[{Thickness[.01],Opacity[.9],Darker[Green],Circle[{-R*ee,0},{R,R*(Sqrt[1-(ee^2)])}]}],
Graphics[{Thickness[.01],Opacity[.3],White,Line[orbit[[1;;Length[orbit]]]]}],Graphics[{Thickness[.009],Opacity[.9],Pink,Line[orbit[[1;;Floor[(t/dt)]+1]]]}],Graphics[Rotate[{Thickness[.009],Dashed,Purple,Circle[{x0[t],y0[t]},{b,a}]},Ω*t]],Graphics[{PointSize[.04],Darker[Red],Point[planet[t]]}],Graphics[{PointSize[.03],Opacity[.7],Lighter[Blue],Point[{x0[t],y0[t]}]}],Graphics[{PointSize[.07],Yellow,Point[{0,0}]}]]],{t,0,T,dt}]]