G=M=1

Rp=1

N[((2 (Sqrt[(2/3)]))/3),32]

Manipulate[ContourPlot[{r^2/(3/2Rp)^2+z^2/Rp^2==1,G M/Sqrt[r^2+z^2]+1/2 Ω^2 r^2==G M/Rp},{r,-3/2Rp,3/2Rp},{z,- Rp, Rp},PlotPoints->64,ContourStyle->{{Black,DotDashed},{Red,Thick}},AspectRatio->2/3],{{Ω,(2/3)^(3/2)},0,((2 (Sqrt[(2/3)]))/3)}]

w1=Block[{Ω=((2 (Sqrt[(2/3)]))/3)},ContourPlot[{r^2/(3/2Rp)^2+z^2/Rp^2==1,G M/Sqrt[r^2+z^2]+1/2 Ω^2 r^2==G M/Rp},{r,-3/2Rp,3/2Rp},{z,- Rp, Rp},PlotPoints->64,ContourStyle->{{Black,DotDashed},{Red,Thick}},AspectRatio->2/3]]

Clear[Rp,Req]

Solve[{G M/Req+1/2 Ω^2 Req^2==G M/Rp,Ω^2 Req==G M/Req^2},{Req,Rp}]

Rp/Req/.%