s[e_,f_]=(2 ArcTan[Sqrt[(1-e)/(1+e)] Tan[f/2]]-e Sqrt[1-e^2] Sin[f]/(1+e Cos[f]))/(2 Pi)

Manipulate[ListPlot[{{1,.243635},{2,.253918},{3,.256396},{4,.246009},{1,Mod[s[e,Pi/2+q]-s[e,q],1]},{2,Mod[s[e,Pi+q]-s[e,Pi/2+q],1]},{3,Mod[s[e,-Pi/2+q]-s[e,-Pi+q],1]},{4,Mod[s[e,q]-s[e,-Pi/2+q],1]}},PlotRange->{{.5,4.25},All}],{e,0,.1},{q,-Pi,Pi}]