ClearAll[aQ,vQ,zQ,Ex,Ez,By,Ax,Az,t,T,Retard,ans]

This notebook numerically solves the equations at the end of the Appendix in the paper "The Path of an Electron Orbiting and Accelerating Nucleus."  q, π, c, (Subscript[ϵ, 0]), and (Subscript[μ, 0]) were set equal to 1 and the charge on the electron was set equal to -1 leaving the mass of the electron, me, as the sole remaining parameter.  The value of me shown below was chosen to make the radius of the electron's orbit 1.0 and its period 10 when the nucleus is at rest.

me:=((Sqrt[625-25 (π^2)])/(π^2))

Select a value for  the terminal velocity of the nuclesu, 0 < v∞ < 1.

v∞=0.995

τ:=250

aQ[t_]:=((2v∞)/τ) (E^((-t)/τ))(1 -(E^((-t)/τ)) )

vQ[t_]=(∫_0^t)aQ[x]\[DifferentialD]x

zQ[t_]=(∫_0^t)vQ[x]\[DifferentialD]x

For times larger than 3000, the nucleus is moving with a speed within 1/1000 of a percent of v∞.  The graphs for the orbit of the electron are drawn using times greater than or equal to 3000.

((v∞-vQ[3000])/(v∞))100

The equation below connects the retarded time T to the regular time t.

Retard:=(∂_t) T[t]-((t-T[t]-x'[t]x[t]-z'[t](z[t]-zQ[T[t]]))/(t-T[t]-vQ[T[t]](z[t]-zQ[T[t]])))

The relevant electric and magnetic fiields are given below.

Ex[t_]:=((x[t])/(((1-((vQ[T[t]](z[t]-zQ[T[t]]))/(t-T[t]))))^3))((1-((vQ[T[t]])^2)+aQ[T[t]](z[t]-zQ[T[t]]))/(((t-T[t]))^3))

Ez[t_]:=(1/(((1-((vQ[T[t]](z[t]-zQ[T[t]]))/(t-T[t]))))^3)){(1/(((t-T[t]))^3))((z[t]-zQ[T[t]]-vQ[T[t]](t-T[t]))(1-((vQ[T[t]])^2))+aQ[T[t]] (((z[t]-zQ[T[t]]))^2))-((aQ[T[t]])/(t-T[t]))}

By[t_]:=((x[t])/(((1-((vQ[T[t]](z[t]-zQ[T[t]]))/(t-T[t]))))^3)){((vQ[T[t]](1-((vQ[T[t]])^2)))/(((t-T[t]))^3))+((aQ[T[t]])/(((t-T[t]))^2))}

These last two equations are the modified equations of motion in the x and z-directions.

Ax:=x''[t](1-((z'[t])^2)) +((x'[t]z'[t])/(1-((x'[t])^2)))(-x''[t]z'[t]x'[t]-(((Ez[t]+x'[t] By[t]))/me)(((1-((x'[t])^2)-((z'[t])^2)))^(3/2)))+(((Ex[t]-z'[t] By[t]))/me)(((1-((x'[t])^2)-((z'[t])^2)))^(3/2))

Az:=z''[t](1-((x'[t])^2)) +((x'[t]z'[t])/(1-((z'[t])^2)))(-z''[t]z'[t]x'[t]-(((Ex[t]-z'[t] By[t]))/me)(((1-((x'[t])^2)-((z'[t])^2)))^(3/2)))+(((Ez[t]+x'[t] By[t]))/me)(((1-((x'[t])^2)-((z'[t])^2)))^(3/2))

ans=NDSolve[{Retard==0,Ax==0,Az==0,T[1]==0,x[1]==1,x'[1]==0,z[1]==0,z'[1]==(π/5)},{T,x,z},{t,1,4000},MaxSteps->100000]

The graph below shows the orbit predicted by the numerical solution.  It is both the orbit "seen" in the lab and also the orbit from the perspective of rocket observers on the assumption that the lab and rocket coordinates are connected by the Galilean transformation equations.  Note that the period was chosen to give exactly one period of motion.

ParametricPlot[{z[t]-zQ[t],x[t]}/.ans,{t,3000,3000+(10/(Sqrt[1-((v∞)^2)]))},AspectRatio->Automatic,PlotLabel->"Figure 3", AxesLabel->{Z-(Subscript[z, q]),X}]

The Manipulate command allows the shape of the orbit to be viewed at different times.  Note that the time interval used is the period at the terminal speed, v∞.  Consequently, for early times and small speeds, the electron makes more than one orbit.

Manipulate[ParametricPlot[{z[t]-zQ[t],x[t]}/.ans,{t,s,s+(10/(Sqrt[1-((v∞)^2)]))}],{s,1,2000}]