For mathematical completeness I must say that +, - ta(P) both signs ara analytic solutions. That logaritmic plot is such that it shows only positive side of this parametric representation of proper time. If you want to find negative part in plotting you must use plottings with abs(ta(P). That last logarithmic plotting command shows log10(ta(P)) axis curve beteen 28 to 16 and u(P)=1/r(P) axis between 0.5*10^(-15) to 3.15*10^(-13). Strong oscillation is in proper time between 20.4 and 16, u(P) is between 10^(-3) and 3.15*10^(-13). Mathematically this is due those three elliptic integrals (1., 2., and 3. kinds). Their shape is triangular if you plot them separately. 0.5*10(-15) corresponds 20*10^9 km. 3.15*10^(-13) corresponds 3.17*10^7 km. I have used ln instead log10, so in my e^28 corresponds to 45830 years, e^20.4 corresponds to 23 years, e^19 corresponds to 5.7 years, e^16 corresponds to 103 days. Largest proper time oscillation seems to be 5.4 years. Oscillation happens from 100 million km to 32 million km. Best Regards Hannu Poropudas. I think thst this analytic solution could be IMPORTANT!