On the stability of corotating and counter rotating P-type orbits around stellar binaries: a numerical study

We here revisit the essential problem of dynamical stability of planetary orbits around stellar binaries. We build on the coplanar three-body system of the Dvorak(1986), extending his stability diagram to both corotation and counter rotation of P-type orbits. His stability diagram express the change of stability across the gap between upper(UCO) and lower(LCO) critical orbits. By radius, this gap has a width of about 8$\%$ in the corotation case and 24$\%$ in the counter rotation case. As the gap of the second lies below the first, counter rotation is more stable, yet by width it is more chaotic. The gap between UCO and LCO follows a transition radius $r^+_g = 2.39 + 2.53e - 1.40 e^2$ and $r^-_g = 0.92 - 2.47e$ for corotation and respectively, counter rotation of the third body (the planet). Our $r^+_g$ agrees with the same of Dvorak to within 0.35$\%$. As a result, we discover $r^+_g/r^-_g \lesssim 2.57$ for all $e$. Around dim binaries, therefore, a relatively close in habitable zone may still be populated with planets on counter rotating orbits. The accurate numerical results presented here based on adaptive integration using MATLAB ODE45 may also serve as a novel benchmark of accurate \textit{N}-body integrators of exosolar systems more generally.