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We here revisit the essential problem of dynamical stability of planetaryorbits around stellar binaries. We build on the coplanar three-body system ofthe Dvorak(1986), extending his stability diagram to both corotation andcounter rotation of P-type orbits. His stability diagram express the change ofstability across the gap between upper(UCO) and lower(LCO) critical orbits. Byradius, 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 gapbetween UCO and LCO follows a transition radius $r^+_g = 2.39 + 2.53e - 1.40e^2$ and $r^-_g = 0.92 - 2.47e$ for corotation and respectively, counterrotation of the third body (the planet). Our $r^+_g$ agrees with the same ofDvorak 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 habitablezone may still be populated with planets on counter rotating orbits. Theaccurate numerical results presented here based on adaptive integration usingMATLAB ODE45 may also serve as a novel benchmark of accurate \textit{N}-bodyintegrators of exosolar systems more generally.

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