Stability of P-type orbits around stellar binaries: An extension to counter-rotating orbits

Continuing advances in observational astronomy may detect exoplanets around exotic systems such as a binary of a compact object - white dwarf, neutron star or black hole - with a stellar companion. Their formation history includes tidal capture, familiar from studies of X-ray binaries, that may put exoplanets on co- or counter-rotating P-type orbits of the resulting binary. We here revisit the essential problem of dynamical stability of planetary orbits around stellar binaries, building on the coplanar three-body system of the Dvorak (1986), extending his stability diagram to include counter rotating P-type orbits, expressing change of stability across a gap between upper (UCO) and lower (LCO) critical orbital radius. A 24% width is found for the counter-rotating case, considerably larger than about 8% for the familiar corotation case. As the gap of the first lies below the second, counter rotation appears more stable, yet by width it is more chaotic. The gap between UCO and LCO follows a transition radius r(g)(-) = 0.92 - 2.47e versus r(g)(+) = 2.39 + 2.53e - 1.40e(2) for the corotation case, the latter in agreement with the same of Dvorak to within 0.35%. This difference in stability may be expressed by r(g)(+)/r(g)(-) less than or similar to 2.57 for all e. Around dim binaries, therefore, a relatively close in habitable zone may still be populated with exoplanets on counter rotating orbits, distinct from the corotating case. The accurate numerical results presented here based on adaptive integration using MATLAB ODE45 may also serve as a novel benchmark of accurate N-body integrators of exosolar systems, e.g., to relax our model problem setting of co-planar orientation and equal massratio binaries.