Hill stability of the coplanar four-body problem with a binary subsystem

We consider a coplanar four-body system including a binary subsystem. The Hill stability is studied using the zero velocity surface of the four-body problem. The value of c²E determines the topology of zero-velocity surfaces and hence regions of possible motions that bifurcates at the collinear equilibrium points. The Hill stability of the system is determined by the relation between the actual and critical values of c²E. The critical value of c²E is determined by the mass of the bodies and other two time-varying parameters associated to the orbit of binary. The influences of the two parameters on the Hill stability are investigated and discussed. Different from the three-body problem, the primary bifurcation may happen at different equilibrium points. Besides, a new temporary Hill stability is defined to illustrate the case that the system may be stable or unstable.