  We elucidate the escape properties of stars moving in the combined
gravitational field of a binary system of two interacting dwarf spheroidal
galaxies. A galaxy model of three degrees of freedom is adopted for describing
the dynamical properties of the Hamiltonian system. All the numerical values of
the involved parameters are chosen having in mind the real binary system of the
dwarf spheroidal galaxies NGC 147 and NGC 185. We distinguish between bounded
(regular, sticky or chaotic) and escaping motion by classifying initial
conditions of orbits in several types of two dimensional planes, considering
only unbounded motion for several energy levels. We analyze the orbital
structure of all types of two dimensional planes of initial conditions by
locating the basins of escape and also by measuring the corresponding escape
time of the orbits. Furthermore, the properties of the normally hyperbolic
invariant manifolds (NHIMs), located in the vicinity of the index-1 saddle
points $L_1$, $L_2$, and $L_3$, are also investigated. These manifolds are of
great importance, as they control the flow of stars (between the two galaxies
and toward the exterior region) over the different saddle points. In addition,
bifurcation diagrams of the Lyapunov periodic orbits as well as restrictions of
the Poincar\'e map to the NHIMs are presented for revealing the dynamics in the
neighbourhood of the saddle points. Comparison between the current outcomes and
previous related results is also made.
