  The escape dynamics in an analytical gravitational model which describes the
motion of stars in a binary system of interacting dwarf spheroidal galaxies is
investigated in detail. We conduct a numerical analysis distinguishing between
regular and chaotic orbits as well as between trapped and escaping orbits,
considering only unbounded motion for several energy levels. In order to
distinguish safely and with certainty between ordered and chaotic motion, we
apply the Smaller ALingment Index (SALI) method. It is of particular interest
to locate the escape basins through the openings around the collinear
Lagrangian points $L_1$ and $L_2$ and relate them with the corresponding
spatial distribution of the escape times of the orbits. Our exploration takes
place both in the configuration $(x,y)$ and in the phase $(x,\dot{x})$ space in
order to elucidate the escape process as well as the overall orbital properties
of the galactic system. Our numerical analysis reveals the strong dependence of
the properties of the considered escape basins with the total orbital energy,
with a remarkable presence of fractal basin boundaries along all the escape
regimes. It was also observed, that for energy levels close to the critical
escape energy the escape rates of orbits are large, while for much higher
values of energy most of the orbits have low escape periods or they escape
immediately to infinity. We hope our outcomes to be useful for a further
understanding of the escape mechanism in binary galaxy models.
