  In the current study, we explore stability of motion of satellite around the
natural moons of planets in Solar system using the novel concept of ER3BP with
variable eccentricity. This concept was introduced earlier when novel type of
ER3BP (Sun-planet-satellite) was investigated with variable spin state of
secondary planet correlated implicitly to the motion of satellite (in the
synodic co-rotating Cartesian coordinate system) for its trapped orbit near the
secondary planet (which is involved in kepler duet <Sun-planet>). But it is of
real interest to explore another kind of aforedescribed problem, ER3BP
(planet-moon-satellite) with respect to investigation of motion of satellite m
around the natural moon m_moon of planet in Solar system with variable
eccentricity of the moon in its motion around the planet. So, we consider here
two primaries, M_planet and m_moon, the last is orbiting around their common
barycenter on quasi-elliptic orbit with slow-changing, not constant
eccentricity (on a large-time scale) due to tidal phenomena. Our aim is to
investigate motion of small dot satellite around the natural moon of planet on
quasi-stable elliptic orbit. Both novel theoretical and numerical findings (for
various cases of trio <planet-moon-satellite>) are presented in the current
research.
