On the Motion of Small Satellite near the Planet in ER3BP

In the current research, a novel approach or solving procedure for equations of the trapped motion for small mass m near the primary mplanet in case of the elliptic restricted problem of three bodies (ER3BP) is presented. We consider two primaries MSun and mplanet which are orbiting around their barycenter on elliptic orbits. Our aim of investigating such the class of motions is to obtain the coordinates of the aforementioned satellite (in the synodic co-rotating Cartesian coordinate system $$ \overrightarrow{r} $$ ={x, y, z}) which will always maintain its orbit located near the second of these primaries, mplanet. We obtain a family of semi-analytical (approximated) solutions as follows: 1) equation for x is given via coordinate y and true anomaly f, with help of the additional variable parameter α, which determines a Riccati-type character of solution for coordinate z (which means possibility of sudden jumping of the magnitude of solution), 2) expression for y is given via coordinate x, true anomaly f, and the chosen parameter α, 3) coordinate z is to be quasi-oscillating with small amplitude depending on true anomaly f which is tracing the motion of small mass (satellite) around primary mplanet. Besides, the additional ways of semi-analytical solving equations of motion are illuminated.