Note on the trapped motion in ER3BP at the vicinity of barycenter

On the Motion of Small Satellite near the Planet in ER3BP

“…on true anomaly f; this is Riccati-type [10] ordinary differential equation (with variable coefficient  (f), determined in ( 7)).…”

“…Equation ( 1) can be transformed to other form (see [8]) in case of loweccentricity orbit e  0 by neglecting the terms of second order of smallness in (1) as follows where in (9), C is the constant of integration having dimension inverse to time; besides, eccentricity e is assumed to be very slowly varying function on long-time scale period; so, it could be considered equal to constant close to zero for the suffitiently large period of changing of true anomaly f. Using expression given in [8], we obtain (here below, e 0 = e(0), a 0 = a p (0 Thus, we obtain from (10) the approximate equation which can be solved by applying of series of Taylor expansions (by neglecting the terms of second order of…”

“…Thus far, we can make a conclusion from (10) (let us remark that we should choose sign "minus" for expression for B in (10)). So, we can use the semi-analytical expression for function e(f) (12) for obtaining numerical solution of Eqns.…”

“…According [8][9][10], equations of ER3BP for small planetoid m can be presented in the synodic co-rotating Cartesian coordinate system {x, y, z} in non-dimensional scaled form (we consider here Cauchy problem with given initial conditions):…”

“…let us remind the denotations in expression for B mentioned above in equation(10): m moon is the mass of moon, M planet is the mass of planet, Q k 2 is the ratio of the Love number k₂ of moon to the quality factor Q (which describes the response of the potential of the distorted body in regard to the influence of current tides); Rmoon is the equatorial radius of the moon.…”