On the analytical solution in non-inertial frame of R2BP

TL;DR

This paper introduces a new analytical approach to solve the restricted two-body problem in a non-inertial frame, incorporating a modified potential to account for the central body's variable velocity, and derives explicit formulas for the motion.

Contribution

It presents a novel analytical solution method for the R2BP in a non-inertial frame using a modified potential and elliptic integrals, extending classical formulations.

Findings

Derived an analytical formula for t(r) using elliptic integrals

Established a method to express initial conditions from Cartesian to polar coordinates

Demonstrated the existence of an analytical solution for the system's equations of motion

Abstract

In this analytical study, we have presented a new type of solving procedure with aim to obtain the coordinates of small mass m, which moves around primary M_Sun, referred to non-inertial frame of restricted two-body problem (R2BP) with modified potential function (taking into account the components of variable velocity of central body M_Sun motion) instead of classical potential function for Kepler formulation of R2BP. Meanwhile, system of equations of motion has been successfully explored with respect to the existence of analytical way for presentation of the solution in polar coordinates with radial distance r = r(t). We have obtained analytical formula for function t = t(r) via appropriate elliptic integral. Having obtained the inversed dependence r = r(t), we can obtain the time-dependence for the polar angle as well. Also, we have pointed out how to express components of solution (including initial conditions) from cartesian to polar coordinates.