# Multiple Scales Asymptotic Solution For The Constant Radial Thrust Problem

## TL;DR

This paper develops an approximate analytical solution for the two-body problem with constant radial thrust using the method of multiple scales, effectively capturing multiple characteristic periods and extending solution validity.

## Contribution

It introduces a novel application of the multiple scales method to the radial thrust problem, avoiding special functions and initial orbit restrictions, with improved accuracy and insight.

## Key findings

- Good agreement with high-precision numerical solutions
- Longer validity range compared to regular expansions
- Provides a simple expression for the long period

## Abstract

An approximate analytical solution for the two body problem perturbed by a radial, low acceleration is obtained, using a regularized formulation of the orbital motion and the method of multiple scales. Formulating the dynamics with the Dromo special perturbation method allows us to separate the two characteristic periods of the problem in a clear and physically significative way, namely the orbital period and a period depending on the magnitude of the perturbing acceleration. This second period becomes very large compared to the orbital one for low thrust cases, allowing us to develop an accurate approximate analytical solution through the method of multiple scales. Compared to a regular expansion, the multiple scales solution retains the qualitative contributions of both characteristic periods and has a longer validity range in time. Looking at previous solutions for this problem, our approach has the advantage of not requiring the evaluation of special functions or an initially circular orbit. Furthermore, the simple expression reached for the long period provides additional insight on the problem. Finally, the behavior of the asymptotic solution is assessed through several test cases, finding a good agreement with high-precision numerical solutions. The results presented not only advance in the study of the two body problem with constant radial thrust, but confirm the utility of the method of multiple scales for tackling problems in orbital mechanics.