#### Date of Award

Summer 2010

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Aerospace Engineering

#### Committee Director

Brett A. Newman

#### Committee Member

Robert Ash

#### Committee Member

Gary Copeland

#### Abstract

This dissertation explores the analytical solution properties surrounding a nominal periodic orbit in two different planes, the plane of motion of the two primaries and a plane perpendicular to the line joining the two primaries, in the circular restricted three-body problem. Assuming motion can be maintained in the plane and motion of the third body is circular, Jacobi's integral equation can be analytically integrated, yielding a closed-form expression for the period and path expressed with elliptic integral and elliptic function theory. In this case, the third body traverses a circular path with nonuniform speed. In a strict sense, the in-plane assumption cannot be maintained naturally. However, there may be cases where the assumption is approximately maintained over a finite time period. More importantly, the nominal solution can be used as the basis for an iterative analytical solution procedure for the three dimensional periodic trajectory where corrections are computable in closed-form. In addition, the in-plane assumption can be strictly enforced with the application of modulated thrust acceleration. In this case, the required thrust control inputs are found to be nonlinear functions in time. Total velocity increment, required to maintain the nominal orbit, for one complete period of motion of the third body is expressed as a function of the orbit characteristics.

#### DOI

10.25777/z4gx-6264

#### ISBN

9781124362076

#### Recommended Citation

Ghazy, Mohammed A..
"Analytic Construction of Periodic Orbits in the Restricted Three-Body Problem"
(2010). Doctor of Philosophy (PhD), Dissertation, Aerospace Engineering, Old Dominion University, DOI: 10.25777/z4gx-6264

https://digitalcommons.odu.edu/mae_etds/61

#### Included in

Aerospace Engineering Commons, Applied Mathematics Commons, Mechanical Engineering Commons