Date of Award

Summer 2019

Document Type


Degree Name

Master of Science (MS)


Mechanical & Aerospace Engineering


Aerospace Engineering

Committee Director

Brett Newman

Committee Member

Colin Britcher

Committee Member

Gordon Melrose


This thesis investigates an approximate analytic construction of halo-type periodic orbits about the collinear equilibrium points in the circular restricted three-body problem. The research follows a parallel approach to the one used by Ghazy and Newman, but the initial assumptions and utilized functions are unique to this thesis. A suppositional base solution constructed using Jacobi elliptic functions and satisfying Jacobi's integral equation is at the core of the analytic construction framework. The locus of this solution is a circle lying in the plane perpendicular to the line joining the primaries. The base solution elicits a closed-form expression for the period in terms of the elliptic integral of the first kind and a frequency-like parameter. The base solution satisfies a combination of third body equations of motion in the y and z axes, and the equation of motion in the x axis is satisfied in an averaged and bounded sense when the vertical plane is located at one of the collinear Lagrange points.

Because the third body cannot traverse this type of path naturally, an analytic correction process is pursued to recover accuracy. An iterated perturbation process is used whereby corrections to the base solution along the axis connecting the primaries is considered first, followed by correction in the other two directions. The iterated approach is followed to exploit the coupling structure inherent in the three-body system to simplify calculations. Linear assumptions are also used in these calculations for simplifying reasons. The non-homogeneous solution excitations for the x and y corrections are in the form of Fourier series expansions of the Jacobi elliptic functions in terms of the nome function. The development assumes the suppositional plane passes through one of the collinear Lagrange points. Only homogeneous correction is needed for the z axis. The modified solution then consists of the base solution plus first order corrections which can be further developed to include second and higher order corrections.

The base solution is compared with a L1 halo orbit example and somewhat rough similarity is observed; the period of the base solution being approximately half of the true orbit. An interesting result is obtained when the truncated version of the correction forcing signal is compared with the exact one. When the frequency-like parameter is greater than or equal to unity, the full and truncated forcing signals become almost identical which justifies the use of the truncated forcing function for the L1 halo orbit test case. The initial conditions for the true orbit are substituted in the truncated series solution and a new value of the frequency-like parameter is obtained using numerical computation depending in which axis is sought. For the test case, a unique value of the parameter is obtained from the y axis velocity initial conditions, which when employed in the x, y, and z solutions gives improved motion compared to the base one, and the corrected orbit reaches closer to the true orbit. The error in the period of this corrected orbit is reduced to zero when compared to the true orbit.


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