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 Some aspects of three and fourbody dynamics
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Some aspects of three and fourbody dynamics Barkham, Peter George Douglas
Abstract
Two fundamental problems of celestial mechanics are considered: the stellar or planetary threebody problem and a related form of the restricted fourbody problem. Although a number of constraints are imposed, no assumptions are made which could invalidate the final solution. A consistent and rational approach to the analysis of fourbody systems has not previously been developed, and an attempt is made here to describe problem evolution in a systematic manner. In the particular threebody problem under consideration two masses, forming a close binary system, orbit a comparatively distant mass. A new literal, periodic solution of this problem is found in terms of a small parameter e, which is related to the distance separating the binary system and the remaining mass, using the two variable expansion procedure. The solution is accurate within a constant error O(e¹¹) and uniformly valid as e tends to zero for time intervals 0(e¹⁴). Two specific examples are chosen to verify the literal solution, one of which relates to the sunearthmoon configuration of the solar system. The second example applies to a problem of stellar motion where the three masses are in the ratio 20 : 1 : 1. In both cases a comparison of the analytical solution with an equivalent numericallygenerated orbit shows .close agreement, with an error below 5 percent for the sunearthmoon configuration and less than 3 percent for the stellar system. The fourbody problem is derived from the threebody case by introducing a particle of negligible mass into the close binary system. Unique uniformly valid solutions are found for motion near both equilateral triangle points of the binary system in terms of the small parameter e, where the primaries move in accordance with the uniformlyvalid threebody solution. Accuracy, in this case, is Q maintained within a constant error 0(e⁸), and the solutions are uniformly valid as e tends to zero for time intervals 0(e¹¹). Orbital position errors near L₄ and L₅ of the earthmoon system are found to be less than 5 percent when numericallygenerated periodic solutions are used as a standard of comparison. The approach described here should, in general, be useful in the analysis of nonintegrable dynamic systems, particularly when it is feasible to decompose the problem into a number of subsidiary cases.
Item Metadata
Title 
Some aspects of three and fourbody dynamics

Creator  
Publisher 
University of British Columbia

Date Issued 
1974

Description 
Two fundamental problems of celestial mechanics are considered: the stellar or planetary threebody problem and a related form of the restricted fourbody problem. Although a number of constraints are imposed, no assumptions are made which could invalidate the final solution. A consistent and rational approach to the analysis of fourbody systems has not previously been developed, and an attempt is made here to describe problem evolution in a systematic manner. In the particular threebody problem under consideration two masses, forming a close binary system, orbit a comparatively distant mass. A new literal, periodic solution of this problem is found in terms of a small parameter e, which is related to the distance separating the binary system and the remaining mass, using the two variable expansion procedure. The solution is accurate within a constant error O(e¹¹) and uniformly valid as e tends to zero for time intervals 0(e¹⁴). Two specific examples are chosen to verify the literal solution, one of which relates to the sunearthmoon configuration of the solar system. The second example applies to a problem of stellar motion where the three masses are in the ratio 20 : 1 : 1. In both cases a comparison of the analytical solution with an equivalent numericallygenerated orbit shows .close agreement, with an error below 5 percent for the sunearthmoon configuration and less than 3 percent for the stellar system.
The fourbody problem is derived from the threebody case by introducing a particle of negligible mass into the close binary system. Unique uniformly valid solutions are found for motion near both equilateral triangle points of the binary system in terms of the small parameter e, where the primaries move in accordance with the uniformlyvalid threebody solution. Accuracy, in this case, is Q maintained within a constant error 0(e⁸), and the solutions are uniformly
valid as e tends to zero for time intervals 0(e¹¹). Orbital position errors near L₄ and L₅ of the earthmoon system are found to be less than 5 percent when numericallygenerated periodic solutions are used as a standard of comparison.
The approach described here should, in general, be useful in the analysis of nonintegrable dynamic systems, particularly when it is feasible to decompose the problem into a number of subsidiary cases.

Genre  
Type  
Language 
eng

Date Available 
20100122

Provider 
Vancouver : University of British Columbia Library

Rights 
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.

DOI 
10.14288/1.0064964

URI  
Degree  
Program  
Affiliation  
Degree Grantor 
University of British Columbia

Campus  
Scholarly Level 
Graduate

Aggregated Source Repository 
DSpace

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Rights
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.