Study of the Potential of Irregular Shaped Bodies and Orbits Around a Non-Spherical Body

Paper number

IAC-07-C1.4.10

Author

Dr. Antonio Prado, National Institute for Space Research, Brazil

Year

2007

Abstract

The purpose of the present work is to determine an analytical form to represent the potential around an irregular shaped body and to obtain a description of the possible orbital evolutions of a particle that travels around a body with those characteristics.
Conventional spherical harmonic representations of the gravitational potential of such bodies require expansions of high degree and order, which are difficult to obtain. The polyhedral method is well suited to evaluate the gravitational field of an irregularly shaped body such as asteroids, comet nucleus, and small planetary satellites. If complete coverage of the surface can be obtained, a polyhedral model of the body can be constructed. Expressions in closed forms are developed for the gravitational potential and for the acceleration due to the polyhedron with constant density. Results are developed in closed forms, instead of an infinite-series expansion, and involve only elementary functions (arc-tangent and logarithm).  
The technique of the determination of the gravitational field through polyhedron is studied from the literature that already exist and, starting from the expressions for the polyhedron, we develop an algorithm to illustrate the equipotential surface of a non-spherical body, which field is not known yet. The results that will be shown consist of sets of analytical equations that give the potential due to the different geometrical forms. 
It is used the polyhedral method to study the gravitational potential of spherical and non spherical three-dimensional bodies (a unity radius sphere, a prolate and an oblate ellipsoids with different values of semi-major axis). The dynamics of the orbit of a test particle around such bodies is studied. In general, when the particle is far from the sphere, its position returns to the initial point after a keplerian orbital period. On the other hand, when the particle gets close to its surface, the effect of its polyhedral form shows short-periodic variations in the semimajor axis and eccentricity of the orbit. The results showed that the orbits close to the ellipsoids become eccentric and precess due to the effects of its potential.  With these results it can be verified that the polyhedral form of the object does work very well and this method is efficient for the trajectory studied. The work generates fundamental theoretical knowledge that can be applied in irregular bodies with more complex forms, such as asteroids.
The polyhedron expressions are approximations to reality, since real bodies are not polyhedra and contain density irregularities, but these expressions are a good representation and can be used to study orbits around such bodies.

Abstract document

IAC-07-C1.4.10.pdf

Manuscript document

IAC-07-C1.4.10.pdf (🔒 authorized access only).

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