Periodic Orbits in the Restricted Full Three-Body Problem for an Ellipsoid-Sphere System.

Paper number

IAC-05-C1.6.02

Author

Ms. Julie Bellerose, University of Michigan, United States

Coauthor

Dr. Daniel J. Scheeres, University of Michigan, United States

Year

2005

Abstract

Over the past few decades, an increasing number of probes have
been sent to some of the small bodies of our solar system.
Generally speaking, the main motivation for the study of asteroids
and comets is that they are minimally processed. Thus,
understanding or bringing samples back from these worlds provides
important information on the original constituents of our solar
system. These small bodies are also of interest for more
fundamental studies in orbital dynamics.

There is now an increasing interest in binary asteroids, as it is
estimated that they may constitute over 15 percent of the
Near-Earth Asteroid population. There are many important issues of
science that can be investigated by studying and sending probes to
investigate these systems. However, the problem of navigating a
spacecraft in a binary system is complex and challenging, as it
requires modeling of the asteroid system in addition to the motion
of the spacecraft about the system. The problem of binary asteroid
orbiters integrates four classical problems of astrodynamics: the
Hill problem, the Restricted Three-Body Problem, the non-spherical
Orbiter Problem and the Full Two-Body Problem.

In this paper, the motion of a satellite in the gravitational
field of two bodies is investigated. The satellite is assumed to
have no influence on the motion of the two primaries. This problem
is referred as the Restricted Full Three-Body Problem as the mass
distribution of the primaries is taken into account. In the
current work, the equations of motion of a particle in the
gravitational field of an ellipsoid-sphere system are derived. As
is common for natural systems, we assume that the ellipsoid
rotates about its maximum moment of inertia and is not synchronous
with its mutual orbit about the sphere.

A previous analysis considered the case of relative equilibrium of
the Full Two-Body Problem, i.e synchronized rotation of the
ellipsoid-sphere system. As with the Restricted Three-Body
Problem, five equilibrium solutions exist. For the Restricted
Three-Body Problem, the Routh criterion provides a range of mass
distribution between the primaries for stability in the vicinity
of the equilateral points. Considering the mass distribution of
one of the bodies, the stability of the equilateral equilibrium
solutions was determined for a range of mass ratio, distance
between the primaries and size parameters of the ellipsoid body.
In general, it was found that the presence of the ellipsoid body
reduces the stability region from the Restricted Three-Body
Problem.

The majority of real asteroid binary systems do not have
synchronized motion between the primary and the secondary orbit.
Incorporation of a rotating primary makes the analysis
fundamentally different than the synchronous case, with the
equations of motion now being time periodic. Despite this, the
results of stability in the synchronous case still provide
insights for the more general case. To study this system we
compute periodic orbits of a spacecraft close to the triangular
libration point regions and analyze their stability properties as
the parameters of the system are modified.  While the stability
behavior of these orbits as a function of the mass ratio, distance
between the primaries, and shape of the primary remains analogous
to the synchronous case, we find that the ratio between the orbit
period of the binary system and the rotation period of the primary
plays a strong role in determining the stability and size of the
resulting periodic orbits.  We provide an analysis and discussion
of this phenomenon and relate our results to the practical design
of spacecraft missions to these bodies.

Abstract document

IAC-05-C1.6.02.pdf

Manuscript document

IAC-05-C1.6.02.pdf (🔒 authorized access only).

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