**DDA 33rd Meeting, Mt. Hood, OR, April 2002**

*Session 1. Brouwer Lecture/Extrasolar Systems*

Monday, April 22, 2002, 8:40-10:30am
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## [1.03] Secular Evolution of Hierarchical Planetary Systems

*M.H. Lee, S.J. Peale (UCSB)*

Among the extrasolar planetary systems discovered to date
are hierarchical systems such as HD 168443 where the orbital
eccentricities of the two known substellar companions are
large and the ratio of the orbital semimajor axes a_{1}/a_{2}
is small. The octupole-level secular perturbation equations,
which are based on an expansion to order (a_{1}/a_{2})^{3},
should be applicable to coplanar hierarchical planetary
systems with a wide range of companion masses and orbital
eccentricities. We find that the octupole approximation
describes well the secular evolution of stable coplanar
hierarchical systems such as HD 168443 by comparison with
direct numerical orbit integrations. The octupole
approximation reduces the secular evolution of coplanar
hierarchical systems to a problem of one degree of freedom,
and the secular evolution of such systems can be understood
by examining how the trajectories in diagrams of the
inner-orbit eccentricity, e_{1}, versus the difference in
the longitudes of periapse, \varpi_{1} - \varpi_{2}, depend on
various parameters. There are usually two libration islands,
one about a point at \varpi_{1} - \varpi_{2} = 0^{\}circ and
another about a point at \varpi_{1} - \varpi_{2} = 180^{\}circ.
The libration islands are large and large oscillations of
both eccentricities are possible if L_{1} \approx L_{2} (where
L_{i} is the maximum possible angular momentum of the ith
orbit if the orbit were circular). The octupole-level
equations do not provide any information on the stability of
hierarchical systems. For hierarchical systems such as HD
168443 where the companion masses are relatively large
compared to the primary mass, we find from direct numerical
orbital integrations that such systems are unstable if the
periapse distance of the outer orbit is less than about 3
times the semimajor axes of the inner orbit. The smallest
periapse distance of the outer orbit for stability decreases
slowly with decreasing companion masses.

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Bulletin of the American Astronomical Society, **34**, #3

© 2002. The American Astronomical Society.