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long-term
integrations
and
stability
of
plaary
orbits
in
our
solar
system
abstract
we
present
the
results
of
very
long-term
numerical
integrations
of
plaary
orbital
tions
over
109
-yr
time-spans
inc露ding
all
nine
plas.
a
quick
inspection
of
our
numerical
data
shows
that
the
plaary
tion，
at
least
in
our
simple
dynamical
del，
seems
to
be
quite
stable
even
over
this
very
long
time-span.
a
closer
look
at
the
lowest-frequency
oscillations
using
a
low-pass
filter
shows
us
the
potentially
diffusive
character
of
terrestrial
plaary
tion，
especially
that
of
mercury.
the
behaviour
of
the
eentricity
of
mercury
in
our
integrations
is
qualitatively
similar
to
the
results
from
jacques
laskars
secular
perturbation
theory
(e.g.
emax～
0.35
over
～±
4
gyr).
however，
there
are
no
apparent
secular
increases
of
eentricity
or
inclination
in
any
orbital
elements
of
the
plas，
which
may
be
revealed
by
still
longer-term
numerical
integrations.
we
have
also
performed
a
couple
of
trial
integrations
inc露ding
tions
of
the
outer
five
plas
over
the
duration
of
±
5
x
1010
yr.
the
result
indicates
that
the
three
major
resonances
in
the
neptune–p露to
system
have
been
maintained
over
the
1011-yr
time-span.
1
introduction
1.1definition
of
the
problem
the
question
of
the
stability
of
our
solar
system
has
been
debated
over
several
混dred
years，
since
the
era
of
newton.
the
problem
has
attracted
many
faus
mathematicians
over
the
years
and
has
played
a
central
role
in
the
development
of
non-linear
dynamics
and
chaos
theory.
however，
we
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