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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
do
not
yet
have
a
definite
answer
to
the
question
of
whether
our
solar
system
is
stable
or
not.
this
is
partly
a
result
of
the
fact
that
the
definition
of
the
term
‘stability’
is
vague
when
it
is
used
in
relation
to
the
problem
of
plaary
tion
in
the
solar
system.
actually
it
is
not
easy
to
give
a
clear，
rigorous
and
physically
meaningful
definition
of
the
stability
of
our
solar
system.
ang
many
definitions
of
stability，
here
we
adopt
the
hill
definition
(gladman
1993):
actually
this
is
not
a
definition
of
stability，
but
of
instability.
we
define
a
system
as
being
unstable
when
a
close
encounter
ours
somewhere
in
the
system，
star
from
a
certain
initial
configuration
(chambers，
wetherill
boss
1996;
ito
tanikawa
1999).
a
system
is
defined
as
experiencing
a
close
encounter
when
two
bodies
approach
one
another
within
an
area
of
the
larger
hill
radius.
otherwise
the
system
is
defined
as
being
stable.
henceforward
we
state
that
our
plaary
system
is
dynamically
stable
if
no
close
encounter
happens
during
the
age
of
our
solar
system，
about
±5
gyr.
incidentally，
this
definition
may
be
replaced
by
one
in
which
an
ourrence
of
any
orbital
crossing
between
either
of
a
pair
of
plas
takes
place.
this
is
because
we
know
from
experience
that
an
orbital
crossing
is
very
likely
to
lead
to
a
close
encounter
in
plaary
and
protoplaary
systems
(yoshinaga，
kokubo
makino
1999).
of
course
this
statement
cannot
be
simply
applied
to
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