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以下是文章内容：

Long-term

integrations

and

stability

of

planetary

orbits

in

our

Solar

system

Abstract

We

present

the

results

of

very

long-term

numerical

integrations

of

planetary

orbital

motions

over

109

-yr

time-spans

including

all

nine

planets.

A

quick

inspection

of

our

numerical

data

shows

that

the

planetary

motion，

at

least

in

our

simple

dynamical

model，

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

planetary

motion，

especially

that

of

Mercury.

The

behaviour

of

the

eccentricity

of

Mercury

in

our

integrations

is

qualitatively

similar

to

the

results

from

Jacques

Laskar's

secular

perturbation

theory

(e.g.

emax∼

0.35

over

∼±

4

Gyr).

However，

there

are

no

apparent

secular

increases

of

eccentricity

or

inclination

in

any

orbital

elements

of

the

planets，

which

may

be

revealed

by

still

longer-term

numerical

integrations.

We

have

also

performed

a

couple

of

trial

integrations

including

motions

of

the

outer

five

planets

over

the

duration

of

±

5

×

1010

yr.

The

result

indicates

that

the

three

major

resonances

in

the

Neptune–Pluto

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

hundred

years，

since

the

era

of

Newton.

The

problem

has

attracted

many

famous

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

planetary

motion

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.

Among

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

becoming

unstable

when

a

close

encounter

occurs

somewhere

in

the

system，

starting

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

planetary

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

occurrence

of

any

orbital

crossing

between

either

of

a

pair

of

planets

takes

place.

This

is

because

we

know

from

experience

that

an

orbital

crossing

is

very

likely

to

lead

to

a

close

encounter

in

planetary

and

protoplanetary

systems

(Yoshinaga，

Kokubo

&

Makino

1999).

Of

course

this

statement

cannot

be

simply

applied

to

systems

with

stable

orbital

resonances

such

as

the

Neptune–Pluto

system.

1.2Previous

studies

and

aims

of

this

research

In

addition

to

the

vagueness

of

the

concept

of

stability，

the

planets

in

our

Solar

system

show

a

character

typical

of

dynamical

chaos

(Sussman

&

Wisdom

1988，

1992).

The

cause

of

this

chaotic

behaviour

is

now

partly

understood

as

being

a

result

of

resonance

overlapping

(Murray

&

Holman

1999;

Lecar，

Franklin

&

Holman

2001).

However，

it

would

require

integrating

over

an

ensemble

of

planetary

systems

including

all

nine

planets

for

a

period

covering

several

10

Gyr

to

thoroughly

understand

the

long-term

evolution

of

planetary

orbits，

since

chaotic

dynamical

systems

are

characterized

by

their

strong

dependence

on

initial

conditions.

From

that

point

of

view，

many

of

the

previous

long-term

numerical

integrations

included

only

the

outer

five

planets

(Sussman

&

Wisdom

1988;

Kinoshita

&

Nakai

1996).

This

is

because

the

orbital

periods

of

the

outer

planets

are

so

much

longer

than

those

of

the

inner

four

planets

that

it

is

much

easier

to

follow

the

system

for

a

given

integration

period.

At

present，

the

longest

numerical

integrations

published

in

journals

are

those

of

Duncan

&

Lissauer

(1998).

Although

their

main

target

was

the

effect

of

post-main-sequence

solar

mass

loss

on

the

stability

of

planetary

orbits，

they

performed

many

integrations

covering

up

to

∼1011

yr

of

the

orbital

motions

of

the

four

jovian

planets.

The

initial

orbital

elements

and

masses

of

planets

are

the

same

as

those

of

our

Solar

system

in

Duncan

&

Lissauer's

paper，

but

they

decrease

the

mass

of

the

Sun

gradually

in

their

numerical

experiments.

This

is

because

they

consider

the

effect

of

post-main-sequence

solar

mass

loss

in

the

paper.

Consequently，

they

found

that

the

crossing

time-scale

of

planetary

orbits，

which

can

be

a

typical

indicator

of

the

instability

time-scale，

is

quite

sensitive

to

the

rate

of

mass

decrease

of

the

Sun.

When

the

mass

of

the

Sun

is

close

to

its

present

value，

the

jovian

planets

remain

stable

over

1010

yr，

or

perhaps

longer.

Duncan

&

Lissauer

also

performed

four

similar

experiments

on

the

orbital

motion

of

seven

planets

(Venus

to

Neptune)，

which

cover

a

span

of

∼109

yr.

Their

experiments

on

the

seven

planets

are

not

yet

comprehensive，

but

it

seems

that

the

terrestrial

planets

also

remain

stable

during

the

integration

period，

maintaining

almost

regular

oscillations.

On

the

other

hand，

in

his

accurate

semi-analytical

secular

perturbation

theory

(Laskar

1988)，

Laskar

finds

that

large

and

irregular

variations

can

appear

in

the

eccentricities

and

inclinations

of

the

terrestrial

planets，

especially

of

Mercury

and

Mars

on

a

time-scale

of

several

109

yr

(Laskar

1996).

The

results

of

Laskar's

secular

perturbation

theory

should

be

confirmed

and

investigated

by

fully

numerical

integrations.

In

this

paper

we

present

preliminary

results

of

six

long-term

numerical

integrations

on

all

nine

planetary

orbits，

covering

a

span

of

several

109

yr，

and

of

two

other

integrations

covering

a

span

of

±

5

×

1010

yr.

The

total

elapsed

time

for

all

integrations

is

more

than

5

yr，

using

several

dedicated

PCs

and

workstations.

One

of

the

fundamental

conclusions

of

our

long-term

integrations

is

that

Solar

system

planetary

motion

seems

to

be

stable

in

terms

of

the

Hill

stability

mentioned

above，

at

least

over

a

time-span

of

±

4

Gyr.

Actually，

in

our

numerical

integrations

the

system

was

far

more

stable

than

what

is

defined

by

the

Hill

stability

criterion:

not

only

did

no

close

encounter

happen

during

the

integration

period，

but

also

all

the

planetary

orbital

elements

have

been

confined

in

a

narrow

region

both

in

time

and

frequency

domain，

though

planetary

motions

are

stochastic.

Since

the

purpose

of

this

paper

is

to

exhibit

and

overview

the

results

of

our

long-term

numerical

integrations，

we

show

typical

example

figures

as

evidence

of

the

very

long-term

stability

of

Solar

system

planetary

motion.

For

readers

who

have

more

specific

and

deeper

interests

in

our

numerical

results，

we

have

prepared

a

webpage

(access

)，

where

we

show

raw

orbital

elements，

their

low-pass

filtered

results，

variation

of

Delaunay

elements

and

angular

momentum

deficit，

and

results

of

our

simple

time–frequency

analysis

on

all

of

our

integrations.

In

Section

2

we

briefly

explain

our

dynamical

model，

numerical

method

and

initial

conditions

used

in

our

integrations.

Section

3

is

devoted

to

a

description

of

the

quick

results

of

the

numerical

integrations.

Very

long-term

stability

of

Solar

system

planetary

motion

is

apparent

both

in

planetary

positions

and

orbital

elements.

A

rough

estimation

of

numerical

errors

is

also

given.

Section

4

goes

on

to

a

discussion

of

the

longest-term

variation

of

planetary

orbits

using

a

low-pass

filter

and

includes

a

discussion

of

angular

momentum

deficit.

In

Section

5，

we

present

a

set

of

numerical

integrations

for

the

outer

five

planets

that

spans

±

5

×

1010

yr.

In

Section

6

we

also

discuss

the

long-term

stability

of

the

planetary

motion

and

its

possible

cause.

2

Description

of

the

numerical

integrations

（本部分涉及比较复杂的积分计算，作者君就不贴上来了，贴上来了起点也不一定能成功显示。）

2.3

Numerical

method

We

utilize

a

second-order

Wisdom–Holman

symplectic

map

as

our

main

integration

method

(Wisdom

&

Holman

1991;

Kinoshita，

Yoshida

&

Nakai

1991)

with

a

special

start-up

procedure

to

reduce

the

truncation

error

of

angle

variables，‘warm

start’(Saha

&

Tremaine

1992，

1994).

The

stepsize

for

the

numerical

integrations

is

8

d

throughout

all

integrations

of

the

nine

planets

(N±1，2，3)，

which

is

about

1/11

of

the

orbital

period

of

the

innermost

planet

(Mercury).

As

for

the

determination

of

stepsize，

we

partly

follow

the

previous

numerical

integration

of

all

nine

planets

in

Sussman

&

Wisdom

(1988，

7.2

d)

and

Saha

&

Tremaine

(1994，

225/32

d).

We

rounded

the

decimal

part

of

the

their

stepsizes

to

8

to

make

the

stepsize

a

multiple

of

2

in

order

to

reduce

the

accumulation

of

round-off

error

in

the

computation

processes.

In

relation

to

this，

Wisdom

&

Holman

(1991)

performed

numerical

integrations

of

the

outer

five

planetary

orbits

using

the

symplectic

map

with

a

stepsize

of

400

d，

1/10.83

of

the

orbital

period

of

Jupiter.

Their

result

seems

to

be

accurate

enough，

which

partly

justifies

our

method

of

determining

the

stepsize.

However，

since

the

eccentricity

of

Jupiter

(∼0.05)

is

much

smaller

than

that

of

Mercury

(∼0.2)，

we

need

some

care

when

we

compare

these

integrations

simply

in

terms

of

stepsizes.

In

the

integration

of

the

outer

five

planets

(F±)，

we

fixed

the

stepsize

at

400

d.

We

adopt

Gauss'

f

and

g

functions

in

the

symplectic

map

together

with

the

third-order

Halley

method

(Danby

1992)

as

a

solver

for

Kepler

equations.

The

number

of

maximum

iterations

we

set

in

Halley's

method

is

15，

but

they

never

reached

the

maximum

in

any

of

our

integrations.

The

interval

of

the

data

output

is

200

000

d

(∼547

yr)

for

the

calculations

of

all

nine

planets

(N±1，2，3)，

and

about

8000

000

d

(∼21

903

yr)

for

the

integration

of

the

outer

five

planets

(F±).

Although

no

output

filtering

was

done

when

the

numerical

integrations

were

in

process，

we

applied

a

low-pass

filter

to

the

raw

orbital

data

after

we

had

completed

all

the

calculations.

See

Section

4.1

for

more

detail.

2.4

Error

estimation

2.4.1

Relative

errors

in

total

energy

and

angular

momentum

According

to

one

of

the

basic

properties

of

symplectic

integrators，

which

conserve

the

physically

conservative

quantities

well

(total

orbital

energy

and

angular

momentum)，

our

long-term

numerical

integrations

seem

to

have

been

performed

with

very

small

errors.

The

averaged

relative

errors

of

total

energy

(∼10−9)

and

of

total

angular

momentum

(∼10−11)

have

remained

nearly

constant

throughout

the

integration

period

(Fig.

1).

The

special

startup

procedure，

warm

start，

would

have

reduced

the

averaged

relative

error

in

total

energy

by

about

one

order

of

magnitude

or

more.

Relative

numerical

error

of

the

total

angular

momentum

δA/A0

and

the

total

energy

δE/E0

in

our

numerical

integrationsN±

1，2，3，

where

δE

and

δA

are

the

absolute

change

of

the

total

energy

and

total

angular

momentum，

respectively，

andE0andA0are

their

initial

values.

The

horizontal

unit

is

Gyr.

Note

that

different

operating

systems，

different

mathematical

libraries，

and

different

hardware

architectures

result

in

different

numerical

errors，

through

the

variations

in

round-off

error

handling

and

numerical

algorithms.

In

the

upper

panel

of

Fig.

1，

we

can

recognize

this

situation

in

the

secular

numerical

error

in

the

total

angular

momentum，

which

should

be

rigorously

preserved

up

to

machine-ε

precision.

2.4.2

Error

in

planetary

longitudes

Since

the

symplectic

maps

preserve

total

energy

and

total

angular

momentum

of

N-body

dynamical

systems

inherently

well，

the

degree

of

their

preservation

may

not

be

a

good

measure

of

the

accuracy

of

numerical

integrations，

especially

as

a

measure

of

the

positional

error

of

planets，

i.e.

the

error

in

planetary

longitudes.

To

estimate

the

numerical

error

in

the

planetary

longitudes，

we

performed

the

following

procedures.

We

compared

the

result

of

our

main

long-term

integrations

with

some

test

integrations，

which

span

much

shorter

periods

but

with

much

higher

accuracy

than

the

main

integrations.

For

this

purpose，

we

performed

a

much

more

accurate

integration

with

a

stepsize

of

0.125

d

(1/64

of

the

main

integrations)

spanning

3

×

105

yr，

starting

with

the

same

initial

conditions

as

in

the

N−1

integration.

We

consider

that

this

test

integration

provides

us

with

a

‘pseudo-true’

solution

of

planetary

orbital

evolution.

Next，

we

compare

the

test

integration

with

the

main

integration，

N−1.

For

the

period

of

3

×

105

yr，

we

see

a

difference

in

mean

anomalies

of

the

Earth

between

the

two

integrations

of

∼0.52°(in

the

case

of

the

N−1

integration).

This

difference

can

be

extrapolated

to

the

value

∼8700°，

about

25

rotations

of

Earth

after

5

Gyr，

since

the

error

of

longitudes

increases

linearly

with

time

in

the

symplectic

map.

Similarly，

the

longitude

error

of

Pluto

can

be

estimated

as

∼12°.

This

value

for

Pluto

is

much

better

than

the

result

in

Kinoshita

&

Nakai

(1996)

where

the

difference

is

estimated

as

∼60°.

3

Numerical

results

–

I.

Glance

at

the

raw

data

In

this

section

we

briefly

review

the

long-term

stability

of

planetary

orbital

motion

through

some

snapshots

of

raw

numerical

data.

The

orbital

motion

of

planets

indicates

long-term

stability

in

all

of

our

numerical

integrations:

no

orbital

crossings

nor

close

encounters

between

any

pair

of

planets

took

place.

3.1

General

description

of

the

stability

of

planetary

orbits

First，

we

briefly

look

at

the

general

character

of

the

long-term

stability

of

planetary

orbits.

Our

interest

here

focuses

particularly

on

the

inner

four

terrestrial

planets

for

which

the

orbital

time-scales

are

much

shorter

than

those

of

the

outer

five

planets.

As

we

can

see

clearly

from

the

planar

orbital

configurations

shown

in

Figs

2

and

3，

orbital

positions

of

the

terrestrial

planets

differ

little

between

the

initial

and

final

part

of

each

numerical

integration，

which

spans

several

Gyr.

The

solid

lines

denoting

the

present

orbits

of

the

planets

lie

almost

within

the

swarm

of

dots

even

in

the

final

part

of

integrations

(b)

and

(d).

This

indicates

that

throughout

the

entire

integration

period

the

almost

regular

variations

of

planetary

orbital

motion

remain

nearly

the

same

as

they

are

at

present.

Vertical

view

of

the

four

inner

planetary

orbits

(from

the

z

-axis

direction)

at

the

initial

and

final

parts

of

the

integrationsN±1.

The

axes

units

are

au.

The

xy

-plane

is

set

to

the

invariant

plane

of

Solar

system

total

angular

momentum.(a)

The

initial

part

ofN+1

(

t

=

0

to

0.0547

×

10

9

yr).(b)

The

final

part

ofN+1

(

t

=

4.9339

×

10

8

to

4.9886

×

10

9

yr).(c)

The

initial

part

of

N−1

(t=

0

to

−0.0547

×

109

yr).(d)

The

final

part

ofN−1

(

t

=−3.9180

×

10

9

to

−3.9727

×

10

9

yr).

In

each

panel，

a

total

of

23

684

points

are

plotted

with

an

interval

of

about

2190

yr

over

5.47

×

107

yr

.

Solid

lines

in

each

panel

denote

the

present

orbits

of

the

four

terrestrial

planets

(taken

from

DE245).

The

variation

of

eccentricities

and

orbital

inclinations

for

the

inner

four

planets

in

the

initial

and

final

part

of

the

integration

N+1

is

shown

in

Fig.

4.

As

expected，

the

character

of

the

variation

of

planetary

orbital

elements

does

not

differ

significantly

between

the

initial

and

final

part

of

each

integration，

at

least

for

Venus，

Earth

and

Mars.

The

elements

of

Mercury，

especially

its

eccentricity，

seem

to

change

to

a

significant

extent.

This

is

partly

because

the

orbital

time-scale

of

the

planet

is

the

shortest

of

all

the

planets，

which

leads

to

a

more

rapid

orbital

evolution

than

other

planets;

the

innermost

planet

may

be

nearest

to

instability.

This

result

appears

to

be

in

some

agreement

with

Laskar's

(1994，

1996)

expectations

that

large

and

irregular

variations

appear

in

the

eccentricities

and

inclinations

of

Mercury

on

a

time-scale

of

several

109

yr.

However，

the

effect

of

the

possible

instability

of

the

orbit

of

Mercury

may

not

fatally

affect

the

global

stability

of

the

whole

planetary

system

owing

to

the

small

mass

of

Mercury.

We

will

mention

briefly

the

long-term

orbital

evolution

of

Mercury

later

in

Section

4

using

low-pass

filtered

orbital

elements.

The

orbital

motion

of

the

outer

five

planets

seems

rigorously

stable

and

quite

regular

over

this

time-span

(see

also

Section

5).

3.2

Time–frequency

maps

Although

the

planetary

motion

exhibits

very

long-term

stability

defined

as

the

non-existence

of

close

encounter

events，

the

chaotic

nature

of

planetary

dynamics

can

change

the

oscillatory

period

and

amplitude

of

planetary

orbital

motion

gradually

over

such

long

time-spans.

Even

such

slight

fluctuations

of

orbital

variation

in

the

frequency

domain，

particularly

in

the

case

of

Earth，

can

potentially

have

a

significant

effect

on

its

surface

climate

system

through

solar

insolation

variation

(cf.

Berger

1988).

To

give

an

overview

of

the

long-term

change

in

periodicity

in

planetary

orbital

motion，

we

performed

many

fast

Fourier

transformations

(FFTs)

along

the

time

axis，

and

superposed

the

resulting

periodgrams

to

draw

two-dimensional

time–frequency

maps.

The

specific

approach

to

drawing

these

time–frequency

maps

in

this

paper

is

very

simple

–

much

simpler

than

the

wavelet

analysis

or

Laskar's

(1990，

1993)

frequency

analysis.

Divide

the

low-pass

filtered

orbital

data

into

many

fragments

of

the

same

length.

The

length

of

each

data

segment

should

be

a

multiple

of

2

in

order

to

apply

the

FFT.

Each

fragment

of

the

data

has

a

large

overlapping

part:

for

example，

when

the

ith

data

begins

from

t=ti

and

ends

at

t=ti+T，

the

next

data

segment

ranges

from

ti+δT≤ti+δT+T，

where

δT?T.

We

continue

this

division

until

we

reach

a

certain

number

N

by

which

tn+T

reaches

the

total

integration

length.

We

apply

an

FFT

to

each

of

the

data

fragments，

and

obtain

n

frequency

diagrams.

In

each

frequency

diagram

obtained

above，

the

strength

of

periodicity

can

be

replaced

by

a

grey-scale

(or

colour)

chart.

We

perform

the

replacement，

and

connect

all

the

grey-scale

(or

colour)

charts

into

one

graph

for

each

integration.

The

horizontal

axis

of

these

new

graphs

should

be

the

time，

i.e.

the

starting

times

of

each

fragment

of

data

(ti，

where

i=

1，…，

n).

The

vertical

axis

represents

the

period

(or

frequency)

of

the

oscillation

of

orbital

elements.

We

have

adopted

an

FFT

because

of

its

overwhelming

speed，

since

the

amount

of

numerical

data

to

be

decomposed

into

frequency

components

is

terribly

huge

(several

tens

of

Gbytes).

A

typical

example

of

the

time–frequency

map

created

by

the

above

procedures

is

shown

in

a

grey-scale

diagram

as

Fig.

5，

which

shows

the

variation

of

periodicity

in

the

eccentricity

and

inclination

of

Earth

in

N+2

integration.

In

Fig.

5，

the

dark

area

shows

that

at

the

time

indicated

by

the

value

on

the

abscissa，

the

periodicity

indicated

by

the

ordinate

is

stronger

than

in

the

lighter

area

around

it.

We

can

recognize

from

this

map

that

the

periodicity

of

the

eccentricity

and

inclination

of

Earth

only

changes

slightly

over

the

entire

period

covered

by

the

N+2

integration.

This

nearly

regular

trend

is

qualitatively

the

same

in

other

integrations

and

for

other

planets，

although

typical

frequencies

differ

planet

by

planet

and

element

by

element.

4.2

Long-term

exchange

of

orbital

energy

and

angular

momentum

We

calculate

very

long-periodic

variation

and

exchange

of

planetary

orbital

energy

and

angular

momentum

using

filtered

Delaunay

elements

L，

G，

H.

G

and

H

are

equivalent

to

the

planetary

orbital

angular

momentum

and

its

vertical

component

per

unit

mass.

L

is

related

to

the

planetary

orbital

energy

E

per

unit

mass

as

E=−μ2/2L2.

If

the

system

is

completely

linear，

the

orbital

energy

and

the

angular

momentum

in

each

frequency

bin

must

be

constant.

Non-linearity

in

the

planetary

system

can

cause

an

exchange

of

energy

and

angular

momentum

in

the

frequency

domain.

The

amplitude

of

the

lowest-frequency

oscillation

should

increase

if

the

system

is

unstable

and

breaks

down

gradually.

However，

such

a

symptom

of

instability

is

not

prominent

in

our

long-term

integrations.

In

Fig.

7，

the

total

orbital

energy

and

angular

momentum

of

the

four

inner

planets

and

all

nine

planets

are

shown

for

integration

N+2.

The

upper

three

panels

show

the

long-periodic

variation

of

total

energy

(denoted

asE-

E0)，

total

angular

momentum

(

G-

G0)，

and

the

vertical

component

(

H-

H0)

of

the

inner

four

planets

calculated

from

the

low-pass

filtered

Delaunay

elements.E0，

G0，

H0

denote

the

initial

values

of

each

quantity.

The

absolute

difference

from

the

initial

values

is

plotted

in

the

panels.

The

lower

three

panels

in

each

figure

showE-E0，G-G0

andH-H0

of

the

total

of

nine

planets.

The

fluctuation

shown

in

the

lower

panels

is

virtually

entirely

a

result

of

the

massive

jovian

planets.

Comparing

the

variations

of

energy

and

angular

momentum

of

the

inner

four

planets

and

all

nine

planets，

it

is

apparent

that

the

amplitudes

of

those

of

the

inner

planets

are

much

smaller

than

those

of

all

nine

planets:

the

amplitudes

of

the

outer

five

planets

are

much

larger

than

those

of

the

inner

planets.

This

does

not

mean

that

the

inner

terrestrial

planetary

subsystem

is

more

stable

than

the

outer

one:

this

is

simply

a

result

of

the

relative

smallness

of

the

masses

of

the

four

terrestrial

planets

compared

with

those

of

the

outer

jovian

planets.

Another

thing

we

notice

is

that

the

inner

planetary

subsystem

may

become

unstable

more

rapidly

than

the

outer

one

because

of

its

shorter

orbital

time-scales.

This

can

be

seen

in

the

panels

denoted

asinner

4

in

Fig.

7

where

the

longer-periodic

and

irregular

oscillations

are

more

apparent

than

in

the

panels

denoted

astotal

9.

Actually，

the

fluctuations

in

theinner

4

panels

are

to

a

large

extent

as

a

result

of

the

orbital

variation

of

the

Mercury.

However，

we

cannot

neglect

the

contribution

from

other

terrestrial

planets，

as

we

will

see

in

subsequent

sections.

4.4

Long-term

coupling

of

several

neighbouring

planet

pairs

Let

us

see

some

individual

variations

of

planetary

orbital

energy

and

angular

momentum

expressed

by

the

low-pass

filtered

Delaunay

elements.

Figs

10

and

11

show

long-term

evolution

of

the

orbital

energy

of

each

planet

and

the

angular

momentum

in

N+1

and

N−2

integrations.

We

notice

that

some

planets

form

apparent

pairs

in

terms

of

orbital

energy

and

angular

momentum

exchange.

In

particular，

Venus

and

Earth

make

a

typical

pair.

In

the

figures，

they

show

negative

correlations

in

exchange

of

energy

and

positive

correlations

in

exchange

of

angular

momentum.

The

negative

correlation

in

exchange

of

orbital

energy

means

that

the

two

planets

form

a

closed

dynamical

system

in

terms

of

the

orbital

energy.

The

positive

correlation

in

exchange

of

angular

momentum

means

that

the

two

planets

are

simultaneously

under

certain

long-term

perturbations.

Candidates

for

perturbers

are

Jupiter

and

Saturn.

Also

in

Fig.

11，

we

can

see

that

Mars

shows

a

positive

correlation

in

the

angular

momentum

variation

to

the

Venus–Earth

system.

Mercury

exhibits

certain

negative

correlations

in

the

angular

momentum

versus

the

Venus–Earth

system，

which

seems

to

be

a

reaction

caused

by

the

conservation

of

angular

momentum

in

the

terrestrial

planetary

subsystem.

It

is

not

clear

at

the

moment

why

the

Venus–Earth

pair

exhibits

a

negative

correlation

in

energy

exchange

and

a

positive

correlation

in

angular

momentum

exchange.

We

may

possibly

explain

this

through

observing

the

general

fact

that

there

are

no

secular

terms

in

planetary

semimajor

axes

up

to

second-order

perturbation

theories

(cf.

Brouwer

&

Clemence

1961;

Boccaletti

&

Pucacco

1998).

This

means

that

the

planetary

orbital

energy

(which

is

directly

related

to

the

semimajor

axis

a)

might

be

much

less

affected

by

perturbing

planets

than

is

the

angular

momentum

exchange

(which

relates

to

e).

Hence，

the

eccentricities

of

Venus

and

Earth

can

be

disturbed

easily

by

Jupiter

and

Saturn，

which

results

in

a

positive

correlation

in

the

angular

momentum

exchange.

On

the

other

hand，

the

semimajor

axes

of

Venus

and

Earth

are

less

likely

to

be

disturbed

by

the

jovian

planets.

Thus

the

energy

exchange

may

be

limited

only

within

the

Venus–Earth

pair，

which

results

in

a

negative

correlation

in

the

exchange

of

orbital

energy

in

the

pair.

As

for

the

outer

jovian

planetary

subsystem，

Jupiter–Saturn

and

Uranus–Neptune

seem

to

make

dynamical

pairs.

However，

the

strength

of

their

coupling

is

not

as

strong

compared

with

that

of

the

Venus–Earth

pair.

5

±

5

×

1010-yr

integrations

of

outer

planetary

orbits

Since

the

jovian

planetary

masses

are

much

larger

than

the

terrestrial

planetary

masses，

we

treat

the

jovian

planetary

system

as

an

independent

planetary

system

in

terms

of

the

study

of

its

dynamical

stability.

Hence，

we

added

a

couple

of

trial

integrations

that

span

±

5

×

1010

yr，

including

only

the

outer

five

planets

(the

four

jovian

planets

plus

Pluto).

The

results

exhibit

the

rigorous

stability

of

the

outer

planetary

system

over

this

long

time-span.

Orbital

configurations

(Fig.

12)，

and

variation

of

eccentricities

and

inclinations

(Fig.

13)

show

this

very

long-term

stability

of

the

outer

five

planets

in

both

the

time

and

the

frequency

domains.

Although

we

do

not

show

maps

here，

the

typical

frequency

of

the

orbital

oscillation

of

Pluto

and

the

other

outer

planets

is

almost

constant

during

these

very

long-term

integration

periods，

which

is

demonstrated

in

the

time–frequency

maps

on

our

webpage.

In

these

two

integrations，

the

relative

numerical

error

in

the

total

energy

was

∼10−6

and

that

of

the

total

angular

momentum

was

∼10−10.

5.1

Resonances

in

the

Neptune–Pluto

system

Kinoshita

&

Nakai

(1996)

integrated

the

outer

five

planetary

orbits

over

±

5.5

×

109

yr

.

They

found

that

four

major

resonances

between

Neptune

and

Pluto

are

maintained

during

the

whole

integration

period，

and

that

the

resonances

may

be

the

main

causes

of

the

stability

of

the

orbit

of

Pluto.

The

major

four

resonances

found

in

previous

research

are

as

follows.

In

the

following

description，λ

denotes

the

mean

longitude，Ω

is

the

longitude

of

the

ascending

node

and

ϖ

is

the

longitude

of

perihelion.

Subscripts

P

and

N

denote

Pluto

and

Neptune.

Mean

motion

resonance

between

Neptune

and

Pluto

(3:2).

The

critical

argument

θ1=

3

λP−

2

λN−ϖP

librates

around

180°

with

an

amplitude

of

about

80°

and

a

libration

period

of

about

2

×

104

yr.

The

argument

of

perihelion

of

Pluto

ωP=θ2=ϖP−ΩP

librates

around

90°

with

a

period

of

about

3.8

×

106

yr.

The

dominant

periodic

variations

of

the

eccentricity

and

inclination

of

Pluto

are

synchronized

with

the

libration

of

its

argument

of

perihelion.

This

is

anticipated

in

the

secular

perturbation

theory

constructed

by

Kozai

(1962).

The

longitude

of

the

node

of

Pluto

referred

to

the

longitude

of

the

node

of

Neptune，θ3=ΩP−ΩN，

circulates

and

the

period

of

this

circulation

is

equal

to

the

period

of

θ2

libration.

When

θ3

becomes

zero，

i.e.

the

longitudes

of

ascending

nodes

of

Neptune

and

Pluto

overlap，

the

inclination

of

Pluto

becomes

maximum，

the

eccentricity

becomes

minimum

and

the

argument

of

perihelion

becomes

90°.

When

θ3

becomes

180°，

the

inclination

of

Pluto

becomes

minimum，

the

eccentricity

becomes

maximum

and

the

argument

of

perihelion

becomes

90°

again.

Williams

&

Benson

(1971)

anticipated

this

type

of

resonance，

later

confirmed

by

Milani，

Nobili

&

Carpino

(1989).

An

argument

θ4=ϖP−ϖN+

3

(ΩP−ΩN)

librates

around

180°

with

a

long

period，∼

5.7

×

108

yr.

In

our

numerical

integrations，

the

resonances

(i)–(iii)

are

well

maintained，

and

variation

of

the

critical

arguments

θ1，θ2，θ3

remain

similar

during

the

whole

integration

period

(Figs

14–16

).

However，

the

fourth

resonance

(iv)

appears

to

be

different:

the

critical

argument

θ4

alternates

libration

and

circulation

over

a

1010-yr

time-scale

(Fig.

17).

This

is

an

interesting

fact

that

Kinoshita

&

Nakai's

(1995，

1996)

shorter

integrations

were

not

able

to

disclose.

6

Discussion

What

kind

of

dynamical

mechanism

maintains

this

long-term

stability

of

the

planetary

system?

We

can

immediately

think

of

two

major

features

that

may

be

responsible

for

the

long-term

stability.

First，

there

seem

to

be

no

significant

lower-order

resonances

(mean

motion

and

secular)

between

any

pair

among

the

nine

planets.

Jupiter

and

Saturn

are

close

to

a

5:2

mean

motion

resonance

(the

famous

‘great

inequality’)，

but

not

just

in

the

resonance

zone.

Higher-order

resonances

may

cause

the

chaotic

nature

of

the

planetary

dynamical

motion，

but

they

are

not

so

strong

as

to

destroy

the

stable

planetary

motion

within

the

lifetime

of

the

real

Solar

system.

The

second

feature，

which

we

think

is

more

important

for

the

long-term

stability

of

our

planetary

system，

is

the

difference

in

dynamical

distance

between

terrestrial

and

jovian

planetary

subsystems

(Ito

&

Tanikawa

1999，

2001).

When

we

measure

planetary

separations

by

the

mutual

Hill

radii

(R_)，

separations

among

terrestrial

planets

are

greater

than

26RH，

whereas

those

among

jovian

planets

are

less

than

14RH.

This

difference

is

directly

related

to

the

difference

between

dynamical

features

of

terrestrial

and

jovian

planets.

Terrestrial

planets

have

smaller

masses，

shorter

orbital

periods

and

wider

dynamical

separation.

They

are

strongly

perturbed

by

jovian

planets

that

have

larger

masses，

longer

orbital

periods

and

narrower

dynamical

separation.

Jovian

planets

are

not

perturbed

by

any

other

massive

bodies.

The

present

terrestrial

planetary

system

is

still

being

disturbed

by

the

massive

jovian

planets.

However，

the

wide

separation

and

mutual

interaction

among

the

terrestrial

planets

renders

the

disturbance

ineffective;

the

degree

of

disturbance

by

jovian

planets

is

O(eJ)(order

of

magnitude

of

the

eccentricity

of

Jupiter)，

since

the

disturbance

caused

by

jovian

planets

is

a

forced

oscillation

having

an

amplitude

of

O(eJ).

Heightening

of

eccentricity，

for

example

O(eJ)∼0.05，

is

far

from

sufficient

to

provoke

instability

in

the

terrestrial

planets

having

such

a

wide

separation

as

26RH.

Thus

we

assume

that

the

present

wide

dynamical

separation

among

terrestrial

planets

(>

26RH)

is

probably

one

of

the

most

significant

conditions

for

maintaining

the

stability

of

the

planetary

system

over

a

109-yr

time-span.

Our

detailed

analysis

of

the

relationship

between

dynamical

distance

between

planets

and

the

instability

time-scale

of

Solar

system

planetary

motion

is

now

on-going.

Although

our

numerical

integrations

span

the

lifetime

of

the

Solar

system，

the

number

of

integrations

is

far

from

sufficient

to

fill

the

initial

phase

space.

It

is

necessary

to

perform

more

and

more

numerical

integrations

to

confirm

and

examine

in

detail

the

long-term

stability

of

our

planetary

dynamics.

——以上文段引自

Ito，

T.&

Tanikawa，

K.

Long-term

integrations

and

stability

of

planetary

orbits

in

our

Solar

System.

Mon.

Not.

R.

Astron.

Soc.

336，

483–500

(2002)

这只是作者君参考的一篇文章，关于太阳系的稳定性。

还有其他论文，不过也都是英文的，相关课题的中文文献很少，那些论文下载一篇要九美元（《Nature》真是暴利），作者君写这篇文章的时候已经回家，不在检测中心，所以没有数据库的使用权，下不起，就不贴上来了。