The orbital evolution of NEAs and the resonances with Jupiter

CHINESE ASTRONOMY AND ASTROPHYSICS Chinese Astronomy

PERGAMON

and Astrophysics

24 (2000)

399-404

The orbital evolution of NEAs and the resonances with Jupiter t * JI Jiang-hui1y4 lPurple

Mountain

Observatory,

“Department 3Shanghai 4National

LIU Lin2t4 Chinese

Astronomical

Chinese

Observatories,

Chinese

Abstract

Abstract

In this paper,

with Jupiter

are studied

(887)

is temporary

this resonance (3552)

Key

210008

210008

Shanghai

of Sciences,

two near-earth-asteroids in the 3:l

could be a source of short-lived

1:l resonant

Nanjing

of Sciences, Academy

Nanjing

200030 3eijing

100012

associated

with

over a time span of lo5 yrs. We found that

trapped

with a large eccentricity

of Sciences,

University,

Academy

resonances asteroid

Academy

Nanjing

of Astronomy,

Observatory,

LIAO Xin-hao3’4

resonance;

NEAs.

thus indicating

that

We also found that asteroid

and a high inclination

is wandering

about

the

region. near-earth-asteroids-arbital

words:

evolution-resonance

1. INTRODUCTION As is well known, resonances

with Jupiter,

Kirkwood

Gaps).

this curious resonance approach

we have concentrations and rarefactions

A great

phenomenon.

and concluded

of main

many famous

belt

at the 2:1,3:1,5:2 papers

Wisdoml’lstudied that chaotic increases

and detailed

the motion

Mars and either collide with Mars or be strongly

of supply for NEAs,

which implies that especially

t Supported by National Received 2000-08-15 * CAA

Natural

around

of fictitious

asteroids

deflected

on

near the 3:l

from their original orbits.

Asteroid)

is temporarily

captured

region can he a source

ones.

Science Foundation

@

(the

would make these asteroids

LETTER

0275-1062/00/Q - see front matter PII: SO275-1062(00)00070-9

regions.

reviews have been written

the bodies in this resonance

the short-lived

the 4:3 and 1:l

and 7:3 resonant

in the eccentricity

In this paper we shall reveal that one NEA (N ear Earth in the 3:l resonance,

asteroids

2000 Elsevier Science B. V. All rights reserved.

400

JI Jiang-hui

et al. / Chinese

At the 1:l resonance, and Trojans existence

that

is now locked libration

of the terrestrial

in a temporary

of the evolution

show that resonance

one NEA

1:l

with a large

the resonant

motion

problem. Pluto),

resonance

with

as follows:

THE

of a NEA moving

The relevant

2 introduces

with Jupiter;

are treated

the

(3753)

coorbital numerical

Here, we will

is trapped

in the 1:l

and Section

all the major

perturbing

The three largest

Accordingly,

in the J2000.0

be uniformly

written

where fi (fj)

are the position The

of Eq. (l),

of orbital

in Liu et a1.18-101. then,

of the major

of the other

rather

(Ceres,

system,

two-body Mercury

the Earth,

to the

than as a combined Pallas and Vesta)

are

1566 (Icarus),

the equations

in the model. of motion

can

planets

(i = l-9),

the Moon (i = lo),

the

= r’, - r,, Y and pi and pj are the perturbing originating

bodies.

(~PN)I

that

is, in the right-sided

from the NEA will not appear

stands

for the post-Newtonian

in

terms

effects of the Sun. is also important

evolution, However,

for calculating

the symplectic

of the motion

as NEAs

often

3. RESONANCES

IN THE

initial values associated

evolution

system,

make close approaches so the stepsize

or the RADAU

ORBITAL

we will use the above dynamical orbital

Hamillton

grows greatly,

integrator[rrl

the motion

algorithm16~71 is advantageous

of the original

when the perturbation

the long-term

(from

such as the asteroid

can be neglected,

terms

the RKF7(8)

In this Section.

asteroids

coordinate

mass of the NEA

of integrator

the whole structure planets,

vectors

the gravitational

of motion

The choice

as a perturbed planets

effects should also be included

and the NEA (.i = 14), &i

in solar units.

in the gravitational

3

as

“big 3” (i = 11-13)

the equations

heliocentric

Section

4 gives a brief discussion.

as it approaches

bodies,

since a few NEAs,

can move very close to the Sun, the relativistic

model;

MODEL

include

as separate

Furthermore,

changed,

made further

the dynamical

the Sun can be regarded

factors

Moreover,

the point

that

with the Earth.

and high inclination

Section

around

perturbing

also taken into account.

masses

the Greeks

investigated

indicates

Apostolosl’l regions

DYNAMICAL

mass at the barycenter.

tigate

which

and, since a NEA can also go near the Moon just

expression

nicknamed

and Innanen[‘]

Earth,

and inclination.

eccentricity

399-404

et al. 13g41found that Cruithne

with coorbital

cases of asteroids

Moon and the Earth

major

of asteroids

Mikkola

Wiegert

.24 (2000)

with Jupiter.

2. The

planets.

of NEAs

This paper is organized describes

groups

with Jupiter.

can occur at large eccentricity

exploration

and Astrophysics

we have the famous

share their orbits

of Trojans

Astronomy

of NEAs.

as it can hold

and it was effectively to one or other of integration

integrato@l

EVOLUTION

model and computational asteroids.

of the

has to be

is preferable.

OF NEAs method

of NEAs over a time span of lo5 yrs. Table

with the two explored

From

to inves1 lists the

Note their large eccentricities.

The

JI Jiang-hui

et al. / Chinese

Astronomy

asteroid (887) is near the 3:l resonant is located in the outer belt.

Table

1

and Astrophysics

region and the asteroid

14 (2000) 399-404

401

(3552) with a high inclination

The Orbital Elements of Two NEAs (Epoch JD 2451400.5)

NO.

a (au)

V)

V)

887

2.4847327

0.5630582

9.30626

110.70680

350.03906

178.08534

3552

4.2324405

0.7140185

30.82683

350.63185

316.70465

302.65404

3.1 The 3:l

Mean

axis a (left)

and

Fig. 1 shows

e

w(“)

W)

Resonance

the variations

in the semi-major

asteroid (887) over the two time spans, i can be written

longitude

t E (0,150OO) and t E (69000,850OO).

(right)

of

The longitude

i = M - 3kfJ + 3(w - ~j~)

where M and MJ are respectively the mean anomalies of the asteroid and Jupiter, and W, WJ, their perihelion longitudes. Throughout this paper, suffix J refers to Jupiter and no suffix, the asteroid.

II 2.40 2.35 2.30 2.25 0

5000 Tim&~)

I

2.60

10000 #887

,

15000

0

5000

10000 Tlme(yr) #887

t

CI 2.40 2.35 2.30 2.25 2.20

’ ’

7000

75000 Time(y)

Fig. 1

80000 #887

The semi-major

Tlme(yr) #887

axis and the longitude

of 3:l resonance

with Jupiter

15000

JI Jiang-hui

402

et al. / Chinese

Astronomy

From the above figures, it is easily noticed

and Astrophysics

that asteroid

24 (2000) 399-404

(887) is temporarily

trapped

in

the 3:l mean resonance with Jupiter, with a executing small vibrations about the resonant value and the resonant argument making the characteristic oscillations. We now present an analytical resonant model to discuss the 3:l resonance of asteroid (887) with Jupiter. For the sake of simplicity, we shall discuss the problem in the framework of Sun-JupiterAsteroid and considering that the inclination and eccentricity of Jupiter are rather small and the inclination of asteroid (887) is also small, we shall use the ideal resonant modelI131, which contains the main secular terms and the 3:l resonant terms in the expansion of the perturbing function. Now we introduce

the action

variable

corresponding

to I,

L=&

(3)

then, in terms of the action-angle can be written as F = Fe(i)

variables

+ F,(L, G) + ~~~~ cos

of (2) and (3), the relevant

Hamiltonian

i

functionI14]

(4)

where

F&i) =

&+3&

F,(LG)

= (5)

F3/1=

G q

( >( &3i+G’,G’=,/m

(P_ro2 [+(I + $e”) + $(l+

5e2)&])

(5)

p.ra3ge2)

In Eq. 4, h J, a J and n J are Jupiter’s mass, semi-major axis and mean motion and G is the gravitational constant. If we take the solar mass & as the unit of mass. UJ as the unit of length, and define the unit of time by [T] = (u;/GM#~ then the gravitational

(6) constant

G will have the value unity.

simplified. As the angular variable g is absent in the hamiltonian formation, g = i, and the integral G = Go (the Delaunay

In these units,

Eq. (5) is further

(4), we have the identical transvariable conjugate to g) in the

Hamiltonian system (4). Accordingly, the eccentricity of the asteroid in Eq.5 can be expressed in terms of de and i, and then (4) b ecomes a one degree of freedom system, and it can be written as F = &(i)

Consequently

+ Fc(i,

60)

+

F3/1 cos

we have the canonical

i

equations

(7) of motion,

(8) From

Eq. 8 we can easily

get the theoretical

phase

diagram

of the 3:l resonance

asteroid

.JI Jiang-hui

(887).

And

agreement

et

we find with

that

al. /

that

Chinese

Astronomy

the phase

of the actual

space

end

Astrophysws

structure

motion;-which

403

24 (SOOO) S99-404

of the analytical indicates

that

model

this asteroid

is in good is indeed

in the resonant state. As mentioned above, the 3:l resonance is associated with one of the Kirkwood gaps in the main belt. As Wisdom pointed out, asteroids in the 3:l resonance will suddenly increase their eccentricities in the course of long-term orbital evolution and leave the resonant region. Fig. 1 also exhibits the repeated entry and exit of the resonance by asteroid (887); this demonstrates to some extent the unstable character of the 3:l resonance scale. Goldstein et al.[15J found the asteroid (887) is exactly synchronized period, which also strongly confirms our results. 3.2

1:l Mean

over a long time at 1:3 Jupiter’s

Resonance

The two left panels

of Fig. 2 display

the variations

in the semi-major

axis and longitude

of asteroid (3552) over the time span t E (0,400OO) and the right panels, the phase diagram, CL- i, with the semi-major axis a instead of 2. When 1 E (13000,37000), a wanders near 5.20. which means during this time the asteroid is temporarily wandering about the 1:l resonance with Jupiter. The right panel shows that the resonant critical argument ,? = M - MJ + (G - GJ) librates slightly during the resonant regime, and that the action variable i correspondingly shows a swinging trend. For t E (37000,40000), the asteroid leaves the resonant region, as is well shown in the right panel. During this time, the longitude goes round the full circle and we have circulation, and the trajectory lies on the farther side of the separatrix, on the edge of the resonance. The movement on the whole is unsteady.

10000

Fig. 2

20000 Tlme(yr) #35X?

The variations

30000

4OGvv

0

90

of the semi-major axis and the longitude resonance with Jupiter

180

270

360

I

of asteriod(3552)

with 1:l

The Trojan asteroids in general have eccentricities less than 0.15 (the average is about 0.0619) and inclinations from a few degrees to 37 degrees, and the orbits of these asteroids are captured relatively steady. But the asteroid (3552) under study is different: it is temporarily by Jupiter and locked in the 1:l resonance, but it will eventually leave the Resonance: this asteroid’s orbit is not as steady as the Trojans’ are.

JI Jiang-hui et al. / Chinese

404

Astronomy

and Astrophysics

24 (2000)

399-404

4. DISCUSSION

The above examples inside

a resonance

approaches asteroid

show that the semi-major

with Jupiter,

to a major

axis of a NEA can sometimes

and so, for a while,

planet.

But the trapping

the asteroid

is generally

be locked

will cease making

temporary;

eventually,

closethe

will leave the resonance.

Our result on asteroid (887) suggests that the 3:l resonance can act as a mechanism of transporting asteroids from the main belt to the near-Earth space, although the actual orbital evolution might be greatly complicated by the overlapping of several main secular resonances[‘6]. Further study will explore the role of overlapping resonances main belt asteroids to Earth-approaching or Earth-crossing asteroids.

pumping

the

References Wisdom

J., Icarus:

1983, 56, 51

Mikkola S., Immnen K. A., AJ, 1992, 104, 1641 Wiegert P. A., Immnen K. A., Mikkola S., Nature 1997, 387, 685 Wiegert P. A., Innanen K. A., Mikkola S., AJ, 1998, 115. 2604 Apostolos

A. Ch.. Icarus, 2000. 144, 1

Feng K., In: Feng K. ed., Proc. Equations,

1984 Beijing Symposium

on Differential

Geometry

and Differential

Science Press, 1985, p. 42

7

Feng K., .I. Comp.

8

Liu L., Ji J. H., Chin. Astron.

9

Liu L., Ji J. H., Liao X. H., Chin. Astron.

Astrophys.,

10

Liu L., Ji J. H., Chin

1999, 23, 273

11

FehIberg E., 1968, NASA TR H-287

12

Everhart E., In: A. Carusi and G. B. VaIsecchi, eds., dynamics of Comtes: Dordrecht:Reidel,

Math., 1986, 4, 279

Astron.

Astrophys., Astrophys.,

1998, 22, 218

Their origin and evolution,

1985, 185

13

Liu L. et al., A&A,

14

Liao X. H., Liu L., The Publications

15

Goldstein

16

FroeschIe Ch., Scholl H., Celest. Mech., 1993, 56, 163

S. J.

1999, 23, 108

1985, SO, 877 of Purple Mountain

, Rudmin J. W., American Astronomical

Observatory,

1989, 8, 229

Society Meeting #193,

1998, 116.02