the role of giant molecular clouds in the evolution of the oort comet cloud

0038-0946/04/3804- © 2004

åÄIä “Nauka

/Interperiodica”0325

Solar System Research, Vol. 38, No. 4, 2004, pp. 325–333. Translated from Astronomicheskii Vestnik, Vol. 38, No. 4, 2004, pp. 372–382.Original Russian Text Copyright © 2004 by Mazeeva.

INTRODUCTION

A spherically symmetric cloud of comets called theOort cloud is assumed to be the source of known com-ets with periods

P

> 200 yr. It is located at a distance of

2

×

10

4

–10

5

AU from the Sun and contains

~10

12

com-ets with nearly parabolic orbits. There is a denser cometcloud inside the Oort cloud (Hills, 1981). Oort cloudcomets are injected into the planetary region under theinfluence of stars and giant molecular clouds passingnear the Sun. The frequency of collisions of the Solarsystem with dense interstellar clouds is much lowerthan the frequency of its encounters with stars, but theirimpact on the orbits of Oort cloud comets is muchstronger.

Molecular clouds are revealed mainly by CO radioemission and ultraviolet interstellar hydrogen absorp-tion lines. A number of papers are devoted to the possi-ble collisions of the Solar system with molecularclouds. Having estimated the influence of interstellarclouds on comets of the Solar system, Biermann (1978)concluded that collisions with interstellar cloudsdeplete significantly the comet cloud at semimajor axesof the cometary orbits

a

> 2.5

×

10

4

AU. Given the forceof such perturbations, the survival of the Oort cloudover the lifetime of the Solar system was called intoquestion. Clube and Napier (1984) hypothesized thatthe existing comet cloud is either Galactic in origin andwas captured when the Sun passed through a giantmolecular cloud or its source is a compact inner cloud(

a

≤

10

3

AU). During each collision with a giant molec-ular cloud, the Roche lobe around the Sun shrinks to

10

4

AU. After a time, the outer Oort cloud is replen-

ished with comets from the inner cloud (

10

3

≤

a

(AU)

≤

2

×

10

4

) (Napier and Staniucha, 1982).

In giant molecular clouds (GMCs), the gravitationalforces are balanced by the large-scale motion of thematter. Dense condensations are formed in molecularclouds under the influence of thermochemical instabil-ity. The mean density in such condensations is

n

(H

2

) =10

4

–10

6

cm

–3

at the mean density over the entire cloudvolume

n

(H

2

) = 10

2

–10

3

cm

–3

. Napier and Staniucha(1982) modeled a GMC as a cluster of 25 condensa-tions. Estimating the gravitational influence of a molec-ular cloud on the evolution of Kuiper belt objects, Stern(1990) pointed out that the dynamical effect of the colli-sion with a dense group of cloudlets would be more sig-nificant than the effect of an extensive tenuous cloud as awhole. The Oort comet cloud has a dynamical analogywith binary stars. Weinberg

et al.

(1987) considered theevolution of soft binary stars under the gravitationalinfluence of a GMC and found the efficiency of such col-lisions to depend on the internal structure of the perturb-ing body. Of greatest importance are collisions duringwhich soft binaries pass through a molecular cloud.

The goal of our study is to compare the two methodsof allowance for the perturbations from GMCs, theimpulse approximation and numerical integration, andto estimate the gravitational influence of molecularclouds with various density distributions on thedestruction of the Oort cloud and the injection of com-ets into the planetary region. In the section entitled“Giant molecular clouds in the Galaxy”, we provide abrief overview of the papers devoted to the possible col-lisions of GMCs with the Solar system and data on the

The Role of Giant Molecular Clouds in the Evolution of the Oort Comet Cloud

O. A. Mazeeva

South-Ural State University, Chelyabinsk, Russia

Received April 8, 2002; in final form, October 21, 2003

Abstract

—We estimated the gravitational influence of giant molecular clouds passing near the Solar system onthe orbital evolution of Oort cloud comets. We performed a comparative analysis of the accuracies of the fol-lowing two methods of allowance for the perturbations from giant molecular clouds: the impulse approximationand numerical integration. The impulse approximation yields fairly accurate estimates of the change in theenergy of Oort cloud comets and the probability of their ejection under the influence of a molecular cloud if thepath of the Solar system does not cross its boundary and if the molecular cloud may be treated as a point per-turbing mass. The comet survival probability in the Oort cloud depends significantly on the internal structureof the perturbing molecular cloud and the impact parameter of the encounter. The most massive injection ofcomets into the planetary region and their ejection from the Oort cloud take place if the Solar system passesthrough a giant molecular cloud composed of several high-mass condensations. In this case, most of the cometsinjected into the planetary region were initially comets of the inner Oort cloud (

a

≤

10

–4

AU) with high orbitaleccentricities.

326

SOLAR SYSTEM RESEARCH

Vol. 38

No. 4

2004

MAZEEVA

distribution and motion of molecular clouds in the Gal-axy. In the section entitled “The Oort cloud”, wedescribe the model of the Solar System comet cloudused here.

GIANT MOLECULAR CLOUDS IN THE GALAXY

After the discovery of GMCs, it emerged that thebulk of the interstellar matter within the solar orbit iscontained in dense massive clouds. Giant molecularclouds are gas condensations with a typical diameter of40 pc and a mean mass of

3

×

10

5

solar masses (

M

S

).Almost half of the mass of the Galactic gas is concen-trated in GMCs whose total number is about 6000. COsurveys of the Milky Way show that the molecular gasin the Galaxy is distributed nonuniformly; it is concen-trated near the Galactic plane. Its distribution along theradius of the Galactic disk is also nonuniform: molecu-lar clouds populate mainly the central disk (

R

G

≤

1

kpc)and the cloud ring (4 kpc

≤

R

G

≤

8

kpc). Moving in cir-cular orbits in the Galactic plane, GMCs, along withyoung stars, have a low dispersion relative to the circu-lar rotation of the Galactic disk. In the solar neighbor-hood, the disk of molecular clouds barely reaches 150 pcin thickness. In addition to the ordered rotation around theGalactic center, molecular clouds also move randomly,mainly under the influence of supernova explosions andstellar winds from O-type and Wolf–Rayet stars.

In the early 1980s, the question of whether the spiralpattern of the Galaxy could be traced in the spatial dis-tribution of GMCs remained unsolved. GMC com-plexes were found to be concentrated in Galactic spiralarm segments (Dame

et al.

, 1986; Cohen

et al.

, 1985).Solomon and Sanders (1985) provided pieces of evi-dence for the presence of molecular clouds between thespiral arms. The question was solved by separating theGMC population into two subsystems, warm and cold.Warm GMCs are concentrated in the Galactic spiralarms, while cold interstellar clouds are encounteredboth in the interarm space and outside the plane of theGalactic disk. Cold clouds account for ? of the entirepopulation of molecular clouds, but warm clouds areconcentrated on the much smaller area occupied by thearms. The heaviest comet showers occurring with aperiod of

~30

×

10

6

yr are assumed (Rampino and Sto-thers, 1984) to be associated with the passage of theSolar system through the Galactic plane, where theprobability of its collisions with GMCs is highest. Thisopinion also has opponents (Bailey

et al.

, 1978).

MOTION OF THE SUN IN THE GALAXYAND ITS COLLISIONS

WITH MOLECULAR CLOUDS

At present, the Sun is very close to the Galacticplane (

~10

pc) and has a velocity component along the

z

axis in the Galactic coordinate system,

v

z

, equal to6 km/s (King, 2002) or 7 km/s (Allen, 1973), which is

approximately equal to the velocity dispersion ofmolecular clouds. Thaddeus and Chanan (1985) arguedthat the distribution of molecular clouds along theGalactic

z

axis is too wide for a periodicity to exist inthe frequency of their collisions with the Solar system.The maximum deviation of the Sun in its verticalmotion relative to the Galactic plane is 80 pc, whichroughly corresponds to the GMC distance limit fromthe Galactic plane at the solar Galactic radius (8.5–10 kpc). It thus follows that the probability of the colli-sion of the Solar system with a molecular cloud near theGalactic plane is slightly higher than that farthest fromit. Talbot and Newman (1977) took into account the factthat the Sun spends most of its lifetime outside theGalactic plane and estimated the number of collisions

in time

t

using the following formula:

N

∝

G

(

ω

,

t

),

where

t

is the time,

σ

||

and

σ

⊥

are the dispersions of thevelocities parallel and perpendicular to the Galacticplane, and

G

(

ω

,

t

)

is a function of the variations in theproperties of the interstellar medium with time andposition of the Sun in the Galaxy. The number of colli-sions is proportional to the velocity of the star (Sun) inthe plane of the Galactic disk, but this numberdecreases if the star is outside the disk for much of itslifetime. Since the interstellar matter in the Galaxy con-tinuously turns into stars, there must have been moregaseous clouds in the past. Talbot and Newman (1977)suggested that the density of the interstellar matterreduced by half over the lifetime of the Sun and con-cluded that the Sun crossed 135 interstellar clouds with

n

(H)

≥

10

2

cm

–3

and about 16 clouds with

n

(H)

≥

10

3

cm

–3

(corresponds to the mean GMC density).

According to the results by Napier (1982), the Solarsystem collides with GMCs with a mean interval of(150–300)

×

10

6

yr. The number of past collisions wasfound to be within the range 10–40. This collision fre-quency was determined by assuming that the distribu-tion of GMCs was random. The Solar system crossesthe Galactic plane with a mean interval of

30

×

10

6

yr.There is a probability of about 25% that the Sun col-lides with GMCs during each crossing of the spiral arm.The mean interval between its passages through the spi-ral arm is (50–60)

×

10

6

yr (Napier and Clube, 1979).Thus,

~20

collisions occur with an interval of

250

×

10

6

yr (Napier and Staniucha, 1982). Scoville andSanders (1986) found the interval between the passagesof the Solar system through GMCs to be

~10

9

yr. Thetypical duration of the passages is

~10

6

yr. Bailey(1983) estimated the number of close collisions overthe lifetime of the Solar system to be ~5 with an uncer-tainty factor of ~2.

A brief review of the previous studies does not givean unequivocal answer to the question of how manytimes the Sun has collided with GMCs over the lifetimeof the Solar system.

tσ||

σ⊥-------

SOLAR SYSTEM RESEARCH

Vol. 38

No. 4

2004

THE ROLE OF GIANT MOLECULAR CLOUDS IN THE EVOLUTION 327

THE INTERNAL STRUCTURE OF GIANT MOLECULAR CLOUDS

Interstellar clouds have a complex nonuniformstructure. There are several gas condensations in thecentral part of GMCs; in addition, they are encounteredover the entire GMC volume. In general, the densityincreases toward the center.

A molecular cloud with evidence of star formationinvestigated by Elmegreen and Lada (1976) is locatednear the Omega emission nebula (M 17). Approxi-mately one third of the cloud mass is contained in con-densations with masses of (5, 10, 10, 15)

×

10

4

M

S

andradii of (3, 3, 4, 6) pc, respectively. Napier (1982)pointed out that a GMC with a mass of

5

×

10

5

M

S

thatpresumably includes 25 condensations with masses of2 × 104MS may be considered to be an acceptablemodel.

To take into account the gravitational influence ofinterstellar clouds on the Oort comet cloud, we used thefollowing three models of the inner GMC structure:

(1) (GMC 1). A spherically symmetric cloud with auniform density distribution (the masses range from 2 ×104 to 5 × 105MS, the radii are RGåC = 5 and 20 pc).

(2) (GMC 2). A cloud composed of 25 condensa-tions (the total GMC mass is MGåC = 5 × 105MS. Thecondensations are uniformly distributed along theradius of the GMC relative to its center (RGåC = 20 pc).

(3) (GMC 3). A cloud composed of 7 condensations(the total GMC mass is MGåC = 1.4 × 105MS). The con-densations in the cloud are concentrated toward thecenter (RGåC = 20 pc).

In turn, the condensations in the molecular cloudwere treated here as homogeneous spheres with radiusRcon = 2 pc and mass Mcon = 2 × 104MS.

A DYNAMICAL MODELWe study the motion of 103 comets initially located

in orbits that correspond to the modeled hypotheticalOort cloud (see below) under the perturbations fromgiant molecular clouds. The molecular cloud movesalong a rectilinear path at a relative velocity of 20 km/s.To model the GMC velocity field in the Galactic coor-dinate system, we used a Schwarzschild triaxial ellip-soidal velocity distribution. The coordinates (xS0, yS0, zS0)that specify the GMC center of mass relative to the Sunin the ecliptic frame of reference were chosen in accor-dance with a random distribution at the time of the clos-est encounter with the Sun. This time is defined as t/2,where t is the lifetime of the molecular cloud (3.3 ×107 yr). The impact parameter (rS0) was varied over therange 0.001 to 0.25 pc.

THE OORT CLOUDOort (1950) hypothesized that the Solar system is

surrounded by a cloud of comets extending to a dis-tance comparable to the distances of the nearest stars.

Having calculated the orbits of 200 long-period cometswith a high accuracy, Marsden et al. (1978) confirmedOort’s hypothesis. Dunkan et al. (1987) modeled theformation of a comet cloud in the Solar system andinvestigated its evolution over a period of 4.6 × 109 yr.The density of the comet distribution is ~r–3.5 ± 0.5; itthus follows that the number of comets in the range[a, a + da] is equivalent to a–1.5 ± 0.5. Let us assume thatthe initial distribution of the orbital semimajor axes forOort cloud comets is f(a) ~ a–2 in the interval ofcometary semimajor axes (3 × 103, 105 AU). The bulk ofthe comet population is located in the inner Oort cloud(3 × 103–2 × 104 AU). According to the results by Hills(1981), the number of comets with perihelia in therange [q, q + dq] is N(q)dq, where N(q) ~ 1 – q/a andq < a. The initial inclinations of the cometary orbitshave a sinusoidal distribution in the interval (0°, 180°);the perihelion argument and the longitude of the ascend-ing node are distributed in the interval (0°, 360°).A comet was considered to be ejected from the sphere ofsolar influence if it passed to a hyperbolic orbit (e > 1)during our calculations or if its semimajor axis was a ≥2 × 105 AU.

NUMERICAL INTEGRATION AND THE IMPULSE APPROXIMATION

AS APPLIED TO THE COLLISION OF THE SOLAR SYSTEM WITH A GMC

The Impulse Approximation

The magnitude of the change in the relative velocityvector of two bodies during their encounter is (Ogorod-

nikov, 1958) ∆v = , where G is the gravitational

constant, r0 is the impact parameter, and M and v arethe mass and relative velocity of the perturbing body,respectively.

It is well known that during the gravitational inter-action between two distant bodies, they may be treatedas point masses. The change in comet velocity relativeto the Sun under the influence of a passing interstellarcloud with a mass MGåC and a relative velocity vGåC is(Torbett, 1986)

(1)

where G is the gravitational constant; ∆vS and ∆vc arethe velocity increments gained by the Sun and thecomet from the molecular cloud; rS0 and rc0 are the min-imum distances from the GMC center of mass to the Sunand the comet, respectively; vGåC is the GMC velocityrelative to the Sun; and MGåC is the GMC mass.

The impulse formula (1) was used to take intoaccount the tidal influence of a distant interstellarcloud. The calculations were performed using the fol-

2GMv r0

-------------

∆v ∆vc ∆vS–2GMGMC

vGMC----------------------

rc0

rc02

------rS0

rS02

-------– ,= =

328

SOLAR SYSTEM RESEARCH Vol. 38 No. 4 2004

MAZEEVA

lowing formula (Biermann, 1979), provided that theSolar system penetrated into the interstellar cloud:

(2)

where ∆v is the velocity increment of the comet or theSun, r0 is the distance from the comet (Sun) to the GMCcenter at the time of the closest encounter, and RGåC isthe GMC radius.

When a spherically symmetric interstellar cloudwith a density uniformly distributed over the entire vol-ume passes near the Sun with an impact parameter rS0 <RGåC, the velocity increment (2) imparted to the pointmass (the Sun or the comet) under the GMC influenceis the sum of two components. The acceleration of thepoint mass inside the molecular cloud is

GMGåCr0/ , and the velocity increment of thepoint mass is

(3)

Outside the molecular cloud, the point mass gains thevelocity increment

(4)

Numerical Integration

A GMC is destroyed through star formation in 107–108 yr (Surdin, 2001). The mean GMC lifetime isdefined as the crossing time of the spiral arm, τ = 3 ×107 yr. Here, we took into account the impact of inter-stellar clouds for 3.3 × 107 yr.

In addition to the impulse formula, we estimated thetidal influence of GMCs on Oort cloud comets bynumerically integrating the equations of relative motionusing the method of Everhart (1974).

We considered two types of interactions between theSun and interstellar clouds: (1) the passage of a GMCnear the Solar system (rS > RGåC) and (2) the collisionof a GMC with the Solar system (rS < RGåC). In the caseof an encounter (rS > RGåC), the GMC is treated as apoint perturbing mass. The equation of motion of acomet relative to the Sun is

(5)

where r is the distance from the comet to the Sun, rS isthe distance from the Sun to the GMC center of mass,and rc is the distance from the comet to the GMC centerof mass.

The Solar system passes through GMCs with a cer-tain interval. When the path of a comet crosses the

∆v2GMGMC

r0vGMC---------------------- 1 1

r02

RGMC2

-------------– 3/2

–

,=

RGMC3

2GMGMCr0

RGMC3

v--------------------------- RGMC

2r0

2– .

2GMGMC

r0RGMCvGMC------------------------------- RGMC RGMC

2r0

2––( ).

d2r

dt2

--------GMSr

r3---------------–

GMGMC rS r–( )

rc3

-------------------------------------GMGMCrS

rS3

------------------------,–+=

GMC boundary (rS < RGåC), Eq. (5) cannot be used todetermine the motion of the comet, because, in thiscase, the molecular cloud cannot be treated as a pointmass. The density distribution inside the GMC beginsto affect the evolution of the cometary orbit. Here, thesize and mass distribution of the perturbing bodyshould be taken into account. No forces act from theouter spherical layer with respect to the comet (Sun),while the inner spherical layer has such attraction as ifits entire mass is concentrated at the center. When apoint mass (the Sun or a comet) penetrate into a molec-ular cloud, it is influenced not by the entire mass of theGMC, but only by its inner layer bounded by a radiusequal to the distance from the center to the point mass.

The mass of the inner layer is = MGåCr3/ ,where RGåC is the GMC radius, and r is the distancefrom the point mass to the GMC center. Taking this intoaccount, we obtain the equation of motion of a cometwhen a GMC passes through the Solar system

(6)

where RGåC is the GMC radius, and is the massof the inner GMC layer.

As was noted above, the condensations in the GMCwere treated as homogeneous spheres. We used Eqs. (5)and (6) to take into account both the tidal influencefrom a GMC with a uniform density distribution andthe influence from an interstellar cloud with an inhomo-geneous internal structure by adding the perturbationsfrom each condensation:

(7)

(8)

(n = 7; 25), (9)

where is the mass of the inner layer of the con-densation, Rcon is the condensation radius, and n is thenumber of condensations.

Comparison of the Efficiencies of the Two Methods of Allowance for Perturbations

We estimated the efficiencies of the impulse approx-imation and the numerical integration used to solve theproblem of the collision between a giant molecularcloud (GMC 1 and 2) and the Solar system. For com-parison, the ejection probability of comets from theOort cloud was plotted against the initial semimajor

MGMC' RGMC3

d2r

dt2

--------GMSr

r3

---------------–GMGMC' r

R3

----------------------,–=

MGMC'

d2ri

dt2

--------- GMSri

ri3

-----------– GMcon( )irci

rci3

----------------------- GMcon( )irSi

rSi3

------------------------,–+=

d2ri

dt2

--------- GMSri

ri3

-----------– GMcon'( )iri

Rcon( )i3

----------------------,–=

d2r

dt2

--------d

2ri

dt2

---------;i 1=

n

∑=

Mcon'



axes (a0) of their orbits (Fig. 1) and the impact parame-ter rS0 (Fig. 2), and the change in the semimajor axes ofcometary orbits ∆a/a0 was plotted against a0 (Fig. 3).

The impulse approximation is based on the assump-tion that the encounter duration is short compared to thecomet orbital period relative to the Sun and that thecomets will not displace in their orbits over the GMCpassage time. Actually, however, this is far from truth.If the solar path (vS = 20 km/s) is assumed to cross thecenter of a GMC with a typical diameter of 40 pc, thena comet (a = 2.5 × 104 AU, P = 4 × 106 yr) will make0.5 of its turn around the Sun in the time spent by theSolar system inside the cloud.

In Fig. 1, the ejection probability of comets from theOort cloud under the tidal influence of a GMC with amass of 5 × 105MS and a radius of 20 pc is plottedagainst the initial semimajor axes (a0) of their orbits.The impulse approximation clearly specifies the bound-ary beyond which the comet cloud will be completelydestroyed. Under the perturbations from a GMC with auniform density distribution (GMC 1), this boundary isspecified at a a ≈ 2 × 104 AU (Fig. 1). Under the pertur-bations from a GMC composed of 25 condensations(GMC 2), all of the comets with semimajor axes a a ≥3 × 104 AU will escape from the sphere of solar influ-ence (Fig. 1). When the impact from an interstellarcloud (GMC 2) is taken into account by numerical inte-gration, there remains a survival probability in the Oortcloud for comets with semimajor axes of their orbitslarger than the limit defined by the impulse approxima-tion. The probability is no more than 50% for cometswith a > 2.5 × 104 AU (Fig. 1).

By varying the impact parameter rS0, we estimatedthe fraction of the comets from the total composition ofthe Oort cloud ejected into interstellar space under the

perturbations from an interstellar cloud (GMC 1 and 2)using both methods of allowance for perturbations. Theimpulse approximation is suitable for estimating theperturbations from GMCs if the path of the Sun passesoutside the GMC limits (rS0 > RGåC) and if the molecu-lar cloud may be treated as a point perturbing mass(Fig. 2). Otherwise, the error of the impulse approxima-tion increases with decreasing impact parameter for bothmodels of the internal GMC structure (GMC 1 and 2).

We can draw the following conclusions from thedata shown in Fig. 2: (1) the impulse approximationoverestimated the comet ejection probability by a factorthat ranges from 2 (rS0 = 0.001 pc) to 1.3 (rS0 = 20 pc)(GMC 1), being, on average, 1.5 (GMC 2); (2) theimpact of an interstellar cloud with an inhomogeneousinternal structure leads to the loss of, on average, twicethe number of comets, from ~3 (rS0 = 0.001 pc) to ~1(rS0 = 20 pc), compared to the impact of a molecularcloud with the same mass, but with a density uniformlydistributed over the entire volume.

The plots in Fig. 3 were constructed for comets thatsurvived in the orbits of the modeled comet cloud afterthe collision of the Solar system with a GMC (rS0 <RGåC). The impulse approximation underestimates thechange in the semimajor axes of cometary orbits in theinner Oort cloud (a < 104 AU) and overestimates theperturbations from the outer cloud, a > 2.5 × 104 AU(Fig. 3). If the entire mass of the perturbing GMC isconcentrated in several massive condensations (GMC2), then the change in the energy (E ~ 1/a) of comets inthe inner (a < 2.5 × 104 AU) Oort cloud will be largerthan that under the perturbations from a sphericallysymmetric interstellar cloud (GMC 1).

The following conclusions are based on the resultsobtained by numerical integration

10

0 20000

20

40000 60000 80000 100000

30405060708090

100

a0, AU

1 2

3

4

rS0 = 5 pc

MGåC = 5 × 105 åSRGåC = 20 pc

Internal GMC structure

2, 3—model GMC 1 (dashed lines)1, 4—model GMC 2 (solid lines)

Method of allowance for perturbations

1, 2—impulse approximation3, 4—numerical integration

Prob

abili

ty o

f co

met

eje

ctio

nfr

om O

ort c

loud

, %

Fig. 1. Probability of comet ejection from the Oort cloud versus initial orbital semimajor axes (a0) under the influence of a GMC(GMC 1 and 2). The two methods of allowance for the perturbations from a GMC were used.

330


MAZEEVA

THE INFLUENCE OF GIANT MOLECULAR CLOUDS ON THE ORBITS

OF OORT CLOUD COMETS

When a molecular cloud with a mass of 5 × 105MS(GMC 1) passes near the Solar system (rS0 = 5 pc), theouter layers of the Oort cloud will be mainly lost. Thetidal influence of a giant molecular cloud composed ofcondensations (GMC 2) will also strongly perturb com-ets of the inner comet reservoir. In this case, ~60% ofthe Oort cloud comets with semimajor axes a = 2.5 ×104 AU will be lost for the Sun (Fig. 1). As regards theinner reservoir of comets (a = 104 AU), 10% of it willbe destroyed, although the region within 104 AU isassumed to be a reliable storage of cometary nuclei. Forthese semimajor axes of cometary orbits, the ejection

probabilities estimated by the two methods of calcula-tion are equal (Fig. 1). If a homogeneous sphere is con-sidered to be a fairly realistic model of an interstellarcloud, at least early in its evolution (before the onset offragmentation), then a comparable loss of the cometpopulation (~60%) under the perturbing influence of aGMC with the above parameters will take place at a ≈4.5 × 104 AU or higher.

Here, we may conclude that the dynamical effect ofthe collision between the Sun and a giant molecularcloud composed of a group of gas condensations will bestronger than the effect of an extensive tenuous cloud asa whole. This is consistent with the conclusion by Stern(1990).

20

0 5

40

10 15 20 25

60

80

100

MGåC = 5 × 105 åSRGåC = 20 pc

GMC density distribution1, 2—Model GMC 1 (solid lines)

3, 4—Model GMC 2 (dashed lines)

Method of allowance for perturbation1, 3—impulse approximatio

2, 4—numerical integration

rS0, pc

µ, %

1

23

4

Fig. 2. µ versus impact parameter rS0; µ is the ratio of the number of ejected comets to the initial number of Oort cloud comets, rS0is the distance between the Sun and the GMC center of mass at the time of the closest encounter. The two methods of allowance forthe perturbations from a GMC were used (GMC 1 and 2).

–3.23.3 3.5 3.7 3.9 4.1 4.3 4.5 4.7 4.9

–2.8

–2.4

–2.0

–1.6

–1.2

–0.8

–0.4

0

0.4

log(a0)

1

2

3

4

MGåC = 5 × 105 åS

MGåC = 25 × 20000 åS

RGåC = 20 pc

MGåC = 5 × 105 åSrS0 = 5 pc

Internal GMC structure

3, 4—model GMC 1 (dashed lines)1, 2—model GMC 2 (solid lines)

Method of allowance for perturbations

1, 3—numerical integration2, 4—impulse approximation

log

(∆a/

a 0)

Fig. 3. Change in the semimajor axes of cometary orbits (∆a/a0) under the influence of a GMC versus initial semimajor axes (a0).



For comparison, let us consider the collision of aless massive molecular cloud (1.4 × 105MS, RGåC =20 pc, rS0 = 0.001 pc) with the Solar system. If theGMC has an inhomogeneous internal structure (GMC 3),then 33% of the region at a = 2.5 × 104 AU (Fig. 4) and17% of the entire Oort cloud population (the table) willbe destroyed. If the density is uniformly distributedover the entire GMC volume, then the inner reservoir ofcomets will survive. Mostly comets of the outer Oortcloud (a ≥ 6 × 104 AU), ~3.7% of its original comet pop-ulation (the table), will be ejected from the sphere ofsolar influence. The greatest loss of the comet cloudcomposition will take place under the tidal impactsfrom the group of small condensations in the GMC(GMC 2 and 3). However, even in this case, the innerOort cloud (a ≤ 104 AU) will not be destroyed com-pletely (Fig. 4).

There are large dense clouds within 1 kpc of the Sunthat are revealed by interstellar extinction (Dame et al.,1987). Ten of the twenty three known molecular cloudshave masses in the interval (3 × 103MS, 4 × 104MS), andonly three molecular clouds have masses higher than5 × 105MS).

Let us assume that a small molecular cloud (MGåC =2 × 104MS, RGåC = 5 pc) passes near the Solar system.The minimum distance rS0 between the Sun and thepath of the GMC center of mass was varied between0.001 to 15 pc. Consider the dependences of µ (the ratioof the number of ejected comets to the initial number ofOort cloud comets) on the impact parameter rS0 (Fig. 5)and the GMC mass MGåC (Fig. 6). The ratio µdecreases linearly with increasing impact parameter fora collision (rS0 < RGåC) and is inversely proportional tothe square of the impact parameter (Fig. 5) for anencounter (rS0 > RGåC). As the GMC (rS0 = 2.5 pc,RGåC = 5 pc) mass increases, µ increases proportionally

to (Fig. 6). Based on the two plots, we can say that

µ is proportional to / when the Solar systemencounters (rS0 > RGåC) with a small molecular cloud.

The comet cloud can be completely lost for theSolar system only if its inner part will be destroyed.Otherwise, the outer part of the Oort cloud will bereplenished with comets from the inner reservoir. Onthe order of 5% of the inner Oort cloud (a ≤ 2.5 ×104 AU) will be destroyed under the gravitational influ-

MGMC0.5

MGMC0.5

rS02

Dependence of µ on the GMC mass and density distribution; µ is the ratio of the number of ejected comets to the initialnumber of Oort cloud comets. rS0 = 0.001 pc

GMC mass, MSGMC densitydistribution µ at a ≤ 105 AU µ at a ≤ 2.5 × 104 AU µ at a ≤ 104 AU

5 × 105 Model GMC 1 0.114 0.005 0.001

25 × (2 × 104) Model GMC 2 0.331 0.145 0.033

1.4 × 104 Model GMC 1 0.037 0.004 0.001

7 × (2 × 104) Model GMC 3 0.17 0.042 0.008

0

10

20000 40000 60000 80000 100000

20

30

40

50

60

70

80

90

100

a0, AU

MGåC = 7 × 20000 åS

MGåC = 25 × 20000 åS

MGåC = 1.4 × 105åS

rS0 = 0.001 pc

1

2

3

RGåC = 20 pcInternal GMC structure

1—model GMC 22—model GMC 33—model GMC 1

Method of allowance for perturbationsnumerical integration

Prob

abili

ty o

f co

met

eje

ctio

nfr

om O

ort c

loud

, %

Fig. 4. Probability of comet ejection from the Oort cloud versus initial orbital semimajor axes under the influence of a GMC(GMC 1, 2, 3).

332


MAZEEVA

ence of a small (RGåC = 5 pc), but massive (MGåC =105MS) molecular cloud with a mean surface density of3.8 × 103 H2/cm3 (rS0 = 0.001 pc). A more massive GMC(MGåC = 5 × 105MS, RGåC = 20 pc, rS0 = 0.001 pc) witha lower density of molecules H2 (n(H2) = 3 × 102 H2/cm3

will destroy 0.5% of the inner Oort cloud (a ≤ 2.5 ×104 AU). The inner Oort cloud is depleted most heavilyduring collisions (rS0 < RGåC) with molecular cloudscomposed of several massive (Mcon = 104–2 × 104MS)gas condensations. During such collisions, the Oortcloud loses ~33% of its total composition (the table),~15% of the comets at a ≤ 2.5 × 104 AU, and ~4% of thecomets from the inner reservoir (a ≤ 104 AU).

The number of collisions between the Solar systemand GMCs is (Bailey, 1986)

(10)

where nGåC is the GMC number density in the Galaxy,T is the lifetime of the Solar system (4.6 × 109 yr), andv is the relative GMC velocity. At MGåC = 5 × 105MS,RGåC = 20 pc, v = 20 km/s, and nGåC = 3.5 × 10–8 pc–3

(for the GMC number density in the Galactic planenGåC(0) = 5 × 10–8 pc–3 and the coefficient of change inthe density of the interstellar medium with time γ = 1.5(Bailey, 1986)), the number of collisions is N = 9.Assume that the comet cloud loses 33% (GMC 2) and11.5% (GMC 1) of its comets during each collision ofthe Sun with a GMC. In this case, 97% (GMC 2) or66% (GMC 1) of the initial composition of the Oortcloud will be destroyed over the lifetime of the Solarsystem through nine collisions with molecular clouds.The large difference between these two values suggeststhat the survival probability of the Oort comet clouddepends significantly on the internal structure of theGMC through which the Solar system passes. Theformer value (97%) was probably overestimated,because the Oort comet cloud exists and has also beensubjected to destruction under stellar, Galactic, andplanetary perturbations over the lifetime of the Solarsystem. The latter value (66%) specifies the lower limit.

The probability of comet ejection from the Oort cloudalso depends on the impact parameter of the encounter;it increases as the latter decreases (Figs. 2 and 5).

The tidal influence of molecular clouds causes theinjection of comets into the planetary region. Hills(1981) determined the fraction of the comets with semi-major axes a that pass within a distance q from the Sununder the influence of passing stars as

(11)

Torbett (1986) used this relation to determine thefraction of the comets injected into the planetary regionunder the influence of a GMC. In this case, χ = 7 × 10–3

for semimajor axes a = 104 AU if the planetary regionis limited by the comet perihelion distance q = 35 AU.Our calculations revealed that when the Solar systemcollides with an interstellar cloud with MGåC = 5 ×105MS, RGåC = 20 pc, and rS0 = 0.001 pc, χ = 9 × 10–3

(GMC 2) and χ = 4 × 10–3 (GMC 1) for comets withsemimajor axes a = 104 AU. The value of χ = 7 × 10–3

calculated using relation (11) is a mean of the abovevalues of χ. The ratio χ decreases with increasingimpact parameter; for example, χ = 5 × 10–3 at rS0 = 10 pcand χ = 1 × 10–3 at rS0 = 20 pc for GMC 2. It thus fol-lows that the most massive comet showers occur whenthe Solar system is closest to the center of mass of aGMC composed of several high-mass condensations.In this case, most of the comets injected into the plane-tary region are comets of the inner Oort cloud (a ≤104 AU) with high orbital eccentricities.

N nGMCπRGMC2

vT 12GMGMC

RGMCv2

----------------------+ ,=

χ 2qa

------.=

1

0 1

4

52 3 4 6 7 8 9 10 11 12 13 14 15

23

56789

10

N ~ r–2

rS0, pc

µ, %MGåC = 2 × 104 MGåC

RGåC = 5 pc

Fig. 5. µ versus impact parameter rS0 (MGåC = 2 × 104MS,GMC 1); µ is the ratio of the number of ejected comets tothe initial number of Oort cloud comets, rS0 is the distancebetween the Sun and the GMC center of mass at the time ofthe closest encounter. The dashed line indicates the bound-ary of the molecular cloud (RGåC = 5 pc) during its colli-sion with the Sun (rS = 0).

8

2 3 4 5 6 7 8 9 10

10

12

14

16

18

20

22

24

MGåC (104 MS)

µ, %

3.1 × 103 H2/cm3

3.89 × 103 H2/cm3

2.3 × 103 H2/cm3

1.6 × 103 H2/cm3

7.78 × 102 H2/cm3

rS0 = 2.5 pcRGåC = 5 pc

N ~ MGåC0.5

Fig. 6. µ versus GMC mass; µ is the ratio of the number ofejected comets to the initial number of Oort cloud comets.The mean GMC H2 densities are indicated. RGåC = 5 pcand rS0 = 2.5 pc.



CONCLUSIONS

Our calculations have led us to the following con-clusions.

The tidal influence of molecular clouds causes theinjection of comets into the planetary region. The mostmassive comet showers occur when the Solar systempasses through a GMC composed of several high-masscondensations. In this case, most of the comets injectedinto the planetary region are initially comets of theinner Oort cloud (a ≤ 104 AU) with high orbital eccen-tricities. The number of comets injected into the plane-tary region under the influence of a passing molecularcloud increases with decreasing impact parameter.

The density distribution inside the giant molecularclouds that perturb the comet cloud has a huge effect onthe number of comets ejected into interstellar space. Theimpact of a GMC with an inhomogeneous internal struc-ture leads to the loss of, on average, twice the number ofcomets compared to the impact of a molecular cloud withthe same mass, but with a density uniformly distributedover its entire volume. The inner Oort cloud is depletedmost heavily during close collisions (rS0 < RGåC) withmolecular clouds composed of several gas condensationswith masses of 104–2 × 104MS). During such collisions,the Oort cloud loses ~33% of its total composition and~4% of the comets of its inner part (a ≤ 104 AU).

The impulse approximation can be used to estimatethe perturbations from a GMC only if the path of the Sunpasses outside the GMC (rS0 > RGåC) and if the molecu-lar cloud may be treated as a point perturbing mass. Oth-erwise, the error of the impulse approximation increaseswith decreasing impact parameter irrespective of theinternal structure of the perturbing interstellar cloud. Theimpulse approximation overestimates the comet ejectionprobability, on average, by a factor of 1.5.

ACKNOWLEDGMENTS

This work was supported by the Russian Foundationfor Basic Research (project no. 01-02-16006).

REFERENCES

Allen, C.W., Astrophysical Quantities, London: Univ. Lon-don, Athlone Press, 1973.

Bailey, M.E., The Mean Energy Transfer Rate to Comets inthe Oort Cloud and Implications for Cometary Origins,Mon. Not. R. Astron. Soc., 1986, vol. 218, pp. 1–30.

Bailey, M.E., The Structure and Evolution of the Solar Sys-tem Comet Cloud, Mon. Not. R. Astron. Soc., 1983,vol. 204, pp. 603–633.

Bailey, M.E., Wilkinson, D.A., and Wolfendale, A.W., CanEpisodic Comet Showers Explain the 30-Myr Cyclicityin the Terrestrial Record?, Mon. Not. R. Astron. Soc.,1987, vol. 227, pp. 863–885.

Biermann, L., Dense Interstellar Clouds and Comets, Astro-nomical Papers Dedicated to Bengt Stromgren, Sympo-sium, Copenhagen, 1978, pp. 327–336.

Clube, S.V.M. and Napier, W.M., Comet Capture FromMolecular Clouds—a Dynamical Constraint on Star andPlanet Formation, Mon. Not. R. Astron. Soc., 1984,vol. 208, pp. 575–588.

Cohen, R.S., Grabelsky, D.M., May, J., et al., MolecularClouds in the Carina Arm, Astrophys. J., 1985, vol. 290,pp. 15–20.

Dame, T.M., Elmegreen, B.G., Cohen, R.S., et al., The Larg-est Molecular Cloud Complexes in the First GalacticQuadrant, Astrophys. J., 1986, vol. 305, pp. 892–908.

Dame, T.M., Ungerechts, H., Cohen, R.S., et al., A Compos-ite CO Survey of the Entire Milky Way, Astrophys. J.,1987, vol. 322, pp. 706–720.

Duncan, M., Quinn, T., and Tremaine, S., The Formation andExtent of the Solar System Comet Cloud, Astron. J.,1987, vol. 94, pp. 1330–1338.

Elmegreen, B.G. and Lada, C.J., Discovery of AnExtended/85 Pc/molecular Cloud Associated with theM17 Star-Forming Complex, Astron. J., 1976, vol. 81,pp. 1089–1094.

Everhart, E., Implicit Single-Sequence Methods for Integrat-ing Orbits, Celest. Mech., 1974, vol. 10, p. 35.

Hills, J.G., Comet Showers and the Steady-State Infall ofComets From the Oort Cloud, Astron. J., 1981, vol. 86,pp. 1730–1740.

King, I.R., An Introduction to Classical Stellar Dynamics,1994. Translated under the title Vvedenie v klas-sicheskuyu zvezdnuyu dinamiku, Editorial URSS, 2002.

Marsden, B.G., Sekanina, Z., and Everhart, E., New Osculat-ing Orbits for 110 Comets and Analysis of OriginalOrbits for 200 Comets, Astron. J., 1978, vol. 83, pp. 64–71.

Napier, W.M. and Clube, S.V., A Theory of Terrestrial Catas-trophism, Nature (London), 1979, vol. 282, pp. 455–459.

Napier, W.M. and Staniucha, M., Interstellar Planetesimals.I—Dissipation of a Primordial Cloud of Comets by TidalEncounters with Massive Nebulae, Mon. Not. R. Astron.Soc., 1982, vol. 198, pp. 723–735.

Napier, W.M., An Interstellar Origin for Comets, Sun andPlanetary System, 1982, pp. 375–378.

Ogorodnikov, K.F., Dinamika zvezdnykh sistem (Dynamicsof Stellar Systems), Moscow: Fizmatgiz, 1958.

Oort, J.H., The Structure of the Cloud of Comets Surround-ing the Solar System and a Hypothesis Concerning ItsOrigin, Bull. Astron. Inst. Neth, 1950, vol. 11, pp. 91–110.

Rampino, M.R. and Stothers, R.B., Terrestrial Mass Extinc-tions, Cometary Impacts and the Sun’s Motion Perpen-dicular to the Galactic Plane, Nature (London), 1984,vol. 308, pp. 709–712.

Scoville, N.Z. and Sanders, D.B., Observational Constraintson the Interaction of Giant Molecular Clouds with theSolar System, The Galaxy and the Solar System, 1986,pp. 69–82.

Solomon, P.M. and Sanders, D.B., Star Formation in a Galac-tic Context—the Location and Properties of MolecularClouds, Protostars and Planets II, 1985, pp. 59–80.

Stern, S.A., External Perturbations on Distant PlanetaryOrbits and Objects in the Kuiper Disk, Celest. Mech.And Dyn. Astron, 1990, vol. 47, pp. 267–273.

Surdin, V.G., Rozhdenie zvezd (The Birth of Stars), Moscow:Editorial URSS, 2001, p. 264.

Talbot, R.J. and Newman, M.J., Encounters Between Starsand Dense Interstellar Clouds, Astrophys. J., Suppl. Ser.,1977, vol. 34, pp. 295–308.

Thaddeus, P. and Chanan, G.A., Cometary Impacts, Molecu-lar Clouds, and the Motion of the Sun Perpendicular tothe Galactic Plane, Nature (London), 1985, vol. 314,pp. 73–75.

Torbett, M.V., Injection of Oort Cloud Comets to the InnerSolar System by Galactic Tidal Fields, Mon. Not. R.Astron. Soc., 1986, vol. 223, pp. 885–895.

Weinberg, M.D., Shapiro, S.L., and Wasserman, I., TheDynamical Fate of Wide Binaries in the Solar Neighbor-hood, Astrophys. J., 1987, vol. 312, pp. 367–389.

the role of giant molecular clouds in the evolution of the oort comet cloud

Documents