Properties of Model Comae around Kuiper Belt and Centaur Objects

ICARUS 135, 415–430 (1998)ARTICLE NO. IS985964

Properties of Model Comae around Kuiper Belt and Centaur Objects

Warren R. Brown and Jane X. Luu

Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, Massachusetts 02138E-mail: [email protected]

Received February 11, 1998; revised May 13, 1998

Chiron, with a diameter of 180 km (Campins et al. 1994,Altenhoff and Stumpff 1995, Bus et al. 1996) and long-The cometary activity of Chiron inspires us to investigate

observable lifetimes of hypothetical dust particles around large, lived cometary activity, has led researchers to investigatedistant Centaur and Kuiper Belt objects (KBOs). Our model its capability to sustain a gravitationally bound coma.computes dust particle trajectories assuming comet nucleus Meech and Belton (1990) assert that Chiron’s long-livedgravity, solar gravity, and solar radiation pressure. We find outbursts are due to coma dust particles on suborbitalthat particle lifetimes are extremely sensitive to the magnitude trajectories which, perturbed by radiation pressure, haveof the initial velocity. Long-lived orbits require an exact combi- residence times of months. Luu and Jewitt (1990) modelnation of nucleus size, heliocentric distance, particle size, and

the surface brightness of Chiron’s coma and find that ainitial velocity. Estimating ejection velocities due to CO subli-bound coma could not contribute more than 10–20% ofmation, we find that particles escape .50,000 km from Cen-the light within 10 of the nucleus; i.e., if the bound comataurs in &6 days and from KBOs in &10 days. Assumingexists, it is not a photometrically important entity. Finally,optimal ejection velocities , vescape , upper limits to bound comaStern et al. (1994) perform numerical simulations of thelifetimes are 50–75 days for Centaurs with Chiron-like activity.

Thus Chiron’s long-lived coma cannot be a result of an outburst. coma grain lifetimes and find that particles do not enterTo observe a KBO coma requires a massive p5 3 109-kg dust the quasi-stable orbits proposed by Meech and Beltoncoma, and its lifetime varies from ,2 months to p1 year de- (1990), but instead have orbital lifetimes of days. However,pending on object size and heliocentric distance. 1998 Aca- the Stern et al. (1994) choice of initial velocities is notdemic Press physically justified, and Meech et al. (1997) argue that

Key Words: comets; Kuiper Belt objects; Centaurs. including rotation and active areas into their model mightgive rise to long-lived orbits.

The cometary activity of Chiron and the fact that Cen-1. INTRODUCTIONtaurs are likely to be precursors of short-period comets,inspires us to investigate the possibility of bound comaeCentaur objects have orbits between Jupiter and Nep-around these large distant bodies and their precursors, thetune. Their planet-crossing orbits are highly unstable, re-KBOs. Previous related work includes photometric obser-sulting in short lifetimes compared to the age of the Solarvations of Centaurs and KBOs, in which we searched forSystem (Hahn and Bailey 1990, Asher and Steel 1993).cometary activity around those objects (e.g., Brown andThe origin of Centaurs is uncertain, but their proximity toLuu 1997). Here, we seek to make predictions about thethe trans-Neptunian Kuiper Belt suggests that they origi-observable lifetimes of hypothetical ‘‘bound’’ dust comaenated in the Belt and were scattered inward and will per-around Centaurs and KBOs and thus the possibility forhaps to turn up as future short-period comets. Their colorsobserving cometary outbursts in these objects.have been found to be similar to those of Kuiper Belt

objects (KBOs), consistent with a Kuiper Belt origin (Luuand Jewitt 1996). The Centaur 2060 Chiron has also exhib- 2. MODELited steady cometary activity for years (Tholen et al. 1988,

The cause of cometary activity is volatiles sublimatingHartmann et al. 1990, Luu and Jewitt 1990, Marcialis andfrom the surface of a comet nucleus. Spacecraft observa-Buratti 1993), proving that Centaurs are icy bodies capabletions of Comet Halley reveal water ice to comprise 80%of outgassing under the right conditions—even at *10 AU.of its volatiles, followed by CO at 12% and CO2 at 2%Unlike known short-period comets, which tend to have(Vanisek 1991), proving that water ice sublimation is thedimensions of 5–10 km in diameter (e.g., Keller 1990),principal driving mechanism of most cometary activity.Centaurs can be as large as hundreds of kilometers in di-

ameter. However, the sublimation of water ice is strongly depen-

4150019-1035/98 $25.00

Copyright 1998 by Academic PressAll rights of reproduction in any form reserved.

416 BROWN AND LUU

dent on temperature and becomes highly inefficient at he- where rSC is the Sun R comet vector. The first term is theliocentric distances greater than 5 AU. At the heliocentric gravitational force due to the comet nucleus. The seconddistance range of the Centaurs and KBOs (R . 5 AU), term is the radiation pressure force due to the Sun, whichcometary activity is most likely driven by CO (e.g., Fanale points in the anti-solar direction. The third term is causedand Salvail 1997), and we focus on mass loss driven by by the divergence of the solar gravitational field, reducedthis volatile. by the radiation pressure force. This force is known as the

In cometary mass loss, a dust particle is entrained in a ‘‘tidal force’’: its magnitude is proportional to rC , and ongas jet emanating from the comet. At some altitude the the Sun–comet line it is directed away from the nucleusgas density falls off sufficiently that the dust particle will (Richter and Keller 1995).decouple from the gas. At a height of 10–20 comet radii, The nucleus of the comet is assumed to be spherical,collisions among dust particles are not likely to be impor- and no higher gravitational moments are included. For 433tant (Grun and Jessberger 1990, Stern et al. 1994). The Eros, with an axis ratio of 2.8, Scheeres (1995) finds solarequation of motion for a particle in the outer coma is thus radiation pressure to dominate the gravity harmonics atdominated by the following forces: comet nucleus gravity, .6 times the effective radius. We note that of the Centaurssolar gravity, and solar radiation pressure. For simplicity, studied so far, only 1995 GO has a significant, nonunitywe ignore electrostatic forces, such as grain charging from axis ratio: a/b $ 1.4 (Brown and Luu 1997). Further, self-the solar UV-driven photoelectric effect. This effect may gravity arguments (Farinella 1987) yield a critical radiuslift charged grains off of a charged comet nucleus, but of p75 km for asteroids and comets, a radius exceeded byshould not alter grain trajectories in a collisionless coma, our sample objects except for 1995 GO. Thus nonsphericalparticularly at large heliocentric distances. The interaction gravitational harmonics should be negligible for Centaursof charged grains with the magnetic field of the solar wind and KBOs, especially at the altitudes we are interested inis also neglected. Furthermore, no consideration is given (.10 comet radii).to particle aggregation or fragmentation in the coma. The Equation (3) is solved using Stoermer’s integrationequation of motion is thus method (Press et al. 1992). We find that Stoermer’s rule

saves a significant amount of computation time over theclassic Runge–Kutta method. In testing various integrationrC 5 2GM( (1 2 b)

rS

r3S

2 GMCrC

r3C

, (1)routines on a critical orbit, the Runge–Kutta method re-quires very small step sizes, taking much longer to repro-duce the results obtained by Stoermer’s rule. The Bulirsch–where G is the gravitational constant, M( is the mass ofStoer method, using the modified midpoint algorithmthe Sun, MC is the mass of the comet, rS is the Sun R(Press et al. 1992), independently yields the same resultsparticle vector, and rC is the comet R particle vector. Thefound with Stoermer’s rule, giving us confidence that theparameter b is the ratio of the solar radiation to the solarintegration routine is reliable. Variable step sizes are cho-gravitational force, and for a spherical particle is given bysen so that there is no significant difference from using(Divine et al. 1986)uniformly small step sizes (1 min of time).

The equations of motion are integrated in a Cartesianb 5

3L(

16fcGM(

Qrd ad

5 0.57 3 1023 Qrd ad

, (2) coordinate system centered on the comet. Prior to eachintegration step, we interpolate the comet’s position fromdaily orbital positions provided by Gareth Williams of the

where L( is the solar luminosity (W), c is the speed ofCenter for Astrophysics.

light (m/s), and rd and ad are the dust particle density(kg/m3) and radius (m), respectively. Q is the scattering

3. INITIAL CONDITIONSefficiency for radiation pressure and depends on the opticalproperties of the particle. Typical values for Q range from

3.1. Nucleus and Dust Properties0.6 to 1.8 (Divine et al. 1986), depending on particle compo-sition and size; here we assume Q Q 1. Since M( @ MC @ The objects under investigation here were selected toMdust , neglecting terms of higher order (rC/rS), Eq. (1) cover a large range of size and heliocentric distance. Tablebecomes (Richter and Keller 1995) I summarizes the physical parameters of the three Centaurs

and three KBOs. A typical 4% geometric albedo is assumedfor all objects other than Chiron, whose albedo was foundrC 5 2GMC

rC

r3C

1 GM( brSC

r3SC

(3)to be 13% (Lebofsky et al. 1984, Sykes and Walker 1991,Altenhoff and Stumpff 1995). The radius of Chiron comesfrom Campins et al. (1994), Altenhoff and Stumpff (1995),2 GM( (1 2 b)

1r3

SCSrC 2 3rSC

rSC 2 rC

r2SC

D ,and Bus et al. (1996), while the radii of the other objects

MODEL COMAE LIFETIMES 417

TABLE IParameters for Test Objects

Physical parameter 2060 Chiron 1995 GO 1997 CU26 1996 TL66 1997 CQ29 1996 TO66

R [AU]a 8.5, 11.3 12.6 13.9 35.2 41.2 45.7Date at R Feb 96, Jan 90 Apr 96 Mar 97 Feb 97 May 97 Jun 97Radius rc [km] 90 54 180 245 170 420Density rc [kg/m3] 1000 1000 1000 1000 1000 1000Albedo pc 0.13 0.04 0.04 0.04 0.04 0.04v*esc [m/s]b 21.3 12.8 42.6 57.9 40.2 99.3vt/v*esc

c 7.0, 6.3 9.9 3.1 1.6 2.1 0.9

a Heliocentric distance.b Escape velocity at launch altitude 10rc .c Ratio of terminal velocity to v*esc .

are estimated from our own photometry. There are no firm (the ‘‘coma’’ being described below). Therefore each0.5 , ad , 10 em particle contributesconstraints on the densities of Centaurs and KBOs, so we

select a typical density of 1000 kg/m3 for all objects.The most efficient scatterers are particles of roughly of

Id 5f1em X pfa2

d Fo R2o

f 4f D2 R2 . (6)the same size as the light wavelength; thus we use the dustradius ad 5 1 em as representative of dust particles in theoptical coma, and ad 5 10 and 100 em for particles for the

The volume scattering function f is taken from Lamyinfrared and submillimeter comae, respectively. The effectset al. (1987), who compute it for astronomical silicate Mieof a dust size distribution are investigated later on. Dustscatterers. The phase function X/4f is assumed to be p1size distributions are usually fit by a power law of the formfor observations made at opposition. The albedo and den-(Lamy et al. 1987)sity of the dust are taken to be p 5 0.04 (Grun and Jess-berger 1990) and 1000 kg/m3, respectively, the same asn(ad) Y a2q

d , (4)the source objects. Equation 6 is also used to estimatehypothetical KBO dust coma (Section 6), which we findwhere n(ad)dad is the number of dust particles with radiusto be good to p1 mag when comparing the Chiron modelin the range ad to ad 1 dad , and q P 3 to 4. Because theagainst observations.dust size distribution for Centaurs and KBOs is unknown,

we assume this typical power law distribution to investigate3.2. Initial Positionthe optical signature of our model dust comae. The bright-

ness at each point in an image of the scattered light from We adopt 10 comet radii as the altitude when the dusta comet dust cloud is given by (Grun and Jessberger 1990) particles decouple from the gas flow and reach terminal

velocity. If a particle drops below an altitude of 10 cometradii, the orbit will quickly decay due to gas drag and weIl 5

Fo (l)R20

R2 El

f(l, u) dlD2 , (5)

consider the particle to ‘‘impact’’ the surface. If, on theother hand, the particle exceeds 50,000 km from the comet,we arbitrarily consider the particle to have ‘‘escaped’’ fromwhere Fo (l) is the irradiance of the Sun at 1 AU (W/m2),

Ro 5 1 AU, R (AU) and D (m) are the heliocentric and the coma. (For example, West (1991) finds the effectivelimit of Chiron’s coma to be 50,000 km in radius.)geocentric distances, respectively, and f(l, u) is the volume

scattering function (m21 sr21). For the distant objects con- We test the effects of launching the coma particles atlatitudes and longitudes uniformly distributed over thesidered here, D is approximately constant over the line of

sight of the integral. While we use 1-em particles to calcu- comet’s surface. It is expected, unless a comet nucleus israpidly rotating, dust will be ejected mainly during thelate characteristic ‘‘optical’’ lifetimes, we allow 0.5 , ad ,

10-em particles to contribute to the optical signature of full sunlit side. However, objects at large heliocentric distancestend to radiate heat from their thermal skins on time scalesmodel comae. Since we are interested in relative brightness

changes, we use f to weight the Mie scattering efficiency longer than their rotation periods, and are described betterby a ‘‘rapid-rotator’’ isothermal latitude model than a stan-of the different radius particles. Finally, we assume opti-

cally thin comae, so that the total brightness of the coma dard thermal model (Sykes and Walker 1991). We findthat the orbital lifetimes are relatively insensitive to initialis just the sum of the brightness of all particles in the coma

418 BROWN AND LUU

FIG. 1. Trajectories for a 10-em particle launched from Chiron at 8.5 AU. Starting at top, left, the velocities are incremented by 0.25%, whichresult in a slow impact trajectory, a long-lived orbit, and a slow escape. Chiron is drawn to scale, the Sun is to 2x.

positions, whether they be on the day or the night side.vesc 5 !8fGrc r3

c

3r(7)We note that under special conditions, orbital lifetimes

depend strongly on initial positions (e.g., Fig. 1). However,averaging over a large number of random day/night initial

for a spherical comet nucleus of density rc and radius rcpositions only produces a p 10% dispersion about theand a particle with the initial launch distance r. For a launch

mean lifetime of a given sized particle. Thus the daytimedistance r 5 10rc , measured from the nucleus center, andconstraint is ignored to sample orbital parameters morea nucleus density rc 5 1000 kg/m3, v*esc 5 0.236rc (m/s) forfully.rc in kilometers. The asterisk serves to remind the readerthat v*esc is not the escape velocity from the nucleus surface,3.3. Initial Velocitybut from an altitude of 10rc . The most interesting range

The most critical parameter, the initial velocity distribu- of velocities lies between the Keplerian velocitytions of different size particles, is not known, so we approxi- 1/Ï2v*esc and escape velocity, since in the absence of radia-mate the initial velocities vo two different ways: (1) tion pressure, these velocities roughly mark the boundaries1/Ï2 v*esc , vo , v*esc and (2) vo P gas velocity. for impacting the surface and escaping from the nucleus.

We note that radiation pressure is strong enough at the3.3.1. Velocity distribution 1/Ï2 v*esc , vo , v*esc . Theescape velocity from the nucleus is given by perihelion distance of Chiron to blow well-placed 1-em


TABLE IIparticles with initial velocities as low as vo 5 0.6v*esc out ofModel Parametersthe coma. On the other hand, 10-em particles launched

with the same vo 5 0.6v*esc all impact within 3 days. GivenParameter Value

the presence of radiation pressure, the Keplerian velocity1/Ï2v*esc is a natural lower limit to consider for long-lived Density rdust 1000 kg/m3

Albedo pdust 0.04 (except 0.13 for Chiron)orbits, while v*esc is a natural upper limit.Particle radius ad 0.5–100 em; power law q 5 4To illustrate how sensitively the orbit lifetime dependsScatt. efficiency Q 1on vo , Fig. 1 shows the effect of incrementing vo by 0.25%.Launch distance 10rcThe three orbits are computed for a 10-em-radius particle Impact r , 10rc

for Chiron at perihelion (R 5 8.45 AU). The initial veloci- Escape r . 50,000 kmVelocity direction Radialties are 0.9275v*esc , 0.9300v*esc , and 0.9325v*esc in the radialVelocity magnitude (a) Determined from CO and CO2 gasdirection. The first launch velocity results in an impact

velocities (with half-Gaussianafter 40 days, the second in a relatively long-lived orbit ofdistribution of HWHM 5 10% vt)

115 days, and the third in a slow escape trajectory, reaching (b) 1/Ï2v*esc , vo , v*esc

50,000 km in 93 days. The long-lived orbit is not unique: No. of test particles 10,000Maximum integration time 365 Daysthere are other long-lived orbits available over the velocity

range 1/Ï2v*esc , vo , v*esc , but Fig. 1 demonstrates thenarrow velocity range needed to achieve them. Initial posi-tion and direction are also critical to produce the long-lived orbits. However, for this scenario, a change of 0.5% equation for a spherical particle barely lifted off the surface

is (Luu and Jewitt 1990)v*esc alters the outcome from an impact to an escape.

3.3.2. vo P gas velocity. The true terminal velocity dis-tribution of coma dust particles is likely to be different d 2r

dt2 53n(R)CDv2

remH

4rad2

43

fGrrc , (8)from 1/Ï2v*esc , vo , v*esc . A proper solution would usea hydrodynamic model to launch particles from the comet

where CD p 1 is the gas drag coefficient, n(R) is the gassurface and calculate their orbital lifetimes. However, pre-density due to sublimation at heliocentric distance R, e isvious works that used hydrodynamic modeling to studythe molecular weight of the gas, mH is the mass of thecometary comae assumed only a single dust size, a singlehydrogen atom, r is the comet and particle density, whichcomet size, and those at a single heliocentric distance (e.g.,are assumed to be equal, ad is the particle radius, rc is theKorosmezey and Gombosi 1990, Kitamura 1986). For ourcomet radius, and G is the gravitational constant. vr Qpurpose we will derive the initial velocities from simplerÏkT/(emH) is the radial speed of the gas, where the tem-considerations.perature T is calculated assuming thermal equilibrium ofThe terminal velocities of dust particles are based onthe comet from solar heating. Setting d2r/dt2 5 0 gives thethe two-phase gas dynamic model of Probstein (1969) andradius of the largest particle which can just be lifted offthe constant effusion model of Weigert (1959). These re-the surface by gas drag.quire knowledge of the thermal gas velocity and the cou-

Shul’man (1977) uses a kinetic equation analysis to findpling scale size L, which in turn require knowledge of thethe terminal gas speed, uy ,gas temperature, the maximum ejected grain size, and the

escape velocity of the gas. For a comet a few AU fromthe Sun, gas molecules escape with typical velocities vr of u2

y &c 1 1c 2 1

u2o , uo 5 0.435vr , (9)

a few 3100 m/s.The largest particle that can be lifted off by the gas flux

where c 5 1.4 (Wallis 1982) is the ratio of specific heatsis determined by the balance between gas drag and thefor the sublimating gas, uo is the initial gas velocity, andgravitational attraction of the comet. The gas flux Fgas isvr is the radial thermal gas speed. The constant effusioncomputed assuming steady-state energy balance on themodel of Weigert (1959) allows us to relate the couplingsublimating surface, (1 2 p)L(/4fR2 5 lFgas , where p isscale size L to the ratiothe object’s albedo, R is the heliocentric distance (m), and

l is the heat of sublimation (J/kg-mol). We use latent heatsL/rmax 5 (vesc/u)2, (10)l tabulated in Brown and Ziegler (1980). We assume that

all the (1 2 p) sunlight absorbed goes into sublimating iceand ignore any effects due to scattered light. The gas den- where rmax is the maximum particle size from Eq. (8), vesc

sity n(R) 5 Fgas/vr assumes that the gas is freely expanding is the gas molecule escape velocity, and u is the gas velocitytaken to be uy from Eq. (9). Wallis (1982) provides a fituniformly from the surface of the comet. The force balance

420 BROWN AND LUU

TABLE IIIParticle Lifetimes about Centaurs and KBOs, 1/Ï2v*esc # vo # v*esc

rdust [em] tmodea t50

a t95a % Escapeb % Impactb

Chiron, 8.5 AU1.0 19 17.4 19.9 88.9 11.110 5 10.7 68.1 32.9 67.1100 5 10.2 221 9.8 89.7

Chiron, 11.3 AU1.0 25 23.1 29.3 79.2 20.810 5 10.4 89.5 24.6 75.4100 5 9.8 163 7.1 89.6

1995 GO, 12.6 AU1.0 27 26.2 30.3 85.5 14.510 5 10.5 97.6 28.6 71.4100 5 10.0 275 7.9 90.3

1997 CU26 , 13.9 AU1.0 5 18.5 37.7 45.3 54.710 5 10.3 133 13.3 86.5100 5 10.2 90.0 6.0 93.1

1996 TL66 , 35.2 AU1.0 5 10.3 113 14.3 85.510 5 10.3 72.1 8.2 91.8100 5 10.3 72.9 8.1 91.9

1997 CQ29 , 41.2 AU1.0 5 10.3 124 14.8 85.110 5 10.0 87.7 5.4 92.8100 5 10.3 103 5.3 95.7

1996 TO66 , 45.7 AU1.0 5 10.3 40.7 14.6 85.410 5 10.3 40.8 14.5 85.5100 5 10.3 40.6 14.5 85.5

a Time to leave coma in days, via impacting the exobase or escaping .50,000 km.b May not sum to 100%, if particles remain after 365 days.

to the particle’s terminal velocity vt from Probstein’s gas escaping perihelion Chiron will remain at 100%, but thetime for half to leave increases by a day and the time fordynamic model,95% to leave nearly doubles; i.e., the escape is moregradual.

ut/uy 51

0.9 1 0.4Ïe 1 1.5Ïad/L, (11)

3.4. Computing Lifetimeswhere e 5 1 is the gas-to-dust ratio of the coma. ForChiron at R 5 8.5 AU, Eq. (11) shows that a 1-em particle To compute typical 1-, 10-, and 100-em particle lifetimes,

10,000 particles of a given size are launched from randomentrained in a CO jet will be accelerated to 150 m/s. Thisis a factor of p4 below velocities calculated for 1-em dust positions at a launch altitude of 10rc . Initial velocities are

randomly selected from 1/Ï2v*esc , vo , v*esc or the half-particles during the Halley encounter (Crifo 1990, Mas-sonne 1990), which gives rough quantitative agreement Gaussian distribution below vo P gas velocity. We follow

Stern et al. (1994) in stating the times for 50% (t50) andconsidering the different heliocentric distances of the twocomets. Note that even if vt is overestimated by a factor 95% (t95) of the particles to leave the coma by either im-

pacting the exobase or escaping beyond 50,000 km. t50 isof 5, at Chiron’s perihelion distance, it will still exceed thenucleus’s escape velocity at launch altitude 10rc . a half-life, while t95 is a typical time for all the particles to

leave. We also present tmode , which is the day during whichFinally, we arbitrarily try adopting a narrow half-Gaussian distribution with maximum velocity vt and a half- the most number of particles leave the coma. The particle

orbits are integrated for a maximum of 365 days, at whichwidth-at-half-maximum (HWHM) equal to 0.1vt to ac-count for the spread of velocities below vt . We find that point the percentage of particles that have impacted, es-

caped, or still remain in orbit is determined.if we increase the HWHM to 0.33vt , the fraction of particles


TABLE IVParticle Lifetimes about Centaurs and KBOs, vo P Gas Velocity

rdust [em] tmodea t50

a t95a % Escapeb % Impactb

Chiron, 8.5 AU1.0 4 4.4 5.0 100 0.010 5 5.1 5.8 100 0.0100 7 7.4 8.4 100 0.0

Chiron, 11.3 AU1.0 4 4.8 5.5 100 0.010 5 5.9 6.7 100 0.0100 9 9.6 11.0 100 0.0

1995 GO, 12.6 AU1.0 5 5.3 6.0 100 0.010 6 6.8 7.7 100 0.0100 11 11.8 13.4 100 0.0

1997 CU26 , 13.9 AU1.0 4 4.9 5.7 100 0.010 5 6.0 7.0 100 0.0100 10 10.4 12.7 100 0.0

1996 TL66 , 35.2 AU1.0 7 7.9 9.7 100 0.010 14 16.2 34.6 95.3 4.7100 2 2.1 2.4 0.0 100

1997 CQ29 , 41.2 AU1.0 8 8.1 9.5 100 0.010 14 15.2 20.5 100 0.0100 2 2.6 3.1 0.0 100

1996 TO66 , 45.7 AU1.0 6 9.0 15.0 0.0 10010 2 2.3 2.6 0.0 100100 0 0.8 0.8 0.0 100

a Time to leave coma in days, via impacting the exobase or escaping .50,000 km.b May not sum to 100%, if particles remain after 365 days.

3.5. Initial Velocity Directions First we investigate the effects of nucleus rotation onparticle lifetimes. Chiron has a well determined rotation

We assume that particle terminal velocities are primarilyperiod of p5.9 h (Bus et al. 1989, Luu and Jewitt 1990,

in the radial direction. Detailed 2D hydrodynamic model- Marcialis and Buratti 1993); thus we try including an arbi-ing by Kitamura (1986) shows that a horizontal pressure trary, large tangential component of 25%v*esc in the initialgradient creates a lateral gas flow in an expanding gas velocity. The resulting t50 and t95 times increase by #20%jet, which accelerates the dust particles in the horizontal for 1- to 100-em particles, compared to lifetimes computeddirection and broadens a jet. More recent work by Koros- for pure radial launch velocities. We do not expect largemezey and Gombosi (1990) shows that the radial pressure tangential motions at 10rc , and this brief exercise showsgradient near the surface is much larger than this azimuthal that rotation of the nucleus does not significantly altergradient and that, beyond 3 comet radii, the dusty gas flow radial velocity particle lifetimes.becomes approximately radial and a jet cone is formed. Next, we investigate lifetimes for particles launched withTherefore, radial ejection of the dust particles is a reason- nonradial velocities randomly distributed inside 108, 258,able assumption in our region of interest, but it is nonethe- 508, and 758 opening angle jets, placed at the subsolar point.less worthwhile to investigate the effects of nonradial ve- For each jet, all the particles impact within 4 days. Welocities. increase the 1/Ï2v*esc , vo , v*esc range up to 3v*esc to sample

3.5.1. Nonradial velocities. We test the effects of non- possible long-lived orbits and find the t50 and t95 lifetimesradial velocities by applying the models to Chiron at peri- to differ by ,5% compared to the corresponding radialhelion. We are interested in how significantly nonradial launch case. The fraction resulting in impacts increases byvelocities will change particle lifetimes and present results a factor &2 relative to the radial launch case.in terms of the baseline radial velocity case (top of Ta- Finally, we investigate a combination of rotation and

jets. Using the same 25%v*esc rotation velocity andble III).

422 BROWN AND LUU

FIG. 2. Lifetime plots for particles launched by CO sublimation (vo P gas velocity). The fraction of particles remaining in the coma is shownversus time. The coma is defined to be 10 comet radii , r , 50,000 km. The solid, dotted, and dashed lines represent 1-, 10-, and 100-em particlelifetimes, respectively. (a) Lifetimes for Chiron (rc 5 90 km) at 8.5 AU, (b) lifetimes for Chiron at 11.3 AU, (c) lifetimes for 1995 GO (rc 5 54km) at 12.6 AU, (d) lifetimes for 1997 CU26 (rc 5 180 km) at 13.9 AU, (e) lifetimes for 1996 TL66 (rc 5 245 km) at 35.2 AU, (f) lifetimes for 1997CQ29 (rc 5 170 km) at 41.2 AU, and (g) lifetimes for 1996 TO66 (rc 5 420 km) at 45.7 AU.

1/Ï2v*esc , vo , v*esc jet velocities, we find that all but a 4. RESULTSfew 100-em particles impact within p5 days. Again, we

The lifetimes of 1-, 10-, and 100-em dust particles in theincrease the upper jet velocities to 3v*esc to sample possiblemodel comae are given in Tables III and IV. Ten thousandlong-lived orbits. We find that t50 and t95 lifetimes to differparticles were used in each Monte Carlo simulation, andby ,8% relative to the radial launch case and the fractionlifetimes were calculated for both the vo P gas velocityof particles resulting in impacts to increase by a factordistribution (assuming CO as the driving volatile) and the&2. Thus nonradial jet velocities may increase the percent-1/Ï2v*esc , vo , v*esc distribution. For the latter distribution,age of impacts, but will not significantly affect the lifetimesvelocities are uniformly, randomly selected within the pre-of particles with terminal velocities .v*esc . For these mod-scribed bounds. Figure 2 plots the fraction of particlesels, the critical parameter is the magnitude, not direction,remaining in the comae as a function of time for the vo Pof the initial velocity. Table II summarizes the model pa-

rameter values. gas velocity distribution. Figure 3 plots the fraction of


FIG. 2—Continued

particles remaining in the comae as a function of time for particles can avoid surface impact because v*esc is low andsolar radiation pressure has time to accelerate the particlesthe 1/Ï2v*esc , vo , v*esc distribution. The lines in Fig. 3

shows two interesting downward ‘‘kinks,’’ e.g., the dotted out of the coma. Increasing object rc at constant R increasesthe percentage of impacts, and decreases the half-lives ofline (for 10-em particles). The first downward drop is

caused by low-velocity particles falling below the 10 comet the particle.The 1/Ï2v*esc , vo , v*esc model t95 times in Fig. 4b areradius altitude, and the second drop occurs when the high-

velocity particles move beyond the 50,000-km coma limit. best understood in terms of the critical radius rcrit , thedistance in the sunward direction at which radiation pres-We extend the 1/Ï2v*esc , vo , v*esc models from Fig. 3

to compute lifetimes over a broad range of object radius sure and the gravitational attraction of the comet sum tozero. Setting rC 5 0 and ignoring the tidal force in Eq. (3),(rc 5 20 to 200 km) and heliocentric distance (R 5 4 to

40 AU). Figure 4a plots the fraction of 1-em particles that Eqs. (2) and (3) yield rcrit Y Ïadr3c R. Figure 4b reveals

how rcrit interacts with the model coma limit r50 5 50,000impact the exobase and Fig. 4b the corresponding time for95% of the particles to leave the coma, as a function of km. For rcrit , 0.5r50 , the t95 lifetime increases linearly with

heliocentric distance R. The maximum values of t95 occurobject radius and heliocentric distance. As expected for alarge object rc at large heliocentric distance R, particles when rcrit P r50 , and so objects with smaller radii rc , at a

given R, will have more long-lived particles. The long-livedalmost always impact the nucleus because of the relativelylarge gravity of the nucleus. However, for small rc objects, particles vanish as rcrit becomes greater than r50 : t95 drops as

424 BROWN AND LUU

FIG. 3. Same as Fig. 2, except with launch velocities 1/Ï2v*esc , vo , v*esc . The initial drops are due to impacts—particles falling below 10comet radii—and the second drops are due to particles escaping the 50,000-km coma limit.

comet gravity dominates radiation pressure, causing most vations and size determinations. We choose 2 Centaursparticles to impact (Fig. 4a). with different sizes but roughly similar heliocentric dis-

tances: 1995 GO (rc 5 54 km) at R 5 12.6 AU and 19975. DISCUSSION CU26 (rc 5 180 km) at R 5 13.9 AU. 1995 GO is p37

times less massive than 1997 CU26 , and one would expectIn this section we examine the effects of nucleus size 1995 GO’s weaker gravity to result in shorter escape times.

and heliocentric distance on dust particle lifetimes, assum- However, 1-em particles escape both comae in 5–6 days.ing the two different velocity distributions: vo P gas veloc- In the case of KBOs, the dust velocities are comparableity and 1/Ï2v*esc , vo , v*esc . to the nuclei’s escape velocities. 1- to 10-em particles

launched from rc p 200-km KBOs can exceed the escape5.1. Effects of Nucleus Size velocity and slowly move out of the KBO coma. However,

100-em particles cannot be launched or, if they are(a) vo P gas velocity. In general, for Centaurs, gas ve-launched, immediately fall back onto the surface. We com-locities exceed the escape velocities, and particles havepare the KBOs 1997 CQ29 and 1996 TO66 to illustrate thetrajectories that reach 50,000 km in 5–10 days. The gravita-effect of different nucleus size on coma lifetimes.tional influence of the nuclei on these short lifetimes is

Table IV and Fig. 2f show that for 1997 CQ29 , 1- andinsignificant. As an illustration, we compare model resultsof previously mentioned objects for which we have obser- 10-em particles have velocities . v*esc and escape within


FIG. 3—Continued

10 and 20 days. For the 15 times more massive 1996 TO66 , Applying the 1/Ï2v*esc , vo , v*esc model to the KBOsCQ29 and 1996 TO66 (different rc , similar R), Figs. 3e andonly 1-em particles are able to achieve orbits that last p15

days. All other particles impact 1996 TO66 within 1–3 days. 3f show that 1997 CQ29 allows much longer t95 lifetimesthan 1996 TO66 . This is because 1997 CQ29 is 2.5 timesThus at large R, where the gas velocities are comparable

to the escape velocity, the nucleus size is an important smaller than 1996 TO66 , and rcrit Y r3/2c is closer to r50 . Of

1997 CQ29’s 10-em particles, 1.8% still remain in orbit afterfactor in the the particle lifetimes. In general, the KBOs’combination of large size and large heliocentric distance 365 days. On the other hand, t95 times for 1996 TO66 are

41 days regardless of particle size since rcrit . 50,000 km.prevents dust particles from having long-lived orbits.

(b) 1/Ï2v*esc , vo , v*esc. Now, applying the 1/Ï2v*esc

, vo , v*esc model to Centaurs 1995 GO and 1997 CU26 5.2. Effects of Heliocentric Distance(different rc , similar R), we find that the particle lifetimesare roughly comparable for both objects. The 10- and (a) vo P gas velocity. Gas velocities decrease with in-

creasing heliocentric distance R, and one expects particles100-em particle t50 times around 1995 GO and 1997 CU26

are all p10 days. However, the larger nucleus of 1997 CU26 to take longer to escape from a coma. Comparing vo Pgas velocity model results for Chiron at R 5 8.5 AU (peri-causes a larger percentage of impacts than 1995 GO, and

the 1997 CU26 1-em half-life is 8 days shorter because helion) and R 5 11.3 AU, the larger R indeed results ina 10–20% increase in the half-lives, though all particlesof this.

426 BROWN AND LUU

FIG. 4. General 1/Ï2v*esc , vo , v*esc model results for 1-em particles. (a) Fraction of particles impacting the exobase, (b) t95 lifetime for agiven heliocentric distance 2.5 , R , 50 AU and object radius 25 , rc , 500 km.

eventually escape. How do these particle escape times com- means thermal gas velocities are slower, and particle life-times longer.pare with observations?

Assuming optically dominant grain sizes of 1 em, Luu (b) 1/Ï2v*esc , vo , v*esc . With this velocity distributionand Jewitt (1990) estimate a p1.4 3 107 kg coma for Chiron at small R, solar radiation pressure accelerates 1-em parti-at 11.3 AU. We find that t95 lifetime for Chiron’s 1-em cles out of the coma, while at large R, most of the vo ,particles to be 5.5 days. If the coma is ejected over the v*esc particles impact after relatively long-lived lifetimes.course of this coma-residence time, then the mean mass Thus 1-em particle lifetimes are controlled primarily byloss rate is 30 kg/s. This is in agreement with the 20 6 10 the strength of solar radiation pressure. Lifetimes of 10-emkg/s mass loss rate determined by West (1991). On the particles are also controlled by radiation pressure, but toother hand, 30 kg/s in large compared to 0.4 kg/s deter- a lesser extent than 1-em particles, and an appropriatemined by Luu and Jewitt (1990) and 4 kg/s determined by combination of R and rc (Y v*esc) will produce the longestMeech and Belton (1990). Note that these works assume t95 residence times.much lower dust velocities and hence longer coma resi- Applying the 1/Ï2v*esc , vo , v*esc model to Chiron atdence times. For example, Luu and Jewitt (1990) assume R 5 8.5 AU and at R 5 11.3 AU, we find that weakera sunward velocity of 100 m/s (from Chiron’s surface) for radiation pressure at 11.3 AU leads to a 40% increase in1-em particles which results in a lifetime of 1 year. The the 1-em lifetimes and an increase from 10 to 20% in theratio of 1 year to 5.5 days fully accounts for the much lower fraction of impacting particles. b is 100 times smaller formass loss estimate. The observed mass loss estimates thus 100-em particles, and 90% of them impact Chiron with aappear to agree with our model values. The short lifetimes characteristic t50 of 10 days, independent of its heliocen-imply that Chiron’s coma is not a result of a single outburst, tric distance.as computed here, but a result of a more continuous outgas- Radiation pressure is weak at KBO distances, and so 1-,sing process. 10-, and 100-em particles in the 1/Ï2v*esc , vo , v*esc KBO

Two KBOs of similar size but at different heliocentric models share similar lifetimes and impact percentages.distance R are 1996 TL66 and 1997 CQ29 . Table IV shows With minimal perturbation from radiation pressure, thethat for both objects, 1- and 10-em particles launched by velocity distribution 1/Ï2v*esc , vo , v*esc generally resultsCO escape, while 100-em particles impact (with different in longer orbital lifetimes, followed by impacts.timescales). Extending the calculations to CO2 , we notethat at the larger distance of 1997 CQ29 , 10-em particles

5.3. Effects of Low Density Nucleilaunched by CO2 have velocities comparable to v*esc . CO2

particle velocities are not comparable to v*esc for 1996 TL66 . Meech et al. (1997), based on observations of Chiron’sexopause distance, find Chiron’s density to be ,103 kg/The slower 1997 CQ29 10-em CO2 velocities allow for a

longer interaction with radiation pressure, and the t95 times m3. Thus we briefly investigate our models with cometdensities of 500 kg/m3. Since the exopause boundary isare 7 times longer. While rcrit , 50,000 km, a larger R


TABLE Vproportional to r0.5c (Meech et al. 1997), the launch distance

Starting R mag of KBO Outburstand exobase are lowered to 7 comet radii for these lowdensity cases. All other parameters remain the same.

Mc [kg]The effect of rc 5 500 kg/m3 on the Centaur and KBO

models is not large. The general results for particles KBO 2.5 3 107 5 3 108 5 3 109 5 3 1010

launched with bound velocities 1/Ï2v*esc , vo , v*esc areTL66 R mag 24.0 20.8 18.3 15.8lifetimes shorter by ,30%, and escape percentages in-CQ29 R mag 24.7 21.4 18.9 16.4creased by ,5% over Table III values. For particlesTO66 R mag 25.2 21.9 19.4 16.9

launched with vo P gas velocity from Centaurs, the t50 andt95 lifetimes increase by no more than 25% and 100% ofthe particles continue to escape. For particles launchedwith vo P gas velocity from KBOs, the lower KBO mass Table V. As described before, only grains in the size range

0.5 , ad , 10 em are assumed to contribute to the optical(and corresponding lower escape velocity) results in life-times shorter by ,30% compared to Table IV values. signature. The brightness of the outbursts are estimated

by finding the number of particles needed for the comamass and summing up the optical contribution using Eq.6. COMA LIFETIMES(6). Using a q 5 3 power law causes the magnitude estimateto become fainter by 2.5 mag. This shows the sensitivityWith the increasing number of known large KBOs and

Centaurs, an interesting question is whether a cometary of the outburst magnitude to the fraction of mass lockedup in a few, larger particles. A dust coma mass of 5 3 1010outburst can be observed on these objects, and, if so, for

how long. For example, cometary activity at large distances kg seems overly optimistic, while a 107-kg Chiron-like comamass will not lead to an observable increase in magnitude.may be triggered by strong impacts, since these objects are

known to suffer from collisions (e.g., Stern 1995). A strong It is interesting, however, that an outburst dust mass of5 3 108 to 5 3 109 kg leads to observable brighteningsimpact may puncture the surface and eject excavated mate-

rial into orbit. Assuming power law size distributions for that would more than double the brightness of currentlyknown KBOs.KBOs, Luu and Jewitt (1996) calculate collision time scales

for 0.1 $ ri $ 1-km impactors to be t p 107 years for a The 1/Ï2v*esc , vo , v*esc velocity distribution favorslong-lived orbits, and so we use this model to find uppersize index of q 5 2.5 and t p 106 years for q 5 3.0.

We can calculate the amount of material that can be limits on the observable lifetimes of cometary outburstson Centaurs and KBOs. Figure 5 shows the time evolutionejected in a collision using scaling laws from laboratory

experiments of cratering in ice at 81 K (Lange and Ahrens of these model (bound) comae. Each simulation starts with20,000 particles in the range 0.5 , ad , 100 em, distributed1987). With object densities of 1000 kg/m3 and a character-

istic impact velocity of 4 km/s (Luu and Jewitt 1996), the in a q 5 4 size distribution. All particles are launched atthe same time. The Sun is in the negative x direction formass ejected by a r 5 0.1- and 1-km impactor will be 5 3

1012 and 5 3 1015 kg, repectively. These ejecta masses are all objects, and the x–y plane is the orbital plane of theobjects. The coma expansion of the smaller objects islarge compared to the observed mass in Chiron’s coma

(p107 kg; Luu and Jewitt 1990), but only a small fraction slower because v*esc is proportional to the size of the object.The ‘‘line’’ appearance in the Centaur plots is due to theof them will enter interesting long-lived orbits. For consis-

tency with the models presented here, we continue to as- projection of a cylindrically symmetric distribution (dueto radiation pressure) onto the x–y plane.sume that the dust coma will decouple from the gas at a

distance of 10 comet radii. If we assume the coma has a brightness comparable tothe nucleus at the outburst, then the coma contributionTheoretical and experimental investigations show that

most of the mass displaced by impact leaves the crater at drops to p0.1 mag in 50 days for Chiron at 8.5 AU and60 days for Chiron at 11.3 AU. Extending the comparablespeeds less than vesc , even for a 6 km/s impact into a hard,

basaltic target (Housen et al. 1983). Thus a fraction of brightness assumption to the other Centaurs, we find thatthe coma contribution drops to p0.1 mag in 75 days forthe ejecta may fall in the interesting velocity range

1/Ï2vesc , vo , vesc . However, only a small fraction of 1995 GO and 70 days for 1997 CU26 . A 0.1-mag increasein brightness is also a 3s detection at a signal-to-noise ratiothe total mass will be in the form of dust grains with sizes

0.5 , ad , 100 em. We thus arbitrarily adopt the mass (S/N) of 30—easy to measure in the case of Chiron, butmore difficult for the smaller Centaurs (e.g., 1995 GO).range 5 3 107–1010 kg for the coma mass, but consider a

coma mass as little as that of Chiron’s 2.5 3 107 kg coma. We thus consider 0.1 mag a practical detection limit for aCentaur outburst. Thus coma particles launched with theDistributing these masses into a q 5 4 power law size

distribution of particles between 0.5 and 100 em, we predict optimistic 1/Ï2v*esc , vo , v*esc will be observable for p2months around typical Centaurs.the magnitudes of the coma outbursts, and list them in

428 BROWN AND LUU

FIG. 5. The evolution of 20,000 particles of 0.5–100 em launched with 1/Ï2v*esc , vo , v*esc . The plots show the bound coma inside 50,000 km.The plots are 120,000 km on a side, which corresponds to p160 at 10 AU and p40 at 40 AU. (a) Outburst for Chiron at 8.5 AU, (b) outburst for1995 GO at 12.6 AU, (c) outburst for 1997 CU26 at 13.9 AU, and (d) outburst for 1996 TL66 at 35.2 AU. Outburst plots for 1997 CQ29 and 1996TO66 appear nearly identical to 1996 TL66 and thus are not shown.


KBO comae disperse more slowly than those of Cen- 4. Cometary activity driven by CO ejecta particles atvelocities near the escape velocities of the KBOs. As ataurs. However, the magnitude estimates of Table V show

that a Chiron-like coma dust mass of 2.5 3 107 kg will be result, an optical outburst is detectable for &10 daysaround 1996 TL66 , 1997 CQ29 , and 1996 TO66 .p3 mag fainter than the KBO itself. A 0.3-mag increase

in brightness is a 3s detection at S/N 5 10. Using this 5. Optical coma lifetimes increase with heliocentric dis-tance for rcrit , 50,000 km. When comet gravity dominatesas a practical detection limit, a 5 3 108-kg coma will be

observable for 12 days around 1996 TL66 , 80 days around radiation pressure in the coma region (rcrit . 50,000 km),long-lived orbits become less probable and finally impossi-1997 CQ29 , and not at all around 1996 TO66 because of

increasing heliocentric distance. A coma 10 times more ble. In general, particles launched near v*esc have the longestinteraction time with radiation pressure, and are moremassive will be observable for 80 days around 1996 TL66 ,

1 year around 1997 CQ29 , and 40 days for 1996 TO66 . Thus, likely to have long-lived orbits that eventually impactthe surface.only a 5 3 109-kg outburst of 0.5- to 100-em particles,

launched with 1/Ï2v*esc , vo , v*esc , might be detectable 6. Using the initial velocity distribution 1/Ï2v*esc , vo ,v*esc , upper limits to the ‘‘bound coma’’ lifetimes for Chiron,around KBOs.1995 GO, and 1997 CU26 are 50–75 days. Since launch

6.1. Comparison with Previous Work velocities are proportional to object size, longer dust or-bital lifetimes are possible with smaller nuclei, with rcrit ,Stern et al. (1994) perform similar calculations of dust50,000 km.lifetimes in Chiron’s coma, and, in agreement with this

7. Impacts may provide an (infrequent) mechanism forwork, conclude that Chiron’s coma is not gravitationallyoutbursts on KBOs. Assuming a (large) 5 3 109 kg coma,bound. However, we are unable to reproduce the Sternwith a q 5 4 dust size distribution and initial velocityet al. particle lifetimes with our code. In attempting to1/Ï2vesc , vo , vesc , a bound coma can be detected forreproduce Fig. 1 of Stern et al., our model does not yield&2 months for 1996 TL66 and 1996 TO66 and 80 days–1the large number of short-lived particles, and instead showyear for the smaller 1997 CQ29 .roughly discrete lifetimes due to the discrete launch veloci-

ties used by Stern et al. As far as we can tell, our calculationsACKNOWLEDGMENTSdiffer from those of Stern et al. only in the integrator and

the exact form of the equation of motion used. We noteSupport for this work was provided by NASA Grant NAG5-4132 tothat the expression vesc 5 Ï(8/3)fGrchiron r, as given by J. Luu.

Stern et al. in their Section 2, is correct only for the surfaceof the object and not for a launch distance of 10 Chiron

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Properties of Model Comae around Kuiper Belt and Centaur Objects