Redistribution of CO at the Location of the CO Ice Line in evolving Gas and Dust Disks

1.2 MB·

by  Sebastian Markus Stammler

Astronomy&Astrophysics (cid:13)cESO2017
February2,2017
Redistribution of CO at the location of the CO ice line in evolving
gas and dust disks
SebastianMarkusStammler1,2 (cid:63),TilmanBirnstiel3,4,OljaPanic´5,6 (cid:63)(cid:63),CornelisPetrusDullemond1,andCarsten
Dominik7
1 HeidelbergUniversity,CenterforAstronomy,InstituteofTheoreticalAstrophysics,Albert-Ueberle-Straße2,69120Heidelberg,
Germany
e-mail:[email protected]
2 InstitutodeAstrofísica,FacultaddeFísica,PontificiaUniversidadCatólicadeChile,Santiago,Chile
3 MaxPlanckInstituteforAstronomy,Königstuhl17,69117Heidelberg,Germany
7 4 Harvard-SmithsonianCenterforAstrophysics,60GardenStreet,Cambridge,MA02138,USA
1 5 SchoolofPhysicsandAstronomy,TheUniversityofLeeds,E.C.StonerBuilding,LeedsLS29JT,UK
0 6 InstituteofAstronomy,UniversityofCambridge,MadingleyRoad,Cambridge,CB30HA,UK
2 7 UniversityofAmsterdam,AstronomicalInstituteAntonPannekoek,Postbus94249,1090GEAmsterdam,TheNetherlands

n
a
J ABSTRACT

1
Context.Icelinesaresuggestedtoplayasignificantroleingraingrowthandplanetesimalformationinprotoplanetarydisks.Evap-
3
orationfrontsdirectlyinfluencethegasandiceabundancesofvolatilespeciesinthediskandthereforethecoagulationphysicsand

efficiencyandthechemicalcompositionoftheresultingplanetesimals.
]
P Aims.Inthiswork,weinvestigatetheinfluenceoftheexistenceoftheCOicelineonparticlegrowthandonthedistributionofCOin
E thedisk.
Methods.WeincludethepossibilityoftrackingtheCOcontentand/orothervolatilesinparticlesandinthegasinourexistingdust
.
h coagulationanddiskevolutionmodelandpresentamethodforstudyingevaporationandcondensationofCOusingtheHertz-Knudsen
p equation.Ourmodeldoesnotyetincludefragmentation,whichwillbepartoffurtherinvestigations.
- Results.Wefindnoenhancedgraingrowthimmediatelyoutsidetheicelinewheretheparticlesizeislimitedbyradialdrift.Instead,
o
wefindadepletionofsolidmaterialinsidetheiceline,whichissolelyduetoevaporationoftheCO.Suchadepressioninsidethe
r
t icelinemaybeobservableandmayhelptoquantifytheprocessesdescribedinthiswork.Furthermore,wefindthattheviscosity
s and diffusivity of the gas heavily influence the re-distribution of vaporized CO at the ice line and can lead to an increase in the
a
COabundancebyuptoafactorofafewintheregionjustinsidetheiceline.Dependingonthestrengthofthegaseoustransport
[
mechanisms,thepositionoftheicelineinourmodelcanchangebyupto∼10AUandconsequently,thetemperatureatthatlocation

2 canrangefrom21to23K.
v Keywords. Protoplanetarydisks–Accretion,accretiondisks–Diffusion–Methods:numerical
5
8
3
21. Introduction itedbyatleasttwoprocesses.Oneistheinwarddriftofparticles
0 on Keplerian orbits losing angular momentum to the pressure-
.Icelinesarelocationsinprotoplanetarydisksatwhichatransi- supported gas orbiting the star with sub-Keplerian velocities
1
tion between the gaseous and the solid phase of an element or
0 (Weidenschilling 1977). The drift velocity is size-dependent.
7a molecular species occurs. Inwards of the ice line, the volatile Therefore,particlesaffectedbydriftquicklyfallontothecentral
speciesiscondensedintheformofice.Outsidetheiceline,the
1 star.Anothermechanismthatpreventsgraingrowthisfragmen-
:material exists in its solid form. Ice lines are of special inter- tation.Ifthecollisionvelocityishighenough,silicateparticles
vest in planetary sciences since they are suggested to affect the
tend to bounce or fragment rather than sticking together (Blum
i
Xformationandcompositionofplanetesimals(Öbergetal.2011). and Wurm 2008). If particles grow to certain sizes (centimeter
Ali-Dib et al. (2014) argue from the chemical composition of
r to decimeter), the streaming instability can take over and form
aUranusandNeptunethattheymayhavebeenformedatthecar- planetesimalsbyclumpingandsubsequentgravitationalcollapse
bonmonoxideiceline.
(see,e.g.,Johansenetal.2007;YoudinandGoodman2005;Bai
Planetesimals are formed from dust and ice surrounding andStone2010;Dra¸z˙kowskaandDullemond2014).
young stars in protoplanetary disks. The dust particles collide
andareheldtogetherbycontactforcesformingeverlargerbod- Howparticlescangrowtosizesforwhichthestreamingin-
ies (see, e.g., Brauer et al. 2008; Birnstiel et al. 2010). How- stability is efficient is unknown. Besides the fact that icy mate-
ever, experiments and simulations show that the growth is lim- rialshavehigherfragmentationvelocitiesthanpuresilicatepar-
ticles (Wada et al. 2009; Gundlach and Blum 2015), and can
therefore grow to larger sizes, several authors argue that ice
(cid:63) MemberoftheInternationalMaxPlanckResearchSchoolforAs-
lines play an important role in the growth process (e.g., Kretke
tronomy and Cosmic Physics at the Heidelberg University. PUC-HD
GraduateStudentExchangeFellow. etal.2008;Braueretal.2008).Also,Kataokaetal.(2013)pro-
(cid:63)(cid:63) RoyalSocietyDorothyHodgkinFellow. posed that fluffy ice aggregates may overcome the drift barrier.
Page1of16
A&Aproofs
StevensonandLunine(1988)found,intheirmodel,asignificant iceline.Themodelisone-dimensionalandthedensitiesarever-
solidmaterialenhancementattheicelineduetodiffusiveredis- tically integrated, assuming that gas and dust are constantly in
tributionofvaporfromtheinnerdiskandsubsequentcondensa- thermal equilibrium, throughout the model. We model the sur-
tiononparticlesoutsidetheiceline.Closelyrelated,RosandJo- face densities of approximately 150 different dust sizes Σ
dust,i
hansen(2013)proposedgraingrowthbyevaporationofinward- andoftwogasspecies,Σ andΣ .OurmodelofCOtransport
H2 CO
drifting particles and subsequent recondensation of backwards- isbuiltuponthegasanddustmodelofBirnstieletal.(2010).
diffusingvapor.CuzziandZahnle(2004)proposedasignificant The gas disk is viscously evolving according the α-disk
vapor enhancement inside the ice line due to rapidly inward- modelofShakuraandSunyaev(1973)whilewestudythediffu-
driftingparticles. sivebehaviorofCObychoosingdifferentvaluesfortheSchmidt
The goal of our work is to include the transport of volatile numberSc,whichistheratioofviscosityoverdiffusivityofthe
materialsintothegraingrowthanddiskmodelofBirnstieletal. gas.
(2010)andtousethismodeltoinvestigatetheinfluenceofpos- We model dust growth by solving the Smoluchowski equa-
sibleparticledensityenhancementsaticelinesondustcoagula- tion. The dust species themselves are subject to radial drift
tionandonthedistributionofCOgas.Wethereforedeveloped and gas drag. In contrast to previous works, we use a two-
amodeltotrackthevolatilecontentofparticlesandthegasin- dimensional scalar field for the particle distribution, where one
cluding the processes of coagulation, radial drift, and viscous dimensionisthesilicatemassandtheseconddimensiontheCO
accretion,aswellasCOevaporationandcondensationattheice icecontentoftheparticle.Wethereforemodelthemigrationof
line. CObothviadriftingparticlesandviadiffusinggasandsimulate
We focus, here, on CO ice near the CO ice line for several the evaporation of CO at the CO ice line by solving the Hertz-
reasons.First,weassumethatthecollisionalphysicsofthepar- Knudsenequation.
ticles is mostly determined by the presence of water ice. If we
further assume that the water ice content is always well mixed
2.1. Evolutionofthegassurfacedensity
within the particle, there will always be enough water ice at
the particle’s surface such that the collisional physics does not Weconsidertwomoleculargasspeciesinourmodel,H ,which
2
changebycrossingtheCOicelineandevaporatingtheCO(but isthemaingasdensityconstituent,andCO,whichisgenerally
seealsoOkuzumietal.2016).Wadaetal.(2009)estimatedthe the second most abundant molecular species in the interstellar
fragmentation velocity of water ice particles to be ∼ 50 m/s. medium(ISM).Thesurfacedensityofthetwospecies,Σ and
Suchhighcollisionvelocitiesarenotreachedinourmodel.We Σ ,andthereforethetotalgasdensity,Σ ,aredirectlyrHe2lated
CO gas
canthereforeneglectfragmentation,since,evenbytotalevapo- throughthecontinuityequation:
ration of the CO, there is still enough water ice in the particles
∂ 1 ∂ (cid:16) (cid:17)
suchthatwecanusethehigherfragmentationvelocityofwater Σ + RΣ u =0, (1)
∂t gas R∂R gas gas
ice.Thegrowthoftheparticlesisthereforesolelylimitedbythe
inwarddrift.ButseealsoGundlachandBlum(2015),whofound where u is the radial gas velocity given by Lynden-Bell and
gas
fragmentationvelocitiesforwatericeintheorderof10m/s.It Pringle(1974);
mightalsobethecasethatthewatericeisnotwellmixedwithin 3 ∂ (cid:16) √ (cid:17)
the particle or that the particle is covered by a layer of CO. In u =− √ Σ ν R , (2)
that case, the fragmentation velocity is that of CO ice, which gas Σgas R∂R gas
mightalsobelowerthanthefragmentationvelocityofwaterice.
whereνistheviscosityofthegas.ShakuraandSunyaev(1973)
We aim to investigate the influence of fragmentation in future
parameterizedtheviscosityintheirα-diskmodeltoaccountfor
works.
anunknownsourceofviscositywith
Another reason for looking at CO is that it has observable
features(Qietal.2015).TheCOicelineattemperaturesofap- ν=αcsHP, (3)
proximately 20 K is in the outer parts of protoplanetary disks where c is the sound speed, H = c /Ω the pressure scale
compared to other volatile species such as water ice. Further- heightofs thedisk,andΩ theKePplerians freKquency.Theviscos-
more, the CO abundance is generally large enough to be well ityparameterαistypicallKyoftheorderof10−2to10−4(e.g.,Jo-
observed.
hansenandKlahr2005).Equation(1)hasnosourcetermsonthe
We investigate the radial distribution of dust particles and
righthandside,becausewedonotconsideranyinfallofmatter
molecularspeciesinthegasandicephasesaroundtheregionof
fromthecommonenvelopeontothedisk.Evaporationandcon-
the ice line in a time dependent manner by solving the Smolu- densation, which would also be a source or sink term for Σ ,
CO
chowskiequationandthere-distributionoftheCOvapororigi-
aretreatedseparately.
natingfromtheevaporatingparticles.
We assume that both gas species evolve separately without
Insection2,wedescribeourmodelofadvection,diffusion, interactingwithoneanother.Wethenobtainaseparatecontinu-
grain growth, and evaporation/condensation. In section 3, we ityequationforeachofthespecies j
show the influence of the ice line on the grain growth and the
influenceoftheadvectionanddiffusionofthegasontheCOva- ∂Σ =−3 ∂ (cid:34)√R ∂ (cid:16)Σ ν√R(cid:17)(cid:35). (4)
pordistributionattheiceline.Insection4,wediscussourresults ∂t j R∂R ∂R j
andtheassumptionsmadeinthemodel.
Following the calculations of Pavlyuchenkov and Dullemond
(2007),thisequationcanbefurthermanipulatedto
2. Model ∂ 3 ∂ (cid:34)√ ∂ (cid:32) Σ √ (cid:33)(cid:35)
Σ =− R Σ j ν R
∂t j R∂R ∂R gasΣ
We constructed a model for simulating the evolution of a vis- gas
couslyevolvingprotoplanetarydiskincludingdustcoagulation, =−3 ∂ (cid:34)√R Σj ∂ (cid:16)Σ ν√R(cid:17)+RΣ ν ∂ (cid:32) Σj (cid:33)(cid:35), (5)
particle drift, condensation, and evaporation of CO at the CO R∂R Σ ∂R j gas ∂R Σ
gas gas
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SebastianMarkusStammler etal.:RedistributionofCOinevolvinggasanddustdisks
whereΣ =(cid:80)Σ isthetotalgassurfacedensity.Bysubstituting 2.2. Radialevolutionofthedustsurfacedensity
gas j
j
thegasvelocityofequation(2)inthisequation,weobtain Thespatialevolutionofthedustparticlesisstronglyinfluenced
byinteractionswiththegas.Althoughitispossible,inrealdisks,
∂ 1 ∂ (cid:16) (cid:17) 3 ∂ (cid:34) ∂ (cid:32) Σ (cid:33)(cid:35) that particles become locally concentrated to gas-to-dust ratios
Σ + RΣ u − RΣ ν j =0. (6)
∂t j R∂R j gas R∂R gas ∂R Σ ofunityorless,especiallyatthedisk’smidplane,andtherefore
gas
triggerstreaminginstabilities,weneglectthishere,andassume
thataccumulationsofdusthappenonlyatsmallerradii(Youdin
Thisequationimpliesthateachgasspeciesisradiallyadvected
withthegasvelocityu andhasadiffusiveterm,whichsmears andShu2002)oronsmallerscales(YoudinandGoodman2005).
gas Inotherwords:thegashasaneffectonthedustparticles,butthe
outconcentrations.
ByintroducingtheSchmidtnumberSc = ν/D,whichisthe dustparticleshavenoback-reactiononthegasinourmodel.
ratioofviscosityνtodiffusivityDofagas,onecandisentangle Todescribethecouplingoftheparticlestothegas,itisuse-
advectionanddiffusion ful to characterize the particles not by size but by their dimen-
sionlessStokesnumber St.TheStokesnumberisdefinedasthe
∂∂tΣj+ R1∂∂R(cid:16)RΣjugas(cid:17)− R1∂∂R(cid:34)RΣgasD∂∂R(cid:32)ΣΣj (cid:33)(cid:35)=0. (7) rtuatrino-oovfetrhteimsteopτpedin;gtimeτsoftheparticlestothelargesteddy’s
gas
τ
ASchmidtnumberofSc = 1/3reproducesequation(6).Phys- St= s . (12)
τ
ed
ically, the Schmidt number is the ratio of momentum transport
topure-masstransport.BychangingScandkeepingαconstant, TheStokesnumbercanbeusedasameasuretodescribethecou-
weinvestigatetheinfluenceofthediffusivity DontheCOdis- plingbetweenthegasandthedust.Sincethiscouplingdepends
tributioninthedisk. notsolelyontheparticlesizebut,amongstotherparameters,also
onthegasdensity,itisconvenienttousetheStokesnumberto
describetheequationsofmotionoftheparticles.
2.1.1. Verticalandradialstructureofthegasdisk
Thestoppingtimeτ inequation(12)istheratiobetweena
s
Weassumethatthegasisalwaysinthermalequilibriumandthat particle’s momentum and the rate of its momentum change by
the temperature profile does not change over time. The vertical thedragforce.Ityieldsthetimescaleonwhichaparticleadapts
structureofthegasdensityisthengivenby tothegasvelocity.Weidenschilling(1977)identifiedfourdiffer-
entregimesofthedragforce.Theimportantregimeinourwork
ρgas(z)=ρgas(z=0) exp−12Hz22, (8) istheEρpsateinregime,wherethestoppingtimeisgivenby
P τ = s , (13)
s ρ u¯
gas
where z is the height above the midplane. Since Σ is the z-
gas
integralofρgas ,itfollowsdirectlyforthemidplanegasdensity with th√e bulk density of the solids ρs and the particle size a.
that u¯ = c 8/πisthemeanthermalvelocityofthegasmolecules.
s
In protoplanetary disks, the Stokes I drag regime is important
ρ (z=0)= √Σgas . (9) only in the inner part of the disk ((cid:46) 1 AU). We therefore only
gas 2πH includetheEpsteinregimeinthiswork.
P
As a first order approximation for the eddy turnover time,
Eventhoughthemodelisone-dimensional,westillneedthe one can use τed = 1/ΩK (Schräpler and Henning 2004), and
verticalstructureofthedisktocalculatethedensitiesatthemid- obtaintheStokesnumber:
planethatareneededforthecoagulationaswellasfortheevap- ρ aπ
orationmethod. St= Σs 2. (14)
In our fiducial model, we used the following power law as gas
initialsurfacedensitydistribution: The advection of the dust surface densities of the different
speciescannowbedescribedwithacontinuityequationasequa-
Σ (R)=250 g ×(cid:18) R (cid:19)−1. (10) tion(7)
gas cm2 1AU
∂ 1 ∂ (cid:32) (cid:34) ∂ Σ (cid:35)(cid:33)
Σ + R Σ u −D Σ dust,i =0,
2.1.2. Temperaturestructureofthedisk ∂t dust,i R∂R dust,i dust,i dust,i gas∂R Σgas
(15)
For the midplane temperature of the disk, we assume a simple
powerlaw; whereD = αc H /(1+St2)isthedustdiffusivityandu
dust,i s P dust,i
theradialvelocityofdustspeciesigivenby
T(R)=150K × (cid:18) R (cid:19)−12 , (11) u 2u
1AU u = gas − P . (16)
dust,i 1+St2 St +St−1
i i i
thatremainsconstantintime.Theviscousheatingrate,whichis
theenergythatiscreatedbytheviscousaccretingofthegas,is Thisvelocityconsistsoftwoterms.Thefirsttermdescribesthe
proportional to the gas density Σ (Nakamoto and Nakagawa dustparticlesthataredraggedalongwiththeinwardsaccreting
gas
1994) and therefore mainly important in the inner part of the gas;itismosteffectiveforsmallparticles(St(cid:28)1)thatarewell-
disk.SinceweareinterestedintheregionoftheCOicelineat coupledtothegasandinsignificantforlargeparticles(St(cid:29)1).
40–50AUwherethedensitiesarelow,thiseffectisnegligible. Thesecondtermdescribesradialdriftduetoapressuregradient
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inthegas.Thegasfeelsanadditionalforceduetoitsownpres- 2.4.1. Smoluchowskiequation
sure gradient. If the pressure gradient is pointing inwards, then
The growth of the dust particles with mass m by hit-and-stick
thegasisonasub-Keplerianorbit.Theaccelerationofthedust
collisionswithoutfragmentationcanbedescribedbytheSmolu-
particles due to the pressure force, on the other hand, is negli-
chowskiequation
gible because of their high bulk density compared to the gas.
Therefore, the particles try to orbit with a Keplerian velocity.
Dheuaedwtointhdi,sloveseloacnitgyudlaiffremreonmceenwtuimth,tahnedgdarsi,ftthienythfeeedliareccotniosntaonft ∂ f (m)=1(cid:90)∞dm(cid:48)(cid:90)∞dm(cid:48)(cid:48)f (cid:0)m(cid:48)(cid:1) f (cid:0)m(cid:48)(cid:48)(cid:1)K(cid:0)m(cid:48);m(cid:48)(cid:48)(cid:1)×
∂t 2
the pressure gradient. The maximal drift velocity (for particles
0 0
withSt=1)isgivenbyWeidenschilling(1977)as ×δ(cid:0)m(cid:48)+m(cid:48)(cid:48)−m(cid:1)
(cid:90)∞
− f (m) dm(cid:48)f (cid:0)m(cid:48)(cid:1)K(cid:0)m;m(cid:48)(cid:1), (19)
∂ P
u =− ∂R gas . (17) 0
P 2ρ Ω where the collision kernel K(m;m(cid:48)) = σ (m;m(cid:48))∆u(m;m(cid:48))
gas K geo
is the product of the geometrical cross section and the relative
velocityofthecollidingparticles.Equation(19)isapartialdif-
2.3. Verticalstructureofthedustdisk ferentialequationforthetimeevolutionofamassdistributionof
particles. The positive term on the right-hand side counts colli-
Sincethedustparticlesdonotfeelanypressureamongstthem- sionsthatleadtoparticlesofmassm.Thenegativetermremoves
selves,wecannotgiveasimplepressure-equilibriumexpression particles of mass m that collided with other particles to create
fortheparticlescaleheightsasforthegasscaleheightH .The largerparticles.
P
particlescaleheightsaregivenbytheequilibriumofverticalset- Tokeeptrackofnotonlythemassoftheparticles,butalsoof
tling by gravity and turbulent stirring and therefore depend on anotherpropertythatrepresentstheCOicecontentofthem,we
the particle’s Stokes number. Therefore, every particle size has addanotherparameterQtoourdistribution.Thefollowingequa-
itsownscaleheight.Wefollow,here,Braueretal.(2008)forthe tionsaregeneralforanarbitraryQthatcouldrepresentanything
descriptionofthescaleheightofspeciesias from the volume of the particle as in Okuzumi et al. (2009) to
theelectricalchargeoftheparticle.Inthefollowingsectionswe
identify Qwiththeparticle’sicefraction.Thetwo-dimensional
Smoluchowskiequationthenreadsasfollows
 
hi = HP·min1, (cid:115)min(cid:16)St,1α(cid:17)·(cid:16)1+St2(cid:17), (18) ∂∂tf (m,Q)=12(cid:90)∞dm(cid:48)(cid:90)∞dm(cid:48)(cid:48)(cid:90)∞dQ(cid:48)(cid:90)∞dQ(cid:48)(cid:48)×
i 2 i
0 0 0 0
× f (cid:0)m(cid:48),Q(cid:48)(cid:1) f (cid:0)m(cid:48)(cid:48),Q(cid:48)(cid:48)(cid:1)×
×K(cid:0)m(cid:48),Q(cid:48);m(cid:48)(cid:48),Q(cid:48)(cid:48)(cid:1)×
which limits the dust scale height to be, at maximum, equal to ×δ(cid:0)m(cid:48)+m(cid:48)(cid:48)−m(cid:1)
the gas scale height HP. From this equation, it can be seen that ×δ(cid:0)Qnew(cid:0)m(cid:48),Q(cid:48);m(cid:48)(cid:48),Q(cid:48)(cid:48)(cid:1)−Q(cid:1)
largerparticles(St(cid:29)1)aresettledtothemidplane,whilesmall
particles (St(cid:28)1) follow the gas scale height. A stronger tur- (cid:90)∞ (cid:90)∞
bulence(i.e.,largerturbulentviscosityparameterα),ingeneral, − f (m,Q) dm(cid:48) dQ(cid:48)f (cid:0)m(cid:48),Q(cid:48)(cid:1)×
increasesthescaleheightuptoamaximumofH .
P 0 0
We assume that the particles are distributed with Gaussian ×K(cid:0)m,Q;m(cid:48)Q(cid:48)(cid:1), (20)
distributions just as the gas in equation (8), but with the dust where Qnew(m(cid:48),Q(cid:48);m(cid:48)(cid:48),Q(cid:48)(cid:48)) is the new Q-value of the particle
particle’sspecificscaleheighthi instead,whichallowsustoan- thatwascreatedinthecollision.Inprinciple,thisequationcould
alyticallyintegratetheSmoluchowskiequationvertically. besolvedasforequation(19),butaddinganotherparameterand
thereforetwoadditionalintegralswouldbecomputationallyvery
expensive.
2.4. Dustcoagulation We therefore follow the moment method introduced by
Okuzumi et al. (2009) to re-write one two-dimensional Smolu-
WeincludedustgrowthinourmodeltostudytheCOredistribu- chowskiequationintotwoone-dimensionalequations.Theidea
tioninprotoplanetarydisks.ThedustcarriestheCOiceinwards isthatinsteadofevolvingthefullQ-distribution,weonlyevolve
through the CO ice line. As seen in equation (16), the Stokes itsfirstmoment,itsmeanQ¯ foragivenmassm.Byintroducing
number,andthereforetheparticlesize,isimportantforthedrift thedistributionfunctionofparticlesofmassm
velocityofthedustparticles. (cid:90)∞
We therefore include hit-and-stick collisions of particles in n(m)= f (m,Q)dQ, (21)
ourmodel.WeneglectfragmentationbecausetheCOicelineis
0
well outside of the water ice line, meaning that, everywhere in
andthemeanQ-valueofparticleswithmassm
ourmodel,theparticleshaveasignificantamountofwaterice,
which is also mixed within the particle in such a way that the (cid:90)∞
1
particlecollisionpropertiesarealwaysdeterminedbythewater Q¯(m)= Qf (m,Q)dQ, (22)
n(m)
ice.Asexplainedabove,wethereforeneglectfragmentation.
0
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SebastianMarkusStammler etal.:RedistributionofCOinevolvinggasanddustdisks
andbyassumingthattheparticledistributionisverysharpinQ isthendeterminedby
f (m,Q)=n(m)δ(cid:16)Q¯(m)−Q(cid:17), (23) Qnew(cid:0)m,Q;m(cid:48),Q(cid:48)(cid:1)= mmitcoet++mmitcoet(cid:48)(cid:48)
Q m+ Q(cid:48) m(cid:48)
= 1−Q 1−Q(cid:48)
whereδistheDiracdeltafunction,equation(20)canberewrit-
1 m+ 1 m(cid:48)
tenandsimplified.Weendupwithonepartialdifferentialequa- 1−Q 1−Q(cid:48)
tionforn(m) (1−Q(cid:48))Qm+(1−Q)Q(cid:48)m(cid:48)
= . (26)
(1−Q(cid:48))m+(1−Q)m(cid:48)
(cid:90)∞
∂n(m)=1 dm(cid:48)n(cid:0)m(cid:48)(cid:1)n(cid:0)m−m(cid:48)(cid:1)× 2.4.3. Relativevelocities
∂t 2
0 Thecoagulationisdrivenbytherelativevelocitiesoftheparti-
×K(cid:16)m(cid:48),Q¯(cid:0)m(cid:48)(cid:1);m−m(cid:48),Q¯(cid:0)m−m(cid:48)(cid:1)(cid:17) clesthatdeterminethecollisionratesandthepossiblecollision
outcome.Weconsider,here,fivedifferentsourcesofrelativeve-
−n(m)(cid:90)∞dm(cid:48)n(cid:0)m(cid:48)(cid:1)K(cid:16)m,Q¯(m);m(cid:48),Q¯(cid:0)m(cid:48)(cid:1)(cid:17), (24) rloadciitailesp:arBtircolwendirainftm∆uoti,ovner∆tiucBaMl,seatztliimngut∆haul pa,rtaincldetdurribfutle∆nucφe,
R sett
0 ∆uturb.
The total relative velocity is then given by the root mean
andoneequationfornQ¯(m)≡n(m)Q¯(m) squareofallsourcesofrelativevelocity
(cid:113)
∆u= ∆u2 +∆u2 +∆u2 +∆u2 +∆u2 . (27)
BM R φ sett turb
(cid:90)∞
∂ (cid:16)nQ¯(m)(cid:17)=1 dm(cid:48)n(cid:0)m(cid:48)(cid:1)n(cid:0)m−m(cid:48)(cid:1)× We assume that all particles of a certain size collide with this
∂t 2 distinctmeanvelocity.Ithasbeenshownthattheuseofaveloc-
0 ity distribution can have a significant influence on dust growth
×K(cid:16)m(cid:48),Q¯(cid:0)m(cid:48)(cid:1);m−m(cid:48),Q¯(cid:0)m−m(cid:48)(cid:1)(cid:17)× by breaking through growth barriers as the bouncing barrier or
×Qnew(cid:16)m(cid:48),Q¯(cid:0)m(cid:48)(cid:1);m−m(cid:48),Q¯(cid:0)m−m(cid:48)(cid:1)(cid:17) t2h0e13fr)a.gSminecnetawtieonarbeanrroiterde(aWliinngdmwaitrhkbeotuanlc.i2n0g1a2n;dG/oarrafuradgmeteanl-.
(cid:90)∞ tationinthismodel,wedonotconsideranyvelocitydistribution.
−nQ¯(m) dm(cid:48)n(cid:0)m(cid:48)(cid:1)K(cid:16)m,Q¯(m);m(cid:48),Q¯(cid:0)m(cid:48)(cid:1)(cid:17). For a detailed description of the various sources of relative
velocity,werefertoBirnstieletal.(2010).
0
(25)
2.5. Evaporationandcondensation
Foradetailedderivationoftheequations,werefertoOkuzumi When particles drift inwards they move from colder regions in
etal.(2009).Bothone-dimensionalequationscanbesimultane- the disk to warmer regions. At some point – at the ice line –
ously solved, which is significantly more efficient than solving thetemperatureishighenoughfortheiceintheparticlestobe
one two-dimensional equation. The coagulation equations (24) evaporated.Thevaporthatiscreatedbyevaporationcanthenbe
and(25)andthedustadvectionequation(7)aresimultaneously diffusedbackwardsthroughtheicelineandcanre-condenseon
solved with the method described in the appendix of Birnstiel the colder particles there. We therefore consider both evapora-
etal.(2010) tionandcondensationalongwithdiffusionandviscousevolution
ofthegasinourmodel.
Weassumethattheparticlesalwaysadaptinstantaneouslyto
2.4.2. TheQ-parameter the new temperature. This might not be the case for very large
particleswithhighheatcapacities.ButasseenlaterinSection3,
Wehavenowthefreedomofchoiceonwhatexactlythephysical
wedonothavesuchlargeparticles.Thetemperaturechangeour
meaningofourQ-parametershouldbe.Onewaytoincludeices
particles experience is solely caused by their drift from colder
inourcoagulationcodeistotakem=mtot asthetotalmassand
regionstowarmerregionsinthedisk.Withtypicaldriftspeeds,
Q= mmice astheicefractionofaparticle.However,thisapproach theparticlesexperienceachangeintemperatureoflessthan1K
hasthedisadvantagethatasaresultofevaporationandconden- in1000yrs.Theassumptionofinstantaneousadaptiontotheam-
sation, both parameters m and Q change. In this case, evapora- bienttemperatureisthereforewelljustified.
tion/condensationwouldbeadvectionofthefunctionQ(m,t)in Evaporation/condensation can be described by the Hertz-
them-coordinatewithanadditionalsourceterm. Knudsen equation which gives the number of CO molecules N
We therefore chose m = msil as the silicate mass and Q = perunitsurfaceareathatleavetheparticle,
mice as the ice fraction of a particle. Here, evaporation and (cid:32) (cid:33)
msil+mice d Peq
condensationonlychangeQandnotm. N =v −n , (28)
By using this approach, the ice mass of a particle is mice = dt therm kBT vap
Q msil , whereas the total mass is mtot = 1 msil. Therefore, where n is the number density of vapor molecules, Peq
1−Q 1−Q vap
wecannotmodelparticlesfullyconsistingofice(Q=1).Qnew, t√he saturation pressure of the volatile species and vtherm =
whichisthenewQ-valueofaparticleresultingfromacollision 8k T/(πm )isthemeanthermalvelocityofgasmoleculesof
B CO
followedbystickingbetweenaparticlewithmassmandanother massm .Iftheambientpressureofthevolatilespeciesequals
CO
particlewithmassm(cid:48) (andicefractionsQandQ(cid:48),respectively), thesaturationpressurethenthereisnonetevaporation.
Page5of16
A&Aproofs
Peq hastobedeterminedexperimentally.Weusethevalues 2.5.1. Condensationmode
fromLegeretal.(1985)as
IfS > 0,thenweareinthecondensationregimeandcondensa-
(cid:34) (cid:35)
dyn 1030K tionwilltakeplaceonallparticles.Inthatcase,wecandirectly
Peq (T)=1 ×exp − +27.37 . (29)
CO cm2 T integrate equations (36) and(33) until the end of the time step.
Ifweassumethattheparticleradiusa isconstantanddoesnot
Evaporation/condensation is, in general, dependent on the i
changeoverthetimestep,thenbothequationshaveanalyticso-
surface curvature of the particle, where convex surfaces have
lutions.Thisassumptionisjustifiedifweuseacanonicalabun-
slightly higher evaporation rates than concave surfaces (Sirono danceofCOof10−4.Byusingadusttogasratioof10−2 ,this
2011).Wedonotconsiderthosesurfacecurvatureeffectsinour
leads to an initial mass ratio of CO (gas and ice phase) to dust
model.Thismeansifequation(28)isnegative,wehaveconden-
of∼ 14%,andthereforeamaximalchangeinradiusof∼ 5%.
sationonallparticlesindependentoftheirsize.Ifequation(28)
Theanalyticsolutionofequation(33)isthen
ispositive,allparticlesevaporateaslongastheystillcarrysome
ice. E (cid:16) (cid:17)
The Hertz-Knudsen equation needs to be solved for every Σvap(t)=Σvap(t0)e−C(t−t0)+ C 1−e−C(t−t0) . (38)
particlesizeateveryradialandverticalpositionofthedisksep-
arately and simultaneously. But since we are not dealing with Usingequation(38)inequation(36)leadsto
theverticalstructureofthediskinourmodel,weneedtotrans-
∂
ferthisexpressionintoanexpressionforsurfacedensities.Ifwe Σ =−E +CΣ
lookatthemidplaneofthedisk,theHertz-Knudsenequationfor ∂t ice,i i i vap
themidplanevapormassdensityreadsas (cid:18) E (cid:16) (cid:17)(cid:19)
d (cid:88) (cid:32)m Peq (cid:33) =−Ei+Ci Σvap(t0)e−C(t−t0)+ C 1−e−C(t−t0)
dtρvap = πa2i ndust,ivtherm CkOT −ρvap , (30) =(cid:16)CiΣvap(t0)−Ei(cid:17)e−C(t−t0), (39)
i B
where a is the particle radius and n the midplane number whereinthelaststep,equation(37)wasused.Thisequationhas
i dust,i
densityofdustparticlesspeciesi.Thissystemisinequilibrium theanalyticalsolution
if
CΣ (t )−E (cid:16) (cid:17)
ρvap = mCkOTPeq ≡ρevqap, (31) Σice,i(t)=Σice,i(t0)+ i vapC0 i 1−e−C(t−t0) . (40)
B
where ρeq is defined as the equilibrium vapor mass density. If Equations(38)and(40)canbeusedtocalculatethesurfaceden-
vap sitiesatanytimestepdirectly.
weassumethatthevaporisalwaysinpressureequilibrium,we
getfortheequilibriumvaporsurfacedensity
√ √ m Peq 2.5.2. Evaporationmode
Σeq = 2πH ρeq = 2πH CO . (32)
vap P vap P kBT If at the start of the time step S < 0, we are in an evaporation
Assuming this, we can transform equation (30) into an expres- regime. Also in this regime, the time evolution is governed by
sionforthevaporsurfacedensity equations(38)and(40).However,weneedtotakeintoaccount
thatbaregrainswillnotcontributetothefluxofmolecules,and
d (cid:88) (cid:16) (cid:17)
Σ = πa2n v Σeq −Σ thatduringthetimestep,grainsmaybecomebare.Therefore,we
dt vap i dust,i therm vap vap splittheintegrationintoanumberofsub-steps.Duringeachsub-
i
(cid:88)(cid:16) (cid:17) step, the same set of grain sizes contributes to the evaporation
= E −CΣ
i i vap flux, and we completely ignore bare grains. For each step, we
i define restricted versions ofC and E. These restricted versions
= E−CΣvap ≡−S, (33) onlysumovernon-baregrains;
with (cid:88) (cid:88)
C = C and E = E. (41)
rest i rest i
C =πa2v n (34)
i i therm dust,i i,notbare i,notbare
E =CΣeq . (35)
i i vap Wedefinebaregrainsasparticlesthathavelessthanonemono-
Therefore, the change in the ice surface density of particle layeroficemoleculesontheirsurface.
speciesiisgivenby Withthesedefinitions,weuseequation(40)tocheckifany
ice component becomes depleted during the time step, that is,
∂
Σ =CΣ −E, (36) wecheckforthefirstrootinanyoftheequations;
∂t ice,i i vap i
wanheicvhapisortahteiodniffteerrmenEcei.bTehtweedeenepaercomnedaennisnagtiboenhtienrdmthCisiΣfvoarpmaanld- Σice,i(t0)+ CiΣvapC(t0)−Ei (cid:16)1−e−C(t−t0)(cid:17)=0. (42)
ism is that we treat the evaporation/condensation rates as if the
particles were in the midplane, while we assume that gas and Ifthatfirstrootisbeforetheendofthetimestep,weevolvethe
particles are always in thermal equilibrium and distributed ac- system to this time, remove the newly bare grain size from the
cordingtotheirpressurescaleheightsH andh. participating set of grain sizes, and continue in the same way.
P i
ForC andE ,thefollowingrelationholds: Eventually, either all grains will be bare, or the first root in the
i i
set of equations (42) will be beyond the end of the time step.
E E
i = . (37) Thenweuseequation(40)withC replacedbyCrest tomovethe
Ci C systemtotheendofthetimestep.
Page6of16
SebastianMarkusStammler etal.:RedistributionofCOinevolvinggasanddustdisks
Table1.Fiducialmodelparameters.
103
] Σgas
Parameter Symbol Value Unit 2m 102 ΣCO
Viscosityparameter α 10−3 – [g/c 101 ΣeCqO
Schmidtnumber Sc 1/3 – y
t
Stellarmass Mstar 1.0 M(cid:12) nsi 100
GGaass--ttoo--dCuOstrraattiioo ffgg22dCO 170000 –– acede 10−1
Silicatebulkdensity ρsil 1.6 g/cm3 urf 10−2
CObulkdensity ρ 1.6 g/cm3 S
CO
10−3
20 25 30 35 40 45 50 55 60
Distancefromstar[AU]
2.5.3. Caveats
We do not consider grain curvature effects on evaporation and Fig.1.Gassurfacedensitiesofthefiducialmodelafter1Myr.Shown
condensation. Thismeans eitherall particles arein evaporation arethetotalgassurfacedensity(solidblackline)andthesurfacedensity
modeorallparticlesareincondensationmode.Ingeneral,this ofgaseousCO(dashedgreenline).Thedottedbluelineistheequilib-
effectismostlyeffectiveforverysmallparticles(< 0.1µm).As riumsurfacedensityofCOgivenbyequation(32).Thelocationwhere
seeninsection3,wegetridofthesmallparticlesveryquickly theCOsurfacedensitydetachesfromtheequilibriumCOsurfaceden-
due to coagulation and do not replenish them since we do not sityiscalledtheiceline.
havefragmentation.
Thesurfaceofarealdiskisdirectlyilluminatedbythestar
andthereforehotterthanthemidplane.Thismeansthatinaddi- tion(Mathisetal.1977)withanuppercutoffat1µm.Weinte-
tion to a radial ice line in the midplane, disks also have a two-
gratetheequations(7),(15),(24),and(25)andusethemethod
dimensional surface called atmospheric ice line towards their forevaporation/condensationasdescribedinsection(2.5)tocal-
surface,wherethetemperaturesarehighenoughforicestoevap- culatethetimeevolutionofthegassurfacedensitiesΣ andΣ ,
orate. Since we are only looking at midplane quantities to de- andthedustsurfacedensitiesofthedifferentgrainsizHe2sΣ CO.
dust,j
cidebetweenevaporationorcondensation,weneglecttheatmo-
sphericiceline.Thisleadstoanoverestimationoftheicecontent Figure 1 shows the gas surface densities after 1Myr of the
oftheparticles.Butsincemostofthedustmassisbasicallybe- simulation. We have chosen this value, because it is the typical
low the vertical ice line, especially the larger grains, the effect age of some of the best studied star-forming regions and thus
is only expected to be important near the radial ice line where alsoofthediskstherein.Also,itisthetypicaltimescalefordisk
theverticalicelinereachesthemidplane.However,thisisara- dispersion(e.g.,Haischetal.2001).Shownisthetotalgassur-
dially narrow region, so it shouldn’t affect the global transport, face density Σgas = ΣH2 +ΣCO and the CO gas surface density
butpossiblymodifiestheshapeofthesnowline. ΣCO.OverplottedistheequilibriumCOgassurfacedensityΣeCqO
Ourparticleshavearelativelysimplestructureinthemodel. given by equation (32), which the system would have if there
TheyaretreatedasparticleswithacoreandamantleofCO.If was no net evaporation/condensation and if there was enough
twoparticleswithmantlescollideinourmodel,theyformanew COavailable.IfthereisnotenoughCOinthesystem,thenthe
particlewithasinglecoreandamantleinsteadofaparticlewith COintheparticlesevaporatesuntilthegrainsarebare.Thera-
two cores connected at their contact point, or even more com- dial distance, where ΣCO detaches from ΣeCqO , is then our defi-
plexfragmentedstructures.Thismayleadtoanartificialgrowth nition of the CO ice line because it is the point where there is
ofsilicateparticles.Ifweconsiderthecasewhereoutsidetheice justenoughCOinourmodeltoprovidethevaporpressurenec-
line,manyparticleswithCOmantlescollide;inourmodelthey essarytocounterbalancetheevaporation.Inourfiducialmodel,
formalargeparticlewithalargesinglecoreandalargeCOman- thishappensat∼ 47AU.Outsidetheiceline,theCOisfrozen
tle.BycrossingtheicelineandevaporatingtheCO,wearethen outontothegrainsuntilΣCO = ΣeCqO.Insidetheiceline,allCO
left with this large core instead of many small cores. Aumatell is in the gas phase. Although it is not a power law, the equilib-
and Wurm (2011) showed, experimentally, that by evaporating riumdensityatthepositionoftheicelinecanbeapproximated
fractaliceparticles,onewouldexpecttobeleftwithmanysmall byΣeq ∝ R−p with p ≈ 20.Sincethisisverysteep,theregion
CO
particlesinsteadofonelargeparticle.Infavorofsimplicity,we aroundtheiceline,wheretheCOispartiallycondensed,isvery
neglectthiseffect. smallandthereforetheicelineiswelldefined.
Furthermore,theevaporationofCOicecanbedecreasedif Figure 2 shows the total dust surface density Σ and the
dust
someoftheCOiceistrappedinsideaporousandspatiallycom- surface densities Σ of grains larger than one millimeter,
a>1mm
plex structure of the dust particle. Or it can be increased if the Σ ofgrainsizesbetween1mmand1µm,andΣ of
dustparticleisveryfluffybecauseitssurfaceareaisthenmuch 1µm<a<mm a<1µm
sub-micrometer-sized grains after 1Myr. The total dust surface
larger than in the case of a spherical particle of the same size.
density outside of 70 AU is dominated by sub-millimeter-sized
Theseeffectsarealsoignoredinthiswork.
grains. The amount of millimeter-sized grains becomes impor-
tant inside of 70 AU. The reason for that is that the maximum
size of the particles is limited by radial drift. As particles grow
3. Results
to larger sizes, their Stokes number increases. In the Epstein
regime of drag, which is relevant in this work, particles of the
3.1. Thefiducialmodel
samesizehavealargerStokesnumberinregionsofsmallergas
The input quantities of our fiducial model are given in table 1. density. This means that particles of a given size drift faster in
The grain sizes are initially distributed with the MRN distribu- the outer region of the disk. When the particles grow and their
Page7of16
A&Aproofs
100
2m] 10−1 ΣΣdau>s1tmm 1.0 15µµmm
g/c 10−2 Σ1µm<a<1mm n 0.8 10µm
nsity[ 1100−−43 0.015 Σa<1µm ractio 0.6 51000µµmm
e f 0.4
ed 10−5 Ice
ac 10−6 0.2
rf 0.005
Su 10−7 40 60 0.0
10−8
20 40 60 80 100 120 40 50 60 70 80 90 100
Distancefromstar[AU] Distancefromstar[AU]
Fig. 2. Dust surface densities of different dust sizes at the CO ice Fig. 3. The CO ice fraction Q of particles with different silicate core
lineafter1Myr.Shownarethetotalsurfacedensity(solidgrayline), sizesdependingontheirradialpositioninthediskafter1Myr.
millimeter-sized (dotted blue line), sub-millimeter-sized (dashed red
line),andsub-micrometer-sized(dash-dottedgreenline)particles.The
verticaldashedlinemarksthelocationoftheCOiceline. tions:
(cid:90)∞
Stokesnumbersapproachunity,theyareheavilyaffectedbyra- σdust,tot(a;R)= n(a;R,z)·m·adz
dialdriftanddriftrapidlytowardsthecentralstar.
−∞
We do not see any enhanced particle growth at the ice line. (cid:90)∞
Thereasonforthatistheso-calleddriftbarrier.Ifparticlesdrift σ (a;R)= nQ(a;R,z)·m·adz. (43)
dust,CO
through the ice line and evaporate their CO, this CO vapor can
diffuse backwards through the ice line and can re-condense on −∞
theparticlesthere.Theseparticlesthengrowinsize.Therefore, The solid line denotes the particles with a Stokes number of
their Stokes number increases and with that their drift velocity, unity.Thedashedlineisananalyticalestimateofthemaximum
makingthemdriftevenfasterthroughtheiceline. drift-limited particle size as given by Birnstiel et al. (2011). At
In fact, the feature we see in Σ in figure 2 at the ice line first,theparticlegrowthismosteffectiveintheinnerregionsof
dust
should rather be seen as a depletion inside the ice line instead thediskbecauseofthelargercollissionratesoftheparticles.The
ofanenhancementoutside,becauseitissolelycausedbyevap- particlegrowthislaterhinderedbyradialdrift.Inthesnapshots
orationoftheCOiceontheinwardsdriftingparticles.Thisde- at 500,000yrs and 1Myr, a region from the ice line at 47AU
pletionapproximatelycorrespondstotheinputratiosof f and outwardscanbeseenwheretherearenosmallparticles(dotted
g2d
f . line).Becauseofthere-condensationofCO,theseparticlesgrow
g2CO
Figure2alsoshowsacompletelackofsub-micrometersized insize.Thedistributionfunctionsσarethereforecompressedto-
grains in a region of approximately 10 AU outside the ice line. wardslargersizesleadingtothedarkerspotatintermediatepar-
Thoseparticlesdonotexisthere,becausethebackwardsdiffus- ticlesizesintheµmrange.Thetotalnumberdensityofparticles
ing CO vapor preferentially re-condenses on the smallest par- hasnodiscontinuityattheiceline.
ticles since they contribute the most to the total surface area. Toinvestigatethetypicalparticlesizea¯mthatcarriesmostof
Therefore, these particles grow in size. The particles with the the CO ice mass and the typical particle size a¯F that transports
smallest silicate cores have ice fractions of Q > 0.999. There- mostoftheCOice,wedefinethefollowingquantities:
fore,theirmassisenhancedbyafactorofatleast1,000andtheir (cid:82)
raadi<us1bµymatfhaecrteofroorefhatavleeaastre1s0u.lPtianrgtircaledsiuwsiothfaasili>ca1teµmco.reof a¯m(R)= (cid:82)a·σdust,COdlna (44)
sil real σ dlna
dust,CO
Figure3showstheseicefractionsQ(R)asafunctionofdis- (cid:82)
tsailnicceatefrcoomrethsiezesstaorfatfhteerpa1rMticylerso.fSitnhceeswimeudloatinoonttfaokrediinffteoreanc-t a¯F(R)= (cid:82)au·uga·sσ·σdust,COdldnlana. (45)
gas dust,CO
countanycurvatureeffectsforourevaporation/condensational-
gorithm, this means the evaporation/condensation rate per unit a¯ is the average particle size weighted by the CO dust distri-
m
surfaceareaisthesameforallparticlesizes.Inthecaseofcon- bution and a¯ is the average particle size weighted by the CO
F
densation, the absolute change in radius would be the same for iceflux.Bothquantitiesareplottedinfigure5.Itcanbeseenby
all particles independent of their size. But the relative gain in comparing figure 5 with figure 4 that a¯ is always close to the
m
particle mass is therefore larger for smaller particles and there- maximum particle size. Furthermore, a¯ is always close to the
m
fore their Q-value increases faster. Figure 3 shows this effect. particle size a¯ that is responsible for the ice flux on the parti-
F
The smaller the particle is, the larger its CO ice fraction Q at clesinthedisk.
the ice line at 47AU. Also, it can be seen that the CO vapor is
redistributedtolargedistancesoutwardsoftheiceline.Evenat
3.2. TransportofCOinthedustphase
R > 70AU, there is a noticeable increase of Q for the small
particles. Figure6showstheexcessintotaldustsurfacedensitycompared
Figure 4 shows different snapshots of the size evolution in toamodelwithoutCO,thedust-to-gasratioandtheice-to-dust
our fiducial model. Plotted are the following distribution func- ratio for different values of the initial gas-to-CO ratio f at
g2CO
Page8of16
SebastianMarkusStammler etal.:RedistributionofCOinevolvinggasanddustdisks
Totaldust COice
101
100 10000yrs 100 10000yrs
100 10−1 10−1
clesize[cm] 1100−−21 2σ[g/cm] 11110000−−−−5432 2σ[g/cm] 11110000−−−−5432
rti 10−3 10−6 10−6
a
P
10−4
10−5
101
0.1Myrs 0.1Myrs
100
m]
[c 10−1
e
z
si 10−2
e
cl
rti 10−3
a
P
10−4
10−5
101
0.5Myrs 0.5Myrs
100
m]
[c 10−1
e
z
si 10−2
e
cl
rti 10−3
a
P
10−4
10−5
101
1Myr 1Myr
100
m]
[c 10−1
e
z
si 10−2
e
cl
rti 10−3
a
P
10−4
10−5
20 40 60 80 100 120 20 40 60 80 100 120
Distancefromstar[AU] Distancefromstar[AU]
Fig. 4. Different snapshots of our fiducial model. Shown are the different distribution functions as defined by equations (43) at 10,000yrs,
100,000yrs,500,000yrs,andat1Myr.ThesolidlinedenotestheparticlesizeswithStokesnumberunity,Thedashedlineisananalyticales-
timationofthedriftbarrier.Thedottedlineencompassesaregionoutsideoftheicelinethatisfreeofsmallparticles.
differenttimesinthesimulation.Weusedvaluesfor f of70, outside the ice line is f /f . The initial value of the gas-
g2CO g2d g2CO
700 (fiducial model), and 7000, which correspond to CO abun- to-dustratiois1/f +1/f .Theice-to-dustratiocompares
g2d g2CO
dancesof10−3,10−4,and10−5.Forguidance,thehorizontallines thesurfacedensityofCOicetothetotaldustsurfacedensity.Its
markthevaluesforthequantitiesonewouldexpectoutsidethe initialvalueis f /(f + f ).
g2d g2d g2CO
icelinesimplyfromtheinitialconditions.Theexcessisdefined
asΣ/Σ −1,whereΣ isthetotaldustsurfacedensityinthemod- It can be seen that the closer the CO ice line is to the star,
elsoifth0edifferentCOi abundancesandΣ isthetotaldustsurface thehighertheCOabundanceis.Further,themoreCOthereisin
0 thesystem, the moresaturated theevaporation/condensationis,
densityinthemodelwithoutCO.Theinitialvalueoftheexcess
causingtheicelinetobepushedinwards.Thisvariestheposition
Page9of16
A&Aproofs
Excess Dust-to-gasratio Ice-to-dustratio
102 10−1 101
initial initial initial fg2CO=70
1− 110001 Σgas 10−2 /Σdust 100 ffgg22CCOO==7700000
Σ/Σ0 10−1 Σ/dust 10−3 ust,CO 10−1
10−2 Σd 10−2
10−3 10−4 10−3
102 10−1 101
10000yrs 10000yrs 10000yrs
1− 110001 Σgas 10−2 /Σdust 100
Σ/Σ0 10−1 Σ/dust 10−3 ust,CO 10−1
10−2 Σd 10−2
10−3 10−4 10−3
102 10−1 101
0.5Myrs 0.5Myrs 0.5Myrs
1− 110001 Σgas 10−2 /Σdust 100
Σ/Σ0 10−1 Σ/dust 10−3 ust,CO 10−1
10−2 Σd 10−2
10−3 10−4 10−3
102 10−1 101
1Myr 1Myr 1Myr
1− 110001 Σgas 10−2 /Σdust 100
Σ/Σ0 10−1 Σ/dust 10−3 ust,CO 10−1
10−2 Σd 10−2
10−3 10−4 10−3
30 40 50 60 70 80 90 100 30 40 50 60 70 80 90 100 30 40 50 60 70 80 90 100
Distancefromstar[AU] Distancefromstar[AU] Distancefromstar[AU]
Fig.6.TheexcessoftotaldustsurfacedensitycomparedtoamodelwithoutCO,thedust-to-gasratioandtheice-to-dustratioatdifferenttimes
forinitialgas-to-COratiosof70(dottedredline),700(fiducialmodel,solidgreenline)and7000(dashedblueline).Thehorizontalgraylinesare
therespectivequantitiesonewouldexpectfromtheinitialconditionsoutsidetheiceline.
oftheCOicelineinthesemodelsfromapproximately40AUto and re-condensing CO vapor increases the ice surface density
50AU,initially. outsidetheiceline.
Thiseffectcanalsobeseenintheice-to-dustratio.At1Myr,
Theleftpanelsshowtheexcesscomparedtoamodelwith-
the ice-to-dust ratio is enhanced above the initial value even at
outCO.Notably,theexcessinsidetheicelineisnonzeroinall
radiilargerthan70AUdependingonthemodel.
cases,meaningthetotalsurfacedensityisstillhigherthaninthe
comparisonmodelwithoutCO.BecauseofthelargerCO-coated
particlesinthemodelswithCO,andthereforethemoreeffective
3.3. TransportofCOinthegasphase
drift,thedustsurfacedensityisenhancedinsidetheiceline.This
isespeciallyvisibleinthehighabundancemodel.Insomemod-
COisradiallytransportedinthediskviathreemechanisms:vis-
els,theexcessattheicelinecanbeafewtimeslargerthanone cous accretion and diffusion in the gas phase and frozen-out
wouldexpectfromtheinitialvalues.Thisisduetoaccumulation
as ice on drifting particles. As, outside of the ice line, most of
andre-distributionofCOattheiceline.
theCOisfrozen-outonparticles,thedominantradialtransport
Ingeneral,thedust-to-gasratiodecreaseswithtime.Thedust mechanismthereisparticledrift.Insidetheiceline,ontheother
surfacedensitydecreaseswhilethegasdensityremainsapprox- hand,alloftheCOexistsinthegasphaseandisradiallytrans-
imately constant over time, as discussed later. The dust-to-gas portedbyviscousaccretionanddiffusion.
ratio decreases from inside out because the particle growth is This means that transport of CO through the ice line from
most efficient in the inner parts of the disk, meaning the par- the outside into the inner part of the disk happens on drifting
ticles start to drift earlier than outside. The dust-to-gas ratio is dustparticles,whiletransportfromtheinsidetotheoutsidetakes
increased at the location of the ice line. This is due to two ef- place in the gas phase. The efficiency of viscous accretion is
fects:first,astheparticlesdriftthroughtheiceline,theyshrink highly dependent on the viscosity parameter α while the drift
becauseofevaporation,makingdriftlessefficientandtherefore velocityofparticlesdependsontheirsizeandisonlyindirectly
leading to a “traffic jam”, and second, the backwards-diffusing relatedtoαthroughthedustgrowth.
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