Lopsided dust rings in transition disks

0.65 MB·English

by  T. Birnstiel

Astronomy&Astrophysicsmanuscriptno.paper˙banana (cid:13)c ESO2013
Friday11thJanuary,2013
LettertotheEditor
Lopsided dust rings in transition disks
T.Birnstiel1,2,C.P.Dullemond3,andP.Pinilla3
1 ExcellenceClusterUniverse,TechnischeUniversita¨tMu¨nchen,Boltzmannstr.2,85748Garching,Germany
2 Harvard-SmithsonianCenterforAstrophysics,60GardenStreet,Cambridge,MA02138,USA
3 Heidelberg University, Center for Astronomy (ZAH), Institute for Theoretical Astrophysics, Albert Ueberle Str. 2, 69120
Heidelberg,Germany
Friday11thJanuary,2013
ABSTRACT
3 Context.Particletrappinginlocalorglobalpressuremaximainprotoplanetarydisksisoneofthenewparadigmsinthetheoryof
1 thefirststagesofplanetformation.However,findingobservationalevidenceforthiseffectisnoteasy.Recentworksuggeststhatthe
0 largering-shapedouterdisksobservedintransitiondisksourcesmayinfactbelopsidedandconstitutelargebanana-shapedvortices.
2 Aims.Wewishtoinvestigatehoweffectivedustcanaccumulatealongtheazimuthaldirection.Wealsowanttofindoutifthesize-
sortingresultingfromthiscanproduceadetectablesignaturesatmillimeterwavelengths.
n
Methods. Tokeepthenumericalcostundercontrolwedevelopa1+1Dmethodinwhichtheazimuthalvariationsaretreatedsepa-
a
ratelyfromtheradialones.Theazimuthalstructureiscalculatedanalyticallyforasteady-statebetweenmixingandazimuthaldrift.
J
Wederiveequilibrationtimescalesandcomparetheanalyticalsolutionstotime-dependentnumericalsimulations.
9
Results.Wefindthatweak,butlong-livedazimuthaldensitygradientsinthegascaninduceverystrongazimuthalaccumulationsof
dust.ThestrengthoftheaccumulationsdependsonthePe´cletnumber,whichistherelativeimportanceofadvectionanddiffusion.
] WeapplyourmodeltotransitiondisksandoursimulatedobservationsshowthatthiseffectwouldbeeasilyobservablewithALMA
P
andinprincipleallowstoputconstraintsonthestrengthofturbulenceandthelocalgasdensity.
E
. Keywords.accretion,accretiondisks–protoplanetarydisks–stars:pre-main-sequence,circumstellarmatter–planetsandsatellites:
h
formation–submillimeter:planetarysystems
p
-
o
r 1. Introduction wardshigherpressure.Inthecaseofvorticesandeddiestheyact
t
s on small scales. Particle traps on a global disk scale have been
a
The process of planet formation is thought to start with the proposedaswell(Whipple1972;Riceetal.2006;Alexander&
[
growthofdustaggregatesinaprotoplanetarydiskthroughcoag- Armitage2007;Garaud2007;Kretke&Lin2007;Dzyurkevich

1 ulation(seee.g.,thereviewby Blum&Wurm2008).Theideais et al. 2010) and shown to be conducive to planet formation
v thattheseaggregatesgetsuccessivelybiggeruntileithergravo- (Brauer, Henning, & Dullemond 2008). Intermediate scale par-
6 turbulent processes set in (Goldreich & Ward 1973; Johansen ticle traps may also occur from magnetorotationally driven tur-
7 et al. 2007; Cuzzi et al. 2008) or planetesimals are formed di- bulence:theso-calledzonalflows(Johansenetal.2009).
9
rectly via coagulation (Weidenschilling 1977; Okuzumi et al.
If we want to observationally test whether this scenario of
1
2012; Windmark et al. 2012). One of the main unsolved prob-
. particle trapping actually occurs in nature, we are faced with a
1 lems in these scenarios is the problem of excessive radial drift
problem.TheEarth-formingregionaroundapre-mainsequence
0 (Weidenschilling 1977; Nakagawa et al. 1986; Brauer et al. starisusuallytoosmallontheskytobespatiallyresolvedsuffi-
3 2007).Theoriginofthisproblemliesinthesub-Keplerianmo-
cientlywelltotestthistrappingscenario.Moreover,theoptical
1
tion of the gas in the disk, caused by the inward pointing pres-
depth of this inner disk region is likely to be too large to be
:
v suregradient.Thedustparticlesinthedisk,however,feelonly abletoprobethemid-planeregionofthedisk.Fortunately,what
Xi the friction with the sub-Keplerian gas and as they grow to constitutes the “meter size drift barrier” at 1 AU is a “cm size
larger sizes, the reduced surface-to-mass ratio causes them to driftbarrier”at∼50AU.Thosediskregionsareopticallythinat
r falltowardthestarwithspeedsupto50ms−1 (Whipple1972;
a millimeter(mm)wavelengthandparticlesinthemmsizerange
Weidenschilling1977).Thus,ifaparticlegrows,itwillsooneror
canbespectroscopicallyidentifiedbystudyingthemmspectral
laterget“flushed”towardthestarbeforeitcangrowverylarge
slope(Testietal.2001;Nattaetal.2004;Riccietal.2010a,b).
(Braueretal.2008;Birnstieletal.2010,2012).Thisiswhatwe
So the goal that has been pursued recently is to identify obser-
calltheradialdriftbarrier,andthisstillposesoneofthemain
vational signatures of dust particle trapping of millimeter-sized
unsolvedproblemsoftheearlyphasesofplanetformation.
particles in the outer regions of disks, as a proxy of what hap-
A possible solution to this problem might lie in the con- pensintheunobservableinnerregionsofthedisk(Pinillaetal.
cept of “particle traps”. This idea has been proposed already 2012b,a).ThePinillaetal.(2012a)papersuggeststhatthehuge
some time ago in the context of anticyclonic vortices by Barge mm continuum rings observed in most of the transition disks
& Sommeria (1995) and Klahr & Henning (1997), as well as (Pie´tuetal.2006,Brownetal.2008,Hughesetal.2009,Isella
in/around turbulent eddies (Johansen & Klahr 2005; Cuzzi & et al. 2010, Andrews et al. 2011), may in fact be large global
Hogan2003).Thesevortexoreddy-relatedlocalpressuremax- particle traps caused by the pressure bump resulting from, for
imaactasparticletrapsbecauseparticlestendtodriftalwaysto- example,amassiveplanetopeningupagap.
1
T.Birnstiel,C.P.Dullemond,P.Pinilla:Lopsideddustringsintransitiondisks
We have focused in these papers on the intermediate scale ∂ug,φ =−Aρ (u −u )− 1Ω u − 1 ∂P, (4)
(zonal-flow-type) and the the large scale (global) pressure ∂t d g,φ d,φ 2 k g,r rρ ∂φ
g
bumps,simplybecausecurrentcapabilitiesofmmobservatories
(includingALMA)arenotyetabletoresolvesmallscalestruc- whereud,randud,φaretherandφcomponentsofthedustveloc-
turessuchasvortices.Thevortextrappingscenariothusappears ity,respectively,ug,r andug,φtherandφcomponentsofthegas,
to remain observationally out of reach. However, a closer look Ω the Keperian frequency, P the gas pressure and ρ and ρ
k d g
at the mm maps of transition disks suggest that some of them the dust and gas densities. A denotes the drag coefficient (see,
may exhibit a deviation from axial symmetry. For instance, the Nakagawa et al. 1986, equations 2.3 and 2.4). Solving above
observations presented in Mayama et al. (2012) or the mm im- equationsforthevelocityalongtheφdirectionatthemid-plane
ages of Brown et al. (2009) suggests a banana shaped lopsided (z=0)gives
ringinsteadofacircularring.Rega´lyetal.(2012)proposedthat
these banana-shaped rings are in fact a natural consequence of u = 1 1 ∂P, (5)
mass piled up at some obstacle in the disk. Once the resulting d,φ St+St−1(1+X)2 ρgVk ∂φ
ringbecomesmassiveenough,itbecomesRossby-unstableand
a large banana-shaped vortex is formed that periodically fades where X = ρd/ρg is the dust-to-gas ratio, Vk the Keplerian ve-
and re-forms with a maximum azimuthal gas density contrast locity,andtheStokesnumber1isgivenby
of a few. Rega´ly et al. (2012) showed that this naturally leads
ρ a
to lopsided rings seen in mm wavelength maps (see also ear- St= s√ Ω , (6)
k
lier work by Wolf & Klahr 2002) on radiative transfer predic- ρ c 8/π
g s
tionsofobservabilityofvorticeswithALMA).Theformationof
suchRossby-waveinducedvorticeswasdemonstratedbeforeby withρs asinternaldensityofthedust,particleradiusa,andthe
Lyraetal.(2009),whoshowedthattheymayinfact(whenthey isothermalsoundspeedcs.
are situated much further inward, in the planet forming region) Thus, dust is advected with the velocity given in Eq. 5 but
leadtotherapidformationofplanetaryembryosofMarsmass. it is also turbulently stirred. Together, the evolution of the dust
Sa´ndoretal.(2011)subsequentlyshowedthatthisscenariomay densityalongtheringisthendescribedby
rapidlyproducea10Earthmassplanetarycore.
(cid:32) (cid:32) (cid:33)(cid:33)
∂ρ ∂ (cid:16) (cid:17) ∂ ∂ ρ
The goal of the current letter is to combine the scenario of d = ρ u − Dρ d , (7)
forming a lopsided gas ring (e.g., Rega´ly et al. 2012) with the ∂t ∂y d d,φ ∂y g ∂y ρ
g
scenario of particle trapping and growth presented by Pinilla
et al. (2012a). In Section 2, we will outline the physical ef- where y = rφ is the coordinate along the ring circumference.
fectsinvolvedandderiveanalyticalsolutionstothedustdistribu- We use a dust diffusion coefficient D according to Youdin &
tion along the non-axisymmetric pressure bump and Section 3, Lithwick (2007), D = Dgas/(1 + St2), where we assume the
we will test the observability of these structures in resolved gas diffusivity to be equals to the gas viscosity, taken to be
(sub-)mm imaging and in the mm spectral index. Our findings ν = αtc2s/Ωk,withαt astheturbulenceparameter(seeShakura
willbesummarizedinSect.4. &Sunyaev1973).Equation7canbeintegratedforwardintime
numerically,butassumingthattheturbulentmixingandthedrift
termhavereachedanequilibriumandalsoassumingalowdust-
to-gas ratio, we can analytically solve for the dust density in a
2. Analyticalmodel
steady-statebetweenmixinganddrifting,whichyields
Dustparticlesembeddedinagaseousdiskfeeldragforcesifthey
(cid:34) (cid:35)
move relatively to the gas. The radial and azimuthal equations ρ (y)=C·ρ (y)·exp −St(y) , (8)
of motion have been solved for example by Weidenschilling d g α
t
(1977) or Nakagawa et al. (1986) for the case of a axisymmet-
ric, laminar disk. It was found that particles drift inward to- whereCisanormalizationconstantandSt(y)istheStokesnum-
wards higher pressure. In this paper, we will focus on the case berwhichdependsonyviathechangesingasdensity.Eq.8thus
where a non-axisymmetric structure has formed a long lived, predicts the distribution of dust for any given profile of the gas
non-axisymmetricpressuremaximuminthedisk.Thispressure density ρg. The contrast between the position of the azimuthal
maximumisabletotrapinward-spirallingdustparticles.Likein pressuremaximumanditssurroundingthengives
theaforementionedworkswecansolveforastationarydriftve-
locity,butunlikeinNakagawaetal.(1986),theradialpressure ρmdax = ρmgax exp(cid:34)Stmin−Stmax(cid:35), (9)
gradient is zero at the pressure maximum while the azimuthal ρmin ρmin α
pressuregradientcanbedifferentfromzero.Thisleadsagaintoa d g t
systematicdriftmotionofthedustparticlestowardsthepressure which is plotted in Fig. 1. Stmax and Stmin are the Stokes num-
maximum,butnowinazimuthalinsteadofinradialdirection.
bersatthepressuremaximumandminimum,respectively(note:
Theequationsofmotioninpolarcoordinatesrelativetothe Stmax <Stmin).ItcanbeseenthatoncetheparticlesStokesnum-
Keplerianmotionbecome(seeNakagawaetal.1986) ber becomes larger than the turbulence parameter α, the dust
t
concentration becomes much stronger than the gas concentra-
∂ud,r =−Aρ (u −u )+2Ω u (1) tion.However,thetimescalesonwhichtheseconcentrationsare
∂t g d,r g,r k g,φ reached can be significant. In order to get an estimate for this
∂u∂dt,φ =−Aρg(ud,φ−ug,φ)− 21Ωkud,r (2) ttihmeedisffcualseio,nwetimcoemspcaalreettdhiffea=dvLe2c/tDio,nwtiimthevseclaolceittyaduv a=ndL/leunagnthd
∂u
g,r =−Aρ (u −u )+2Ω u (3) 1 St=1typicallycorrespondstoparticlesofmmtocmsizesinthe
∂t d g,r d,r k g,φ outerdisk.Fortypicaldiskconditionsa(cid:39)0.4cm·St·Σ /1gcm−2.
g
2
T.Birnstiel,C.P.Dullemond,P.Pinilla:Lopsideddustringsintransitiondisks
104 101 gas 4 cm, numerical, 1×tadv
1 µm sized dust, analytical 4 cm, numerical, 2×tadv
4 cm sized dust, analytical 4 cm, numerical, 5×tadv
103
dustcontrast102 [normalized]Σ 100
101 10-1
gas contrast: 1.1
gas contrast: 1.5
gas contrast: 2
100 0 π π 3π 2π
10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 2 2
St φ
Fig.1.Contrastbetweenthedustdensityintheazimuthalmax- Fig.2.Azimuthalsteadystatesolutionsforsmall(redline)and
imum and its surroundings for different gas density contrasts large dust grains (black line) for a given sinusoidal gas profile
(ρmax/ρmin),StokesnumbersSt,asderivedinEq.9.Turbulence (blackline,identicalwithredline).Dashedlinesshownumerical
g g
parametersareα =10−2(solid),andα =10−3(dashed). solutionsat1,2,and5advectiontimescales.
t t
scaleL.Theratioofthesetimescales,knownasthePe´cletnum- AsradialgassurfacedensityprofileΣg(r)andtemperatureT(r)
weusetheresultspresentedinPinillaetal.(2012a).Thesesim-
berPe,which,usingEq.5,canbewrittenas
ulationsrepresentadiskofmassM =0.05M aroundasolar
disk (cid:12)
Pe= tdiff = St L ∂ρg (cid:39) St∆ρg. (10) massstarwitha15Jupiter-massplanetat20AU.Forsimplicity
t α ρ ∂y α ρ thetwo-dimensionalgassurfacedensityisthentakentobe
adv t g t g
It describes the relative importance of advection and diffusion Σ (r,φ)=Σ (r)·(cid:20)1+A(r)·sin(cid:18)φ− π(cid:19)(cid:21) (13)
g g
and confirms that dust accumulations occur only for particles 2
(cid:34) (cid:35)
with St (cid:38) α, because otherwise, diffusion dominates over ad- c−1 (r−R )2
t A(r)= ·exp − s (14)
vection which means that variations in the dust-to-gas ratio are c+1 2H2
being smeared out. It also shows that for those large particles
theadvectiontimescaleistheshorterone,thussettingthetime wherec = Σg,max/Σg,min isthelargestcontrastofthegassurface
scaleoftheconcentrationprocess,whichcanbewrittenas density,takentobe1.5.Rs isthepositionoftheradialpressure
bump.ThedustsizedistributionΣ (r,a)wasalsotakenfromthe
d
π2 (cid:18)H(cid:19)−2 1 simulationsofPinillaetal.(2012a),anddistributedazimuthally
t = , (11)
adv δSt r Ω usingtheanalyticalsolutionfromEq.8(seeFig.2).Wealsocon-
k
firmed the analytical solution and time scales by solving Eq. 7
whereH = c /Ω isthepressurescaleheightandwehaveused
ameanvelocsityuk= 1(cid:12)(cid:12)(cid:12)(cid:82)πudφ(cid:12)(cid:12)(cid:12)= c2s δSt,with nounmlyehroicladlslya,tatsheshpoowsintiionnFoifg.th2e.rTahdiisalanparelysstiucraelbsuomlupti,obnusttrsiicntclye
π 0 πVk
δSt=(cid:12)(cid:12)(cid:12)arccot(cid:16)Stmin(cid:17)−arccot(cid:0)Stmax(cid:1)(cid:12)(cid:12)(cid:12), (12) mthoesstimofutlhaetiomnmsoefmPiisnsiilolanectomal.es(2f0ro1m2a)thaerelatrrgaeppgerdainnseawrhthicehrian-
(cid:12) (cid:12)
dialpressuremaximum,thisshouldbeareasonableapproxima-
which forSt < 1 simplifiesto δSt = Stmin −Stmax. Further,we tion.Fulltwo-dimensionalsimulationswillbeneededtoconfirm
definethepressuremaximumandminimumtobeatφ = 0and thisandtoinvestigatetheeffectsofshear.
φ=π,respectively.Asanexample,at35AU,forH/r =0.07,a To compare directly with current ALMA observations, we
Stokesnumberof0.2andagasdensitycontrastofρmax/ρmin = calculate the opacities for each grain size at different wave-
g g
lengths and assume spherical silicate grains with optical con-
2, the time scale is 3×105 years, but could be as short as 102
stants for magnesium-iron grains from the Jena database2. The
orbitsforoptimalconditions.Anygasstructurethereforehasto
continuum intensity maps are calculated assuming that in the
be long-lived to cause strong asymmetries in the dust, making
sub-mm regime, most of the disk mass is concentrated in the
e.g. asymmetries caused by a planet or long lived vortices the
opticallythinregion.Weassumethesamestellarparametersas
bestcandidates(Meheutetal.2012).Ifsuchanaccumulationis
inPinillaetal.(2012a),azimuthallyconstanttemperatureT(r),
formed and the gas asymmetry disappears, it still takes tdiff to typicalsourcedistances(d=140pc)andzerodiskinclination.
“remove”it,whichat35AUisoftheorderofMyrs.Itremains
We run ALMA simulations using CASA (v. 3.4.0) at 345 GHz
tobeshownwhethershort-lived,butreoccurringstructureslike
(band 7) and 675 GHz (band 9), shown in Fig. 3. We consider
zonalflowsareabletoinducestrongdustaccumulations.
2 hours of observation, the most extended configuration that is
currentlyavailablewithCycle1,genericvaluesforthermaland
3. SimulatedObservations atmosphericnoisesandabandwidthof∆ν = 7.5GHzforcon-
tinuum.Atthesetwodifferentfrequencies,itispossibletodetect
Inthefollowing,wewillevaluatewhethertheduststructureswe
expectwouldbeobservableinresolvedmmimagesofALMA. 2 http://www.astro.uni-jena.de/Laboratory/Database/databases.html
3
T.Birnstiel,C.P.Dullemond,P.Pinilla:Lopsideddustringsintransitiondisks
Fig.3.ALMAsimulatedimagesat345GHzwithanobservation Fig.4.Spectralindexmapα usingsimulatedimagesatband7
mm
time of 2 hours. The total flux of the source is 0.13 Jy and the and9.Theantennaconfigurationischosensuchthattheangular
contourlinesareat{2,4,6,8}therms(σ = 0.22mJy). resolutionissimilarforbothbands∼0.16”(∼22AUat140pc).
and resolve regions where the dust is trapped creating a strong
Birnstiel,T.,Dullemond,C.P.,&Brauer,F.2010,A&A,513,79
azimuthalintensityvariation. Birnstiel,T.,Klahr,H.,&Ercolano,B.2012,A&A,539,148
The spectral slope α of the spectral energy distribution Blum,J.&Wurm,G.2008,ARA&A,46,21
mm
F ∝ να is directly related to the dust opacity index at these Brauer,F.,Dullemond,C.P.,Johansen,A.,etal.2007,A&A,469,1169
ν Brauer,F.,Henning,T.,&Dullemond,C.P.2008,A&A,487,L1
longwavelengths(e.g.Testietal.2003),anditisinterpretedin
Brown,J.M.,Blake,G.A.,Qi,C.,Dullemond,C.P.,&Wilner,D.J.2008,ApJ,
terms of the grain size (αmm (cid:46) 3 implies mm sized grains). 675,L109
WiththesimulatedimagesofFig.3,wecomputetheα map Brown,J.M.,Blake,G.A.,Qi,C.,etal.2009,ApJ,704,496
mm
(Fig. 4), considering an antenna configuration that provides a Cuzzi,J.N.&Hogan,R.C.2003,Icarus,164,127
Cuzzi,J.N.,Hogan,R.C.,&Shariff,K.2008,ApJ,687,1432
similarresolutionof∼0.16”(∼22AUat140pc)foreachband.
Dzyurkevich,N.,Flock,M.,Turner,N.J.,Klahr,H.,&Henning,T.2010,A&A,
Thisresolutionisenoughtodetectα variationsalongtheaz-
mm 515,70
imuth,confirmingthatthoseareregionswheredustaccumulates Garaud,P.2007,ApJ,671,2091
andgrowsduetothepresenceofanazimuthalpressurebump. Goldreich,P.&Ward,W.R.1973,ApJ,183,1051
Hughes,A.M.,Andrews,S.M.,Espaillat,C.,etal.2009,ApJ,698,131
Isella,A.,Natta,A.,Wilner,D.,Carpenter,J.M.,&Testi,L.2010,ApJ,725,
4. Summaryandconclusions 1735
Johansen,A.&Klahr,H.2005,ApJ,634,1353
Wehaveshownthatweak,butlong-livedazimuthalasymmetries Johansen,A.,Oishi,J.S.,Low,M.-M.M.,etal.2007,Nature,448,1022
inthegasdensitycancauseveryefficientaccumulationofdust Johansen,A.,Youdin,A.,&Klahr,H.2009,ApJ,697,1269
Klahr,H.H.&Henning,T.1997,Icarus,128,213
atthepositionoftheazimuthalpressuremaximum.Wehavede- Kretke,K.A.&Lin,D.N.C.2007,ApJ,664,L55
rivedanalyticalsteady-statesolutionsforthedustdistributionfor Lyra,W.,Johansen,A.,Klahr,H.,&Piskunov,N.2009,A&A,493,1125
anygivenazimuthalgasdensitydistributionandthetimescales Mayama,S.,Hashimoto,J.,Muto,T.,etal.2012,ApJL,760,L26
Meheut, H., Keppens, R., Casse, F., & Benz, W. 2012, Astronomy &
onwhichthesedistributionsdevelop.Goodagreementhasbeen
Astrophysics,542,9
foundbetweenthesolutionsandnumericalsimulations.
Nakagawa,Y.,Sekiya,M.,&Hayashi,C.1986,Icarus,67,375
For this dust concentration mechanism to work, particles Natta,A.,Testi,L.,Neri,R.,Shepherd,D.S.,&Wilner,D.J.2004,A&A,416,
musthavegrowntolargersizes(St>α)suchthattheazimuthal 179
t
drift becomes stronger than the turbulent diffusion. This typi- Okuzumi,S.,Tanaka,H.,Kobayashi,H.,&Wada,K.2012,ApJ,752,106
Pie´tu,V.,Dutrey,A.,Guilloteau,S.,Chapillon,E.,&Pety,J.2006,A&A,460,
callycorrespondstoparticlesofsub-mmtocmsizes.Thestrong
L43
concentrationofthelargestgrainsleadstoasize-sortingwhich Pinilla,P.,Benisty,M.,&Birnstiel,T.2012a,A&A,545,81
is observable via low spectral indices at mm wavelengths and Pinilla,P.,Birnstiel,T.,Ricci,L.,etal.2012b,A&A,538,114
alsoinlopsidedbananashapedstructuresinresolved(sub-)mm Rega´ly,Z.,Juha´sz,A.,Sa´ndor,Z.,&Dullemond,C.P.2012,MNRAS,419,1701
Ricci,L.,Testi,L.,Natta,A.,&Brooks,K.J.2010a,A&A,521,66
images. Finding the size range where the bifurcation between
Ricci,L.,Testi,L.,Natta,A.,etal.2010b,A&A,512,15
concentration and diffusion happens would put constraints on
Rice,W.K.M.,Armitage,P.J.,Wood,K.,&Lodato,G.2006,MNRAS,373,
theturbulencestrengthandthelocalgasdensityofthedisk. 1619
Sa´ndor,Z.,Lyra,W.,&Dullemond,C.P.2011,ApJL,728,L9
Acknowledgements. We thank Ewine van Dishoeck, Simon Bruderer, Nienke Shakura,N.I.&Sunyaev,R.A.1973,A&A,24,337
vanderMarel,GeoffreyMathews,HuiLiandtherefereeforusefuldiscussions. Testi,L.,Natta,A.,Shepherd,D.S.,&Wilner,D.J.2001,ApJ,554,1087
Testi,L.,Natta,A.,Shepherd,D.S.,&Wilner,D.J.2003,A&A,403,323
Weidenschilling,S.J.1977,MNRAS,180,57
References Whipple,F.L.1972,FromPlasmatoPlanet,211
Windmark,F.,Birnstiel,T.,Ormel,C.W.,&Dullemond,C.P.2012,A&A,544,
Alexander,R.D.&Armitage,P.J.2007,MNRAS,375,500 L16
Andrews,S.M.,Wilner,D.J.,Espaillat,C.,etal.2011,ApJ,732,42 Wolf,S.&Klahr,H.2002,ApJ,578,L79
Barge,P.&Sommeria,J.1995,A&A,295,L1 Youdin,A.N.&Lithwick,Y.2007,Icarus,192,588
4