Probing the turbulent mixing strength in protoplanetary disks across the stellar mass range: no significant variations

0.48 MB·English

by  Gijs D. Mulders

Astronomy&Astrophysicsmanuscriptno.18127 c ESO2012
(cid:13)
January9,2012
Probing the turbulent mixing strength in protoplanetary disks
across the stellar mass range: no significant variations.
GijsD.Mulders1,2 andCarstenDominik1,3
1 AstronomicalInstitute“AntonPannekoek”,UniversityofAmsterdam,POBox94249,1090GEAmsterdam,TheNetherlands
2 SRONNetherlandsInstituteforSpaceResearch,POBox800,9700AV,Groningen,TheNetherlands
3 DepartmentofAstrophysics/IMAPP,RadboudUniversityNijmegen,P.O.Box90106500GLNijmegenTheNetherlands
Preprintonlineversion:January9,2012
2
ABSTRACT
1
0
Context.Dust settlingandgraingrowtharethefirststepsintheplanet-formationprocess inprotoplanetary disks. Thesedisksare
2
observedaroundstarswithdifferentspectraltypes,andthereareindicationsthatthedisksaroundlowermassstarsaresignificantly

n flatter,whichcouldindicatethattheysettleandevolvefaster,orinadifferentway.
a Aims.Weaimtotestthisassumptionbymodelingthemedianspectralenergydistributions(SEDs)ofthreesamplesofprotoplanetary
J disks:aroundHerbigstars,TTauristarsandbrowndwarfs.Wefocusontheturbulentmixingstrengthtoavoidastrongobservational

6 biasfromdiskandstellarpropertiesthatdependonstellarmass.
Methods.WegeneratedSEDswiththeradiativetransfercodeMCMax,usingahydrostaticdiskstructureandsettlingthedustina

] self-consistentwaywiththeα-prescriptiontoprobetheturbulentmixingstrength.
P Results.Weareabletofitallthreesampleswithadiskwiththesameinputparameters,scalingtheinneredgetothedustevaporation
E radiusanddiskmasstomillimeterphotometry.TheHerbigstarsrequireaspecialtreatmentfortheinnerrimregions,whiletheT-Tauri
. starsrequireviscousheating,andthebrowndwarfslackagoodestimateofthediskmassbecauseonlyfewmillimeterdetectionsexist.
h Conclusions.Wefindthattheturbulentmixingstrengthdoesnotvaryacrossthestellarmassrangeforafixedgrainsizedistribution
p
andgas-to-dustratio.Regionswiththesametemperaturehaveaself-similarverticalstructureindependentofstellarmass,butregions
- atthesamedistancefromthecentral starappear moresettledindisksaround lowermassstars.Wefindarelativelylowturbulent
o
mixingstrengthofα = 10 4 forastandardgrainsizedistribution,butourresultsarealsoconsistentwithα = 10 2 foragrainsize
r − −
t distributionwithfewersmallgrainsoralowergas-to-dustratio.
s
a Keywords.protoplanetarydisks-stars:pre-main-sequence-radiativetransfer-dustsettling-graingrowth
[

1
v 1. Introduction fromdiskmodelstryingtomatchaccretionratesisα =0.01
viscous
3 (e.g. Hartmannetal. 1998). Comparison of steady-state accre-
5 Protoplanetary disks are the main sites of planet formation. tiondiskmodelstomillimeterimagesyieldsα =0.5...10 4
4 Withinthem,thesmallsub-micronsizeddustgrainsgrowtomil- (Isellaetal.2009). viscous −
1 limeter and centimeter sizes and settle to the midplane, where
Theturbulentmixingstrengthmanifestsitself inthedegree
.
1 they eventually form kilometer-sized planetesimals and proto- ofdustsettling.Whendustgrainsgrowtolargersizes,theyde-
0 planets that are the building blocks of planetary systems. How
couple from the gas and start settling toward the midplane. At
2 thedustgrowsandsettlesdependsnotonlyonthegrainsizeand
what size and height they decouple depends not only on the
1 gasdensity,butalsoontheturbulentmixingstrengthofthegas
gas density and temperature but also on the turbulent mixing
: that mixes small grains back up into the disk atmosphere and
v strength:alowerturbulentmixingstrengthleadstoflatterdisks
i makesthem collideinthe midplane.Withoutturbulentmixing, (e.g.Dullemond&Dominik2004b).Asdustdecouples,itsrel-
X
alldiskswouldbeflatanddustwouldgrowslower.Additionally,
ativevelocityalsoincreases,whichleadstofragmentation,pro-
r turbulentmixingisanimportantdriverfortheviscousevolution
a vidinganaturalstoptograingrowthandreplenishingthesmall
of the disk, for radial mixing of dust, and it may also play an
dustgrainpopulationthatisthenmixedupintothedisksurface
importantroleinthelaterstagesofplanetformation.
(Dullemond&Dominik2005).
Although turbulent mixing is fundamental for our under-
Thereisbynowawealthofobservationalevidencesupport-
standingofdiskevolutionandplanetformation,thecauseofdisk
inggraingrowthanddustsettlinginprotoplanetarydisks.Most
turbulenceisnotcompletelyunderstood(e.g.Balbus&Hawley
notably, mid-infrared spectroscopic observations have shown
1998; Armitage 2011). Turbulent mixing in disks is often im-
that grains in the warm surface layers have grown beyond the
plemented using the α prescription from Shakura&Sunyaev
sizeofamicron(e.g.Henning&Meeus2009).Inaddition,the
(1973)andPringle(1981),butthevalueoftheturbulentmixing
spectral index at (sub)millimeter and centimeter wavelengths
strength α is hard to predict from theory. Although it could
turb shows that grains grow up to centimeter sizes in the cold disk
be empirically derivedfrom line-broadeningwith future obser-
midplane(e.g.Beckwith&Sargent1991;Mannings&Emerson
vatoriessuch as ALMA, this is challengingwith currentfacili-
1994). This picture has been confirmedwith hydrostatic radia-
ties(Hughesetal.2011).Thevaluetypicallyusedandinferred
tivetransfermodelsbyD’Alessioetal.(2006)andFurlanetal.
(2005) for T Tauri stars, who require a depletion of small dust
Sendoffprintrequeststo:GijsMulders,e-mail:[email protected] grains at the disk surface and a population of big grains in the
1
Mulders&Dominik:Probingturbulentmixingacrossthestellarmassrange
disk midplane. Without grain growth and dust settling, hydro- T_initial T_dust T_final
staticdiskmodelsoverpredictobservedfar-infraredfluxes.
Interestinglyenough,thisseemsnottobethecaseforHerbig
Ae and Be stars, the intermediatemasscounterpartsof T Tauri
vertical radiative
stars.Hydrostaticdiskmodelsdowellinexplainingfar-infrared
fluxes without the need to deplete the upper layers of the disk structure transfer
(Dominiketal.2003;Dullemond&Dominik2004a;Ackeetal.
2009),especiallyinMeeusgroupIsources1.However,the(sub)
millimeter slopes show that a population of millimeter-sized
grains exist in these disks as well (e.g. Ackeetal. 2004), indi- rho rho
catingthatthegrain-sizepopulationisnotthatdifferentfromT gas dust
Tauristars.Thisopensupthepossibilitythatthesedisksareless
settled,andthatdiskproperties-inparticulartheturbulentmix- dust
ingstrength- couldvarystronglyas a functionofstellar mass.
settling
Therecentdiscoverythatdisksaroundbrowndwarfsappearto
bemoresettledthanTTauristarsfitswellintothispicture(e.g.
Ladaetal.2006;Szu˝csetal.2010),althoughthisisnotalways Fig.1. Artist impression of vertical structure iterations in
found(Furlanetal.2011),andindicatesthattheobserveddegree MCMax to determine the vertical structure. Model parameters
ofsettlingcouldbeinverselyproportionaltostellarmass. aredisplayedinblackonwhite,partsofthecodeasblueblocks.
However,the degreeofsettling doesnotdirectlyreflectthe Treferstothetemperature,rhotothedensity.
turbulent mixing strength. Disks around T Tauri stars, Herbig
starsand browndwarfsvaryin disk mass (e.g. Beckwithetal.
1990; Scholzetal. 2006; Vorobyov 2009) and stellar mass,
and observations at the same wavelength probe different ra- combined with the method of continuous absorption by Lucy
dial regions in the disk because of the changing stellar lumi- (1999). The code has been benchmarked against other radia-
nosity. Comparing the degree of settling using a fixed height tivetransfercodesformodelingprotoplanetarydisks(Pinteetal.
or a lower scale height for bigger grains (e.g. D’Alessioetal. 2009).Theradialgridaroundtheinnerradiusanddiskwallisre-
2006; Sauter&Wolf 2011) is therefore less suitable for com- finedtosampletheopticaldepthlogarithmically.TheSEDand
parison along the stellar mass range. Radiative transfer mod- images are calculated by integrating the formal solution to the
els that describe dust settling using turbulent mixing exist equationofradiativetransferbyray-tracing.
(Dullemond&Dominik2004b;Hasegawa&Pudritz2010),but Inadditiontoradiativetransfer,thecodecanexplicitlysolve
havenotbeenusedtocomparetoobservationsdirectly. for the verticalstructure of the disk. With the implicitassump-
To constrain the turbulent mixing strength across the stel- tionthatthegastemperatureissetbythedusttemperature,the
lar mass range, we focused primarily on median spectral en- vertical structure of the gas can be calculated by solving the
ergydistributions(SEDs).Wedidthisbecausemeasuringthede- equation of hydrostatic equilibrium. On top of that, the dust
greeofsettlinginindividualdisksisnotstraightforwardbecause can be settled with respect to the gas, which we will describe
of a number of degeneracies in disk modeling (Sauter&Wolf in section 2.2. After calculating the new structure for the dust,
2011),andrequiresdetailedmultiwavelengthmodeling,includ- theradiativetransfercodecanberuntoobtainanewdusttem-
ing scattered-light images (Pinteetal. 2008), but conclusions perature,andthewholeprocedure(radiativetransfer,gasstruc-
can be drawn from larger samples (Furlanetal. 2005, 2006). ture, dust settling) can be iterated until convergenceis reached
TheyareavailableforbothTTauristars(e.g. Furlanetal.2005; (Figure 1). The temperature structure usually convergeswithin
D’Alessioetal. 2006), and brown dwarfs (Szu˝csetal. 2010), fiveiterations.Theexactnumberofiterationsdependsondisks
andasubstantialsampleisavailabletoconstructthemforHerbig parameters.Ingeneral,verysettledor’flat’diskstakelongerto
starsaswell(e.g.Ackeetal.2009;Juha´szetal.2010). convergebecausetheydeviatemorefromtheinitialguess.
We will describe how dust settling is implemented in our
radiative transfer code in section 2, and compare the available
approachesthatuse theturbulentmixingstrength.Insection3, 2.2.Dustsettling
wewillusethismodeltoconstraintheturbulentmixingstrength
To settle the dust with respect to the gas, we used the for-
fromthemedianSEDsofTTauristars,Herbigstarsandbrown
malism byDullemond&Dominik(2004b). We willbrieflyde-
dwarfs, which were constructed in appendixA and B. We will
scribe the basic assumptions, and refer to the previously men-
discusswhatthemediandiskmodellookslikeinsection4,and
tioned paper for details. We compare our approach to that of
whetherdisksaroundlowermassstarsareflatterormoresettled
Hasegawa&Pudritz(2010)insection2.2.2,whousedthemid-
thantheirhigh-masscounterparts.Wesummarizeourfindingsin
plane formalism by Dubrulleetal. (1995) in combination with
section5.
radiativetransfermodels.
2. Model
2.2.1. Theory
2.1.Diskmodel
The vertical motion of dust particles within the disk is a bal-
ThediskmodelusedinthispaperisMCMax(Minetal.2009),
ance of the downwardmotioncaused by the gravityof the star
a2DMonteCarloradiativetransfercode.Itisbasedontheim-
versus the mainly upward motion caused by turbulent mixing.
mediatere-emissionprocedurefromBjorkman&Wood(2001),
Comparing the timescales for both processes (the friction time
1 WenotethatforgroupIIsources,somesettlingmayberequiredto tfric versustheeddyturn-overtimetedd)resultsinanexpression
settletheouterdiskintotheinnerdiskshadow. fortheStokesnumberSt,whichdescribeshowwellthedustis
2
Mulders&Dominik:Probingturbulentmixingacrossthestellarmassrange
coupledtothegasateverylocationinthedisk:
t 3m Ω
St= fric = k , (1)
t 4σρ c
edd gas s
wherem/σis the mass-to-surfaceratio ofthe dustgrain,Ω =
k
GM /r3 the Keplerian frequency, ρ is the local gas den-
gas
sity an∗d c = kT/µm the local sound speed for the mean
s proton
p
molecular weight µ. A Stokes number smaller than one means
the dust is well coupled to the gas, whereas particles decou-
ple from the gas at larger Stokes numbers. Using this expres-
sion for the dust-gas coupling and the standard α prescription
from Shakura&Sunyaev (1973) in the form of Pringle (1981)
(ν=αc H ),thediffusioncoefficientforadustparticleatevery
s p
locationinthediskcanbewrittenas2
c H
s p
D=α , (2)
turb1+St2
whereH = c /Ω isthelocalpressurescaleheightandα is
p s k turb
theturbulentmixingstrength.
Taking the gas density and temperature structure from the
radiative transfer and hydrostatic structure calculation, we can
nowsolvefortheverticalstructureofadustgrainofcertainsize
ρ at every location in the disk by solving a vertical diffusion
a
equation:
∂ρ (z) ∂ ∂ ρ (z) ∂
a = D (z)ρ a ρ v (z) , (3)
∂t ∂z a gas∂z ρ − ∂z a sett,a
gas !!
(cid:0) (cid:1)
where v (z) is the local settling speed for that dust grain in
sett,a
the absence of turbulent mixing. This equation is then solved
in a time-dependent way using implicit integration. Because
Fig.2. Dust-to-gasratio versus relative height (z/r) in the mid-
the timescalesonwhichthedisk settles towardequilibriumare
infraredemitting regionof the T Tauri star at 10 AU using the
shorter than one million years(Dullemond&Dominik2004b),
self-consistentsettling(top)andthemidplanesettlingapproach
wejumptowardequilibriumintenrelativelylargetimesteps.
(bottom)withthesamegasdensitydistribution.Theblacksolid
We assumed that the turbulent mixing strength is constant
line representsthe total dust-to-gasratio, the gray lines denote
throughout the disk. There is no a priori reason to do so, nor
the individual contributions of grains of different sizes, from
a computationalone,and there are indicationsfor variationsin
0.02micron(dotted)to6mm(solid).Thedottedlinerepresents
both the vertical (Fromang&Nelson 2009; Simonetal. 2011)
a fully mixed disk, with a dust-to-gas ratio of 0.01. The radial
and radial (Isellaetal. 2009) direction. However, because we
τ=1 surface in the optical is located at a relative height of ap-
cannotconstrainthesevariationsfromthedatausedinthispaper,
proximately0.16and0.2.
wekeptαconstant.
2.2.2. Impactondiskstructure The well-mixed model has a uniform dust-to-gas ratio of
0.01,whereasthesettledmodelsshowacleartrendfromanin-
The main difference between this approach and that by
creaseddust-to-gasrationearthemidplanetoa decreasedratio
Dubrulleetal. (1995) is that we used the local sound speed
near and above the surface, which is located around z/r 0.18
and gas density to calculate the dust-gas coupling. Their ap- ∼
(radialτ = 1surfaceintheoptical).Thisiscausedbystratifica-
proach is an analytical solution for modeling the dust distri-
tion,thesettling ofdifferentdustspeciestodifferentheightsin
bution in the isothermal midplane, and uses only the midplane
thedisk.Inthemidplane,themidplanesettlingapproachagrees
density and temperature. Hence, we will refer to the latter as
verywellwith the self-consistentsettling, whichis also the re-
midplane settling, though it is sometimes applied outside this
gion that the former has been constructed for (Dubrulleetal.
regime (Hasegawa&Pudritz 2010). To compare the different
1995).
approaches,we havedisplayedthe verticalstructureatthe first
Closer to the disk surface, the two settled models start to
iterationinaverticalslabcenteredat10AUinFigure2.Because
deviate.Withincreasingheightabovethemidplane,thegasden-
thismodelhasnotbeeniterated,theverticalstructureofthegas
sity decreases. For the self-consistent settling, this means that
iscalculatedfromtheopticallythindusttemperature,andcorre-
smaller particles will decouple from the gas. For each particle
spondstoaGaussian.
size, there is a characteristic height above the midplane where
2 WeusedaSchmidtnumberof1+St2followingYoudin&Lithwick the coupling is so weak that they are almost fully depleted
(2007),ratherthan1+StfromDullemond&Dominik(2004b).Because (Dullemond&Dominik2004b).Forparticlesdowntoasizeof
thisdoesnotchangethepointwhereparticlesdecoupleatSt=1,ithasa roughly1micron,thisdecouplingtakesplacebelowthedisksur-
verylimitedinfluenceonourmodel. face,butsmallergrainscanstillbefoundabovethedisksurface.
3
Mulders&Dominik:Probingturbulentmixingacrossthestellarmassrange
Fig.3.Dusttemperaturesversusrelativeheightinaverticalslab Fig.4. Dust opacities for the different grain sizes used in this
centeredat10AU foraT Tauristar,showingthatstratification paper.
in dustsize leadsto a stratification in dusttemperature.Shown
aretheself-consistentmodel(solidline),thesettlingrecipefrom
Dubrulle(dottedline)anda well-mixedmodelwithoutsettling
forcomparison(dashedline). fraction of 15 % carbon, a value found from modeling this ra-
tio in Herbig Ae/Be stars (Meijeretal. 2005, 2009). The fi-
nal composition, with reference to the optical constants, is
Themidplanesettling approachshowsa differentbehavior, 11.73 % MgFeSiO4 (Dorschneretal. 1995), 32.6 % Mg2SiO4
(Henning&Stognienko1996),36.5%MgSiO (Dorschneretal.
becausethedust-gascouplingiscalculatedinthemidplaneonly. 3
1995), 1.5 % NaAlSi O (Mutschkeetal. 1998), 15% C
Particles do not decouplefrom the gas at a specific height, but 2 6
(Preibischetal. 1993). The shape of our particles is irregular,
keep following a Gaussian distribution all the way throughthe
andapproximatedusingadistributionofhollowspheres(DHS,
disksurface.Thisgivesanalmostconstantdust-to-gasratiofor
Minetal.2005).Weusedavacuumfractionof0.7,thoughthis
thesmallestparticles(<1µm),andamuchweakerdepletionfor
choicedoesnotinfluenceourresultsbecausewedonotfocuson
theintermediatesizes.
detailedfeatureshapes.
Settling the dust also impacts the temperature structure of
thedisk(Fig.3),andthereforeitsdensitystructure.Non-settled
disk modelshave a more or less constanttemperaturedistribu-
2.3.2. Size
tionbelowthedisksurface(Fig.3,dash-dottedline).Biggrains,
however,areefficientcoolers,andsettlingthemleadstoacolder Observations of protoplanetary disks show evidence for
midplane and warmer surface. The stratification in dust sizes (sub)micronandmillimeter-sizedorlargergrains.Intermediate
also leads to a gradual increase in temperature from the mid- sizedgrainsarenotobserved,butarepredictedbydustcoagula-
plane to the disk surface (see also Hasegawa&Pudritz 2010). tionmodels(e.g. Weidenschilling1984;Birnstieletal.2010a).
Becausethetemperatureinandabovethemidplaneisnolonger We therefore used a grain size distribution ranging from sub-
isothermal,itbecomesimportanttosolvefortheverticalstruc- microntomillimetersizesthatfollowsapowerlaw, f(a)=a q,
−
tureofthediskusingthelocalscaleheight,whichincreaseswith andusedadefaultindexofq=3.5similartotheMathis,Rumpl
height.Usingonlythemidplanetemperaturetocalculatethever- and Nordsieck (MRN) distribution (Mathisetal. 1977). To al-
ticalstructureofthediskunderestimatesthescaleheightwithin low settling of different grain size to a different height in the
themidplanebyuptoafactoroftwo(Fig.3).Thisleadstoflat- disk,wedividedoursizedistributionintotwobinsperorderof
terdiskswith-inturn-evenlowermidplanetemperatures(Fig. magnitude.Wedefinethegrainradiusforthesettlingcalculation
3,dottedline). withthelogarithmicmeanoftheminimumandmaximumgrain
sizesto(i.e.a=1.78micronforgrainsizebinbetween1.00and
3.16micron).Theopacitiesofthese grainswere calculatedus-
2.3.Dustgrainproperties
ing10sizesperbintoavoidstrongresonantfeaturesthatcould
Dustgrainsinourmodelarecharacterizedbythreedifferentpa- resultfromusingasinglegrainsize.
rameters:composition,sizeandshape.Differentgrainsizesset-
tletodifferentheightsinthedisk(SeeFig.2),andtheiropacities
2.4.Diskparameters
aredisplayedinfigure4.
Because our disks are in vertical hydrostatic equilibrium, the
onlyfreefittingparametertocontroltheverticalstructurefora
2.3.1. Compositionandshape
given mass is the turbulent mixing strength α. The radial dis-
Our dust grains consist mainly of amorphous silicates, mea- tribution of dust and gas is parametrized by a power-law as
suredforthecompositionofinterstellardusttowardtheGalactic Σ(r) = r p with the commonly used p = 1 for a steady-state
−
center by Minetal. (2007), co-added with with a continuum accretiondisk. The inner radiusis placed at a dust evaporation
opacity source, for which we used amorphous carbon. The ra- temperatureof1500K,whereastheouterradiusisfixedto300
tio of amorphous carbon / amorphous silicates influences the AU in accordance with D’Alessioetal. (2006). The total disk
peakstrengthofthe10-micronsilicatefeature.Weusedamass mass is a fixed fraction of 1 percent of the stellar mass (e.g.
4
Mulders&Dominik:Probingturbulentmixingacrossthestellarmassrange
Parameter Value Range
p 1.0 0...2.0
M 0.01M 0.001...0.1
disk star
dust-to-gas 0.01 0.01...1
T 1500 1000...2500
evap
R [AU] 300 50...1000
out
a [µm] 0.01 0.01...10
min
a [µm] 1000 10...105
max
q -3.5 -2.5...-4.5
α 0.01 10 6...1.0
turb −
Table1.Parametersofourdiskmodel,andrangesexplored.
Parameter BrownDwarf TTauri HerbigAe
T [K] 3000 4000 8500
eff
L [L ] 0.025 0.9 21
M∗ [M⊙ ] 0.08 0.5 2.0
R∗[AU⊙] 0.011 0.07 0.30 Fig.5. Observed median SED of T Tauri stars in Taurus (dia-
in
dMisdtuastn[cMe⊙[p]c] 816×5 10−6 514×0 10−5 2140× 10−4 mmoonddels)wwitihthaqtuurabrutilleenst(mgriaxyinagresatr)e.nOgvtehropfloαtte=d1is0o4uarnbdesatn-fiatcdcirsek-
−
Mτhaalcocr[M⊙/yr] 00 30 × 10−8 00.14 wtioitnhrαate=o1f032×(d1o0t−t8edMl⊙in/yer)(asnodlidwliitnheo)u.tAalcscoreptliootnte(ddaasreheadmliondee)l.
−
Table2.Stellar-dependentmodelparameters. The gray line denotes the stellar photosphere.The inset shows
themillimeterregime.
Scholzetal.2006;Williams&Cieza2011),assumingadust-to- diskbysettlingthedust.Aturbulentmixingstrengthofα=10 2
−
gasratioof1:100. - a value commonly used for viscous accretion disks - over-
predicts the mid-infraredfluxes longward of 20 micron, which
arises from the region around 10 AU. We need to lower the
3. FitstomedianSEDs ∼
turbulentmixingstrengthtoachieveagoodfit,whichresultsin
To compare our disk model to the three different data sets, we α=10−4.
used median SEDs. Instead of fitting all individual sources or Becausethe20micronfluxmainlyprobestheheightofthe
comparing them to a model grid, we only fitted our model to disksurface,therearealsootherwaystolowerit,see alsosec-
onemedianSEDtoextract’typical’fitparameters.Furlanetal. tion4.5.Wecanreducethenumberofsmallgrains,whichwill
(2005,2006) haveshownthatthisapproachworkswell, taking lowertheopticaldepthintheupperlayersanddecreasethevisi-
intoaccounttheobservedspreadinmid-infraredcolors. bledisksurface.Decreasingthetotaldiskmasswouldbeincon-
When fitting the data, we tried to keep asmanyparameters sistentwiththemillimeterflux,butwecanmodifythegrainsize
fixedaspossible,seeTable1.Parametersthatcouldnotbefixed distribution because it is dominated by big grains. Decreasing
because of the different stellar properties are given in Table 2. thepower-lawindexofthegrainsizedistributionremovessmall
These are the inner radius, which we kept at 1500 Kelvin and grainsfromthedisksurface,butdoesnotsignificantlyaffectthe
the dust mass, which is a fixed fraction of the stellar mass of massinbiggrainsnearthediskmidplane.Changingthepower-
10 4M . lawindexto3.25issufficienttofitthemedianSEDwithatur-
−
∗ bulentmixingstrengthofα=10 2.
−
3.1.TTauristars
3.2.Herbigstars
The Taurus median SED has been fitted using a hydrostatic
disk modelby D’Alessioetal. (2006), which demonstratedthe SpectralenergydistributionsforalargenumberofHerbigstars
need for grain growth and dust settling. Our goal is to re- areavailable,butamedianSEDhasnotbeenconstructedbefore.
produce these results, but now in the context of a more self- WeconstructedoneinappendixB,whichwepresentinfigure6.
consistent settling approach to constrain the turbulent mixing Totesttheassumptionthatmoremassivestarshavelesssettled
strength, and not an arbitrary height for the midplane layer disks, we retained as many disks parameters from the T Tauri
of big grains. We have extended the existing data set from starsaspossible.We changed(seealsotable2)thediskmass-
Furlanetal. (2006) with sub-millimeter fluxes at 850 µm us- whichscaleswiththestellarmassandisincreasedwithafactor5
ingthecatalogfromAndrews&Williams(2005).Thedatafrom withrespecttotheTTauristar,consistentwiththesubmillimeter
D’Alessioetal. (1999) include some non-detectionsand upper data-andtheinnerradius-whichalsoincreasestobeconsistent
limits(seeappendixA).Theupdatedmillimetermedianiswell withdustevaporation.
fittedbyadustmassof5 10 5M ,consistentwiththeresult Although accretion does not play a role in the SEDs
−
fromAndrews&Williams×(2005). ⊙ of Herbig stars because of the increased stellar luminos-
TofitthemedianSEDinthenear-infrared,weneededtoin- ity and larger inner radius, the near-infrared fluxes (2-8 mi-
cludeviscousheating(Fig.5).Wedidnotself-consistentlysolve cron) of hydrostatic disk models underpredictthe observations
the radialstructureof the disk, butcalculatedthe viscousheat- (Vinkovic´etal. 2006, see also Fig. 6). Different mechanisms
ingwithamassaccretionrateof3 10 8 M /yr,thesameas have been proposed to explain the difference, from an in-
−
Furlanetal.(2006).Tofitthemid-in×fraredflux⊙,weflattenedthe creasedrimscaleheight(Verhoeffetal.2010;Ackeetal.2009)
5
Mulders&Dominik:Probingturbulentmixingacrossthestellarmassrange
Fig.6. Observed median SED of Herbig stars (diamonds) with Fig.7. Observedmedian SED of browndwarfsin Chamaeleon
quartiles(grayarea).Overplottedisourbest-fitdiskmodelwith I(diamonds),wherethe24micronandmillimeterpointshould
a turbulent mixing strength of α = 10 4 and a dust shell with betreatedasupperlimits.Overplottedisourbest-fitdiskmodel
−
opticaldepthτ=0.14(solidline).Alsoplottedareamodelwith with a turbulentmixingstrengthof α = 10 4 (solid line). Also
−
α = 10 2 (dotted line) and without a dust shell (dashed line). plottedfor comparisonis a modelwith α = 10 2 (dottedline).
− −
The gray line denotes the stellar photosphere.The inset shows Thedashedlinedenotesthestellarphotosphere.Theinsetshows
themillimeterregime. themillimeterregime.
to halos (Vinkovic´etal. 2006; Verhoeffetal. 2011) to mate- Unlike the Herbig and T Tauri stars, hydrostatic disk mod-
rial within the dust evaporation radius (e.g. Krausetal. 2008; els of brown dwarfs do not show a flux deficit in the near-
Tannirkulametal. 2008a). Although an artificially increased infrared compared to the observed median. We scaled the disk
scale height works well for the mid-infrared SED (Ackeetal. massandinnerradiustothebrowndwarfregime(seeTable2).
2009), it does not work well when fitting simultaneously the The stellar photosphere was fitted by a NextGen stellar atmo-
near-infrared region of the Herbig median SED (see appendix sphere(Hauschildtetal. 1999),andwe adjustedtheluminosity
C). Because an inner gaseous disk is beyond the current capa- downto 0.025L .We fitted themedianshortwardof8micron
bilitiesofourcode,wemodeledthenear-infraredfluxdeficitof withoutanyaddi⊙tionaladjustments(seefigure7).Theassumed
the hydrostaticmodelwith an optically thin sphericalhalo. We dust mass of 8 10 6 M is consistent with the upper limit
−
emphasize that this halo is a parametrization of optically thin fromKleinetal.×(2003), an⊙damountsto only2-3earthmasses
emissionfromtheinnerregions,includingdustorgaswithinthe ofsolidmaterial.
innerrim,andrefertheinterestedreadertotheappendix. Wefoundthataturbulentmixingstrengthofα=10 4iscon-
−
TheopticallythinhaloplaystheroleofaccretioninTTauri sistentwiththe24micronupperlimit,whereasamixingstrength
stars: it provides near-infrared flux without affecting the flux ofα = 10 2 clearlyoverpredictsthisupperlimit.Wefoundthe
−
longward of 20 micron, in contrast to the puffed-up inner rim. sameturbulentmixingstrengthasfortheearlierspectraltypes.
Weusedasmallhaloof0.3...1microngrainsbetween0.14and
0.3AU3thathasanopticaldepthof0.14inthevisualandadust
massof4 10 12M (Figure6).Asaresultofthefittingproce- 4. Discussion
−
dure,we fo×undthe sa⊙me turbulentmixingstrengthforthe disk
Although we found the same turbulent mixing strength for the
of α = 10 4 as for the T Tauristars. A higher mixing strength
− threedifferentsamples,thisdoesnotnecessarilymeanthatthese
of α = 10 2 overpredictsthe Spitzer IRS fluxes from 20 to 40
− disksareequallyflat.Theverticalstructureisaproductofmore
micronandtheIRASfluxat60micron.
than just the turbulent mixing strength, most notably the local
density, temperature and Keplerian frequency.In the following
3.3.Browndwarfs sectionwewilldescribethediskproperties,andshowthatdisks
aroundlowermassstarsappearmoresettledataspecificradius,
Disks around brown dwarfs are more difficult to detect owing
consistent with Szu˝csetal. (2010), but that the structure in the
totheirlowluminosity.Completesamplesdowntophotospheric
mid-infraredemittingregionisremarkablyself-similar.
valuesonlyexistupto8micron,andareincompleteat24micron
Wewillalsodiscusswhyourchoiceofgrainsizedistribution
(e.g. Ladaetal. 2006; Scholzetal. 2007; Luhmanetal. 2008).
doesnotaffectourconclusionthattheturbulentmixingstrength
Furthermore, very few millimeter detections exist (Kleinetal.
doesnotvaryalongthestellarmassrangeinsection4.3andits
2003;Scholzetal.2006).WeusedthemedianSEDconstructed
implicationfortheplanet-formingpotentialofthediskinsection
by Szu˝csetal. (2010) for Chameleon I, where the median 24
4.4. The dependence of the fitted turbulent mixing strength on
micronfluxshouldbetreatedasanupperlimit.Forconvenience,
otherdiskparametersisdiscussedinsection4.5.
wealsoplotthemillimeterupperlimitfromKleinetal.(2003),
scaledtothedistanceoftheChameleonIstar-formingregion.
4.1.Diskpropertiesatfixedradius
3 Becausethehalospansaverynarrowtemperaturerange,itsnear-
infraredspectrumisvirtuallyindistinguishablefromthatofapuffed-up To compare the dust and gas distribution in our best-fit mod-
innerrim(seeFig.C.1). els, we display their pressure scale heightand the radialτ = 1
6
Mulders&Dominik:Probingturbulentmixingacrossthestellarmassrange
Fig.8.Locationofthedustandgasinthediskofthebest-fitmodelofaTTauristar(center),Herbigstar(left)andabrowndwarf
(right). Displayed in gray is the gas located within one (dark) and two (light) pressure scale heights. The solid lines marks the
locationoftheradialτ=1surfaceinthevisual.
surfaces in figure 8. The pressure scale heightis highestin the By scaling the radius to the intensity of the radiation field, we
brown dwarf disk and lowest in the Herbig star, reflecting the canretrieveaself-similarsolutionfortheopticallythindusttem-
differencesinstellarmass.Althoughtemperaturesarealsolower peratureprofile
forlessmassivestars,thisisnotenoughtocompensatethelower
0.5
gravitationalpotentialinwhichthediskresides,andthegasre- T (r)=307K r /AU L /L −
dust T
mainsmoreflaring. ∗ ∗ ∗ ⊙ (5)
The dust, on the other hand, follows a different path: the =315K∗(cid:16)(rT/AU)−0p.5 =Tˆdus(cid:17)t(rT),
τ = 1surfaceintheouterdiskreachesarelativeheight(z/r)of wherethetemperatureat1AUfortheTTauristarisclosetothe
0.17,0.15and0.14fortheHerbigstar4,TTauristarandbrown selfconsistentsolutionbecauseitsluminosityiscloseto1L .As
dwarf,respectively.Thisindicatesthatdisksaroundlessmassive can be seen in figure9b, these scaled solutionsagree to w⊙ithin
starsarealittleflatter,thoughthedifferenceremainswithin20%. 20%fortheTTauristar,Herbigstarandbrowndwarf.
Comparedtothemuchhigherpressurescaleheightatthesame
radius,thediskoflowermassstarsthereforeappearsmoreset-
4.2.1. Self-similartemperatureandsurfacedensity
tled,becausethedisksurfacereachesdowntoonepressurescale
heightforthe Herbigstar, butto less than halfa pressurescale With the same procedure,we can also comparedifferentquan-
heightinthebrowndwarf.Thiscanbeexplainedbylower(sur- titiestoseeiftheyarealsoself-similarinallthreesamples.We
face)densitiesand temperatures(see figure10a, 11a andTable find that also the midplane temperatures for all three samples
3)thatmakethedustdecoupleclosertothemidplane,aswellas (seeFig.10b)canbedescribedwithaself-similarfunction:
aloweropticaldepth.
The shape of the disk surface can also be viewed in Tˆmid(rT)=90K (rT/AU)−0.44, (6)
∗
scattered-lightimages.Wedisplaytheradialprofilesofsynthetic
which is a strong indication that this must be a consequence
scattered-lightimagesat0.5µminfigure12a.Theradialprofiles
ofaself-similarverticalstructureacrossthestellarmassrange,
arescaledtothephotosphericfluxatthesamewavelength,such
which will we explore below in more detail. Interestingly,this
thatittracesonlytheshapeofthedisksurface,andnottheprop-
scaling law also works for the surface density (Fig. 11), such
ertiesofthecentralstar.Acleartrendisvisibleatallradii,with
thatregionswithsimilartemperaturesalsohavesimilarsurface
higher fluxes and shallower slopes for Herbig stars and lower
densities:
fluxes and steeper slopes for brown dwarfs (see also Table 3). Σˆ(r )=0.25g/cm2 (r /AU) 1, (7)
T T −
Thisindicatesthatthe disk surfaceat a specific locationis less ∗
flaring. This is because across the three samples, the luminosity scales
roughlywithstellarmasssquared5.Sincethediskmassisafixed
fractionofthestellarmass,thesurfacedensityscalesas
4.2.Self-similarsolutionsintheMIRemittingregion
M L
Because the mid-infraredemission probesa different region in Σˆ1AU = ⊙ Σ1AU = ⊙ Σ1AU, (8)
M L
all three samples, it is also interestingto compareregionswith ∗! r ∗
similartemperatures,whicharelocatedatdifferentradiifordif- However,theregioncorrespondingtothesametemperaturelies
ferentstars.Theseregionscanbeidentifiedbyusingthatthedis- fartherawayfromthestar,compensatingforthechangeindisk
tancer,atwhichaspecificfluxfromthestarisreceived,scales mass:
withthesquarerootofthestellarluminosityL .Byscalingthe
craodnisutsrutcottdhiemveanlsuieonolfestsheeqsuteivllaalrenfltusxo,frTou=r dri/sk∗√pLr∗o/pLe⊙r,tiwese(cdaen- Σˆ(rT)=Σˆ1AU rT−1 = √ΣL1A/UL √Lr/L −1 (9)
! !
notedbyaˆ),andcheckiftheyareself-similaracrossthestellar =Σ r 1 =Σ(r), ∗ ⊙ ∗ ⊙
1AU −
massrange.
For example, the optically thin dust temperature for the T Thus the self-similar solutions for the surface density agree to
Tauristarcanbedescribedas(seeFig.9a) within20%.
T (r)=307K (r/AU) 0.5, (4) 5 Although the main-sequence mass luminosity relation for solar
dust −
∗ mass stars follows L M3.5...4, the T Tauri stars in our sample are
4 Ithastobenotedthatthehaloincreasesthecalculatedτ=1surface pre-mainsequenceand∝donotfollowthisrelation,whereasthebrown
fortheHerbigstar. dwarfsareinadifferentscalingregimealtogether.
7
Mulders&Dominik:Probingturbulentmixingacrossthestellarmassrange
a) b)
Fig.9.Radialtemperatureprofileoftheopticallythindustforthebest-fitmodelofaTTauristar(solidline),Herbigstars(dotted
line)andabrowndwarf(dash-dottedline).Theleftpanelshowstherealtemperature,therightpanelshowstheself-similarsolutions.
a) b)
Fig.10.Radialtemperatureprofileinthemidplaneforthebest-fitmodelofaTTauristar(solidline),Herbigstars(dottedline)and
abrowndwarf(dash-dottedline).Theleftpanelshowstherealtemperature,therightpanelshowstheself-similarsolutions.
a) b)
Fig.11. Radial surface density profile for the best-fit model of a T Tauri star (solid line), Herbig stars (dotted line) and a brown
dwarf(dash-dottedline).Theleftpanelshowstherealsurfacedensity,therightpanelshowstheself-similarsolutions.
a) b)
Fig.12. Radial intensity profile in scattered light at 0.5 micron for the best-fit model of a T Tauri star (solid line), Herbig stars
(dottedline)andabrowndwarf(dashedline).Theleftpanelshowstherealintensityasfractionofthestellarfluxatthatwavelength
F (r)/F versusr),therightpanelshowstheself-similarsolutions(F (r)/F (L /L )versusr/√L /L
scat scat
∗ ∗· ∗ ⊙ ∗ ⊙
8
Mulders&Dominik:Probingturbulentmixingacrossthestellarmassrange
4.2.2. Self-similarscaleheight
If we now look at a vertical slab in the disk located at r =
T
r/√L /L ,wefindthattwoofthemainparametersfortheset-
tlingc∗alcu⊙lationinequations(1)and(2)-thegassurfacedensity
(8)andtemperature(6)-areself-similar.Thethirdimportantpa-
rameter,theKeplerianfrequency,isnotthesame,butscaleswith
√L /L
∗ ⊙
Ωˆk(rT)= sGrMT3∗ = sG √LM/⊙L √Lr/L !−3 (10)
∗ ⊙ ∗ ⊙
GM
= r r3⊙ L∗/L⊙ =Ωk(r) L∗/L⊙,
p p
whichisequivalenttor/r .Thegaspressurescaleheightatthis
T
locationisthereforenotself-similar,buttherelativescaleheight
Hp/ris: Fig.13. Dust-to-gasratio at rT = 10 AU for the best-fit model
ofaTTauristar(solidline),Herbigstar(dottedline)andbrown
Hˆp(rT) = cˆs(rT) = cs(r) = cs(r) = Hp(r). dwarf(dashedline).Thegraylinedenotesa well-mixedmodel
rT rTΩˆk(rT) rT(Ωk(r)r/rT) rΩk(r) r with a dust-to-gas ratio of 0.01. The surface temperature at
(11) thislocationcorrespondstoroughly100K,andthereforeemits
Thismeansthatifweexpresstheheightabovethemidplane(z) mainlyinthemid-infrared.
in terms of the relative height(z/r), the local scale height, and
thereforetheverticalstructureofthegas,isself-similar:
Diskproperties
Hˆ (r ,z /r )= H (r ,z /r ). (12) Parameter Brown TTauri Herbig Self-
p T T T p T T T dwarf Ae/Be similar
T [K] 36 86 185 90
1AU
4.2.3. Self-similardustsettling q 0.45 0.44† 0.43 0.44
Σ [gr/cm2] 0.04 0.24 0.95 0.25
1AU
The vertical structure of the dust in the mid-infrared emitting I [mJy/AU2] 1.110 9 1.310 6 4.210 4 2.910 5
regionatrT = 10AUisshowninfigure13.Likethatofthegas, o100AU 3.1· − 2.9· − 2.6· − 2.9· −
the verticalstructureof the dustalso appearsself-similar when
Table 3. Disk propertiesfor our best-fit models. The midplane
expressedintermsofitsrelativeheightz/r.Thisisbecausealso
temperatureis fitted by T (r)=T r q. The intensity in scat-
theStokesnumber(Eq.(1)) mp 1AU −
teredlightbyI(r)=I r ooutwardsof70AU. Themidplane
100AU − †
3m Ωˆ (r ) temperatureinTTauristarsshowsaclearbreakat2AUcaused
Sˆt(r ,z /r )= k T
T T T 4σρˆ (r ,z /r )c (r ,z /r ) by viscous heating (fig 10a), and at smaller radii it is best de-
gas T T T s T T T scribedasT(r)=125r 0.93.
−
3m Ω (r)/√L /L (13)
= k ∗ ⊙
4σρ (r,z/r)/√L /L c (r,z/r)
gas s
∗ ⊙
=St(r,z/r)
is self-similar because the gas density scales with 1/Hp. The The self-similarity of verticalstructure is furtherillustrated
sameistrueforthediffusioncoefficient(2) by synthetic scattered-light images that trace the shape of the
dust disk surface (Fig. 12). If we plot the radial intensity pro-
cˆ (r ,z /r )Hˆ (r ,z /r )
Dˆ(rT,zT/rT)=αturbcs(rT1,+zTS/ˆrt2T()rHT,p(zrTT/,rzTT)/rT) (14) ttfiihvleeetahhsereiagehfdut6inffc(eFtirsoecnant/toFsft∗at∗hrseLl∗sin/cLael⊙ue)dp,wrnaiedceisuleyse(rtFTh,iagtcutohrreere1rca2tdbeid)a,lfaopnrrdothfithelearstetflhoae-r
s T T T p T T T
=α shapeofthedisksurfaceisindeedself-similar.
turb 1+St2(r ,z /r )
T T T Since the amount of radiation reprocessed at every radius
= D(r,z/r) is set by the coveringfraction (z/r) - which is the same for all
stars - even the SED in the mid-infrared becomes self-similar
because it is only a productof self-similar quantitieswhen ex-
(figure 14). The scaled model SEDs line up to within 20% in
pressedinunitsofr andz /r .Consequently,theverticalstruc-
T T T themid-infraredat20-30micronwherewefittedtheSEDs.The
tureofbothgasanddustinthemid-infraredemittingregionsof
similarityextendsintothefar-infrared(within40%)andeventhe
protoplanetarydisksareself-similar.
near-infraredupto3micron(withinafactorof2).Thelatteris
ThisresultisconsistentwiththatofFurlanetal.(2011),who
surprisingandprobablycoincidental,giventhe differentnature
alsofounndnovariationinthedegreeofsettlingbetweenbrown
oftheemissionmechanismatthatwavelength:viscousheating
dwarfs and T Tauri stars, but partly contradicts Szu˝csetal.
for the T Tauri star, optically thin emission for the Herbig star
(2010).Theseauthorsalsofoundthesamerelativescale height
andmostlyphotosphericfluxforthebrowndwarf.
(H/r)forthedustinthemid-infraredemittingregion.However,
the required reduction in dust scale height compared to their
fully flared models implies a difference in gas scale heightbe- 6 The relative height z/r scales with r /r = 1/√L /L , while the
T
tweenbrowndwarfsandTTauristars,whichwedonotfind. intensityinscatteredlightscaleswith(r /r)2=1/(L /L∗ ).⊙
T
∗ ⊙
9
Mulders&Dominik:Probingturbulentmixingacrossthestellarmassrange
4.4.Planet-formingpotential
If the same disk properties are found at a different location in
the disk aroundheavier stars, does this mean that the resulting
planetarysystemsaresimplyscaledversionsofeachother?
To comparethe later stagesof planetformation,we looked
at the Hill sphere and feedingzone of a formingplanet, to see
howmuchmassisavailableforitsformation,inawaysimilarto
Raymondetal.(2007).TheHillsphereisgivenby
m
R (r)=r 3 P , (16)
H
3M
r ∗
wherem isthemassoftheplanet.Expressedinthesameunits
P
thatresultinself-similarsolutionsfortheverticalstructure(r =
T
r/√L /L )itbecomes
∗ ⊙
m R (r)
Fig.14.Best-fitmodelSEDsfortheTaurusmedian(solidline), RˆH(rT)=r/ L /L 3 P = H , (17)
Herbig median (dotted line) and brown dwarf median (dashed ∗ ⊙ 3M /√L /L √3L /L
line),scaledtoaluminosityof1L andadistanceof140pc. p r ∗ ∗ ⊙ ∗ ⊙
⊙ So the size of the Hill sphere is not self-similar, thoughits de-
pendenceonluminosityisweak.Itdecreasesinsizeforlesslu-
minous stars, despite the lower gravity of the central star. The
4.3.Grainsizedistributions
totalmassavailableinthefeedingzonedeviatesmorebecauseit
scaleswithanextrafactor2πr :
Throughoutthisarticle,wehaveassumedthattherearenoradial T
gradientsin the grainsize distribution.Modelsof graingrowth
Mˆ (r )=2πr Σˆ(r ) 2Rˆ (r )
predictthatgrainsizesdecreasewithradius(e.g. Birnstieletal. feed T T T ∗ H T
2010b),whichhasalsobeenobservedbyGuilloteauetal.(2011) =2πΣ(r)r/ L /L 2 RH(r) (18)
andBanzattietal. (2011).Additionally,disksaroundlatertype ∗ ⊙∗ √3L /L
sMtaordsealriengcothldeegrraanindsleizsesmdiastsrsiibvuet,iolenadisinbgeytoonsdmtahlelesrcgorpaeinosfiztheiss. = L /L−5/6Mpfeed(r). ∗ ⊙
∗ ⊙
paper, though a recipe for doing so is available (Birnstieletal.
Thesizeofthefeedingzonethereforescalesalmostlinearlywith
2011).Fornow,wewillonlydiscussifweexpecttoseediffer-
luminosity. This means that although the initial conditions for
encesin the grainsize distributionin the mid-infraredemitting
planetformationarethesameacrossthestellermassrange,this
regionbetweenthethreedifferentsamples.
isnottrueforthelaterstagesofplanetformation,inagreement
Because the vertical structure in the mid-infrared region is
withRaymondetal.(2007).
self-similar, we investigated if this is also true for the grain-
sizedistributionanditsplanet-formingpotential.Grainsinpro-
toplanetary disks are in a coagulation-fragmentation equilib- 4.5.Dependenceondiskparameters
rium (Weidenschilling 1984), driven by turbulence. According
Themethodwepresenthereisusefulforconstrainingvariations
toBirnstieletal.(2011),themostimportantparametersthatde-
in the turbulentmixing strength,but it is more difficultto con-
fine the size distribution are the turbulent mixing strength, the
strain its absolute value. The mid-infrared flux with which we
gas surface density, the temperature, and three parameters that
fitted our models traces the height of the disk surface, which
refertothemicrophysicsofdust:thefragmentationvelocityu ,
f canalsobeinfluencedbyotherfactorsthantheturbulentmixing
thesoliddensityofdustρ andthesizedistributionoffragments
s strength. However, because our solutions for the mid-infrared
ξ.Weassumedthelastthreetobethesameacrossthethreesam-
emitting region are self-similar, these uncertainties will affect
ples.
our estimate of the turbulent mixing strength in the same way
We findnovariationsin theturbulentmixingstrength(sec-
forallstars,anddonotinfluenceourconclusionthatitdoesnot
tion3).Regionswithsimilartemperaturealsohaveasimilarsur-
varyacrossthestellarmassrange.
facedensity(section4.2).Thegrainsizedistributionshouldthen
Inthissectionwewillexplorehowotherparameterscanaf-
alsobeself-similar.Wecanseewhythisisthecasebylookingat
fect our estimate of the turbulent mixing strenght. In particu-
theexpressionforthemaximumgrainsizeforturbulence-driven
lar, we will explore how to keep the turbulent mixing strength
growth7-asgivenbyequation(12)ofBirnstieletal.(2009):
fixed at α = 0.01 and fit the three median SEDs by vary-
turb
ingtheseotherparameters.Asalreadymentionedinsection3.1,
aˆ (r )= Σˆ(rT)u2f = Σ(r)u2f =a (r), (15) smallchangesinthegrainsizedistributionalsoaffecttheheight
max T πα ρ cˆ2(r ) πα ρ c2(r) max ofthedisksurface.Usingaflatterslopeforthisdistributionre-
turb s s T turb s s
moves small grains from the disk surface, and we can fit the
which is also self-similar when looking at regionswith similar medianSEDs with a power-lawindexof 3.25.Anotherparam-
temperature (r = r/√L /L ). This means that also the grain eter that directly influences the SED in the mid infrared is the
T
size distribution in the m∗id-i⊙nfrared emitting region is not ex- gas-to-dustratio.Alowerratioleadstoaweakercouplingofthe
pectedtovaryacrossthestellarmassrange. dustandhencetoflatterdisks.Wefittedagas-to-dustratioof1
forα =0.01.
turb
7 which istherelevant regimeforparticlesizessmallerthanamil- One parameterthat we cannotchangeis the dustmass, be-
limeter causethiswouldbeinconsistentwiththemillimeterphotometry.
10