Effects of photophoresis on the dust distribution in a 3D protoplanetary disc

2.4 MB·

by  Nicolas Cuello

MNRAS000,1–??(2015) Preprint23March2016 CompiledusingMNRASLATEXstylefilev3.0
Effects of photophoresis on the dust distribution in a 3D
protoplanetary disc
N. Cuello1, J.-F. Gonzalez1 and F.C. Pignatale1
1Université de Lyon, F-69003 Lyon, France; Université Lyon 1, Observatoire de Lyon, 9 avenue Charles André,
F-69230 Saint-Genis Laval, France; CNRS, UMR 5574, Centre de Recherche Astrophysique de Lyon;
Ecole Normale Supérieure de Lyon, F-69007 Lyon, France
6
1 Accepted...Received...
0
2
ABSTRACT
r
a Photophoresis is a physical process based on momentum exchange between an illu-
M minated dust particle and its gaseous environment. Its net effect in protoplanetary
discs (PPD) is the outward transport of solid bodies from hot to cold regions. This

2 process naturally leads to the formation of ring-shaped features where dust piles up.
2 In this work, we study the dynamical effects of photophoresis in PPD by including

thephotophoreticforceinthetwo-fluid(gas+dust)smoothedparticlehydrodynamics
]
P (SPH) code developed by Barrière-Fouchet et al. (2005). We find that the conditions
E of pressure and temperature encountered in the inner regions of PPD result in impor-
tant photophoretic forces, which dramatically affect the radial motion of solid bodies.
.
h Moreover, dust particles have different equilibrium locations in the disc depending
p
on their size and their intrinsic density. The radial transport towards the outer parts
-
o of the disc is more efficient for silicates than for iron particles, which has important
r implications for meteoritic composition. Our results indicate that photophoresis must
st be taken into account in the inner regions of PPD to fully understand the dynamics
a and the evolution of the dust composition.
[
Key words: protoplanetarydiscs–planetsandsatellites:formation–hydrodynam-
2
ics – methods: numerical.
v
2
6
6
3 1 INTRODUCTION the ratio between its stopping time and its orbital time, is
0 close to 1. The stopping time is defined as the time it takes
1. Thousands of extra-solar planets have been detected in the for a particle moving at a given initial velocity to reach the
past 20 years (Batalha et al. 2013), which undoubtedly
0 samevelocityasthegas.Foraparticleofsizesanddensity
shows that planets are common byproducts of star forma-
6 ρd intheEpsteinregime,theoptimalsizeforradialdriftin
tion. As a matter of fact, it is widely accepted that planets
1 the midplane is given by (Fouchet et al. 2010):
: form in PPD around young stellar objects in a few Myr as
v the result of the accretion of kilometre-sized bodies called s (r)= √Σg(r) , (1)
Xi planetesimals (Chiang & Youdin 2010). These solids are opt 2πρd
formed through collisional growth of dust particles, with where Σ is the gas surface density and r is the distance to
r g
a sizes ranging from ∼ 0.1 µm to ∼ 10 cm. Interferometric thestar.s dependsontheconsidereddiscmodel:forthe
opt
observationsatdifferent wavelengthsprobethesedustpop- MinimumMass Solar Nebula (MMSN)models s ∼ 1 m
opt
ulations and suggest that grains grow in the disc (Natta at1au(Weidenschilling1977;Hayashi1981),whileinclas-
et al. 2007, and references therein). Thus, in order to allow sical T Tauri star protoplanetary discs s ∼1 dm at 1 au
opt
collisional growth to occur, dust grains must survive in the (Laibe et al. 2012). If the radial drift is fast enough and
disc without being accreted by the central star. the particle cannot decouple from the gas, then the solid
However, the study of the motion of solids inside the body will be accreted by the star in time-scales of the or-
disc leads to the so-called radial-drift barrier (Weiden- der of hundreds of orbital periods. This is a severe problem
schilling 1977). A solid body orbiting a star in a Keple- for planet formation theory since it prevents dust grains to
rian fashion loses angular momentum because of the aero- reach planetesimal sizes.
dynamic drag of the gas, which orbits at a sub-Keplerian Several mechanisms have been proposed to break the
velocity.Thisisbecausethegasphaseispressuresupported radial-drift barrier such as radial mixing (Keller & Gail
while the dust is not. The radial drift of a solid body is the 2004,andreferencestherein);particletrapscausedbyplan-
most efficient when the Stokes number of a grain, which is ets (Paardekooper & Mellema 2004; Fouchet et al. 2007,
(cid:13)c 2015TheAuthors
2 N. Cuello et al.
2010; Ayliffe et al. 2012; Gonzalez et al. 2012; Pinilla et al. pressure gaseous environments. Coupling their results to
2012; Gonzalez et al. 2015); dead zones (Kretke & Lin a one-dimensional disc model, they predicted that pho-
2007; Dzyurkevich et al. 2010; Armitage 2011) and anticy- tophoresis would be able to stop the inward drift for large
clonicvortices(Barge&Sommeria1995;Meheutetal.2010; bodieswithsizesrangingfromamillimetretoametre.Based
Regály et al. 2012); meridional circulation (Fromang et al. on their measurements, they also demonstrated that poros-
2011) and radiation pressure effects (Vinković 2014). How- ity dramatically increases the photophoretic force felt by
ever, it remains unclear if a single mechanism or a combi- illuminated dust particles.
nationofthemcanactuallypreventaccretion.Inthiswork, The expression of the photophoretic force depends on
weconsiderphotophoresisasapossiblemechanismtobreak the properties of the particle such as its size (Rohatschek
the radial-drift barrier. 1995), chemical composition (Loesche & Wurm 2012) and
Photophoresis has long been known (Ehrenhaft 1918), porosity(Duermannetal.2013).Consequently,aradialsort-
howeveritisonlyrecentlythatithasbeenconsideredfrom ing among grains of different properties may occur in the
anastrophysicalpointofview(Krauss&Wurm2005).This disc. This is experimentally demonstrated in Wurm et al.
phenomenon is due to the temperature gradient across an (2010) by measuring the photophoretic effects on different
illuminateddustparticleinalow-pressuregaseousenviron- materials(glass,steelandchondrule-likeparticles).Theirre-
ment.Inthisconfiguration,amomentumexchangebetween sults show that photophoresis is less efficient for good heat
the dust particle and the surrounding gas molecules takes conductorssincethetemperaturegradientbetweentheillu-
place,whichresultsinamotionoftheparticleawayfromthe minated and the shadowed side of the particle is lower.
radiationsource(cf.Section2).Intheopticallythinregions The sorting of grains with different thermal properties
of PPD dust particles are illuminated by the star and sur- may have played an important role in determining the ob-
rounded by low-pressure gas. Then, according to Krauss & servedpropertiesofmeteorites,suchascarbonaceouschon-
Wurm(2005),photophoresismightleadtotheformationof drites, where high and low temperature material coexist in
ring-shapeddustdistributionswithsharpinneredgeswhere the same object (Scott & Krot 2014). Indeed, photophore-
particles pile up. sis naturally transports high-temperature forming material
However,itmightbearguedthatphotophoresiscaused from the inner regions of PPD outwards (Wurm & Krauss
by stellar radiation only affects a small fraction of the disc. 2006; Mousis et al. 2007). The analysis of meteorites shows
Thisiscertainlytrueforyoungsystemswherethediscisstill that,amongthechondrites,mostofthegroups1aredepleted
verymassiveandthestellarradiationcannotpenetratevery in iron relative to bulk solar system abundances (CI chon-
deep into it. Nevertheless, in transitional discs at later evo- drites). However, there are some groups, like EH and EL,
lutionarystagestheamountofgasdecreasesasthegasdissi- which show an overabundance of iron (Scott & Krot 2014).
patesandtheinnerregionsbecomeopticallythin(Augereau This indicates that a mechanism has to operate in the in-
et al. 1999; Calvet et al. 2002; Wahhaj et al. 2005). ner regions of the solar nebula in order to separate iron-
Thefactthattransitionaldiscstendtodepletetheinner rich and silicate particles. Moudens et al. (2011) computed
regionsofgasinshorttime-scalescouldbeseenasanissue. the efficiency of the photophoretic transport for different
In fact, without gas the photophoretic effects completely grain sizes over the disc lifespan in one-dimensional MMSN
vanish.However,Wurm&Krauss(2006)suggestthatthere models including turbulence. They found that photophore-
isastage,justbeforethegasclearing,wherethediscisstill sis leads to accumulation regions in the disc in agreement
dense in gas but transparent for the stellar radiation. This with the ring-shaped dust distributions first proposed by
phase might have existed in the late solar nebula. Consid- Krauss & Wurm (2005). McNally & Hubbard (2015) com-
eringtransitionaldiscs,Takeuchi&Krauss(2008)compute putedthephotophoreticforceonopaquesphericalparticles
the location of the τ = 1 line, which defines the transition inadilutegasintheopticallythickregime.Inthiscase,the
between the optically thin and the optically thick regimes. temperature gradient is caused by the energy dissipation
By doing so, they estimate the locations in the disc which in the accretion disc, which is highly localized and results
canbereachedbystellarradiation.Giventhesmallamount in strong vertical temperature fluctuations (McNally et al.
of µm-sized particles remaining at this epoch, they neglect 2014). Then, non-illuminated dust particles can be locally
the dust absorption and consider the Rayleigh scattering transportedbyphotophoresisintheverticaldirection.How-
by H as the main extinction mechanism. Their analysis ever,sinceinthisworkwefocusonopticallythintransitional
2
leads to the important conclusion that these discs can be discs, this effect will not be considered here.
considered as optically thin for distances as large as 10 au To date, no studies have explored the complex effects
(cf. their fig. 4). Considering one-dimensional transitional of photophoresis in PPD through hydrodynamical simula-
disc models, they computed, for different particle sizes, the tions. The aim of this paper is twofold: on the one hand,
equilibrium distances at which the photophoretic and the we study the efficiency of the outward transport of solids
dragforcecanceleachotherout.Theirresultsindicatethat due to photophoresis and, on the other hand, we attempt
photophoresis has a significant effect on the dust particles to connect the resulting dust dynamics with the chemical
located from 0.01 au up to several au and that it might be composition of the grains. We give an overview of the pho-
able to stop their inward drift. tophoreticforceinSection2,detailthemethodinSection3,
Furthermore, microgravity experiments on illuminated show our results in Section 4, discuss the radial transport
mm-sized particles surrounded by gas show that pho- duetophotophoresisinSection5andconcludeinSection6.
tophoresis efficiently transports solids away from the ra-
diation source (Wurm et al. 2010). Moreover, Duermann
et al. (2013) measured the strength of the photophoretic 1 A chondritic group contains chondrites with similar chemical
force on illuminated plates with given porosities in low- properties,presumablyformedfromthesameparentbody.
MNRAS000,1–??(2015)
Effects of photophoresis on the dust in a 3D PPD 3
The expression for the photophoretic force, F , used
ph
in this work is derived in Rohatschek (1995) through the
study of illuminated spherical particles in aerosols. It is a
semi-empiricalformulawhichcoverstheentirerangeofpres-
sures,fromthefreemolecularflowregimeatlowpressureto
hydrodynamicflowathighpressure.Therearethreeregimes
for the photophoretic force according to the pressure, p, of
thegaseousenvironment:thelow-pressureregimeforwhich
F ∝ p, the high-pressure regime for which F ∝ 1/p,
ph ph
and the transition regime when F reaches its maximum.
ph
TheseregimesaredefinedbythevalueoftheKnudsennum-
Figure 1.Momentumexchangebetweenthedustgrainandthe ber defined as Kn = λ/s, where λ is the mean free path of
gasparticles(radiationfromtheleft). the gas molecules and s is the radius of the solid particle.
√
Its expression is given by λ = k T/ 2πd2p, where k is
B B
the Boltzmann constant and d is the diametre of the gas
molecules. In the low-pressure regime Kn (cid:29) 1, whereas in
the high-pressure regime Kn (cid:28) 1. The expression for the
photophoretic force reads as follows:
2F
F = max (2)
ph p/p +p /p
max max
with
s2 (cid:114)αI
F = D , (3)
max 2 2 k
Figure 2.Netforceonthedustgrain(radiationfromtheleft). (cid:114)
3T 2
p = D , (4)
max πs α
2 THE PHOTOPHORETIC FORCE
(cid:114)
πc¯η πκ
2.1 A model for all pressures D= t , (5)
2T 3
ThesketchinFig.1showsanilluminateddustparticleem-
and
beddedinalow-pressuregaseousenvironmentwiththeradi-
(cid:114)
ationcomingfromtheleft.Inthisconfiguration,atempera- 8R T
c¯= g (6)
turegradientdevelopsonitssurface:thehottersideisonthe πµ
left and the cooler side is on the right. We assume that all
where I is the irradiance of the incident beam of light, α is
the gas molecules have the same mass and thermal velocity
the thermal accommodation coefficient (dimensionless and
closetothedustgrain.Whenthesemoleculesarediffusively
taken equal to 1 (Rohatschek 1995)), k is the thermal con-
scattered2bytheparticlesurface,theyequilibrateandreach
ductivity of the solid particle, T is the gas temperature, η
thesametemperature.Thus,theirnewvelocitiesdependon
istheviscosityofthegas,κ isthethermalcreepcoefficient
wheretheyhitthesurface:moleculesleavingthehotterside t
(equal to 1.14 (Rohatschek 1995)), R is the universal gas
willhavelargervelocitiescomparedtotheonesonthecooler g
constant and µ is the molar mass of the gas. All the quan-
side. Therefore, the momentum exchange between the par-
tities on which Eqs. (3) to (6) depend are functions of the
ticleandthegasenvironmentisanisotropicandresultsina
disc local conditions and the distance to the star.
netforceonthedustgraindirectedawayfromtheradiation
The photophoretic force is often split into two compo-
source as shown in Fig. 2.
nents: the ∆T-force mainly driven by the temperature gra-
The rotation of the dust particles could lower the tem-
dient between the illuminated and shadowed sides of the
perature gradient over the surface of the grain and conse-
solid particle, and the ∆α-force which depends on the in-
quently the intensity of the photophoretic force. Loesche &
homogeneities in composition and shape of the particle. In
Wurm (2012) identified three main mechanisms that can
this work we consider spherical and homogeneous dust par-
leadtorotationoftheparticlesinthedisc:randomrotation
ticles, so we neglect the ∆α-term and take constant values
processes,collisionswithothergrainsandBrownianmotion.
for α and κ . Since the dust in the disc is composed of dif-
However, under the typical conditions of pressure and den- t
ferentspecies,itisparticularlyinterestingtostudyhowthe
sityinPPD,thetime-scalesforrotationareseveralordersof
∆T-term varies with the chemical composition and the size
magnitudelongerthanthetime-scaleforthermalconduction
of the solid bodies.
(Loesche & Wurm 2012). Hence, the temperature gradient
over the grain surface re-establishes itself much faster than
itcanbesuppressedbyrotation.Thisindicatesthatwecan 2.2 Effects of porosity on thermal conductivity
safely neglect rotation effects on the photophoretic force.
As already mentioned, porosity is an important parameter
given that grain growth strongly depends on the composi-
2 Diffusivescatteringdoesnotincludespecularreflection. tions and relative velocities of the impacting bodies (Blum
MNRAS000,1–??(2015)
4 N. Cuello et al.
2010). Laboratory experiments support the fact that colli-
sionsbetweensubmm-andmm-sizedSiO aggregatesresult
2
in dust grains with high porosities (Meisner et al. 2013).
In addition, Kataoka et al. (2013) computed the evolution
of porous dust grains in MMSN-like disc models and found
that porosity has strong effects on their radial motion. In
fact, porous grains grow more efficiently compared to com-
pact grains, which eventually allow the former to decouple
from the gas and prevent accretion.
Porosityalsoaffectsthethermalpropertiesoftheparti-
cles.Atamicroscopiclevel,heatconductionistreatedasthe
propagationofpacketsofvibrationalenergy,calledphonons,
through a crystal lattice (Presley & Christensen 1997). For
solids bodies ranging from millimetre to decimetre sizes
madeofaggregatedgrains,theheatconductionstronglyde-
pends on the contact surface between these grains. Since
a porous body has a smaller contact surface compared to
a bulk solid, the resulting thermal conductivity is signifi- Figure 3. Thermal conductivity for a porous dust particle at
cantly lower (Friedrich et al. 2008). Therefore, an increase T =280Kwithour1Dmodel.Thecolourbarcorrespondstothe
in porosity translates into an increase in the magnitude of logarithmofthethermalconductivity.
the photophoretic force given by Eq. (3).
Due to the lack of thermal conductivity data for large
perature: k = 0.01 Wm−1K−1, k = 1.147 Wm−1K−1
cm- and dm-bodies, it is difficult to fix a value for k which g sil
and k = 80 Wm−1K−1. We let q vary from 0 to 1, i.e.
wouldsimultaneouslydependonsize,composition,porosity iron
frompureirontopuresilicateparticles.Theresultingther-
andtemperature.Inthiscontext,porositycanbedefinedas
mal conductivity is plotted in Fig. 3 as a function of P and
thevolumeofemptyspaceinagivenparticledividedbythe
q.Asmallamountofporosityisenoughtodramaticallyde-
total volume of the particle. The thermal conductivity for
bulk silicates is of the order of magnitude of 1 Wm−1K−1 crease the thermal conductivity: for instance, if P ≥ 10 %
then k≤0.1 Wm−1K−1, regardless the composition of the
(Opeil et al. 2012), while if we consider the same silicates
particle. Despite this simplified approach, these values are
withafewpercentofporosity,k dropsbyatleastoneorder
in agreement with the thermal conductivities measured for
of magnitude (Loesche & Wurm 2012).
small porous chondrules (Opeil et al. 2012). However, our
A very simple one-dimensional model allows to under-
model might not be appropriate in a very low pressure en-
stand the effect of porosity on thermal conductivity. As
vironment where there is no gas within the pores. In this
a first approximation, we consider a porous solid body of
case, the heat transfer in pores might be dominated by ra-
length L as a superposition of n layers of length l and
0 j
diationor,iftheporesaresmall,byconductionthroughthe
thermalconductivitiesk .Thisisequivalenttoacomposite
j
dust itself (Presley & Christensen 2010a,b). Other studies
solid body of length L with a total thermal resistance, R,
0
have been recently conducted considering two-layered par-
equaltothesumofthethermalresistanceofeachlayer.The
ticlesmadeofacompactcorecoveredbyafine-grainedrim
thermal conductivity, k, is defined as the length divided by
withalowerconductivity:Loesche&Wurm(2012)andMc-
the thermal resistance: k=L /R. Thus, the resulting ther-
0
Nally&Hubbard(2015)solvetheheatequationinorderto
mal conductivity for the porous particle is given by:
compute the resulting conductivity and find that the outer
L
k= (cid:80)n 0l /k . (7) porous layer dramatically decreases the thermal conductiv-
j=1 j j ity of the bulk particle. These detailed calculations are also
We can then represent the pores inside the dust particle in agreement with our estimate for the thermal conductiv-
as voids filled with gas at the surrounding temperature. In ity. This fact motivates our choice of a constant thermal
this case, we obtain alternate layers of dust and gas with conductivity for iron and silicate particles. We choose to
thermal conductivities k and k respectively. If we define takek=0.1 Wm−1K−1,whichisequivalenttoconsidering
d g
theporosityP asthelengthcorrespondingtoporesdivided particles with an average porosity of the order of 10%.
bythetotallengthoftheparticle,wecanexpressEq.(7)in
the following form:
L k k 3 NUMERICAL SIMULATIONS
k= 0 = d g . (8)
PL0 + (1−P)L0 P(kd−kg)+kg 3.1 Code description
kg kd
As an illustrative example, we consider a solid body made We study the effect of photophoresis in PPD by means of
of a mixture of silicate and iron at 280 K, i.e. the fiducial the Lagrangian code of Barrière-Fouchet et al. (2005). We
temperature at 1 au in the MMSN models. We define the treatthegasandthedustastwoseparatefluidsinteracting
iron to silicate ratio, q, as the ratio between the amount throughaerodynamicaldrag.Thecodesolvestheequations
of iron and the amount of silicate by mass. In this case, of motion for each phase through the SPH formalism. Our
k =qk +(1−q)k , where k and k are the ther- goal is to assess the efficiency of the radial transport due
d iron sil iron sil
malconductivitiesforironandsilicateparticlesrespectively. to photophoresis. We add the photophoretic force to the
AccordingtotablesinTouloukianetal.(1970),atthistem- equation of motion for the dust particles, which contains
MNRAS000,1–??(2015)
Effects of photophoresis on the dust in a 3D PPD 5
in addition the stellar gravity and the gas drag term. The
equation of motion for the gas particles also contains three 0.02 -10
]
Gas t = 0 yr 3
terms: the gas pressure, the stellar gravity and the back- m-
reaction drag due to the interaction with the dust phase. c
0.01 g
Barrière-Fouchetetal.(2005)givesadetaileddescriptionof [
the code and its limitations. The criterion for the Epstein -11 ) 𝞺
(
regime, λ ≥ 4s/9, establishes the upper limit for which we U] og
can compute the gas drag using the equation: A 0 l
[
z
4π
F = ρ c s2∆v , (9) -12
d 3 g s
- 0.01
whereρ isthegasdensity,c thesoundspeedand∆visthe
g s
differentialvelocitybetweendustandthemeangasmotion.
Forthetypeofdiscconsideredinthisstudythislimitisap- - 0.02 -13
proximately equal to 1 metre (fig. 5 in Laibe et al. (2012)), 0 1 R [AU] 2 3
therefore the drag for particles with s ≤ 1 m can be com-
puted using Eq. (9). For larger sizes, we enter the Stokes
regime and this equation is not valid anymore. Figure 4.Meridianplanecutofthegasdistributionattheend
AsexplainedinSection2,thephotophoreticforcefora oftherelaxationphase.Thecoulourbarrepresentsthelogarithm
givenparticleofindexi,F ,dependsuponthedustgrain ofthevolumetricdensityofthefluid.
ph,i
properties, the local properties of the disc and the distance
tothestar.Thesequantitiesareknownforanygivenparti- Σ (r)=25(r/1au)−1/2 gcm−2.Theverticaldensityprofile
g
cle of our simulations. Thus, Fph,i can be straightforwardly is a Gaussian profile ρ(r,z) = ρ(r,0)exp(−z2/2h2), where
calculated by using Eq. (2) as follows: h is the disc scale height. This density profile corresponds
2F to the vertical hydrostatic solution for a thin disc. We set
Fph,i(ri,si,Ti,(cid:104)ρg,i(cid:105))= (cid:104)p (cid:105)/p m+axp,i /(cid:104)p (cid:105) , (10) h(r=1au)=3.33×10−2auandwehaveh(r)∝r5/4.This
i max,i max,i i
density profile is taken as the initial state of the disc for
where ρ is the gas density and r is the distance to the
g,i i oursimulations.Weonlyfocusintheinnerpartsofthedisc
starforparticlei.F dependsontheradiantfluxdensity
max from0.1to5ausincethisiswhereweexpectphotophoretic
I(r )whichissimplycomputedastheluminositydividedby
i transport to be more efficient (Takeuchi & Krauss 2008).
the surface of the sphere of radius r , and the quantities in
i
brackets are computed through SPH interpolations.
We apply the SPH prescription for the viscosity in the 3.3 Simulation setup
disc with α = 0.1 and β = 0.2, which control the
SPH SPH
We start with 100.000 gas particles distributed in such a
strength of the viscosity (Fouchet et al. 2007). Arena &
way that we obtain the expected profile for the density (cf.
Gonzalez(2013)demonstratedthattheSPHformalismade-
Section 3.2). The initial velocity of the gas particles is set
quatelyreproducestheturbulenceexpectedinthedisc.The
equaltotheKeplerianvelocity.Particlesmovinginteriorto
chosen values for α and β correspond to a uniform
SPH SPH
0.1auandoutside5auareconsideredaslost.Afterarelax-
valueoftheShakura&Sunyaev(1973)parameterα∼0.01,
ation phase of approximately 16 yr, the gas disc reaches an
as suggested by observations of PPD (King et al. 2007).
equilibrium state (Fig. 4). We then inject an equal number
It is worth noting that, instead of taking a constant
of dust particles on top of the gas particles with the same
valueforthegasviscosityη,wecomputeitinthefollowing
velocity. This state constitutes the t = 0 configuration for
way:
all the simulations. We let the system evolve for approxi-
η(r )= (cid:104)ρg(cid:105) λcs(ri) , (11) mately 250 years, i.e. 250 orbits at 1 au. We run a series of
i 2 10 simulations with 200.000 SPH particles to study the de-
where c only depends on r in the case of a vertically pendenceofphotophoresisontheintrinsicdensity(silicates,
s i
isothermal disc. Since the gas pressure at r also depends iron) and on the size of the solid bodies (1 mm to 1 m) as
i
on the height above the midplane, the photophoretic force showninTable1.PandNPstandforsimulationswithand
varies with height in our simulations. withoutphotophoresisrespectively.Byvaryingtheintrinsic
densityweareabletosimulatedifferentdustspecies:wetake
ρsil =3.2gcm−3 andρiron =7.8gcm−3 forthesilicateand
d d
3.2 Disc model
theironparticlesrespectively.GiventheLagrangiannature
We choose to study the same transitional disc model as of the SPH formalism, we are able to track the motion of
Takeuchi & Krauss (2008) to be able to consider the in- the SPH particles in 3D throughout the disc evolution.
ner regions of the disc as optically thin (cf. Section 1). The
disc extends from 0.01 to 100 au and has a total mass of
gas equal to M = 1.2×10−2M . It orbits a 1M star 4 RESULTS
g (cid:12) (cid:12)
of 1L luminosity and it is composed of 99% gas and of
(cid:12) 4.1 Photophoretic outward transport
1% dust by mass. The disc is non-self-gravitating, geomet-
rically thin and vertically isothermal where the tempera- InFig.5weshowthemeridianplanecutofthedustdistri-
ture follows the radial power-law T(r) = 280(r/1au)−1/2 butionfordifferentgrainsizes(1mm,1cm,10cm,1m)and
K. The surface density for the gas also follows a power-law intrinsic densities (ρsil, ρiron) at the end of the simulations
d d
MNRAS000,1–??(2015)
6 N. Cuello et al.
0.02 -10 0.02 -1 0
Sil-1cm-P t = 250 yr 3m-] Iron-1cm-P t = 250 yr 3m-]
c c
0.01 g 0.01 g
-11) [𝞺 -1 1 ) [𝞺
AU] 0 log( AU] 0 log(
z [ z [
-12 -12
- 0.01 - 0.01
a) b)
- 0.02 -13 - 0.02 -13
0 1 R [AU] 2 3 0 1 R [AU] 2 3
0.02 -10 0.02 -10
Sil-10cm-P t = 250 yr 3] Sil-10cm-NP t = 250 yr 3]
m- m-
0.01 g c 0.01 g c
-11) [𝞺 -1 1 ) [𝞺
U] g( U] g(
A 0 o A 0 o
z [ l z [ l
-12 -12
- 0.01 - 0.01
c) d)
- 0.02 -13 - 0.02 -13
0 1 R [AU] 2 3 0 1 R [AU] 2 3
0.02 -10 0.02 -1 0
Sil-1m-P t = 250 yr 3m-] Iron-1mm-P t = 250 yr 3m-]
0.01 g c 0.01 g c
AU] 0 -11log() [𝞺 AU] 0 -1 1 log() [𝞺
z [ z [
-12 -12
- 0.01 - 0.01
e) f)
- 0.02 -13 - 0.02 -13
0 1 R [AU] 2 3 0 1 R [AU] 2 3
Figure 5.Meridianplanecutofthedustdistributionforthefinalstateafter250yearsofevolutionforSil-1cm-P(a),Iron-1cm-P(b),
Sil-10cm-P (c), Sil-10cm-NP (d), Sil-1m-P (e), Iron-1mm-P (f). The coulour bar represents the logarithm of the volumetric density of
thefluid.
after250yr.Thestateofthegasaftertherelaxationphase, NP simulations. The inner rim is defined as the location of
shown in Fig. 4, determines the initial position of the dust the sharp decrease in the dust surface density in the inner
particles and it is the same for all the simulations. The gas regions of the disc. In addition, in P simulations the dust
distribution remains very close to the initial state through- accretion by the star is completely stopped, while in NP
outthesimulation.InPsimulationsweobserveinwarddrift simulations the dust is continuously accreted. Thus, pho-
for particles located at distances larger than approximately tophoresis prevents the accretion by the star for particles
2 au, whereas all the NP simulations show radial drift for that otherwise would have been lost.
allthedustparticlesinthedisc(nomattertheirdistanceto
InFig.5d,weshowthefinalstateofthedustdiscinthe
thestar).Thisisduetothefactthatdustparticlescloseto
Sil-10cm-NPsimulationtobecomparedwiththefinalstate
the star feel strong photophoretic effects which result into
ofthedustdiscinthesimulationSil-10cm-PinFig.5c.For
an efficient outward transport. This phenomenon produces
the former case, the inner rim is at 0.1 au and we observe
a sharp inner rim located at ∼ 1 au, which is not seen in
a continuous dust accretion onto the star. Figure 6 shows
MNRAS000,1–??(2015)
Effects of photophoresis on the dust in a 3D PPD 7
Table 1.ListofPandNPsimulations.
3
Name ρ (gcm−3) s(cm) Photophoresis
d
Sil-1cm-P 3.2 100 ON
2.5
Sil-1cm-NP 3.2 100 OFF
Sil-10cm-P 3.2 101 ON
Sil-10cm-NP 3.2 101 OFF 2
Sil-1m-P 3.2 102 ON
Sil-1m-NP 3.2 102 OFF U)
Iron-1mm-P 7.8 10−1 ON r(A 1.5
Iron-1mm-NP 7.8 10−1 OFF
Iron-1cm-P 7.8 100 ON
Iron-1cm-NP 7.8 100 OFF 1
0.5
the evolution of the radial distance for some selected parti-
cles of the simulations Sil-10cm-P and Sil-10cm-NP. Parti-
cles which are located at the same initial position have dif- 0
0 50 100 150 200 250
ferent radial motions with and without photophoresis. For
t(yr)
r ≤ 1.5 au, the photophoretic force on the dust particles
dominates over the gas drag and the solid bodies migrate
outward and accumulate at the equilibrium position; while, Figure 6.Timeevolutionofthedistancetothestarforselected
withoutphotophoresis,thedustparticlesdriftinwarddueto particles of different initial position Sil-10cm-P (thick lines) and
Sil-10cm-NP(thinlines).
the aerodynamic drag exerted by the gas phase. These be-
haviours and the corresponding velocities are in agreement
with previous computed radial velocities for dust particles
subjecttophotophoresisinthesolarnebula(fig.11inDuer- 10
t=0 yr (Sil-10cm-P)
8
mann et al. (2013)). t=50 yr (Sil-10cm-P)
6 t=150 yr (Sil-10cm-P)
In the following discussion, Sil-10cm-P will be consid- t=250 yr (Sil-10cm-P)
t=250 yr (Sil-10cm-NP)
ered as the reference case for photophoretic transport since 4 t=250 yr (Gas/100)
it shows the strongest effects (cf. Fig. 5). We observe three
types of behaviors for particles in P simulations: -2m)
kg 2
(i) Outwardmotionfordustparticleslocatedbetween0.1 Σ(
and ∼ 1 au from the star for Sil-10-cm-P. Without pho-
tophoresis, these particles tend to be accreted into the star 1
in a few hundreds of orbits. The average outward radial ve-
locities v¯ range from 10 to 30 ms−1.
R
0.5
(ii) Stableorbitsforparticlesatr∼1.5aufromthestar 0 0.5 1 1.5 2 2.5 3
for Sil-10cm-P. This effect is clearly due to photophoresis r(AU)
since it is not observed in NP simulations. In this case, the
forces are balanced and v¯ ∼0.
R Figure 7. Dust surface density for Sil-10cm-P at different evo-
(iii) Inward drift for particles in the outer regions of the
lutionarytimeswithphotophoresis(thicklines),Sil-10-cm-NPat
discoutsideof1.5auforSil-10cm-P.Consequently,v¯ <0.
R t=250yr(thindotdashedline)andgassurfacedensitydivided
The radial motion is slower than the case without pho- by100att=250yr(thindottedline).
tophoresis indicating that photophoresis has a decreasing
effect with increasing radial distance, as expected.
distribution. The value of the peak of the dust distribution
In Fig. 7, we show the time evolution of the surface
inSil-10cm-NPislowerthantheinitialvaluebecauseofthe
density for 10 cm silicate grains. With increasing time, the
dust accretion.
positionofthepeakmovesoutwardsandthepeakvaluein-
creases. This accumulation is due to the combined effect of
the outward motion of particles interior to the equilibrium
4.2 Accumulation regions for different grain sizes
position and to the inward drift of particles beyond this lo-
and densities
cation. This leads to an efficient pile-up of dust at a given
radialdistance.Thepeakpositionevolveslittlewithtimeaf- We expect particles to accumulate at the equilibrium posi-
tert=250yrandislocatedat∼1.3auatt=250yr.The tionatwhichthedragandphotophoreticforcescanceleach
increase of the peak value of the dust surface density is ob- otherout.InTable.2wereporttheinnerrimlocations,r ,
rim
servedforallthesimulationswithphotophoresisasshownin andthepeakpositioninthedustsurfacedensity,Σpeak,for
d
Fig.8.ThepeakofthedustsurfacedensityforSil-10cm-NP different grains species and sizes. For a given dust species,
in Fig. 7 remains very close to the peak of the gas surface the slight discrepancy between r and Σpeak is due to the
rim d
densitydividedby100.Infact,intheabsenceofphotophore- factthatthemaximumaccumulationdoesnotoccursatthe
sis, dust accumulates at the pressure maximum of the gas rim itself.
MNRAS000,1–??(2015)
8 N. Cuello et al.
10 Table 2.Summaryoftheresultsatt=250yr.
Iron-1mm-P
8
Iron-1cm-P
6 SiSl-il1-01ccmm--PP Simulation rrim (au) Σpdeak (au)
Sil-1m-P
4 Sil-1cm-P 0.8 1.25
-2m) SSili-l1-10mcm-P-P 10..16 11..03
kg 2 Iron-1mm-P 0.3 0.7
Σ(
Iron-1cm-P 0.6 1.0
1
4.3 Vertical settling
0.5
0 0.5 1 1.5 2 2.5 3
Weobservethatforsilicateparticles,thelargerthegrainsize
r(AU)
themoreefficienttheverticalsettlingisintheinnerregions,
asshowninBarrière-Fouchetetal.(2005).Intheouterdisc,
Figure 8.Dustsurfacedensityatt=250yrforPsimulations. 1mparticlesdonotsettlebecausetheyarealmostdecoupled
from the gas and remain on inclined orbits. In addition,
comparing Sil-1cm-P in Fig. 5a and Iron-1cm-P in Fig. 5b,
wealsoseethattheverticalsettlingismoreefficientforthe
denser particles. In fact, particles with the same product
The drag force in the Epstein regime is proportional ρdshavethesameStokesnumber,i.e.dynamicalbehaviour.
to ρ s, whereas the photophoretic force solely depends on Thus,a1cmironparticlefeelsthegasdraginthesameway
d
s. In the radial direction, the drag force points towards the as a silicate particle of size equal to ρidron/ρsdil ≈ 2.44 cm.
star,whilethephotophoreticforcepointsintheoppositedi- This justifies the stronger settling observed for 1 cm iron
rection. Therefore, if we consider two particles of the same particles compared to 1 cm silicate particles. Despite the
sizebutdifferentintrinsicdensities,thedenserparticlefeels fact the photophoretic force varies with height, we do not
a stronger drag force compared to the lighter one. Conse- observe differential vertical transport due to photophoresis.
quently,theaccumulationregionfordenserparticlesiscloser Infact,ifweconsiderparticlesatafixedradialdistancewith
to the star compared to lighter particles. This is in agree- different heights, they all settle towards the midplane in a
ment with the inner rim locations and Σpeak of Sil-1cm-P few orbital periods and then vertical photophoretic effects
d
and Iron-1cm-P reported in Table 2 and Fig. 8. are suppressed.
For particles with the same intrinsic density but dif-
ferent size (1 cm, 10 cm, 1 m), analytical calculations in
one-dimensional disc models show that larger particles ac-
cumulate closer to the star compared to smaller ones, as 5 DISCUSSION
shown in fig. 1 of Takeuchi & Krauss (2008). In this sense,
5.1 Photophoretic accumulation region
theinnerrimofSil-1cm-Pisindisagreementwiththisstate-
ment. However, the authors also estimate the time that it Ourresultsindicatethatphotophoresiscantransportdusty
takes for a given particle to reach its equilibrium location, grains from the inner region of the disc to the outer re-
τ , and found that it increases with decreasing size. For gion with different efficiency according to the grain density
mig
example,accordingtotheircalculations,for10cmparticles andsize.Thedustaccumulationoccursintheopticallythin
τ ∼ 100 yr, while for 1 cm τ ∼ 5000 yr. In parallel, regions of the disc and it is due to the combined effect of
mig mig
Moudensetal.(2011)solvedtheequationfortheradialve- stellarradiationandmomentumexchangebetweendustand
locityforparticlesofdifferentparticlesizesin1+1Dturbu- gasparticles.Thepeakvalueofthesurfacedensityprofiles,
lentdiscsandfoundthat10cmparticlespresentthelargest Σpeak dramatically increases in time-scales of the order of
d
radial velocities when compared to smaller particles. This 100 yr for cm-sized particles (cf. Fig. 7) and longer time-
implies that the former reach the equilibrium location on scales for smaller particles. For example, for 10 cm silicates
shorter time-scales (in agreement with Takeuchi & Krauss particles Σpeak(t = 150yr)/Σpeak(t = 0yr) ≈ 2. We define
d d
(2008)).AfterafewMyrsmallerparticlesreachtheirrespec- the dust-to-gas ratio, (cid:15), as the ratio between the dust and
tive equilibrium distances, which are larger than for 10 cm gas volume densities in the midplane. In Fig. 9 we plot (cid:15) as
particles.Thissuggeststhatinoursimulationsmm-andcm- afunctionofthedistancetothestarforSil-10cm-P.Atthe
sized particles have not reached their equilibrium location beginning of the simulation (cid:15) = 0.01 by definition, and we
yet. Unfortunately, the high computational cost of hydro- expect(cid:15)tobecomecloseto1inthemidplanewithtimeevo-
dynamical simulations prevent us to extend this study for lution due to the vertical settling of dust (Barrière-Fouchet
longertimes.Nevertheless,ourresultscorrespondtoanevo- et al. 2005). However, we observe that in the inner regions
lutionarystageofthediscforwhich10cmparticleshaveal- photophoretic effects results into a dramatic increase of (cid:15)
ready reached their equilibrium location and smaller grains which reaches values close to 40 between 1 and 1.7 au. In
are slowly migrating outwards. The dynamics involved in the outer parts (cid:15) remains close to 1 because photophoresis
this process have important implications for the chemical has little effect at large distances. The effective pile-up of
separation of particles in the disc, which are discussed in dust caused by photophoresis results into favourable condi-
Section 5. tionsforgraingrowthintheinnerregionsofthedisc.Conse-
MNRAS000,1–??(2015)
Effects of photophoresis on the dust in a 3D PPD 9
100 1.1
t=0 yr (Sil-10cm-P) t=0 yr
t=250 yr (Sil-10cm-P) t=250 yr
1
ε = 1
10 0.9
Sil)]
Σ( 0.8
+
1 e)
ε Σ(F 0.7
e) / [ 0.6
0.1 Σ(F 0.5
0.4
0.01
0.3
0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3
r(AU) r(AU)
Figure 9.Dust-to-gasratiointhediscmidplaneforSil-10cm-P. Figure 10. Fe/(Fe+Sil) ratio for 1cm-sized particles with pho-
tophoresis.
quently,solidbodiesmayrapidlygrowatthisaccumulation
islowerthan0.5becausethisiswhere1cmsilicateaccumu-
region and decouple from the gas in shorter time-scales.
late(cf.Fig8).Ithastobenotedthatweareonlyshowing
The dust accumulation caused by photophoresis is a
twospecies:ironasthedenserextremecaseandsilicatesas
process which differs from the particle traps discussed ear-
themostabundantdustspecieswithinthesnow-line.Wedo
lier (cf. Section 1) where a pressure maximum is required
notconsidericeparticlesinthisstudysinceweassumetobe
inordertocollectdustatagivenlocationinthedisc.Pho-
interiortothesnow-lineintheregionofthediscconsidered.
tophoretic accumulation regions are defined as the location
In a model with more grain species, we would then expect
where drag and photophoretic forces cancel each other out.
a gradual separation between lighter and denser particles.
Interestingly, in our disc model, photophoresis prevents the
Thechemicalsortingbetweenironandsilicateparticles
accretion by the star for particles ranging from millimetre
shown in Figs. 8 and 10 results in important variations in
to metre sizes. This is particularly relevant since these par-
the composition in the first 2 au from the star. We demon-
ticles are expected to be subject to a strong radial drift,
strate that photophoresis differentially transports grains of
whichcompromisestheirsurvivalaroundthestar.However,
different intrinsic density, i.e. different chemical composi-
other disc models would show accumulation regions at dif-
tion. Moreover, for a given density, we see that grain size
ferent radial distances. For instance, the optically thin re-
also affects the efficiency of photophoresis (cf. Fig. 8). As
gion in a younger and more massive disc is much less ra-
a consequence, photophoresis may lead to size and density
diallyextended.Consequently,photophoresiswillonlyhave
sorting of grains. Such a sorting is generally observed in
an observable effect on the particles very close to the gas
carbonaceous chondrites where iron-rich grains are smaller
inner edge. In this configuration, the particle survival will
thansilicate-richchondrules(Kuebleretal.1999).Thus,we
still hold but the size and density sorting will operate over
suggestthatphotophoresiscanbeoneofthepossiblemech-
shorterlengthscales.However,regardlessofthemodelcon-
anisms responsible for grain sorting such as those found in
sidered,ourresultsstronglysuggestthatphotophoresiscan
chondrites.Providedthatthesesolidsarethenincorporated
effectivelybreaktheradial-driftbarrierintheopticallythin
into large planetesimals, Wurm et al. (2013) showed that
regions of PPD.
photophoresiscannaturallyexplainthemeasuredvariations
inthebulkchemicalcompositionoftherockybodiesofthe
Solar System with pure metal grains close to the star and
5.2 Dust chemical composition
silicate particles farther out.
In Fig. 8 we report the dust surface density profiles at
the end of the simulation for Sil-(1cm, 10cm, 1m)-P and
5.3 Outward transport of high-temperature
Iron-(1mm,1cm)-Pwhichshowsthedifferencebetweeniron
forming material
and silicate particles. In addition, in Fig 10 we show the
ΣFe/(ΣFe+ΣSil) ratio as a function of the radial distance Calcium- and aluminium-rich inclusion (CAIs) and amoe-
d d d
for cm-sized particles in P simulations. We assume that, at boid olivine aggregates (AOAs) are submm- to cm-sized
thebeginningofthesimulation,thediscismadeofanhomo- refractory inclusions characterized by a lack of volatile el-
geneous mixture of iron and silicates particles, which gives ements. They are thought to be the products of high-
ΣFe/(ΣFe+ΣSil) = 0.5 at t = 0 yr. After 250 years of evo- temperature processes occurring in the inner hot regions
d d d
lution,weseethattheratiobecomescloseto1intheinner of the solar nebula such as condensation, evaporation and
regionsindicatinganalmostiron-purecomposition,whilein melting (Scott & Krot 2014). However, these inclusions are
theouterpartsitiscloseto0.5.From0.4to1.4au,theratio observedincarbonaceouschondritesmixedwithchondrules
decreasesbecauseoftheincreasingamountofsilicateparti- and amorphous material, presumably formed at lower tem-
cles. This is basically due to the fact that silicate particles peratures (MacPherson et al. 2012). It has been suggested
are more efficiently transported outwards by photophoresis that an efficient chemical mixing occurred at some evolu-
comparedtoironparticles.Between1.2and2.2autheratio tionary stage of the PPD. In addition, the Stardust mis-
MNRAS000,1–??(2015)
10 N. Cuello et al.
sion recently returned grains from Comet 81P/Wild 2 with inordertocaptureandacquirethelocalchemicalcomposi-
high-temperature minerals, such as forsterite and enstatite tionofthedisc.Ifthisisthecase,theresultingbulkchemical
(Brownleeetal.2006).Thisindicatesthathigh-temperature compositionmightalsovarywiththeheightabovethemid-
formingmaterialformedintheinnerregionshastobetrans- plane. It is hard to predict a priori the resulting chemical
ported towards the outer parts of the solar nebula in order compositionateachlocationofthediscduetothecomplex
to be incorporated into larger bodies. process of grain growth. Indeed, the outcome of grain col-
Mousis et al. (2007) showed that photophoresis can re- lisions strongly depends on the composition, structure and
sult into an influx of refractory material to the outer re- relativevelocityofthecollidingbodies(Blum&Wurm2008;
gions of the disc. Our 3D simulations show an efficient ra- Wadaetal.2009;Blum2010;Zsometal.2010;Wadaetal.
dialtransportofparticleslocatedclosetothestar(∼0.5au) 2013). The study of grain growth in the optically thin re-
todistancesbeyond1au,wheremeteoriteparentbodiesare gions of the disc taking into account photophoretic effects
thoughttohaveformed(Scott&Krot2014).InMousisetal. will be addressed in detail in future work.
(2007), the particles of the inner regions are transported
evenfartheroutsincetheauthorsconsiderdiscmodelswith
larger inner cavities (1 and 2 au). Hence, our simulations
including photophoresis are in agreement with the scenario ACKNOWLEDGEMENTS
where “hot minerals” are injected in the outer parts of the
We thank the anonymous referee for useful comments. We
disc and mixed with solids formed at lower temperatures.
alsothankGerhardWurmandMarkusKuepperforenlight-
ening discussions about photophoresis experiments. This
research was supported by the Programme National de
6 CONCLUSION Physique Stellaire and the Programme National de Plané-
tologie of CNRS/INSU, France. The authors are grateful
Photophoresis is a physical process which transports illu- to the LABEX Lyon Institute of Origins (ANR-10-LABX-
minated particles embedded in gaseous environments away 0066) of the Université de Lyon for its financial support
from the radiation source. In this study, we have quan- within the program “Investissements d’Avenir" (ANR-11-
tified the effects of photophoresis on the inner regions of IDEX-0007) of the French government operated by the Na-
transitional discs via 3D hydrodynamical simulations. Our tional Research Agency (ANR). All the computations were
results show that photophoresis dramatically affects the performedattheCommonComputingFacility(CCF)ofthe
structure of the dusty disc, while the gaseous disc remains LABEXLIO.Figs.4and5weremadewithSPLASH(Price
unchanged. We have considered grains with different sizes 2007).
(1 mm to 1 m) and intrinsic densities (ρsil = 3.2 g cm−3,
d
ρiron = 7.8 g cm−3). In our simulations, the dust piles up,
d
duetophotophoresisopposingradialdrift,atdifferentradial
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