Photophoresis in protoplanetary disks: a numerical approach
Nicolas Cuello, Francesco C. Pignatale, Jean-François Gonzalez
ee-mail: [email protected]
SF2A 2014
J. Ballet, F. Bournaud, F. Martins, R. Monier and C. Reyl´e (eds)
PHOTOPHORESIS IN PROTOPLANETARY DISKS: A NUMERICAL APPROACH
N. Cuello , F. C. Pignatale and J.-F. Gonzalez Abstract.
It is widely accepted that rocky planets form in the inner regions of protoplanetary disks(PPD) about 1 - 10 AU from the star. However, theoretical calculations show that when particles reachthe size for which the radial migration is the fastest they tend to be accreted very efficiently by the star.This is known as the radial-drift barrier. We explore the photophoresis in the inner regions of PPD as apossible mechanism for preventing the accretion of solid bodies onto the star. Photophoresis is the thermalcreep induced by the momentum exchange of an illuminated solid particle with the surrounding gas. Recentlaboratory experiments predict that photophoresis would be able to stop the inward drift of macroscopicbodies (from 1 mm to 1 m in size). This extra force has been included in our two-fluid (gas+dust) SPH codein order to study its efficiency. We show that the conditions of pressure and temperature encountered inthe inner regions of PPD result in strong dynamical effects on the dust particles due to photophoresis. Oursimulations show that there is a radial and a vertical sorting of the dust grains according to their sizes andtheir intrinsic densities. Thus, our calculations support the fact that photophoresis is a mechanism whichcan have a strong effect on the morphology of the inner regions of PPD, ultimately affecting the fate ofplanetesimals.Keywords: protoplanetary disk, photophoresis, hydrodynamics, planet formation
Thousands of extra-solar planets have been detected in the past 20 years (Batalha et al. 2013), which un-doubtedly shows that planets are common byproducts of stars. It is currently accepted that planets form inprotoplanetary dusty disks around young stellar objects. This means that, throughout the evolution of theprotoplanetary disk, dust grains have to grow from µ m sizes to kilometric sizes to form planetesimals. However,the study of the motion of solids inside the disk leads to the so-called radial-drift barrier. This issue was firstpointed out by Weidenschilling (1977) by solving the equation of motion for dust particles embedded in thegas disk of a PPD. A solid body orbiting a star in a keplerian fashion loses angular momentum through theinteraction with the gas which orbits at a subkeplerian velocity. This is because the gas phase, which is assumedto be in hydrostatic equilibrium, is pressure supported while the dust phase is not. The difference between theorbital velocities of each phase can be interpreted as a headwind in the rest frame of the particle, which causesthe particle to spiral down into the star.Several mechanisms have been proposed to break the radial-drift barrier such as radial mixing (Keller & Gail2004, and references therein), magnetic braking and dead zones (Armitage 2011), particle traps (Fouchet et al.2010; Pinilla et al. 2012), meridional circulation (Fromang et al. 2011) and radiation pressure force (Vinkovi´c2014). However, it remains unclear if a single mechanism or a combination of them could actually preventaccretion. The aim of this work is to consider an extra mechanism called photophoresis, which has been firstintroduced by Rohatschek (1995) through the study of illuminated particles in aerosols. Duermann et al. (2013)measured the strength of the photophoresis force on illuminated plates under similar temperature and pressureconditions that those found in PPD. Then, via an interpolation of their results, they computed the accelerationfelt by solids of different sizes and porosities orbiting a 1 solar mass star. Plugging this extra force into theequations of motion for dust particles in a 1D model, they predicted that photophoresis might be able to Universit´e de Lyon, Lyon, F-69003, France; Universit´e Lyon 1, Observatoire de Lyon, 9 avenue Charles Andr´e, Saint-GenisLaval, F-69230, France; CNRS, UMR 5574, Centre de Recherche Astrophysique de Lyon; Ecole Normale Sup´erieure de Lyon,F-69007, France c (cid:13)
Soci´et´e Francaise d’Astronomie et d’Astrophysique (SF2A) 2014 a r X i v : . [ a s t r o - ph . E P ] D ec
38 SF2A 2014stop the inward drift for large bodies ranging from a millimeter to a meter. Since the force is size dependent(Duermann et al. 2013), the photophoresis effects would ultimately lead to a radial sorting among different grainpopulations. However, to date, no studies have explored the complex dynamics obtained when considering thisextra mechanism. This work is the first attempt to understand how photophoresis shapes the inner regions ofPPD between 0.1 and 5 AU through numerical simulations.In section 2, we give a brief overview of the photophoresis force; in section 3, we explain how we model thePPD by means of SPH simulations; in section 4, we present our results, and our conclusions are reported insection 5.
Rohatschek (1995) derived the first semi-empirical model for the photophoretic force which connects low andhigh pressure regimes based on experiments and theoretical calculations. Indeed, there are three regimes forthe photophoretic force, F ph , according to the pressure, p , of the gaseous environment around an illuminatedparticle: the low pressure regime for which F ph ∝ p , the high pressure regime for which F ph ∝ /p , and, inbetween, the transition regime when F ph reaches its maximum. These regimes are determined by the value of aquantity called the Knudsen number defined as Kn = λ/ a, where λ is the mean free path of the gas moleculesand a is the radius of the solid particle. Thus, Kn (cid:29) (cid:28) F ph = 2 F max pp max + p max p , (2.1)with F max = a D (cid:114) α Ik , (2.2) p max = 3 Tπa D (cid:114) α , (2.3) D = π ¯ cη T (cid:114) πκ , (2.4)¯ c = (cid:115) RTπµ , (2.5)where I is the irradiance of the incident beam of light, α is the thermal accommodation coefficient (dimensionlessand often taken equal to 1), k is the thermal conductivity of the solid particle, T is the gas temperature, η isthe viscosity of the gas, κ is the thermal creep coefficient, which is equal to 1.14 (Rohatschek 1995), R is theuniversal gas constant and µ is the molar mass of the gas particle.The photophoretic force can also be split into two components: the ∆ T -force mainly driven by the gradient oftemperature between the illuminated and shadowed sides of the solid particle, and the ∆ α -force which dependson the differences in composition of the particle. In this work, we consider homogeneous particles for whichwe vary the chemical composition and the size. Thus, we only need to compute the ∆ α -force. Due to thelack of thermal conductivity data for large bodies, it is difficult to fix a value for k which would depend onthe temperature and the size. However, Opeil et al. (2012) and Loesche & Wurm (2012) showed, throughmeasurements in chondrules and heat transfer calculations respectively, that a low porosity in dust aggregatessuffices to lower thermal conductivities. For instance, the thermal conductivity for bulk silicates is of the orderof magnitude of 1 W m − K − (Opeil et al. 2012), while if we consider the same silicates with a few percent ofvoid, k drops by at least one order of magnitude. This motivates our choice of a constant thermal conductivity k = 0 . − K − as in Duermann et al. (2013). All the other quantities on which equations (2.2) to (2.5)depend are functions of the disk local conditions and can be easily computed in our code as showed in the nextsection. In order to be able to compare our results to Duermann et al. (2013) we chose to study the same disk model,namely the Minimum Mass Solar Nebula (MMSN) of Hayashi (1981). We consider a protoplanetary disk ofhotophoresis in PPD: a numerical approach 2390.013 M (cid:12) mass with 1% of dust by mass around a 1 M (cid:12) star and a radial extension from 0.1 to 36 AU. Thedisk is vertically isothermal and T ( r ) ∝ r − q with q = 1 /
2. We focus on the inner regions from 0.1 to 5 AU sinceit is the range for which the photophoretic force is the more efficient (Duermann et al. 2013).We adapt the code of Barri`ere-Fouchet et al. (2005) to our study by including an extra term in the equationof motion for the dust particles. Gas and dust are considered as two separated fluids, which interact throughaerodynamical drag. The code solves the equations of motion for each phase through the SPH formalism. Price(2012) reviews the method and presents the recent developments. The equation of motion for the gas SPHparticles reads: d v a dt = P ab + M aj + D aj + G a , (3.1)which is the sum of the pressure term P ab , the mixed pressure term M aj , the drag term D aj and the gravity ofthe central star G a . The subscripts a and b refer to gas particles and i and j for dust particles. For the dustparticles the equation of motion is the following:d v i dt = M ib + D ib + G i + F photo i , (3.2)where, in addition to the mixed pressure M ib , the drag D ib and the gravity G i terms, we add the photophoresisforce given by Equation (2.1). It is important to note that the gas feels the pressure force whereas the dustdoes not, and that only the dust phase feels the photophoresis force. The photophoretic force depends upon thelocal properties of the gas at the position r of the dust particle: the temperature, the pressure, the viscosityand the amount of energy received from the star. In our simulations we consider that the medium is opticallythin so that the radiant flux density at a given position r is simply computed as the luminosity over the surfaceof the sphere of radius r . This constitutes the main limitation of our calculations since photophoresis has noeffect in optically thick regions. We are currently developing a more detailed model to include this effect.In the simulations, we let a gaseous disk evolve from an initial surface density distribution given by Σ ∝ r − / .The initial velocity is keplerian, given by v k = (cid:112) GM (cid:12) /r . The reference values at 1 AU for the pressure and thetemperature power laws are given by the MMSN model of Hayashi (1981). Once the gas disk reaches equilibrium,we inject the dust particles on top of the gas particles with the same velocity. This state constitutes our initialstate (Fig. 1-a). Then, we let the system evolve for an evolutionary time of 30 years approximately, i.e. We show the initial state of the dust phase at t = 0 (Fig. 1-a), the final state of the dust phase for 10 cm silicategrains with photophoresis (Fig. 1-b), for 10 cm silicate grains without photophoresis (Fig. 1-c) and for 10 cmiron grains with photophoresis (Fig. 1-d). We observe that there is a strong dust sedimentation as alreadyobserved in the simulations by Barri`ere-Fouchet et al. (2005). Radial migration is dramatically affected by theinclusion of photophoresis in the equations of motion. In fact, whithout photophoresis, the dust particles startto spiral down to the star and there is no outward motion. On the contrary, when we take photophoresis intoaccount, the particles which are very close to the star between 0.1 and 1.8 AU tend to move outwards until theyreach a stable orbit at around 1.8 AU for 10 cm grains. The location of the inner rim, i.e. the radial migration,depends on the grain size: its position is at 0.5, 1.9, 1.5 AU for 1 cm, 10 cm and 1 m particles respectively.We see the strongest effect for 10 cm while the grains of 1 cm and 1 m are less affected by photophoresis. Thisresult matches with the analytical calculation by Duermann et al. (2013).Fig. 1-b and Fig. 1-d show the different behavior for two different chemical compositions for a grain size of 10cm: we notice that the vertical sedimentation is more efficient for iron grains than for silicates. This phenomenonis also observed in simulations without photophoresis since it solely depends on the intrinsic density ( ρ sil = 3 .
2g cm − and ρ iron = 7 . − ). Nevertheless, if we consider a more realistic disk made of a mixture of differentspecies with a given distribution of sizes, then the sedimentation coupled with the different inner rim locationpatterns will have an effect on the mixing of solids in the inner regions of PPD. This will be the subject of afuture work.40 SF2A 2014 a) initial state b) 10 cm silicatesc) 10 cm silicates d) 10 cm iron z [ A U ] z [ A U ] l o g ( d en s i t y ) [ k g / m ] l o g ( d en s i t y ) [ k g / m ] r [AU] r [AU]t=30 yrt=30 yrt=30 yrt=0 yr with photophoresiswith photophoresiswithout photophoresis Fig. 1.
Meridian plane cut of the dust distribution for the initial state of the dust phase (a) and the final state after 30years of evolution for 10 cm silicate grains with photophoresis (b), without photophoresis (c) and with photophoresis foriron grains (d).
We have included photophoresis in our simulations in order to understand its effects on the inner regionsof PPD. Even though our calculations present some limitations, our results show that photophoresis affectsthe structure of the inner rim of the dusty disk whereas the gaseous disk remains unchanged. Moreover,these preliminary results are in accordance with the predictions made by Duermann et al. (2013) concerningthe different accumulation zones for different grain populations. Future work will explore the effect of thisaccumulation on the growth of large solids in the inner regions of PPD where we expect terrestrial planets toform. In fact, even if photophoresis is mainly effective for centimeter and meter sized bodies, it might leadto an efficient pile-up of particles close to the star. Chatterjee & Tan (2014) recently proposed an inside-outplanet formation scenario based on magneto-rotational instabilities (MRI). In this case, pebbles collect at thepressure maximum associated with the transition from a dead zone to an inner MRI-active zone. Alternatively,taking into account photophoresis effects, solid bodies could accumulate and grow at the stable point definedby the transition between the optically thin and optically thick regions. In the optically thin part, particlesmove outwards since they mainly feel photophoresis, whereas in the optically thick one they move inwards dueto the radial drift. If the dust-to-gas ratio is high enough in the accumulation zone, a planet could form atthis location. Both mechanisms lead to systems with tightly-packed inner planets. It worths noticing that thisplanetary architecture seems to be one of the principal outcomes of planet formation since it has been detectedin a large fraction of targets (more than 10% of the stars) by the
Kepler mission (Batalha et al. 2013).
This research was supported by the Programme National de Physique Stellaire, the Programme National de Plan´etologie ofCNRS/INSU. The authors are grateful to the LABEX Lyon Institute of Origins (ANR-10-LABX-0066) of the Universit´e de Lyonfor its financial support within the program ”Investissements d’Avenir” (ANR-11-IDEX-0007) of the French government operatedby the National Research Agency (ANR). All the computations were performed at the Service Commun de Calcul Intensif del’Observatoire de Grenoble (SCCI). Figure 1 was made with SPLASH (Price 2007). hotophoresis in PPD: a numerical approach 241