Constraining turbulence mixing strength in transitional discs with planets using SPHERE and ALMA
M. de Juan Ovelar, P. Pinilla, M. Min, C. Dominik, T. Birnstiel
MMon. Not. R. Astron. Soc. , 1– ?? (2014) Printed 18 April 2018 (MN L A TEX style file v2.2)
Constraining turbulence mixing strength in transitionaldiscs with planets using SPHERE and ALMA
M. de Juan Ovelar (cid:63) , P. Pinilla , M. Min , , C. Dominik and T. Birnstiel Astrophysics Research Institute, Liverpool John Moores University, 146 Brownlow Hill, Liverpool L3 5RF, UK Leiden Observatory, Leiden University, P.O. Box 9513, 2300RA Leiden, The Netherlands SRON Netherlands Institute for Space Research, Sorbonnelaan 2, 3584 CA Utrecht, The Netherlands Anton Pannekoek Institute for Astronomy, University of Amsterdam, 1090 GE Amsterdam, The Netherlands Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA
Accepted 2016 March 21. Received 2016 March; in original form 2015 September 16
ABSTRACT
We investigate the effect that the turbulent mixing strength parameter α turb playson near-infrared polarimetric and sub-millimetre interferometric imaging observationsof transitional discs with a gap carved by a planet. We generate synthetic observa-tions of these objects with ALMA and VLT/SPHERE-ZIMPOL by combining hy-drodynamical, dust evolution, radiative transfer and instrument models for values of α turb = [10 − , − , − ]. We find that, through a combination of effects on the vis-cosity of the gas, the turbulent mixing and dust evolution processes, α turb stronglyaffects the morphology of the dust distribution that can be traced with these ob-servations. We constrain the value of α turb to be within an order of magnitude of10 − in TD sources that show cavities in sub-mm continuum images while featuringcontinuous distribution of dust or smaller cavities in NIR-polarimetric images. Key words:
Planet-disc interactions. Techniques: high angular resolutions, polari-metric, interferometric. Hydrodynamics, radiative transfer. Methods: numerical
The field of transitional discs (TDs) has recently experienceda paramount push thanks to the technical advancementsin high contrast imagers and interferometers. Originally de-tected and characterised thorough spectral energy distribu-tion (SED) fitting, these protoplanetary discs (PPDs) ap-peared to be depleted of material in the inner regions andare considered a transition between a full protoplanetarydisc and a planetary system (Strom et al. 1989). This tran-sitional stage may be caused by processes of planet-disc in-teraction (e.g. Rice et al. 2003; Papaloizou et al. 2007), ordisc evolution (e.g. Dullemond & Dominik 2005; Alexander& Armitage 2007).With resolutions of a few AUs at 140 pc, currentfacilities such as the Atacama Large Millimeter/sub-millimeter Array (ALMA) or the new planet imager Spectro-Polarimetric High-contrast Exoplanet Research SPHERE)on the Very Large Telescope (VLT) are providing the com-munity with a plethora of images of TDs showing very differ-ent and complex structures such as rings, azimuthal asym-metries, dips and spiral arms (e.g. Quanz et al. 2012; Garufiet al. 2013; van der Marel et al. 2013; Casassus et al. 2013; (cid:63) [email protected]
P´erez et al. 2014; Zhang et al. 2014; Walsh et al. 2014;Benisty et al. 2015; Canovas et al. 2016). This has triggereda large number of theoretical studies to explore the poten-tial mechanisms responsible (e.g. Reg´aly et al. 2012; Ataieeet al. 2013; Birnstiel et al. 2013; Zhu & Stone 2014; Juh´aszet al. 2015; Flock et al. 2015).In particular, a group of TDs seem to feature gaps in870 µ m interferometric images (Andrews et al. 2011) whileshowing smaller or non-existent gaps in H-band (1 . µ m) po-larimetric images (i.e. the “missing cavities” problem Donget al. 2012). With the former tracing the emission of rel-atively large ( ∼ ∼ µ m) ones, these observa-tions suggest that a filtration mechanism is causing the lo-calised depletion of large dust grains, leaving small ones un-affected. Theoretical studies such as Zhu et al. (2012) orPinilla, Benisty, & Birnstiel (2012), show that this prefer-ential filtration of certain sizes of dust grains can be causedby a planetary-mass companion while it remains difficult toexplain by disc evolution processes. Based on the modelspresented in the latter, de Juan Ovelar et al. (2013) addedradiative transfer and instrument modelling to produce syn-thetic observations of this scenario showing that images atNIR and sub-mm wavelengths would indeed show this ap-parent dichotomy and that their combination can be used to c (cid:13) a r X i v : . [ a s t r o - ph . S R ] M a r M. de Juan Ovelar et al. estimate the mass of the companion. However, parameterssuch as the turbulence mixing strength ( α turb ) are knownto have an important effect on the (hydro)dynamical, dustevolution, and radiative transfer processes that govern theevolution of PPDs, and their response to external perturba-tions (e.g. Lynden-Bell & Pringle 1974; de Juan Ovelar et al.2012; Rosotti et al. 2014), but its effect on such observationsremains to be investigated.In this letter we explore this issue with general TD mod-els instead of using particular sources. We focus on howsuch observations can be used to constrain the value of α turb within the range of 10 − to 10 − , currently assumed inthe literature and supported by recent observational studies(e.g. Mulders & Dominik 2012; Flaherty et al. 2015).The letter is organised as follows: In § § § Following the same methodology and models presented in deJuan Ovelar et al. (2013), we combine 2D-hydrodynamical,1D-dust evolution, radiative transfer, and instrument sim-ulations to produce synthetic observations of a disc host-ing a planet of masses M p = [1 , , ,
15] M jup . The valuesof all parameters are given in Appendix A together witha brief description of the modelling procedure. For moredetails of our method we refer the reader to the abovementioned paper. We run all cases with three values of α turb = [10 − , − , − ]. In the interest of space we describe the effect of α turb onthe gas and dust distribution in all cases while discussion onimages is focussed on models run with 1 and 9 M Jup plan-ets only, which are representative of our sample. Syntheticimages of cases [5 , M Jup are shown in Appendix B. α turb on the gas and dust densitydistribution Figure 1 shows the radial profiles of the gas and (binnedby size from 1 µ m to > α turb studied for each planetary mass con-sidered. The gap-opening power of a planet is determinedby the balance between the mutually counteracting gravita-tional and viscous torques that arise from its presence in thedisc and the viscous conditions of the gas, respectively(e.g.Crida et al. 2006). Thus, as the value of α turb decreases(i.e. upper to lower panels), the gap opened by a planet of acertain mass in the gas distribution is significantly deeper.The gap also becomes wider as planet mass increases (fora fixed α turb ). Additionally, the pressure gradient becomespositive at the outer edge of the gap opened and a pressuremaximum appears (e.g. Paardekooper & Mellema 2004).The characteristics of the new pressure gradient distribu-tion (e.g. steepness) and those of the pressure maximum (e.g. amplitude) control the filtering/trapping of dust parti-cles of different size in the disc (see Equation 11 in Pinillaet al. 2012).Turbulent mixing also plays an essential role on dustgrowth and evolution. For instance, fragmentation occursbecause of high relative velocity collisions between dustgrains, with main sources being turbulent motion and radialdrift. If radial drift is reduced by a positive pressure gradi-ent, turbulent motion dominates and the maximum grainsize that particles can reach before fragmentation ( a max )depends directly on α turb (Birnstiel et al. 2010, 2012). If α turb is high, a max can be much lower than the size of par-ticles that can be trapped in pressure bump, preventing ac-cumulation of mm-sized particles in the pressure bump (e.g.Pinilla et al. 2015). In addition, turbulence also drives dif-fusion of particles within pressure bumps, and therefore if α turb is high, particles can more easily escape a dust trap.This is the case of models with α turb = 10 − (upper row)where we see no dust traps for mm- or even cm-sized par-ticles, even for very massive planets. In summary, there arethree reasons why trapping is weaker for higher values of α turb : less deep planetary gaps and hence lower pressuregradient at the outer gap edge; more effective fragmenta-tion of particles which leads to smaller grains that are moredifficult to trap; and higher diffusion or mixing of dust thatallows the particles to escape from the trap.Additionally, when α turb = 10 − , and also indepen-dently from planet mass, small grains pile up at the po-sition of the planet even surpassing the surface density val-ues of the gas. This is an effect of our 2D (gas)+1D (dust)approximation, in particular from assuming the gas veloc-ity using viscous accretion and assuming the averaged gassurface density for the dust evolution as in Pinilla et al.(2012); Pinilla et al. (2015). To test this, we re-run the dustcalculations in the 1 M Jup and α turb = 10 − case includ-ing only dynamics and neglecting coagulation, fragmenta-tion and grain growth processes. The enhancement of smallparticles then remains suggesting that it is indeed a numer-ical artifact and not the result of dust evolution processes.We then run another simulation for this case, where the dustvelocities are assumed to be v gas ∗ Σ gas / Σ gas instead, with v gas and Σ gas being the azimuthally and time (over the last100 orbits) averaged values from the hydrodynamical sim-ulations. In this case, accretion rate throughout the gap isalmost constant and the pile up of small dust at the loca-tion of the planet disappears (see details in Appendix B).Because in our dust evolution models we assume that thegas velocity comes from viscous evolution that tries to closethe gap, the particles that feel these velocities are pushedinto the gap. Since the viscous velocities are proportional to α turb , in the case of lower values (i.e. 10 − , 10 − ) they arenegligibly small, and radial drift becomes the dominatingcontribution for the dust velocity, which moves dust up thepressure gradient and prevents this artifact from appearing.This is, therefore, an inherent limitation of our modellingprocedure that affects high turbulence cases in the regionof the disc near the planet. To treat this issue, 2D gas anddust evolution models that include grain growth and dynam-ics simultaneously are needed, which are beyond the scopeof our study. For our analysis of these cases we therefore ig-nore this feature and base our conclusions on the otherwisecontinuously decreasing distribution of dust. c (cid:13) , 1–, 1–
15] M jup . The valuesof all parameters are given in Appendix A together witha brief description of the modelling procedure. For moredetails of our method we refer the reader to the abovementioned paper. We run all cases with three values of α turb = [10 − , − , − ]. In the interest of space we describe the effect of α turb onthe gas and dust distribution in all cases while discussion onimages is focussed on models run with 1 and 9 M Jup plan-ets only, which are representative of our sample. Syntheticimages of cases [5 , M Jup are shown in Appendix B. α turb on the gas and dust densitydistribution Figure 1 shows the radial profiles of the gas and (binnedby size from 1 µ m to > α turb studied for each planetary mass con-sidered. The gap-opening power of a planet is determinedby the balance between the mutually counteracting gravita-tional and viscous torques that arise from its presence in thedisc and the viscous conditions of the gas, respectively(e.g.Crida et al. 2006). Thus, as the value of α turb decreases(i.e. upper to lower panels), the gap opened by a planet of acertain mass in the gas distribution is significantly deeper.The gap also becomes wider as planet mass increases (fora fixed α turb ). Additionally, the pressure gradient becomespositive at the outer edge of the gap opened and a pressuremaximum appears (e.g. Paardekooper & Mellema 2004).The characteristics of the new pressure gradient distribu-tion (e.g. steepness) and those of the pressure maximum (e.g. amplitude) control the filtering/trapping of dust parti-cles of different size in the disc (see Equation 11 in Pinillaet al. 2012).Turbulent mixing also plays an essential role on dustgrowth and evolution. For instance, fragmentation occursbecause of high relative velocity collisions between dustgrains, with main sources being turbulent motion and radialdrift. If radial drift is reduced by a positive pressure gradi-ent, turbulent motion dominates and the maximum grainsize that particles can reach before fragmentation ( a max )depends directly on α turb (Birnstiel et al. 2010, 2012). If α turb is high, a max can be much lower than the size of par-ticles that can be trapped in pressure bump, preventing ac-cumulation of mm-sized particles in the pressure bump (e.g.Pinilla et al. 2015). In addition, turbulence also drives dif-fusion of particles within pressure bumps, and therefore if α turb is high, particles can more easily escape a dust trap.This is the case of models with α turb = 10 − (upper row)where we see no dust traps for mm- or even cm-sized par-ticles, even for very massive planets. In summary, there arethree reasons why trapping is weaker for higher values of α turb : less deep planetary gaps and hence lower pressuregradient at the outer gap edge; more effective fragmenta-tion of particles which leads to smaller grains that are moredifficult to trap; and higher diffusion or mixing of dust thatallows the particles to escape from the trap.Additionally, when α turb = 10 − , and also indepen-dently from planet mass, small grains pile up at the po-sition of the planet even surpassing the surface density val-ues of the gas. This is an effect of our 2D (gas)+1D (dust)approximation, in particular from assuming the gas veloc-ity using viscous accretion and assuming the averaged gassurface density for the dust evolution as in Pinilla et al.(2012); Pinilla et al. (2015). To test this, we re-run the dustcalculations in the 1 M Jup and α turb = 10 − case includ-ing only dynamics and neglecting coagulation, fragmenta-tion and grain growth processes. The enhancement of smallparticles then remains suggesting that it is indeed a numer-ical artifact and not the result of dust evolution processes.We then run another simulation for this case, where the dustvelocities are assumed to be v gas ∗ Σ gas / Σ gas instead, with v gas and Σ gas being the azimuthally and time (over the last100 orbits) averaged values from the hydrodynamical sim-ulations. In this case, accretion rate throughout the gap isalmost constant and the pile up of small dust at the loca-tion of the planet disappears (see details in Appendix B).Because in our dust evolution models we assume that thegas velocity comes from viscous evolution that tries to closethe gap, the particles that feel these velocities are pushedinto the gap. Since the viscous velocities are proportional to α turb , in the case of lower values (i.e. 10 − , 10 − ) they arenegligibly small, and radial drift becomes the dominatingcontribution for the dust velocity, which moves dust up thepressure gradient and prevents this artifact from appearing.This is, therefore, an inherent limitation of our modellingprocedure that affects high turbulence cases in the regionof the disc near the planet. To treat this issue, 2D gas anddust evolution models that include grain growth and dynam-ics simultaneously are needed, which are beyond the scopeof our study. For our analysis of these cases we therefore ig-nore this feature and base our conclusions on the otherwisecontinuously decreasing distribution of dust. c (cid:13) , 1–, 1– ?? onstraining α turb in transitional discs with planets. − − − Σ [ g × c m − ] α turb = 10 − M Jup M Jup M Jup M Jup − − − Σ [ g × c m − ] α turb = 10 − R [AU] − − − Σ [ g × c m − ] α turb = 10 −
10 20 30 40 50 60 70 80 R [AU]
10 20 30 40 50 60 70 80 R [AU]
10 20 30 40 50 60 70 80 R [AU] gas a = [10 − − − ] cm a = [10 − − − ] cm a = [10 − − − ] cm a = [10 − −
1] cm a >
Figure 1.
Radial profile of the gas (black diamonds) and dust (coloured lines) distributions for α turb = 10 − , − , and 10 − (frombottom to top). The blue-plus, green-cross, yellow-vertical-triangle, red-horizontal-triangle and red-dashed lines correspond to dust binsizes of a = [10 − − − ] cm, a = [10 − − − ] cm, a = [10 − − − ] cm, a = [10 − −
1] cm, and a > α turb = 10 − cases is a numerical artifact (see details in Section 3.1) We also note that our dust evolution treatment cannotfollow processes occurring when dust-to-gas ratios are largerthan 1 which can trigger instabilities and fast growth toplanetesimals (e.g. Johansen et al. 2007).In the cases where α turb is low ( α turb = 10 − , lower pan-els in Figure 1), the effect on the gas distribution would inprinciple favour trapping. However, dust growth is increasedbecause turbulent relative velocities are low: growth domi-nates over fragmentation and particles can continue grow-ing to even meter-sized objects inside the trap. Additionally,turbulent diffusion is not effective. As a result, only particleslarger than cm-sizes accumulate in a very narrow ring (red-dashed line in Fig. 1) and lower dust grain sizes are depleted.The waves that appear with low values of α turb (e.g. >
40 AUfeatures in 5 M Jup , α turb = 10 − case) are an artifact. Thesefeatures come from the spiral waves in the hydrodynami-cal simulations that appear as fixed density bumps to thedust evolution code because we assume a stationary gas dis-tribution azimuthally averaged after 1000 orbits. However,they have a pattern speed equal to the planet, and thus areunable to trap dust.All results are compared after 1 Myr of dust evolution. α turb The combination of effects of α turb on the gas and dust dis-tribution of the disc has a clear impact on scattering and emission flux images. Figure 2 shows synthetic SPHERE-ZIMPOL R-band (0 . µ m) polarimetric and ALMA Band7 (850 µ m) continuum emission observations (first and sec-ond columns of each panel, respectively) of a disc with anembedded 1 (left panel) and 9 M Jup (right panel) planet andfor α turb = [10 − , − , − ] (bottom to top rows). α turb = 10 − In the 1 M Jup and α turb = 10 − case (top row) there is noeffective trapping and dust grains of all sizes populate theregion with approximately constant surface density up tothe location of the planet where small particles are arti-ficially enhanced due to the limitations of our model (seeprevious subsection). Unfortunately, our simulations of po-larimetric observations of SPHERE-ZIMPOL at short wave-lengths ( ∼ . µ m) are dominated by this feature in thisparticular case. These images trace starlight scattered bysmall ( ∼ µ m, blue and green lines in Fig. 1) dust grainsat the surface of the disc, and therefore show the abruptenhancement in density of the small grains in this locationas a narrow ring. When ignoring the pile-up, the distribu-tion of small ( < M p > M Jup and this value of α turb c (cid:13) , 1– ?? M. de Juan Ovelar et al. − . − . . . . δ D EC [ a rc s e c ] α = −
30 AU
SPHERE-ZIMPOL (R-band) α = − ALMA (Band 7) − . − . . . . δ D EC [ a rc s e c ] α = −
30 AU α = − − . − . . . . δ RA [arcsec] − . − . . . . δ D EC [ a rc s e c ] × α = −
30 AU − . − . . . . δ RA [arcsec] × α = − . . . . F ν × ν [ W × m − ] × − F ν × ν [ W × m − ] × − M p =1 M Jup M p =1 M Jup M p =1 M Jup − . − . . . . δ D EC [ a rc s e c ] α = −
30 AU
SPHERE-ZIMPOL (R-band) α = − ALMA (Band 7) − . − . . . . δ D EC [ a rc s e c ] α = −
30 AU α = − − . − . . . . δ RA [arcsec] − . − . . . . δ D EC [ a rc s e c ] × α = −
30 AU − . − . . . . δ RA [arcsec] × α = − . . . . F ν × ν [ W × m − ] × − F ν × ν [ W × m − ] × − M p =9 M Jup M p =9 M Jup M p =9 M Jup
Figure 2.
R-band SPHERE and ALMA Band 7 synthetic observations of a protoplanetary disc with a 1 and 9 M Jup planet embeddedat 20 AU for the three values of α turb considered. The flux in the case of α turb = 10 − has been increased by of 2 and 5 for the SPHEREand ALMA images, respectively. Note that the ring-like feature in the case of 1 M Jup and α turb = 10 − is a numerical artifact (see detailsin Section 3.1). show the inner region of the disc because the gradient of thesmall grains distribution here is very high. It is this gradientinstead of the pile-up feature what dominates the image inthese cases.ALMA images trace thermal emission from ∼ M Jup panel in Fig.2) the gap becomes deeper andwider but the pressure maximum at the outer edge is stillnot strong enough to trap efficiently and, therefore, no ring-like feature appears in these images. α turb = 10 − For an intermediate value of α turb = 10 − (middle rows inFig. 1 and 2) the situation changes. Trapping here is effec-tive for dust grains of sizes around a ∼ M Jup , small particles – still cou-pled to the gas – flow freely through the gap to the innerregions of the disc ( r < r planet ). SPHERE-ZIMPOL imagestherefore show two components for the disc separated by thegap at the location of the planet, where small grains are par-tially depleted and cannot scatter as much radiation. For a9 M Jup planet the effects are amplified in both images. The trapping power of the planet is much larger, the pressurebump moves outwards and the inner regions to the positionof the planet are strongly depleted. This causes SPHERE-ZIMPOL images to show very clearly the position of thewall (outer edge of the gap) in the disc whose surface iscovered with small grains well coupled to the gas and ef-fectively scattering starlight. The pressure bump trappinglarge grains shines in ALMA continuum emission images asa wide ring at around ∼
45 AU. α turb = 10 − The models with the lowest value α turb we consider ( α turb =10 − ) tell the story of dust coagulation. Despite the fact thatturbulence here is very low and therefore the effect of theplanet on the gas distribution is amplified (which favours ef-fective trapping), relative velocities between dust grains arevery low and coagulation processes dominate over fragmen-tation ones in the dust trap. This results in a distribution ofgrain sizes dominated by larger than cm sizes (red-dashedline in Fig. 1) which leads to low fluxes at sub-mm wave-lengths. The dust trap affecting mm grains is present inimages of ALMA in both planet mass cases, but it is anextremely faint feature. The gap opened by a 1 M Jup planetappears in the SPHERE-ZIMPOL image thanks to the factthat it traces scattering radiation instead of emission, whichwill be affected by the strong depletion of small grains due to c (cid:13) , 1– ?? onstraining α turb in transitional discs with planets. coagulation. Indeed, few grains still scatter efficiently fromthe surface of the disc and wall, and are therefore able totrace the gap. Note that in NIR and sub-mm images ob-tained for this value of α turb the flux has been increased byfactors of [2 , M Jup is opening the gap however, the depletion becomesvery strong and the image, although still tracing the wall,becomes much fainter. Here the NIR-scattering image showsa secondary ring corresponding to one of the artifacts men-tioned in the previous section.
We perform (2D-)hydrodynamical, (1D-)dust evolution, ra-diative transfer and instrument simulations to obtain syn-thetic SPHERE-ZIMPOL and ALMA observations of TDswhere a gap is opened by a planet of different masses andwith three different values of the turbulent mixing strengthparameter α turb = [10 − , − , − ]. In this work we donot have simultaneous evolution of gas and dust, but as-sume the gas density from hydrodynamical simulations ofplanet disc interaction after 1000 orbits. The gas densityprofile remains then static for the dust evolution and wedo not include planet accretion or migration (see AppendixA for details). Under these assumptions, we find that α turb has a major impact on observations of dust in the disc. Inparticular our results show that: • We confirm that, as shown in Pinilla et al. (2012);Pinilla et al. (2015), for α turb = 10 − the trapping mech-anism is weak, resulting in SPHERE-ZIMPOL and ALMAimages showing continuous distributions of ∼ µ m and ∼ mm dust grains, respectively (see text for details on thelimitations of our models and the 1 M Jup image in this case). • For values of α turb = 10 − , growth is favoured over frag-mentation, and dust grains grow to sizes of (cid:38) • Current observations of TDs showing continuous dis-tributions (or small gaps/cavities) in NIR-polarimetric im-ages, and large gaps/cavities and ring-like features in sub-mm images (i.e. the ”missing cavities” effect) are only re-produced when α turb = 10 − . Since, to our knowledge, nomechanism other than planet-disc interaction has been pro-posed to cause this effect, it is reasonable to assume thatsuch combination of images is indicative of the presence ofa planet. Then, according to our results, the value of α turb can be constrained to 10 − within an order of magnitude,and the mass estimator presented in de Juan Ovelar et al.(2013) can be used to estimate the mass of the companion inthese sources. Note that ALMA images on their own couldalso be used to constrain the value of α turb but this is onlyif one knows for sure that a planet is causing the gap andthen SPHERE-ZIMPOL images would still be needed to usethe planet-mass estimator. The authors are thankful to the anonymous referee for athorough review, and to J. M. D. Kruijssen and G. P. Rosotti for useful comments on the manuscript. T. B. issupported by the NASA Origins of Solar Systems grantNNX12AJ04G and the Smithsonian Institution Pell Grantprogram. P. P. is supported by Koninklijke NederlandseAkademie van Wetenschappen (KNAW) professor prize toEwine van Dishoeck.
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Table A1.
Basic parameters of the simulationsTemperature of the star ( T star ) 4730 KRadius of the star ( R star ) 1 . R (cid:12) Mass of the star ( M star ) 1 M (cid:12) Mass of the disc ( M disc ) 0 . M (cid:12) Position of the planet ( R p ) 20 AUFragmentation velocity ( v f ) 10 m / sInner disc radius ( R disc , inn ) 0 . R p Outer disc radius ( R disc , out ) 7 . R p Solid density of dust particles ( ρ dust ) 1 . / cm Alpha viscosity ( α turb ) [10 − , − , − ] APPENDIX A: MODELLING APPROACHA1 Gas and dust models
We use the hydrodynamical code FARGO (Masset 2000) tostudy the evolution of the gas distribution on a 2-D (ra-dial+azimuthal) disc set up with an embedded planet at R p = 20 AU. Table A1 shows the parameters used as inputto the code (these are the same as in de Juan Ovelar et al.2013). We select open boundary conditions. The logarithmi-cally extended radial grid has a resolution of 512 in radiusby 1024 in azimuth and is taken from R disc , inn = 0 .
05 AU to R disc , out = 140 AU. Here we consider Σ ∝ r − , a kinematicviscosity of ν = α turb c s / Ω (Shakura & Sunyaev 1973), with c s and Ω being the sound speed and Keplerian frequencyrespectably, and with α turb = [10 − , − , − ]. The diskis assumed to be a flared disk with h/r ∝ r / , such thatthe temperature scales as T ∝ r / . And the aspect ratio atthe location of the planet is 0.05. We set the mass of the discto M disc = 0 . M (cid:12) . Stellar parameters are those typicalof a T-Tauri star.Figure A1 shows the 2-D surface density distributionof gas in all simulations run after 1000 orbits of evolution.This distribution is azimuthally averaged and fed as inputto the 1-D dust evolution code. Note that this is an impor-tant caveat of our method since by doing this we are as-suming that the gas distribution remains quasi-static after1000 orbits, which may not be the case in particular for thecases with very massive planets where strong spiral shocksand/or vortices still exist in the disk after this time. Theshape, depth and width of the gap is clearly dependent onthe mass of the planet. In addition, azimuthal structures arealso affected by the viscosity, as for instance the presence ofa vortex or eccentric shape of the outer edge of the plane-tary gap, which existence depends on the planet mass anddisc viscosity (Kley & Dirksen 2006; Ataiee et al. 2013; Zhu& Stone 2014; Fu et al. 2014).Density waves are clearly appearing in almost all casesbut one should bare in mind that this is just a snapshot andthat these waves cannot act as dust traps.To obtain the dust distribution, as we mentioned ear-lier, we take the evolved gas surface density from the hy-drodynamical simulations and then compute grain growthand fragmentation in the dust due to radial drift, turbulentmixing, and gas drag forces. We use the 1-D (radial) codedescribed by Birnstiel et al. (2010). The dust is initially dis- tributed such that the dust mass is 1% that of the gas (after1000 orbits), with an initial size for the grains of 1 µ m andevolving the distribution for 1 Myr. For the dust, we have alogarithmically spaced grid of the grain size with 180 cellscovering sizes from 1 µm to 200 cm, and dust densities foreach grain size.Note that we do not take into account feedback fromthe dust onto the gas which becomes non-negligible whenthe dust-to-gas ratio approaches values close to unity. Thiseffect could cause a secondary ring at mm-wavelengths (re-sult of a secondary pressure bump) even in the case of a dischosting a single planet, as shown in Gonzalez et al. (2015).Other processes that we do not take into account are planetmigration and accretion of dust onto the planet. Regardingthe latter, Owen (2014) showed that the luminosity createdby the process could affect the SED of the source, but to ourknowledge no study has investigated its effect on the gen-eral distribution of dust in the disc. In the absence of suchstudy, other than the distribution of dust in the surround-ings of the planet and possibly streams connecting the edgesof the gap with the planet, we have no reason to believe thatthe overall morphology of the distribution of dust in the disc(i.e. radial location of pressure maxima, gap wall, etc...) willbe drastically affected. We do not expect any changes dueto migration either, since the timescales for planet migra-tion for such high mass planets will be similar to viscoustimescales, meaning that the trap will move as the planetmigrate (as explained in Pinilla et al. 2015) which will causethe relative radial location of small and big grains to remainconstant. We also performed a test run using typical pa-rameters of a Herbig star instead of a T-Tauri one, findingthat the basic morphology of the gas/dust distribution re-mains the same as in the cases presented here and only thebrightness of the images changes. A2 Radiative transfer and synthetic observations
We compute full resolution emitted and scattered flux im-ages of the dust distribution in the disc at wavelengthsof λ = [0 . , µ m to use them as a model input forSPHERE-ZIMPOL and ALMA simulators in bands R and7, respectively. The radiative transfer is carried out usingthe Monte-Carlo radiative transfer code MCMax (Min et al.2009), which self-consistently solves the temperature andvertical structure of the distribution of gas and dust in thedisc, provided as an input, including the effect of dust set-tling.For the opacities of the dust we use a mixture of sili-cates ( ∼ ∼ ∼ ρ mix = 3 . / cm (Min et al. 2011) and a porosity of p = 0 . ρ = 1 . / cm used in the dust evolu-tion simulations. The indices of refraction are obtained from:Dorschner et al. (1995); Henning & Stognienko (1996) for thesilicates, Begemann et al. (1994) for the iron sulphide, andPreibisch et al. (1993) for the carbonaceous dust grains. For (In these images the resolution is not limited by the capabilitiesof a particular instrument; they are the direct output from theMC Radiative transfer simulation)c (cid:13) , 1– ?? onstraining α turb in transitional discs with planets. Figure A1. each value of the viscous turbulence ( α turb ), MCMax self-consistently solves the vertical structure of the gas in thedisc iteratively assuming vertical hydrostatic equilibrium.The vertical structure of the dust is then computed consid-ering settling and turbulent mixing.We simulate polarimetric and interferometric observa-tions with SPHERE and ALMA at λ = [0 . , µ m withthe SPHERE-ZIMPOL (Thalmann et al. 2008) and CASA(McMullin et al. 2007) simulators, respectively. We assumeone hour of total observing time with both instruments. Inthe SPHERE-ZIMPOL simulator we select the RI filter andprocess the full resolution Stokes Q , and U images obtainingthe final polarised-intensity image as P I = (cid:112) Q + U . Toobtain synthetic observations with ALMA in Band 7 we pro-cess the 850 µ m images with the CASA simulator specifyinga center frequency of ν obs = 345 GHz, and a band width of∆ ν obs = 7 . , (cid:48)(cid:48) and include the effectof atmospheric noise. http://casa.nrao.edu/ The CASA simulator accepts this as an input and then searchesfor the available configuration that achieves the specification.
APPENDIX B: ADDITIONAL SYNTHETICIMAGES
Figure B1 shows how the cases of planets of masses 5 and15 M Jup follow the same trends pointed out in Section 3.Only the cases of α turb = 10 − show the missing cavitiesfeature, i.e. no gap (or small gap) in NIR-scattering imageswhile large gap in sub-mm ones.Figure B2 show the synthetic observations for the caseof 1 M Jup and α turb = 10 − when the gas velocity consideredfor the dust evolution inside the gap is v gas ∗ Σ gas / Σ gas in-stead of the one set by viscous accretion (see Section 3.1).This prescription prevents the small grains from piling-upat the position of the planet as a consequence of the vis-cous velocities pushing them towards this position. However,this is a manual adjustment, and a physically reasonable ap-proximation for cases of a low mass companion ( (cid:46) M Jup planet), while it becomes more of a challenge for more mas-sive planets because of the large velocity fluctuations closeto the planet. For consistency with our modelling approach,we keep the results from all our models in the main textand only show here the alternative image for clarificationpurposes. When the artifact is removed, the observationsclearly show the otherwise continuous dust distribution in c (cid:13) , 1– ?? M. de Juan Ovelar et al. − . − . . . . δ D EC [ a rc s e c ] α = −
30 AU
SPHERE-ZIMPOL (R-band) α = − ALMA (Band 7) − . − . . . . δ D EC [ a rc s e c ] α = −
30 AU α = − − . − . . . . δ RA [arcsec] − . − . . . . δ D EC [ a rc s e c ] × α = −
30 AU − . − . . . . δ RA [arcsec] × α = − . . . . F ν × ν [ W × m − ] × − F ν × ν [ W × m − ] × − M p =5 M Jup M p =5 M Jup M p =5 M Jup − . − . . . . δ D EC [ a rc s e c ] α = −
30 AU
SPHERE-ZIMPOL (R-band) α = − ALMA (Band 7) − . − . . . . δ D EC [ a rc s e c ] α = −
30 AU α = − − . − . . . . δ RA [arcsec] − . − . . . . δ D EC [ a rc s e c ] × α = −
30 AU − . − . . . . δ RA [arcsec] × α = − . . . . F ν × ν [ W × m − ] × − F ν × ν [ W × m − ] × − M p =15 M Jup M p =15 M Jup M p =15 M Jup
Figure B1.
R-band SPHERE and ALMA Band 7 synthetic observations of a protoplanetary disc with a 5 and 15 M Jup planet embeddedat 20 AU for the three values of α turb considered. The flux in the case of α turb = 10 − has been increased by of 2 and 5 for the SPHEREand ALMA images, respectively. both SPHERE-ZIMPOL and ALMA observations at NIRand sub-mm wavelengths, respectively. − . − . . . . δ RA [arcsec] − . − . . . . δ D EC [ a rc s e c ] α = −
30 AU
SPHERE-ZIMPOL (R-band) − . − . . . . δ RA [arcsec] α = − ALMA (Band 7) . . . . F ν × ν [ W × m − ] × − F ν × ν [ W × m − ] × − M p =1 M Jup
Test run withgas velocities from hydrodynamical simulations
Figure B2.
R-band SPHERE and ALMA Band 7 syntheticobservations of 1 M Jup planet and α turb = 10 − case with the v gas ∗ Σ gas / Σ gas prescription considered for the gas velocity inthe gap instead of the viscous accretion one (see Section 3.1).c (cid:13) , 1–, 1–