We investigate the interaction of gas and dust in a protoplanetary disk in
the presence of a massive planet using a new two-fluid hydrodynamics code. In
view of future observations of planet-forming disks we focus on the condition
for gap formation in the dust fluid. While only planets more massive than 1
Jupiter mass (MJ) open up a gap in the gas disk, we find that a planet of 0.1
MJ already creates a gap in the dust disk. This makes it easier to find
lower-mass planets orbiting in their protoplanetary disk if there is a
significant population of mm-sized particles.Comment: 5 pages, 3 figures, accepted for publication in A&A Letter

Bryden

Eulderink

G. Mellema

Godon

Klahr

Kley

Mathis

Miyake

Papaloizou

S.-J. Paardekooper

Shakura

Strang

Takeuchi

Wolf

English

arXiv

Wetensch. publicatieFaculteit der Wiskunde en Natuurwetenschappe

Paardekooper, S.-J.

Mellema, G.

Leiden University Scholary Publications

A&A 425, L9–L12 (2004)DOI: 10.1051/0004-6361:200400053c© ESO 2004Astronomy&AstrophysicsPlanets opening dust gaps in gas disksS.-J. Paardekooper1 and G. Mellema2,11 Leiden Observatory, Postbus 9513, 2300 RA Leiden, The Netherlandse-mail: paardeko@strw.leidenuniv.nl2 ASTRON, Postbus 2, 7990 AA Dwingeloo, The Netherlandse-mail: gmellema@astron.nlReceived 7 July 2004 / Accepted 10 August 2004Abstract. We investigate the interaction of gas and dust in a protoplanetary disk in the presence of a massive planet using anew two-fluid hydrodynamics code. In view of future observations of planet-forming disks we focus on the condition for gapformation in the dust fluid. While only planets more massive than 1 Jupiter mass (MJ) open up a gap in the gas disk, we findthat a planet of 0.1 MJ already creates a gap in the dust disk. This makes it easier to find lower-mass planets orbiting in theirprotoplanetary disk if there is a significant population of mm-sized particles.Key words. hydrodynamics – stars: planetary systems1. IntroductionResolved images of protoplanetary disks can reveal densitystructures in gas and dust hinting at the presence of embeddedplanets. It is therefore important to investigate the response ofa dusty circumstellar disk on planets, as opposed to for exam-ple radiation pressure effects in optically thin disks (Takeuchi& Artymowicz 2001).The most dramatic disk structure a planet can create isan annular gap (Papaloizou & Lin 1984), and these planet-induced gaps can be used as tracers for recent giant planet for-mation. Recently, Wolf et al. (2002) showed that the AtacamaLarge Millimeter Array (ALMA) will be able to detect a gap at5.2 AU caused by the presence of a Jupiter mass planet in a disk140 pc away in the Taurus star-forming region. However, theyassumed a constant dust-to-gas mass ratio of 1:100, whereas inthe presence of large pressure gradients the gas and the dustevolution will decouple to some extent (Takeuchi & Lin 2002).Due to radial pressure support the gas orbits at sub-Keplerianvelocity, and when the dust particles are slowed down by dragforces they move inward.In this letter we present the first results of two-dimensionalnumerical simulations where we treat the gas and the dust asseparate but coupled fluids. These two-fluid calculations takeinto account the full interaction of gas and dust, and can beused to simulate observations in a better way.In Sect. 2 we briefly discuss the physics of gas-dust inter-action. We describe the numerical method in Sect. 3, and theinitial conditions in Sect. 4. We present the results in Sect. 5and we conclude in Sect. 6.2. Basic equationsProtoplanetary disks are fairly thin, i.e. the vertical thickness His small compared with the distance r from the centre of thedisk. Typically we use H/r = h = 0.05. Therefore it is conve-nient to average the equations of motion vertically, and workwith vertically averaged quantities only. The governing equa-tions as well as the gas disk model are the ones described indetail in Kley (1999).2.1. DustWe treat the dust as a pressureless fluid, and its evolution isgoverned by conservation of mass and conservation of lin-ear and angular momentum. The major difference between theequations for the gas and the dust is the absence of pressure inthe latter. In this sense the dust fluid behaves like a gas that ismoving always with supersonic velocity. This implies that nearshock waves, where the gas goes from sonic to supersonic flowand large density and velocity gradients are present, dust andgas behave in a very different way.The interaction between the gas and dust fluids occurs onlythrough drag forces. The nature of the drag force depends onthe size of the particles with respect to the mean free path ofthe gas molecules. We consider only spherical particles with ra-dius s, and for the particle sizes we are interested in (s ≈ 1 mm)we can safely use Epsteins drag law (Takeuchi & Artymowicz2001). In the limit of small relative velocities of gas and dustcompared to the local sound speed, the drag forces are writ-ten as:f d = −ΣΩKTs(u − ug)(1)Letter to the EditorL10 S.-J. Paardekooper and G. Mellema: Planets opening dust gaps in gas diskswhere ΩK is the Keplerian angular velocity, and Ts is the non-dimensional stopping time (Takeuchi & Lin 2002):Ts =√π8ρp svKρgrcs· (2)Here, ρp is the particle internal density, s is the radius of the par-ticle and cs is the sound speed. We have used ρp = 1.25 g cm−3,and the gas density ρg is found by using ρg = Σg/2H. Duringsimulations we have always checked whether Eq. (1) applied.3. Numerical methodFor the evolution of the gas component, we use the methoddescribed by Paardekooper & Mellema (2004). It is a second-order Eulerian hydrodynamics code which uses an approximateRiemann solver (Roe 1981). It derives from a general relativis-tic method (Eulderink & Mellema 1995), from which it inheritsthe capability to deal with non-Cartesian, non-inertial coordi-nate systems. We work in a cylindrical coordinate frame (r, φ),corotating with the embedded planet that has a Keplerian angu-lar velocity. A module for Adaptive Mesh Refinement (AMR)can be used to obtain very high resolution near the planet.We use basically the same method to follow the evolution ofthe dust, but without the pressure terms. This is a very dif-ferent approach than for example in Klahr & Henning (1997),who solve the equations of motion of particles instead of thefluid equations. All the source terms except the drag forcesare integrated using stationary extrapolation (Paardekooper &Mellema 2004). The drag forces are incorporated separatelyusing an ordinary differential equation:ddt(Σu) = f d. (3)Separating the drag forces from the advection in this wayputs constraints on the time step. Basically the assumptionis that no drag force acts on the dust during the advection,and this is clearly not correct when the drag force sourceterms are very stiff. The time step is set by the hydrodynamicCourant-Friedrichs-Lewy (CFL) condition, and therefore onehas to make sure that the typical time scale for the drag forces(the stopping time) is not much smaller than this time step.Switching the order of integration every time step yields a sec-ond order accurate update for the state (Strang 1968). If thestopping time is very small (well-coupled particles) the dustcan not be assumed to move free of drag during a dust up-date. This puts a lower limit on the particle size that we canput into the simulations. For a typical disk, this limit is about0.1 mm. The time step can however be reduced to includesmaller particles.4. Initial and boundary conditionsThe standard numerical resolution is (nr, nφ) = (128, 384). Thisway, the cells close to the planet have equal size in the radialand in the azimuthal direction. We do not resolve the Rochelobe at this resolution, but since we are interested only in thelarge scale evolution of the disk, this is not important. Wewill discuss the flow within the Roche lobe and accretion pro-cesses in future papers. Using a low resolution allows us to runlonger simulations, which is important because the drift veloc-ities can be very low. We take a constant initial surface densityΣ0 = 34 g cm−2. This surface density corresponds to the valueat 5.2 AU in a disk with 0.01 M within 100 AU and a sur-face density profile that falls off as r−1/2, or to about 13 AU inthe Minimum Mass Solar Nebula. The initial dust-to-gas ratiois 0.01. Using the radius of the planet’s orbit as our unit dis-tance, the inner boundary is at r = 0.4 and the outer boundaryat r = 2.5. The gas rotates with a slightly sub-Keplerian veloc-ity due to the radial pressure gradient. The dust rotates exactlywith the Keplerian velocity, initially. All radial velocities aretaken to be zero. The constant kinematic viscosity is set so thatthe Shakura & Sunyaev (1973) α parameter equals 0.004 atr = 1. For a disk described above, the critical planet mass forgap opening is about 1 Jupiter mass (1 MJ) (Bryden et al. 1999).We varied the mass of the planet between 0.001 MJ and 0.5 MJ.The boundary conditions are as in Paardekooper & Mellema(2004), deriving from Godon (1996). They are non-reflecting,so that all waves generated in the simulated region leave thecomputational domain without further influencing the interior.5. ResultsThe evolution of the gas component is described in detail inPaardekooper & Mellema (2004), and the drag forces on thegas are too small to alter the flow. Therefore we focus on theevolution of the dust particles.5.1. General flow structureFor the standard case we use a planet of 0.1 MJ, and we con-sider dust particles of 1 mm. In the left panel of Fig. 1 we cansee that the dust particles are opening a gap, which leads tovariations in the dust-to-gas ratio of more than an order of mag-nitude (middle panel). For this small planet these variations indust density are so large that the spiral waves are no longer vis-ible. The radial cut (right panel) shows that this planet is notable to open up a gap in the gas, while the dust reacts morestrongly and shows a very dense ring at r = 1.3 just outside avery low density gap.Also shown in the right panel of Fig. 1 are positions ofthe 3:2, the 2:1 and the 3:1 mean motion resonances. Especiallythe 3:2 resonance (at r = 1.25) plays a role in structuring thedisk, while only faint features can be seen at the other reso-nances. There also exists a small density bump at corotation(r = 1.0).Figure 2 shows a close-up on the gas density near the planetin a high-resolution simulation. The high-density core can beseen, as well as the spiral shocks. The arrows indicate the rel-ative velocity of the dust compared to the gas, showing thatthe dust decouples from the gas near the spiral shocks. In theshocks the gas velocity is reduced, and the gas is also deflected.The dust particles do not feel the shock directly, but notice itspresence by the drag forces in the post-shock region. Figure 2shows that it takes a certain distance before the velocity differ-ence between dust and gas has disappeared again. Because ofLetter to the EditorS.-J. Paardekooper and G. Mellema: Planets opening dust gaps in gas disks L11Fig. 1. Dust flow near a 0.1 MJ planet. Left panel: grey-scale plot of the logarithm of the dust surface density after 400 planetary orbits.Middle panel: logarithm of the dust-to-gas ratio. Right panel: radial cut at φ = 0 (opposite to the planet). Solid line: gas surface density, dashedline: dust surface density ×100. The filled circles indicate the 3:2, the 2:1 and the 3:1 mean motion resonances.Fig. 2. Close-up on the gas density near the planet for a high resolutionAMR simulation (3 levels of refinement, so a resolution that is 8 timeshigher than our standard resolution). The arrows indicate the relativevelocity of the dust compared to the gas. The typical relative velocityacross the shocks is 0.035 cs. Note that the color scale is reversed withrespect to Fig. 1.this, the flux of dust particles streaming into the spiral wave islarger than in the case of a perfectly coupled gas-dust mixture.Since the wave is responsible for gradually pushing the gas anddust particles away from the orbit of the planet this leads toa depletion of dust particles, and hence a deeper gap in thedust distribution. The dust density is affected most where thewaves are the strongest, which is approximately at a radial dis-tance h from the orbit of the planet. This is where a low densitygap starts to form, leaving the particles at r = 1 largely unaf-fected. The particles moving to larger radii seem to get trappedin the 3:2 and the 2:1 mean motion resonances, creating thedense rings at r = 1.3 and r = 1.6. Interestingly, it has beenfound that the Kuiper Belt Objects also tend to accumulate atthe the 3:2 and 2:1 mean motion resonances with Neptune (Luu& Jewitt 2002). Whether this is related to the effect we find,requires further study. Inside corotation, the combined inwardforce due to the spiral waves, the imposed radial temperaturegradient and the viscous gas flow is so large that basically allparticles move over the resonances quickly and end up near thewave-damping boundary where they are absorbed completely,leaving an almost featureless inner disk. Simulations with dif-ferent planetary masses showed that the minimum planetarymass for this mechanism to operate is 0.05 MJ with particles of1 mm.5.2. Simulated observationsTo investigate the observational effects of the gas-dust decou-pling we simulate an image taken at a wavelength of 1 mm. Thedisk is optically thin at this wavelength, and we put the planetat 5.2 AU. Calculations with different particle sizes showed thatthe minimum grain size for gap formation is approximately0.1 mm, with a very sharp transition. We therefore assumethat all particles larger than this critical size create a gap, andsmaller particles follow the gas exactly. To estimate the rela-tive amount of mass in these two families of particles we takean MRN size distribution (Mathis et al. 1977), in which theLetter to the EditorL12 S.-J. Paardekooper and G. Mellema: Planets opening dust gaps in gas disksFig. 3. Logarithm of flux densities at 1 mm, normalized by the maximum and convolved with a Gaussian of FWHM 2.5 AU, corresponding toa resolution of 12 mas at 140 pc. Left panel: all particles follow the gas exactly (static dust evolution). Middle panel: particles larger than thecritical size decouple from the gas (dynamic dust evolution). Right panel: the corresponding radial flux densities.number density of particles is a power law in s with expo-nent −3.5. For a maximum particle size of 5 mm most massis in particles larger than our critical size, and as a conserva-tive assumption we take an equal amount of mass in the parti-cles smaller than 0.1 mm and in the larger particles. But sincethe opacity at 1 mm is dominated by particles with s ≈ 1 mm(Miyake & Nakagawa 1993), the emission will be dominatedby the larger particles. We used the size-dependent opacity datafrom Miyake & Nakagawa (1993) to calculate the flux densi-ties. In Fig. 3 the resulting flux densities are shown, convolvedwith a Gaussian of FWHM 2.5 AU, which corresponds to anangular resolution of 12 mas at 140 pc (the Taurus star formingregion). This resolution is comparable to the maximum reso-lution of 10 mas that ALMA will achieve. In the left panel ofFig. 3 we computed the flux under the assumption that all par-ticles move with the gas, as was done for example by Wolfet al. (2002). There is a little dip visible due to the decrease indensity in the gas, but only when we include the density dis-tribution of Fig. 1 for the larger particles a clear gap emerges(middle panel of Fig. 3). The right panel of Fig. 3 shows theradial dependence of the flux density. From this panel we cansee that the contrast between the inside of the gap and the gapedges is enhanced from almost a factor of two (dashed line)to more than an order of magnitude (solid line). So the impactof the dynamics of the larger particles on the appearance of alow-mass planet in a disk is considerable.6. SummaryWe have presented the first two-fluid calculations of a circum-stellar disk with an embedded planet. These models allow us tomake predictions on the dust distributions that will be observedin protoplanetary disks in the near future.For a reasonable disk mass (0.01 M within 100 AU) thestrong spiral shocks near the planet are able to decouplethe larger particles (≥0.1 mm) from the gas. This leads to theformation of an annular gap in the dust, even if there is nogap in the gas density. Because the opacity at millimeter wave-lengths is dominated by these larger particles, the signaturesof low-mass planets in disks can be stronger than previouslythought. The minimum mass for a planet to open a gap thisway was found to be 0.05 MJ for 1 mm particles.Acknowledgements. We wish to thank Carsten Dominik for carefullyreading the manuscript. S.P. and G.M. acknowledge support fromthe European Research Training Network “The Origin of PlanetarySystems” (PLANETS, contract number HPRN-CT-2002-00308).ReferencesBryden, G., Chen, X., Lin, D. N. C., Nelson, R. P., & Papaloizou,J. C. B. 1999, ApJ, 514, 344Eulderink, F., & Mellema, G. 1995, A&AS, 110, 587Godon, P. 1996, MNRAS, 282, 1107Klahr, H. H., & Henning, T. 1997, Icarus, 128, 213Kley, W. 1999, MNRAS, 303, 696Luu, J. X., & Jewitt, D. C. 2002, ARA&A, 40, 63Mathis, J. S., Rumpl, W., & Nordsieck, K. H. 1977, ApJ, 217, 425Miyake, K., & Nakagawa, Y. 1993, Icarus, 106, 20Paardekooper, S.-J., & Mellema, G. 2004, A&A, in preparationPapaloizou, J., & Lin, D. N. C. 1984, ApJ, 285, 818Roe, P. L. 1981, J. Comp. Phys., 43, 357Shakura, N. I., & Sunyaev, R. A. 1973, A&A, 24, 337Strang, G. 1968, Siam. J. Num. Analysis, 5, 506Takeuchi, T., & Artymowicz, P. 2001, ApJ, 557, 990Takeuchi, T., & Lin, D. N. C. 2002, ApJ, 581, 1344Wolf, S., Gueth, F., Henning, T., & Kley, W. 2002, ApJ, 566, L97Letter to the Editor

Planets opening dust gaps in gas disks

We investigate the interaction of gas and dust in a protoplanetary
 disk in the presence of a massive planet using a new two-fluid 
 hydrodynamics code. In view of future observations of planet-forming 
 disks we focus on the condition for gap formation in the dust fluid. 
 While only planets more massive than 1 Jupiter mass (${{M}_{\rm J}}$) open up a 
 gap in the gas disk, we find that a planet of 0.1 ${{M}_{\rm J}}$ already creates
 a gap in the dust disk. This makes it easier to find lower-mass planets 
 orbiting in their protoplanetary disk if there is a significant population 
 of mm-sized particles.

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Planets opening dust gaps in gas disks

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