A&A 425, L9-L12 (2004)
S.-J. Paardekooper 1 - G. Mellema 2,1
1 - Leiden Observatory, Postbus 9513, 2300 RA Leiden, The Netherlands
2 - ASTRON, Postbus 2, 7990 AA Dwingeloo, The Netherlands
Received 7 July 2004 / Accepted 10 August 2004
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 ( ) open up a gap in the gas disk, we find that a planet of 0.1 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.
Key words: hydrodynamics - stars: planetary systems
The most dramatic disk structure a planet can create is an annular gap (Papaloizou & Lin 1984), and these planet-induced gaps can be used as tracers for recent giant planet formation. Recently, Wolf et al. (2002) showed that the Atacama Large Millimeter Array (ALMA) will be able to detect a gap at 5.2 AU caused by the presence of a Jupiter mass planet in a disk 140 pc away in the Taurus star-forming region. However, they assumed a constant dust-to-gas mass ratio of 1:100, whereas in the presence of large pressure gradients the gas and the dust evolution will decouple to some extent (Takeuchi & Lin 2002). Due to radial pressure support the gas orbits at sub-Keplerian velocity, and when the dust particles are slowed down by drag forces they move inward.
In this letter we present the first results of two-dimensional numerical simulations where we treat the gas and the dust as separate but coupled fluids. These two-fluid calculations take into account the full interaction of gas and dust, and can be used to simulate observations in a better way.
In Sect. 2 we briefly discuss the physics of gas-dust interaction. We describe the numerical method in Sect. 3, and the initial conditions in Sect. 4. We present the results in Sect. 5 and we conclude in Sect. 6.
The interaction between the gas and dust fluids occurs only through
drag forces. The nature of the drag force depends on the size of the
particles with respect to the mean free path of the gas molecules. We
consider only spherical particles with radius s, and for the particle
sizes we are interested in (
mm) we can safely use Epsteins
drag law (Takeuchi & Artymowicz 2001). In the limit of small relative
velocities of gas and dust compared to the local sound speed, the drag
forces are written as:
|Figure 1: Dust flow near a 0.1 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 (opposite to the planet). Solid line: gas surface density, dashed line: dust surface density . The filled circles indicate the 3:2, the 2:1 and the 3:1 mean motion resonances.|
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Also shown in the right panel of Fig. 1 are positions of the 3:2, the 2:1 and the 3:1 mean motion resonances. Especially the 3:2
resonance (at r=1.25) plays a role in structuring the disk, while
only faint features can be seen at the other resonances. There also
exists a small density bump at corotation (r=1.0).
|Figure 2: Close-up on the gas density near the planet for a high resolution AMR simulation (3 levels of refinement, so a resolution that is 8 times higher than our standard resolution). The arrows indicate the relative velocity of the dust compared to the gas. The typical relative velocity across the shocks is 0.035 . Note that the color scale is reversed with respect to Fig. 1.|
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Figure 2 shows a close-up on the gas density near the planet in a high-resolution simulation. The high-density core can be seen, as well as the spiral shocks. The arrows indicate the relative velocity of the dust compared to the gas, showing that the dust decouples from the gas near the spiral shocks. In the shocks the gas velocity is reduced, and the gas is also deflected. The dust particles do not feel the shock directly, but notice its presence by the drag forces in the post-shock region. Figure 2 shows that it takes a certain distance before the velocity difference between dust and gas has disappeared again. Because of this, the flux of dust particles streaming into the spiral wave is larger than in the case of a perfectly coupled gas-dust mixture. Since the wave is responsible for gradually pushing the gas and dust particles away from the orbit of the planet this leads to a depletion of dust particles, and hence a deeper gap in the dust distribution. The dust density is affected most where the waves are the strongest, which is approximately at a radial distance h from the orbit of the planet. This is where a low density gap starts to form, leaving the particles at r=1 largely unaffected. The particles moving to larger radii seem to get trapped in the 3:2 and the 2:1 mean motion resonances, creating the dense rings at r=1.3 and r=1.6. Interestingly, it has been found that the Kuiper Belt Objects also tend to accumulate at the 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 inward force due to the spiral waves, the imposed radial temperature gradient and the viscous gas flow is so large that basically all particles move over the resonances quickly and end up near the wave-damping boundary where they are absorbed completely, leaving an almost featureless inner disk. Simulations with different planetary masses showed that the minimum planetary mass for this mechanism to operate is 0.05 with particles of 1 mm.
|Figure 3: Logarithm of flux densities at 1 mm, normalized by the maximum and convolved with a Gaussian of FWHM 2.5 AU, corresponding to a resolution of 12 mas at 140 pc. Left panel: all particles follow the gas exactly (static dust evolution). Middle panel: particles larger than the critical size decouple from the gas (dynamic dust evolution). Right panel: the corresponding radial flux densities.|
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To investigate the observational effects of the gas-dust decoupling we simulate an image taken at a wavelength of 1 mm. The disk is optically thin at this wavelength, and we put the planet at 5.2 AU. Calculations with different particle sizes showed that the minimum grain size for gap formation is approximately 0.1 mm, with a very sharp transition. We therefore assume that all particles larger than this critical size create a gap, and smaller particles follow the gas exactly. To estimate the relative amount of mass in these two families of particles we take an MRN size distribution (Mathis et al. 1977), in which the number density of particles is a power law in s with exponent -3.5. For a maximum particle size of 5 mm most mass is in particles larger than our critical size, and as a conservative assumption we take an equal amount of mass in the particles smaller than 0.1 mm and in the larger particles. But since the opacity at 1 mm is dominated by particles with mm (Miyake & Nakagawa 1993), the emission will be dominated by the larger particles. We used the size-dependent opacity data from Miyake & Nakagawa (1993) to calculate the flux densities. In Fig. 3 the resulting flux densities are shown, convolved with a Gaussian of FWHM 2.5 AU, which corresponds to an angular resolution of 12 mas at 140 pc (the Taurus star forming region). This resolution is comparable to the maximum resolution of 10 mas that ALMA will achieve. In the left panel of Fig. 3 we computed the flux under the assumption that all particles move with the gas, as was done for example by Wolf et al. (2002). There is a little dip visible due to the decrease in density in the gas, but only when we include the density distribution of Fig. 1 for the larger particles a clear gap emerges (middle panel of Fig. 3). The right panel of Fig. 3 shows the radial dependence of the flux density. From this panel we can see that the contrast between the inside of the gap and the gap edges is enhanced from almost a factor of two (dashed line) to more than an order of magnitude (solid line). So the impact of the dynamics of the larger particles on the appearance of a low-mass planet in a disk is considerable.
For a reasonable disk mass (0.01 within 100 AU) the strong spiral shocks near the planet are able to decouple the larger particles (0.1 mm) from the gas. This leads to the formation of an annular gap in the dust, even if there is no gap in the gas density. Because the opacity at millimeter wavelengths is dominated by these larger particles, the signatures of low-mass planets in disks can be stronger than previously thought. The minimum mass for a planet to open a gap this way was found to be 0.05 for 1 mm particles.
We wish to thank Carsten Dominik for carefully reading the manuscript. S.P. and G.M. acknowledge support from the European Research Training Network "The Origin of Planetary Systems'' (PLANETS, contract number HPRN-CT-2002-00308).