Density distribution of dust particles

The simulations have been done with a disk of mass 0.05M, with a surface density de- scribed by equation (9.1) from 0.1 AU to 300 AU, around a star with one solar mass. Figure (9.1) shows the vertically integrated dust density distribution, taking into account: coagulation, radial mixing, radial drift and fragmentation, after 1 Myr of the evolution of the protoplanetary disk. The solid white line shows the particle size corresponding to a Stokes number of unity.

Both plots of Figure (9.1) have the same amplitude of the sinusoidal perturbation

A= 0.1. The factor f which describes the width of the perturbation, is taken to be f = 1 for the top plot, and f = 3 for the bottom plot of Figure (9.1). This result shows that: first, the amplitude A = 0.1 of the perturbation is not high enough to have a pressure gradient such that particles can be retained in the outer regions of the disk. Instead the dust particles are still affected by radial drift, they move inwards and the high relative velocities lead to reach the fragmentation velocity such that the particles do not grow over mm size particles in the outer regions. Second, taking a greater value of the factorf, that implies a longer wavelength for the perturbation, at the same amplitude, implies that the retention of particles is even weaker. We show that the trapping efficiency is higher when the wavelength of the perturbation is shorter, however if we want to compare these kind of perturbations with future observations (e.g. ALMA), the width of the perturbation has to be large enough to have detectable structures. In this case, due to the assumption of the presence of a bumpy surface density, will be regions on the disk where the dust accumulate. The separations of these ring structures depend of f, if the width of the perturbation is too short, so the rings will be very closed each other such that it would be very difficult to resolve them.

On the other hand, for longer widths, we should have therefore higher values of the amplitudes, however the disk becomes easily unstable when the amplitude goes up, this is the reason why we keep our results only forf = 1 and we take two different possible values for the amplitude.

Figure (9.2) compares the surface density distribution for two different values of the amplitude of the perturbationA = 0.1 andA= 0.3, at different times of evolution. Taking

A= 0.3, we can notice that in the pressure bumps, there is high density of dust particles, even after 5 Myr of evolution for a maximum radius around 100 AU.

To break through the meter-size barrier, it is important to take into account the two sources of relative velocities: radial drift and turbulence. In the bumps the radial drift relative velocities are zero, but there are still relative velocities due to the turbulence.

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Figure 9.4: Comparison of the observed fluxes at mm-wavelengths of the Taurus (red dots) and the Ophiuchus (blue dots) star forming regions (Ricci et al., 2010a,b, 2011) with the results of the simulations at different times of the disk evolution (star-dots). Results for

Figure 9.5: Disk image at 2 Myrs and observing wavelength of 0.45 mm, the amplitude of the perturbation isA= 0.3 and the factorf = 1 for: disk model with parameters of Table 9.1 (left) , simulated image using full configuration of ALMA (right) with a maximum value of baseline of around 3km and with an observing time of 4 hours.

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Therefore, it is necessary get a constraint on the turbulence. The largest turbulent relative velocity between particles is given by (Ormel et al., 2007a),

∆umax '2αc2s (9.17)

Therefore, to break through the meter size barrier, we must have that

uf >

√

2αcs. (9.18)

This last condition is satisfied in the outer region of the disk when the conditions of the perturbation areA≥0.3 andf = 1. With the turbulence parameter fixed toα= 10−3 and with the low temperatures at the outer regions of the disk, T ∼ 10K, the relative velocities due to the turbulance are lower than the taken fragmentation velocity, allowing to the particle to grow over mm sizes.

Figure 9.3 shows the radial dependence of the dust-to-gas ratio for different times of the simulation and two values for the perturbation amplitude. For A = 0.1 (top plot of Figure 9.3), we can see that the dust-to-gas ratio decreased significant after few Myrs of the dust evolution, this implies that the dust particles does not grow considerably. Due to fragmentation the dust particles collide and become even smaller. Then, because of the turbulence and radial drift, the dust particles mix and move inwards. Since the dust particles remain vary small, the dust-gas ratio decreases and become almost constant with time, which implies that the dust particles are small enough to be well coupled to the gas. Whereas, due to the strong over pressures at A = 0.3 (bottom plot of Figure 9.3), the dust-to-gas ratio remains almost constant with time for r < 100AU, oscillating radially between 10−3 to 10−1. This oscillating behavior, even after 5 Myr of dust evolution, is possible thanks to the fact that the particles are retained in the bumps and grow enough to make the dust-gas ration higher inside the bumps. Only around ∼ 100AU from the star, the dust to gas ratio decreases significantly. Summarizing, due to the presence of the pressure bumps with an amplitude A = 0.3, the dust-gas mass ratio increases locally with the dust evolution. This implies that the drift is counteracted by the positive local pressure gradient, allowing that the time scales for the growth are comparable with the disk evolution times, i. e. with the drift time scales.