On the Dynamics of Pebbles in Protoplanetary Disks with Magnetically Driven Winds

The formation of planetary systems is a long-standing quandary and has attracted considerable interest in recent years (Blum & Wurm 2008; Armitage 2011; Winn & Fabrycky 2015; Nayakshin 2017). Depending on the physical properties of protoplanetary disks (PPDs), several mechanisms have been proposed to explain planet formation in these complex systems. Gravitational instability, for instance, may eventually lead to giant planet formation in the outer parts of a PPD (Boss 1997, 2017; Rafikov 2005; Forgan & Rice 2013), whereas core accretion is believed to be the dominant planet formation mechanism for the inner regions of a PPD (Pollack et al. 1996; Matsuo et al. 2007; Boley 2009). Both of these planet formation theories, however, have their own weaknesses and are subject to controversy.

In the core accretion theory, for instance, the core mass requirement to trigger the accretion of gas is debated because the growth timescale of a massive solid core can be longer than the lifetimes of the PPDs (Rafikov 2011). The typical size of the solids that can be accreted to form planetary cores is rather controversial (Kobayashi et al. 2011; Alibert 2017). Various physical factors, including disk metallicity, the location in the disk, and the accretion rate of solids, can dramatically affect the efficiency of planet formation in the core accretion scenario (Fortier et al. 2013; Piso & Youdin 2014). Planet formation by gravitational instability, on the other hand, depends on the cooling efficiency of a disk, which is still a challenging issue (Meru & Bate 2011; Rice et al. 2015; Takahashi et al. 2016). Under typical conditions in the gravitationally unstable disks, however, Kratter et al. (2010) showed that disk-born fragments are able to grow beyond the deuterium-burning planetary mass limit. This implies that wide orbit gravitational instabilities typically produce stars, not planets. In recent years a mixed planet formation scheme known as tidal-downsizing has also been developed,; it supplements giant gaseous protoplanet formation in the outer parts of a PPD due to gravitational instability and simultaneous migration of this newly formed protoplanet and sedimentation of the grains deep inside it to form a solid core (Helled et al. 2008; Nayakshin 2010, 2015).

While the accretion of grains onto a planetary embryo is not very efficient, it has been shown that for particles with sizes from centimeters to meters, the accretion rate onto a planetary embryo is enhanced due to the drag force exerted by the ambient gas (Ormel & Klahr 2010; Lambrechts & Johansen 2012; Visser & Ormel 2016). Centimeter-sized to meter-sized particles, which migrate efficiently in a PPD, are called pebbles. Therefore, it is very important to understand how micron-sized particles are able to grow to this size range and how these pebbles, once formed, are delivered throughout the PPD (Lambrechts & Johansen 2014; Levison et al. 2015; Moriarty & Fischer 2015; Chambers 2016; Ida & Guillot 2016; Ida et al. 2016; Krijt et al. 2016; Sato et al. 2016; Picogna et al. 2018). Lambrechts & Johansen (2014; hereafter; LJ2014) investigated the production of pebbles and their transport in a gaseous disk with power-law profiles for the surface density and temperature. Their analytical effort showed that the pebble production line spreads outward and the radial pebble flux through the disk has a weak time dependence. Ida et al. (2016) investigated the radial dependence of the pebble accretion rate and showed that the outcome of the pebble accretion scenario strongly depends on the physical conditions in the disk, the drag law, and the dimension of the accretion model (i.e., 2D or 3D). Ida & Guillot (2016) presented an analytical model for exploring the effect of the snow line on the inward flow of pebbles and showed that the accumulation of particles may occur interior to the snow line due to the sublimation of the migrating pebbles. While most analytical models for the dynamics of pebbles are based on the steady-state gaseous disk model, Bitsch et al. (2015) studied pebble production and delivery in an evolving gaseous disk model. They found that the pebble accretion scenario is able to resolve many of the challenging issues about ice and gas-giant planet formation in evolving PPDs.

To the best of our knowledge, none of the previous works on pebble formation and delivery through a PPD have considered the potential role of the presence of a wind. There are, however, a number of observational and theoretical studies that suggest that a PPD can lose its mass and/or angular momentum through winds or outflows, significantly affecting its dynamical structure (Blandford & Payne 1982; Combet & Ferreira 2008; Guilet & Ogilvie 2014; Bai et al. 2016; Bjerkeli et al. 2016; Wang & Goodman 2017). The reduction of mass in PPDs by winds has been traditionally attributed to photoevaporation (Hollenbach et al. 1994; Clarke et al. 2001; Ruden 2004; Gorti et al. 2015), but in recent years, due to extensive studies of the role of non-ideal magnetohydrodynamics (MHD) effects in PPDs, magneto-driven winds have been at the forefront of theoretical developments (Salmeron et al. 2011; Bai & Stone 2013; Bai et al. 2016). While it is commonly believed that photoevaporative winds are active and effective at the outer parts of a PPD, especially during the final stages of its evolution, theoretical arguments show that magnetic fields can also play a key role in a wind-launching (Ramsey & Clarke 2011; Salmeron et al. 2011; Guilet & Ogilvie 2014; Bai et al. 2016). Given the existing observational evidence about PPDs, however, it is difficult to determine which of these mechanisms dominates. Most of the previous (semi)analytical models for a disk with a magnetically driven wind are constructed based on the same simplifying assumptions of the standard accretion disk model, with some modifications due to mass-loss and angular momentum transport by winds (Combet & Ferreira 2008; Bai et al. 2016; Suzuki et al. 2016).

Although the dynamics of dust particles in an evolving PPD with winds driven by photoevaporation can be studied according to the current models made for such systems, in this study we plan to study the dynamics of pebbles in a PPD with magnetically driven winds. We use a steady-state model for the structure of a gaseous disk with a magnetically driven wind, as provided by Hasegawa et al. (2017) on the basis of the results of numerical simulations (Fromang et al. 2013; Simon et al. 2013; Zhu et al. 2015). In this model, the net accretion rate is due to stresses resulting from MHD turbulence within the disk and a magnetically driven wind. Both of these stresses are parameterized as a function of β₀, the ratio between gas and magnetic pressure at the disk midplane (Simon et al. 2013; Zhu et al. 2015).

Zhu et al. (2015) showed that the vertical diffusion coefficient of dust particles is actually a function of the parameter β₀. Using three-dimensional global unstratified MHD simulations, they explored the properties of magnetorotational instability (MRI)-driven turbulence in the ideal MHD case and a non-ideal case with ambipolar diffusion. Dust transport in the ideal MHD simulations can be reproduced with analytical models (Youdin & Lithwick 2007); however, ambipolar diffusion qualitatively changes dust diffusion coefficients. The difference is due to the nature of turbulence in each of these cases. Turbulence in the simulations with ambipolar diffusion exhibits different temporal autocorrelation functions and power spectra compared to the generated turbulence in the ideal MHD simulations. Efficient vertical dust diffusion, for instance, is found in the ambipolar-diffusion-dominated simulations (Zhu et al. 2015). Considering how the stress tensor components and dust diffusion coefficient depend on the parameter β₀, some authors have investigated the structure of an accretion disk with magnetically driven winds in the steady-state and even time-dependent cases (Armitage et al. 2013; Hasegawa et al. 2017). Just recently, Hasegawa et al. (2017) constructed a gaseous disk model with magnetically driven winds by considering both radially and vertically angular momentum transport via MHD turbulence and disk winds, respectively, and found that a high accretion rate and efficient dust settling in the HL Tau disk can all be explained with this model. Since the mass accretion in the HL Tau disk occurs at a high rate, the estimated dust thickness is larger than what is inferred using a conventional viscous disk model (Akiyama et al. 2016; Carrasco-González et al. 2016; Pinte et al. 2016). Hasegawa et al. (2017) suggested that HL Tau disk accretion is driven by the disk turbulence and wind-launching, which then leads to a high accretion rate, whereas the vertical dust scale height is maintained by the turbulence and the dust diffusion. Their model is able to explain the presence of a thin dust layer in HL Tau disk with a high-mass accretion rate.

A key finding of the MHD simulations of a disk with magnetically driven winds is that the diffusion coefficient driving radial angular momentum transport is not necessarily the same as the coefficient mediating the vertically diffusion of dust particles (Zhu et al. 2015). Most previous analytical works about pebble production and its delivery assumed that these coefficients are the same (Lambrechts & Johansen 2014; Ida & Guillot 2016; Ida et al. 2016). Since the growth rate from micron-sized particles to pebbles strongly depends on the turbulent coefficient driving angular momentum transport and the vertical diffusion coefficient of the particles, and given that in a disk with a magnetically driven wind these coefficient are not the same, it is reasonable to expect that the formation and transport of pebbles are affected in the presence of winds.

In this paper, we extend the work of LJ2014 on the dynamics of pebbles in a PPD by considering the effect of magnetically driven winds. In the next section, we first present the steady-state model for an accretion disk with winds. In Section 3, we explore the effect of the wind on pebble formation and their transport through the PPDs. We obtain the radial mass flux of the pebbles and their surface density as functions of time and the initial plasma parameter β₀ at the disk midplane and other model parameters. In Section 4, we investigate the effect of winds on the accretion rate of the pebbles onto a planetary embryo. We discuss the implications of our findings in Section 5.

In most of the previous analytical models for analyzing the delivery of pebbles in PPDs, the gas component is described using power-law functions of the radial distance. For instance, the most widely adopted gaseous model is the minimum mass solar nebula (Hayashi 1981) or a modified version of the standard disk model, which is constructed considering physical properties of the PPDs (e.g., Ida et al. 2016). These solutions are applicable to PPDs without winds. However, it is reasonable to expect that the disk structure is significantly modified due to the mass and angular momentum loss by winds (Suzuki et al. 2016). Our goal is not to present a detailed model for the gas component in the presence of winds, but rather to explore the implications of adopting a semi-analytical model motivated by the results of recent numerical simulations. For instance, it has been shown that the steady-sate disk accretion rate in the presence of the magnetically driven winds can be approximated as seen in Fromang et al. (2013), Suzuki et al. (2016):

$$\begin{eqnarray}&&\dot{M}=\displaystyle \frac{2\pi {{\rm{\Sigma }}}_{{\rm{g}}}{c}_{{\rm{s}}}^{2}}{{{\rm{\Omega }}}_{{\rm{K}}}}\left({W}_{r\phi }+\displaystyle \frac{2r}{\sqrt{\pi }{H}_{{\rm{g}}}}{W}_{z\phi }\right),\end{eqnarray} \tag{ 1 }$$

where W_rϕ and W_zϕ are the normalized accretion stresses in the radial and the vertical directions, defined by Suzuki et al. (2016) and Hasegawa et al. (2017) as

$$\begin{eqnarray}&&{W}_{r\phi }=\displaystyle \frac{{\int }_{-H}^{H}(\overline{\rho {u}_{{\rm{r}}}\delta {u}_{\phi }}-\overline{{B}_{{\rm{r}}}{B}_{\phi }}/4\pi ){dz}}{{{\rm{\Sigma }}}_{{\rm{g}}}{c}_{{\rm{s}}}^{2}},\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&{W}_{z\phi }=\displaystyle \frac{{(\overline{\rho {u}_{{\rm{z}}}\delta {u}_{\phi }}-\overline{{B}_{{\rm{z}}}{B}_{\phi }}/4\pi )}_{z=-H}^{z=H}}{2{\rho }_{0{\rm{g}}}{c}_{{\rm{s}}}^{2}}.\end{eqnarray} \tag{ 3 }$$

Here, u_i and B_i are the ith component of the gas velocity and the magnetic field, respectively, Σ_g is the gas surface density, c_s is the sound speed, and ρ_0g is the gas density at the disk midplane. Furthermore, the disk height of the wind base is represented by H. The overbar denotes time-averaged quantities. Thus, W_rϕ represents the radial angular momentum transport via MHD turbulence, whereas W_zϕ can be regarded as the stress exerted by the wind, which quantifies the angular momentum transport by the wind. The thickness of the gas component is Σ_g and H_g. The disk is rotating with a Keplerian profile, i.e., ${{\rm{\Omega }}}_{{\rm{K}}}\,=\sqrt{{{GM}}_{\star }/{r}^{3}}$, where r is the radial distance and M_⋆ denotes the mass of the central star.

Using the above relations in the time-dependent case and considering more detailed physical processes, Suzuki et al. (2016) did an extensive analysis of the PPD evolution in the presence of magnetically driven winds and found a large variety of surface density profiles (also see Bai et al. 2016). Although they appreciated the role of the wind in the enhancement of the dust-to-gas ratio, they did not focus on understanding the role of the winds in dust transport or pebble production in a PPD.

The MHD shearing-box simulations show that the accretion stresses W_rϕ and W_zϕ can be fitted by the following relations (Simon et al. 2013):

$$\begin{eqnarray}&&\mathrm{log}{W}_{r\phi }=-2.2+0.5{\tan }^{-1}\left(\displaystyle \frac{4.4-\mathrm{log}{\beta }_{0}}{0.5}\right),\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&\mathrm{log}{W}_{z\phi }=1.25-\mathrm{log}{\beta }_{0},\end{eqnarray} \tag{ 5 }$$

where β₀ is the plasma parameter defined as ${\beta }_{0}=8\pi {P}_{\mathrm{mid}}/{B}_{{\rm{z}}}^{2}$, where P_mid is the gas pressure at the midplane of the disk and B_z is the vertical component of the magnetic field. In addition, according to the results of Zhu et al. (2015), the normalized vertical diffusion coefficient of dust particles, α_D = D_z/(c_sH_g), can be approximated by the following relation:

$$\begin{eqnarray}&&\mathrm{log}{\alpha }_{{\rm{D}}}=1.1-\mathrm{log}{\beta }_{0},\end{eqnarray} \tag{ 6 }$$

where D_z represents the vertical diffusion coefficient. We note that the general trend of α_D is closer to the vertical stress coefficient W_zϕ instead of following the radial stress coefficient W_rϕ. Despite various efforts to constrain the radial evolution of the net vertical magnetic field threading the disk (Lubow et al. 1994; Rothstein & Lovelace 2008; Beckwith et al. 2009; Lovelace et al. 2009; Okuzumi et al. 2014; Bai & Stone 2017), there are significant uncertainties about this problem that should be addressed in future works. However, following a previous phenomenological approach (Armitage et al. 2013), we assume that the midplane plasma parameter β₀ is constant.

Using Equation (1), and for a given accretion rate, the surface density becomes

$$\begin{eqnarray}&&{{\rm{\Sigma }}}_{{\rm{g}}}=\displaystyle \frac{\dot{M}{{\rm{\Omega }}}_{{\rm{K}}}}{3\pi {c}_{{\rm{s}}}^{2}}\displaystyle \frac{1}{{\alpha }_{\mathrm{eff}}({\beta }_{0},r)},\end{eqnarray} \tag{ 7 }$$

where the effective turbulent coefficient is defined as

$$\begin{eqnarray}&&{\alpha }_{\mathrm{eff}}=\displaystyle \frac{2}{3}\left({W}_{r\phi }+\displaystyle \frac{2r}{\sqrt{\pi }{H}_{{\rm{g}}}}{W}_{z\phi }\right).\end{eqnarray} \tag{ 8 }$$

With the above definition for the turbulent coefficient, Equation (7) resembles the classical one-dimensional viscous disk model. The accretion rate in a viscous disk model is written as

$$\begin{eqnarray}&&\dot{M}=\displaystyle \frac{6\pi }{r{\rm{\Omega }}}\displaystyle \frac{\partial }{\partial r}({r}^{2}{\rm{\Omega }}\nu {{\rm{\Sigma }}}_{{\rm{g}}}),\end{eqnarray} \tag{ 9 }$$

where ν is the kinematic viscosity. Various mechanisms can be proposed for generating a disk turbulence, however, the kinematic viscosity is obtained via some further assumptions without discussing the true origin of disk turbulence. It has generally been assumed that the radial-azimuthal component of the stress tensor is proportional to the gas pressure where the constant of proportionality is the dimensionless parameter α_SS (Shakura & Sunyaev 1973). The kinematic viscosity thus can be written as ν = α_SSc_sH_g. Since the dynamics of dust particles is dependent on the disk properties, the turbulent viscosity parameter appears in expressions that describe motion of a dust particle, such as its relative velocity to the background gas or other dust particles. Upon comparing Equations (1) and (9), one can easily show that ${W}_{r\phi }=(3/2)(\nu {\rm{\Omega }}/{c}_{{\rm{s}}}^{2})$ when wind does not exist. Therefore, the conventional viscosity parameter α_SS can be written as ${\alpha }_{\mathrm{SS}}=(2/3){W}_{r\phi }\simeq {W}_{r\phi }$. Using this result, our effective viscosity parameter becomes α_eff = (2/3)W_rϕ ≡ α_SS when angular momentum removal by wind is neglected. The advantage of writing Equation (7) similar to the classical viscous disk model is to include the contribution of wind angular momentum transport using an effective viscosity parameter. The viscosity parameter α_SS is generally treated as a fixed input parameter, but the above formulation shows that once this parameter takes the form seen in Equation (8), the structure of a disk with winds can still be described within the framework of the standard disk model. It is a simplified approach, but we think its simplicity enables us to explore the role of angular momentum transport by wind and disk turbulence using a single parameter. Upon introducing the turbulent viscosity as Equation (8) and replacing it with the usual parameter α_SS into the standard disk equations, one can then investigate disk quantities in the presence of winds.

For a given temperature profile, we can determine the surface density using the above equation. Following most of the previous studies, we also assume that the temperature follows a power-law profile of the radial distance, i.e.,

$$\begin{eqnarray}&&T={T}_{0}{\left(\displaystyle \frac{r}{{r}_{0}}\right)}^{-q},\end{eqnarray} \tag{ 10 }$$

where T₀ = 100 K, r₀ = 1 au, and q = 1/2 (Alexander et al. 2004). Although this temperature profile is not the most general case and a detailed energy balance for the thermodynamics of the disk has not been considered in the present work, its simplicity enables us to investigate the role of magnetically driven winds in pebble delivery without being involved with the complexities due to the thermodynamics of the disk. Figure 1 displays our adopted gas surface density profile for different values of the parameter β₀ and the accretion rate $\dot{M}$. Each curve is marked by the corresponding value of β₀. The solid lines (black) represent disk surface density for the accretion rate ${10}^{-8}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$ and dotted lines (blue) display surface density profiles for ${10}^{-7}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$. For a given accretion rate, as the net magnetic field strength increases (i.e., plasma parameter β₀ becomes smaller), the surface density reduces significantly. The gas surface density profile of LJ2014, i.e., ${{\rm{\Sigma }}}_{{\rm{g}}}=500\,{\rm{g}}\,{\mathrm{cm}}^{-2}{(r/\mathrm{au})}^{-1}$, is shown by a dashed line in Figure 1.

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We note that the presence of the wind is included only via its role of transporting angular momentum but excluding the possibility of wind-driven mass-loss. The accretion rate is thus a fixed given input parameter and does not vary with the distance, due to the wind mass-loss. A more realistic model, which is beyond the scope of the present study, would incorporate angular momentum removal and mass-loss by the wind (Khajenabi et al. 2018). Since the accretion rate in the present disk model is a given value, as a wind becomes stronger, surface density reduces further and this trend is mainly attributable to the angular momentum removal by the wind.

Note that disk surface density Equation (7) was introduced by Hasegawa et al. (2017), and novel developments only become apparent when this solution is implemented to investigate pebble dynamics in the subsequent sections.

Using the presented gaseous disk model, we can investigate the dynamics of the dust particles, including pebbles. We first calculate the growth rate from micron-sized particles to pebbles. Once the growth timescale becomes comparable to the drift timescale, the radial migration of the pebbles becomes efficient. Considering the radial motion of the pebbles, we can obtain the radial mass flux of these particles and their surface density as a function of time.

3.1. Growth Rate of the Dust Particles

The dust surface density is assumed to be a constant fraction of the gas surface density, i.e.,

$$\begin{eqnarray}&&{{\rm{\Sigma }}}_{{\rm{d}}}={Z}_{0}{{\rm{\Sigma }}}_{{\rm{g}}},\end{eqnarray} \tag{ 11 }$$

where Z₀ is the dust-to-gas mass ratio and is assumed to be equal to the typical ISM value, i.e., Z₀ = 0.01 (assumed to be a fixed model parameter). We note, however, that both observational evidence and theoretical arguments suggest that Z₀ is not necessarily a fixed given value (e.g., de Gregorio-Monsalvo et al. 2013; Birnstiel & Andrews 2014).

Using the above gaseous disk model and following the approach of LJ2014, we investigate pebble production and its delivery in the presence of winds. In doing so, the first step is to estimate the growth rate of dust particles. For simplicity, we assume that a dust particle is a sphere with radius R. So, its growth rate is given by Ormel & Cuzzi (2007) as

$$\begin{eqnarray}&&\displaystyle \frac{{dR}}{{dt}}=\displaystyle \frac{1}{4}\displaystyle \frac{{\rho }_{0{\rm{d}}}}{{\rho }_{{\rm{m}}}}{\rm{\Delta }}{v}_{{\rm{t}}},\end{eqnarray} \tag{ 12 }$$

where ρ_0d is the volume density of the particles at the disk midplane and ρ_m is the material density of a particle. A key quantity in the above equation is the relative turbulent velocity between particles, i.e., Δv_t.

3.2. Relative Velocity between Dust Particles

Understanding the processes that determine the relative velocity between dust particles in a turbulent medium is a long-standing problem in the fluid mechanics community (e.g., Meek & Jones 1973). In the astrophysical context, a pioneering model was developed by Voelk et al. (1980) with subsequent developments and refinements to determine the relative velocity of the particles in a turbulent gaseous medium (Markiewicz et al. 1991; Cuzzi & Hogan 2003; Ormel & Cuzzi 2007). Using a Kolmogorov power spectrum, the relative velocity of the particles is obtained in terms of the gas Reynolds number and Stokes number that quantifies the dust-gas coupling strength (Cuzzi & Hogan 2003; Ormel & Cuzzi 2007). The Reynolds number is defined as Re = LV_L/ν_m, where L is the largest spatial scale, V_L is the associated velocity and ν_m is the molecular viscosity. Since available relations for the relative dust velocity depend on the Reynolds number, it is important to have reasonable estimates for Re. One approach for determining V_L is based on the α−formalism, in which the angular momentum is transported only due to a turbulent viscosity ν_T = αc_sH_g. Under theses circumstances, we have ${V}_{L}={c}_{{\rm{s}}}\sqrt{\alpha }{({{\rm{\Omega }}}_{L}/{{\rm{\Omega }}}_{{\rm{K}}})}^{1/2}$ and $L={H}_{{\rm{g}}}\sqrt{\alpha }{({{\rm{\Omega }}}_{{\rm{K}}}/{{\rm{\Omega }}}_{L})}^{1/2}$, where Ω_L is the large-eddy frequency (see Equations (1) and (2) in Cuzzi et al. 2001). Defining the Stokes number as St = (ρ_mR/ρ_0gc_s)Ω_K (Epstein regime) and assuming that Ω_L ≃ Ω_K, the relative velocity in the α−disk framework becomes Ormel & Cuzzi (2007)

$$\begin{eqnarray}&&{\rm{\Delta }}{v}_{{\rm{t}}}={(3\alpha \mathrm{St})}^{1/2}{c}_{{\rm{s}}}.\end{eqnarray} \tag{ 13 }$$

In obtaining Equation (13), we used the α−formalism for estimating Re. There is, however, another approach to determining V_L using the turbulent kinetic energy per unit mass k and an associated α_k. Cuzzi et al. (2001) provided a detailed comparative study between the two approaches above for estimating Re. Following the second approach, they showed that without imposing any restriction on Ω_K and Ω_L, the Reynolds number becomes Re = α_kc_sH_g/ν_m. We therefore can still use Equation (13) by simply substituting α = α_k. Invoking energy balance between the energy released due to the accretion and the turbulent kinetic energy dissipated, Cuzzi et al. (2001) showed that

$$\begin{eqnarray}&&{\alpha }_{k}=\displaystyle \frac{3\dot{M}{\xi }_{T}}{2\pi {c}_{{\rm{s}}}{H}_{{\rm{g}}}{{\rm{\Sigma }}}_{{\rm{g}}}},\end{eqnarray} \tag{ 14 }$$

where ξ_T < 1 is the turbulent energy conversion efficiency. Using Equation (7) for the disk surface density and Equation (14), we obtain α_k = (9/2)α_effξ_T ≃ α_eff. This suggests that we can substitute α ≡ α_eff into Equation (13) to obtain the relative velocity between dust particles in a PPD with magnetic winds.

Note that in order to relate α−models with the underlying physical processes driving the turbulence, it is necessary to rely on numerical simulations. In the context of the shearing box, the α parameter can be obtained in the saturated state in turbulent disks subject to the MRI or gravitational instability. Having said this, however, most previous studies of pebble dynamics, including LJ2014, relied on the α−formalism for describing disk turbulence. Magnetically driven winds, on the other hand, contribute to the disk turbulence and the resulting mass accretion rate. Our disk model encapsulates this key feature too.

We provided some physical motivations for introducing α_eff to incorporate disk-wind effects within the framework of the α−formalism; however, available non-ideal MHD disk simulations with magnetically driven winds are useful for addressing this issue. Simon et al. (2013), for instance, investigated a portion of a PPD threaded by a net vertical magnetic field using a set of shearing-box simulations. They found that there is a strong correlation between the strength of the imposed vertical magnetic field and the generated disk turbulence. For a sufficiently strong field, however, the main accretion-driving mechanism was found to be due to wind-launching and not MRI turbulence. They also reported the associated viscosity coefficient using simulations in their Table 1, which is in good agreement with our effective viscosity coefficient α_eff. Simon et al. (2013) found that the viscosity coefficient due to the combined effects of MRI turbulence and wind-launching is α ≃ 0.077, 0.03, and 0.005 for ${\beta }_{0}={10}^{3},{10}^{4}$, and 10⁵, respectively (see their models AD30AU1e3−AD30AU1e5). In Figure 2, we present the radial profile of α_eff for different values of β₀. Although the spatial dependence of α_eff cannot be contrasted directly with these shearing-box simulations, our effective viscosity parameter at 100 au is α_eff ≃ 0.1, 0.03, and 0.004 for ${\beta }_{0}={10}^{3},{10}^{4}$, and 10⁵, respectively.

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In another approach, Zhu et al. (2015) studied ideal and non-ideal MRI-driven turbulence using global and local simulations. They also found that the emergence of winds is a consequence of considering non-ideal terms, including ambipolar diffusion (see also Gressel et al. 2015). They found that disk turbulence is strongly suppressed when ambipolar diffusion is included and it has a longer correlation time for the vertical velocity. They also calculated the power spectrum characterizing the turbulence in both the ideal and the non-ideal MHD cases (see their Figure 9) and the corresponding dust diffusion coefficients (see their Figure 10). The resulting radial and vertical dust diffusion coefficients in the ideal MHD runs were found to be in good agreement with the analytical relation of Youdin & Lithwick (2007), who computed these coefficients in a gaseous background with a power spectrum similar to Kolmogorov turbulence. In the non-ideal runs, however, Zhu et al. (2015) found that the dust diffusion coefficients for the particles with St ≳ 1 exhibit noticeable discrepancies with the analytical calculations (Youdin & Lithwick 2007). Furthermore, Zhu et al. (2015) also computed the viscosity coefficient based on their non-ideal MHD simulations (see their Table 3). Although the value of the viscosity coefficient according to their study is smaller than ours, we suggested that the relative velocity Δv_t can be calculated in terms of the effective viscosity coefficient using Equation (13).

3.3. Growth Time of the Dust Particles

Youdin & Lithwick (2007) found a relation between the thickness of the gaseous disk, H_g, and the thickness of the dust component, H_d, which can be written as

$$\begin{eqnarray}&&{H}_{{\rm{d}}}\simeq {H}_{{\rm{g}}}\sqrt{\displaystyle \frac{{\alpha }_{{\rm{D}}}}{\mathrm{St}}}.\end{eqnarray} \tag{ 15 }$$

This relation is valid as long as the particles are well-coupled to the gas (St < 1).

If we define the growth timescale of a dust particle as ${t}_{{\rm{g}}}=R/\dot{R}$, then it becomes

$$\begin{eqnarray}&&{t}_{{\rm{g}}}=\displaystyle \frac{4}{\sqrt{3}{\epsilon }_{{\rm{g}}}Z{{\rm{\Omega }}}_{{\rm{K}}}}{\left[\displaystyle \frac{{\alpha }_{{\rm{D}}}({\beta }_{0})}{{\alpha }_{\mathrm{eff}}({\beta }_{0},r)}\right]}^{\tfrac{1}{2}},\end{eqnarray} \tag{ 16 }$$

where the parameter _g is introduced to control the growth efficiency of the particles. Using Equation (16) for the growth rate of a pebble, one can easily obtain the time t needed for a particle to grow from its initial size (R_initial) to its final size (R_drift). Thus, following LJ2014, we obtain

$$\begin{eqnarray}&&t=\displaystyle \frac{4}{\sqrt{3}{\epsilon }_{{\rm{d}}}{Z}_{0}{{\rm{\Omega }}}_{{\rm{K}}}}{\left[\displaystyle \frac{{\alpha }_{{\rm{D}}}({\beta }_{0})}{{\alpha }_{\mathrm{eff}}({\beta }_{0},r)}\right]}^{\tfrac{1}{2}},\end{eqnarray} \tag{ 17 }$$

where

$$\begin{eqnarray}&&{\epsilon }_{{\rm{d}}}={\epsilon }_{{\rm{g}}}{\left[\mathrm{ln}\left(\displaystyle \frac{{R}_{\mathrm{drift}}}{{R}_{\mathrm{initial}}}\right)\right]}^{-1},\end{eqnarray} \tag{ 18 }$$

and we have ln (R_drift/R_initial) ≃ 10. In our model, the dust sticking efficiency is set to _g = 0.5.

3.4. Dominant Size of the Pebbles

After time t the growth timescale becomes comparable to the drift timescale and a pebble forms and starts its journey toward the central star. The radial drift velocity of a particle is given by Weidenschilling (1977) and Nakagawa et al. (1986) as

$$\begin{eqnarray}&&{v}_{{\rm{r}}}=-2\displaystyle \frac{\mathrm{St}}{{\mathrm{St}}^{2}+1}\eta {v}_{{\rm{K}}},\end{eqnarray} \tag{ 19 }$$

where v_K = rΩ_K is the Keplerian velocity and η is given by

$$\begin{eqnarray}&&\eta =-\displaystyle \frac{1}{2}{\left(\displaystyle \frac{{H}_{{\rm{g}}}}{r}\right)}^{2}\displaystyle \frac{d\mathrm{ln}P}{d\mathrm{ln}r}.\end{eqnarray} \tag{ 20 }$$

Therefore, the radial drift timescale is defined as t_drift = r/v_r. If we set the growth timescale equal to the radial drift timescale, i.e., t_g = t_drift, in the case with St ≲ 1, the dominant size of the pebbles is obtained as follows:

$$\begin{eqnarray}&&{\mathrm{St}}_{{\rm{p}}}=\displaystyle \frac{\sqrt{3}}{8}\displaystyle \frac{{\epsilon }_{{\rm{p}}}}{\eta }\displaystyle \frac{{{\rm{\Sigma }}}_{{\rm{p}}}}{{{\rm{\Sigma }}}_{{\rm{g}}}}{\left[\displaystyle \frac{{\alpha }_{\mathrm{eff}}({\beta }_{0},r)}{{\alpha }_{{\rm{D}}}({\beta }_{0})}\right]}^{\tfrac{1}{2}},\end{eqnarray} \tag{ 21 }$$

where Σ_p is the pebble surface density, and _p is the coagulation efficiency between pebbles, following LJ2014, we assume _p = _g = 0.5. Note that Equation (21) is obtained using disk quantities with smooth power-law profiles. Thus, one cannot, for example, have pressure bumps trapping particles in the disk, or the drift-growth equilibrium breaks down. Particle-trapping, however, may occur at certain locations in a disk with magnetic winds when angular momentum removal and mass-loss are properly considered (Suzuki et al. 2016).

3.5. Pebble Production Front

Our analysis clearly demonstrates that not only the growth rate but also the dominant size of the pebbles depend on the ratio of the radial and the vertical turbulent coefficients. If we set these coefficients as equal to each other, then our equations reduce to the LJ2014 analysis. However, as we pointed out above, numerical simulations of disks with magnetically driven winds suggest that these coefficients are not equal. A few previous studies have also mentioned that the turbulent coefficient that is used for estimating the relative velocity of the particles is not necessarily equal to the global turbulent coefficient for the viscous evolution of the disk (Carrera et al. 2017). This theoretical argument, however, to the best of our knowledge, has not been implemented in the analytical models for pebble production to explore the possible consequences.

Since the ratio α_D/α_eff plays a key role in our model, we show the radial profiles of α_eff and α_D/α_eff for various values of plasma parameter β₀ in Figure 2. Using Equations (4) and (5), the accretion stresses W_rϕ and W_zϕ are obtained for a given β₀, and thereby, the effective turbulent coefficient α_eff is obtained from Equation (8). The top panel of Figure 2 displays not only the effective viscosity coefficient using Equation (8) as solid curves but also contributions of its first term (dotted, blue) and the second term (dashed, red) are shown to have a better insight on the dominant physical mechanism in transporting angular momentum for a given β₀. As the magnetic field increases (lower β₀), the contribution of the radial turbulence reduces compared to the angular momentum transport by the winds. For ${\beta }_{0}={10}^{3}$ the dominant angular momentum transport is provided by the winds, whereas for ${\beta }_{0}={10}^{5}$, radial angular momentum transport is most efficient.

The bottom panel of Figure 2 depicts the radial profile of the ratio α_D/α_eff, which is always less than unity for the allowed range of plasma parameter β₀; however, the ratio increases as β₀ decreases. This typical trend implies that in a disk with magnetically driven winds, vertical turbulent diffusion is weaker than the radial turbulent coefficient. But as the winds get stronger (lower β₀), the enhancement of α_D is larger than the enhancement of α_eff, implying a higher value for the ratio α_D/α_eff. Furthermore, Equation (16) implies that the growth timescale is longer than a case with a weak magnetic wind (i.e., smaller β₀). This general trend eventually affects the location of the pebble production front.

If we set α_D = α_eff, then Equation (16) reduces to Equation (10) of LJ2014, which gives the pebble production line, r_g, as a function of time:

$$\begin{eqnarray}&&{r}_{{\rm{g}}}(t)=122.7\,\mathrm{au}{\left(\displaystyle \frac{{M}_{\star }}{{M}_{\odot }}\right)}^{1/2}{\left(\displaystyle \frac{t}{{10}^{6}\mathrm{yr}}\right)}^{2/3}.\end{eqnarray} \tag{ 22 }$$

In the presence of winds, Equation (17) can be solved numerically to obtain the time evolution of r_g(t). In order to do so, it is convenient to introduce the following dimensionless quantities:

$$\begin{eqnarray}&&x=\displaystyle \frac{r}{{r}_{0}},{M}_{1}=\displaystyle \frac{{M}_{\star }}{{M}_{\odot }},{\dot{M}}_{1}=\displaystyle \frac{\dot{M}}{{\dot{M}}_{0}},{t}_{6}=\displaystyle \frac{t}{{10}^{6}\,\mathrm{yr}},\end{eqnarray} \tag{ 23 }$$

where we take r₀ = 1 au and ${\dot{M}}_{0}={10}^{-8}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$. Thus, we can rewrite Equation (8) as

$$\begin{eqnarray}&&{\alpha }_{\mathrm{eff}}({\beta }_{0},r)=\displaystyle \frac{2}{3}[{W}_{r\phi }({\beta }_{0})+{\lambda }_{0}\,{M}_{1}^{1/2}{x}^{(q-1)/2}{W}_{z\phi }({\beta }_{0})],\end{eqnarray} \tag{ 24 }$$

where the dimensionless parameter λ₀ depends on the reference values as ${\lambda }_{0}=(2/\sqrt{\pi })({r}_{0}{{\rm{\Omega }}}_{0})/{c}_{0{\rm{s}}}\simeq 53.6$. Here, we have ${{\rm{\Omega }}}_{0}=\sqrt{{{GM}}_{\odot }/{r}_{0}^{3}}=2\times {10}^{-7}$ s⁻¹ and ${c}_{0{\rm{s}}}\,=\sqrt{{k}_{{\rm{B}}}{T}_{0}/\mu {m}_{{\rm{H}}}}=627.3$ m s⁻¹, where k_B is the Boltzmann constant, μ = 2.1 is the mean molecular weight and m_H is the mass of hydrogen. We also write the gas surface density Equation (1) as follows:

$$\begin{eqnarray}&&{{\rm{\Sigma }}}_{{\rm{g}}}={{\rm{\Sigma }}}_{0}\displaystyle \frac{{\dot{M}}_{1}\,{M}_{1}^{1/2}{x}^{q-\tfrac{3}{2}}}{{W}_{r\phi }({\beta }_{0})+{\lambda }_{0}\,{M}_{1}^{1/2}{x}^{(q-1)/2}{W}_{z\phi }({\beta }_{0})},\end{eqnarray} \tag{ 25 }$$

where ${{\rm{\Sigma }}}_{0}={\dot{M}}_{0}{{\rm{\Omega }}}_{0}/2\pi {c}_{0{\rm{s}}}^{2}\simeq 5\,{\rm{g}}\,{\mathrm{cm}}^{-2}$. With these definitions, the location of the pebble production line x_g as a function of time is obtained from the dimensionless form of Equation (17), i.e.,

$$\begin{eqnarray}&&{t}_{6}=\displaystyle \frac{9\times {10}^{-4}}{{M}_{1}^{1/2}}{x}_{{\rm{g}}}^{3/2}{\left[\displaystyle \frac{{\alpha }_{{\rm{D}}}({\beta }_{0})}{{W}_{r\phi }({\beta }_{0})+{\lambda }_{0}{M}_{1}^{1/2}{x}_{{\rm{g}}}^{(q-1)/2}{W}_{z\phi }({\beta }_{0})}\right]}^{\tfrac{1}{2}}.\end{eqnarray} \tag{ 26 }$$

Figure 3 displays the time evolution of r_g(t) for different values of the parameter β₀. It shows that the pebble formation front spreads outward, with an expansion rate that strongly depends on the adopted value for β₀. As the winds become stronger, the outward propagation of the pebble formation line becomes slower. For comparison, the evolution of r_g(t) according to LJ2014's model is shown by a dashed curve (red). We emphasize that in a weak field case, say, ${\beta }_{0}={10}^{5}$, our time evolution of pebble production front does not converge to the LJ2014 solution. This is simply because the growth rate depends on the ratio α_D/α_eff, which does not tend to unity even in a weak magnetic configuration, whereas LJ2014 assumed this ratio to be always equal to one. Furthermore, our density profile is different from the adopted LJ2014 density profile. However, our treatment enables us to explore the role of magnetic winds in the outward movement of the pebble production line. Slower outward motion of r_g(t) in the presence of the winds is understandable in terms of the growth timescale for pebble formation. We showed that for a particle with a given initial small size, it takes a longer time for the particle to reach the size at which drift is efficient in the presence of a wind. As the wind becomes stronger, this timescale becomes even longer, thus the pebble formation line spreads at a slower rate. Consequently, Figure 3 shows that the pebble formation line reaches a given radial distance at a much later time as the wind becomes stronger.

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3.6. Pebble Accretion Rate

Now we can calculate the radial mass flux of the pebbles as

$$\begin{eqnarray}&&{\dot{M}}_{{ \mathcal P }}=2\pi {r}_{{\rm{g}}}\displaystyle \frac{{{dr}}_{{\rm{g}}}}{{dt}}{{\rm{\Sigma }}}_{{\rm{d}}}({r}_{{\rm{g}}}),\end{eqnarray} \tag{ 27 }$$

where, upon substituting Equation (25) into the above equation, the radial mass flux of the pebbles is given by

$$\begin{eqnarray}{\dot{M}}_{{ \mathcal P }} & = & (1.2\times {10}^{-8}\,{M}_{\oplus }\,{\mathrm{yr}}^{-1})\displaystyle \frac{{{dx}}_{{\rm{g}}}}{{{dt}}_{6}}\\ & & \times \ \left[\displaystyle \frac{{\dot{M}}_{1}\,{M}_{1}^{1/2}{x}_{{\rm{g}}}^{q-(1/2)}}{{W}_{r\phi }({\beta }_{0})+{\lambda }_{0}{M}_{1}^{1/2}{x}_{{\rm{g}}}^{(q-1)/2}{W}_{z\phi }({\beta }_{0})}\right].\end{eqnarray} \tag{ 28 }$$

Using Equation (26), we finally achieve the following relation for the radial pebble flux, i.e.,

$$\begin{eqnarray}{\dot{M}}_{{ \mathcal P }} & = & (8.8\times {10}^{-6}\,{M}_{\oplus }\,{\mathrm{yr}}^{-1}){\dot{M}}_{1}\,{M}_{1}^{3/2}{[{\alpha }_{{\rm{D}}}({\beta }_{0})]}^{-1/2}{x}_{{\rm{g}}}^{q-1}\\ & & \times \ \left\{\displaystyle \frac{{[{W}_{r\phi }({\beta }_{0})+{\lambda }_{0}{M}_{1}^{1/2}{x}_{{\rm{g}}}^{(q-1)/2}{W}_{z\phi }({\beta }_{0})]}^{1/2}}{{W}_{r\phi }({\beta }_{0})+(\tfrac{7-q}{6}){\lambda }_{0}{M}_{1}^{1/2}{x}_{{\rm{g}}}^{(q-1)/2}{W}_{z\phi }({\beta }_{0})}\right\}.\end{eqnarray} \tag{ 29 }$$

The top panel of Figure 4 shows the evolution of the radial flux of pebbles for different values of the parameter β₀ and a fixed accretion rate $\dot{M}={10}^{-8}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$. The central mass is M_⋆ = M_⊙. This figure shows that the time dependence of ${\dot{M}}_{{ \mathcal P }}$ can be described as a power-law function, i.e., ${\dot{M}}_{{ \mathcal P }}\propto {t}^{\xi }$, where the exponent ξ is labeled on the right side. To put it more precisely, the stronger the wind, the more accretion rate of the pebbles is drastically reduced; however, the exponent ξ has a negligible dependence on the changes of parameter β₀. However, the time dependence of the pebble accretion rate becomes slightly weaker as the winds gets stronger. For comparison, the radial pebble flux based on the LJ2014 model is presented as a dashed line (red). As noted earlier, due to the differences in the radial and vertical turbulent coefficients, our model with a weak magnetic wind does not reduce to the LJ2014 pebble accretion rate. However, our results show that the presence of the magnetically driven winds in a PPD can significantly reduce the radial mass flux of the pebbles.

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In the bottom panel of Figure 4, we explore the role of the gas accretion rate in the evolution of the radial flux of pebbles. Our adopted disk surface density distribution is directly proportional to the gas accretion rate. A higher accretion rate therefore leads to a larger gas surface density at a given radius. Here, the plasma parameter is fixed, i.e., ${\beta }_{0}={10}^{4}$, whereas in addition to our nominal accretion rate, i.e., ${10}^{-8}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$, a disk with a higher accretion rate, ${10}^{-7}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$, is also considered. The pebble radial flux increases with the accretion rate due to enhanced gas surface density. Although the disk surface density is comparable to the LJ2014 profile for an accretion rate ${10}^{-7}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$, the pebble radial flux is not the same as that in LJ2014. This trend can be explained in terms of the effective viscosity and the vertical dust diffusion coefficients, which are not equal in our model.

3.7. Pebble Surface Density

We now calculate the surface density of the pebbles. From Equations (19) and (21), the following radial velocity can be obtained:

$$\begin{eqnarray}&&{v}_{{\rm{r}}}=\displaystyle \frac{\sqrt{3}}{4}{\epsilon }_{{\rm{p}}}\displaystyle \frac{{{\rm{\Sigma }}}_{{\rm{p}}}}{{{\rm{\Sigma }}}_{{\rm{g}}}}{\left[\displaystyle \frac{{\alpha }_{{\rm{D}}}({\beta }_{0})}{{\alpha }_{\mathrm{eff}}({\beta }_{0},r)}\right]}^{\tfrac{1}{2}}{v}_{{\rm{K}}},\end{eqnarray} \tag{ 30 }$$

where Σ_p is the surface density of the pebbles and we assumed that St < 1. Therefore, the mass conservation of the pebbles gives the surface density Σ_p as

$$\begin{eqnarray}&&{{\rm{\Sigma }}}_{{\rm{P}}}={\left(\displaystyle \frac{2{\dot{M}}_{{ \mathcal P }}{{\rm{\Sigma }}}_{{\rm{g}}}}{\sqrt{3}\pi {\epsilon }_{{\rm{p}}}{{rv}}_{{\rm{K}}}}\right)}^{\tfrac{1}{2}}{\left[\displaystyle \frac{{\alpha }_{{\rm{D}}}({\beta }_{0})}{{\alpha }_{\mathrm{eff}}(r,{\beta }_{0})}\right]}^{\tfrac{1}{4}},\end{eqnarray} \tag{ 31 }$$

where we can substitute the pebble accretion rate from Equation (29) and the gas surface density from Equation (25) into the above equation. We do not write here the final result of these replacements, which are tedious. However, the profile of the pebble surface density evolution is explored in Figure 5. Two extreme configurations are considered: a disk with a strong wind (i.e., ${\beta }_{0}={10}^{3}$, solid curves, black) and a disk with a weak wind (i.e., ${\beta }_{0}={10}^{5}$, dashed curves, red). Each curve is marked with its elapsed time t₆. We note that pebble surface density is shown beyond the pebble production line and the curves are limited to the pebble production line. Although we did not consider the reduction in mass through the wind, this figure shows that as the power of the wind increases, the surface density of the pebbles is slightly increased compared to a disk with a weak wind. LJ2014 found the following equation for the ratio of pebble and gas surface densities, i.e.,

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{{\rm{\Sigma }}}_{{\rm{p}}}}{{\rm{\Sigma }}}\right)}_{\mathrm{LJ}2014}=7.8\times {10}^{-4}{\left(\displaystyle \frac{t}{{10}^{6}\mathrm{yr}}\right)}^{-\tfrac{1}{6}}{\left(\displaystyle \frac{r}{\mathrm{au}}\right)}^{\tfrac{1}{4}}.\end{eqnarray} \tag{ 32 }$$

This ratio of the surface densities is smaller than what we have found by a factor of 10.

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Having obtained the surface density and the radial mass flux of the pebbles, we are now able to calculate the mass accretion rate onto the core of a planetary embryo with mass M_c. In the case of St < 1, the accretion rate ${\dot{M}}_{{\rm{c}}}$ can be written as Lambrechts & Johansen (2012),

$$\begin{eqnarray}&&{\dot{M}}_{{\rm{c}}}=2{\left(\displaystyle \frac{\mathrm{St}}{0.1}\right)}^{\tfrac{2}{3}}{r}_{{\rm{H}}}{v}_{{\rm{H}}}{{\rm{\Sigma }}}_{{\rm{p}}},\end{eqnarray} \tag{ 33 }$$

where ${r}_{{\rm{H}}}=r{({M}_{{\rm{c}}}/3{M}_{\star })}^{1/3}$ is the Hill radius, and we have v_H = r_HΩ_K. Using Equations (20) and (21), the above equation becomes

$$\begin{eqnarray}{\dot{M}}_{{\rm{c}}} & = & 5.79\times {10}^{-2}{\left(\displaystyle \frac{{M}_{\star }}{{M}_{\odot }}\right)}^{\tfrac{1}{12}}{\left(\displaystyle \frac{r}{\mathrm{au}}\right)}^{\tfrac{8q-7}{12}}{\left(\displaystyle \frac{{{\rm{\Sigma }}}_{{\rm{g}}}}{100{\rm{g}}{\mathrm{cm}}^{-2}}\right)}^{\tfrac{1}{6}}\\ & & \times \ {\left(\displaystyle \frac{{\alpha }_{\mathrm{eff}}}{{\alpha }_{{\rm{D}}}}\right)}^{-\tfrac{1}{12}}{\left(\displaystyle \frac{{M}_{{\rm{c}}}}{{M}_{\oplus }}\right)}^{\tfrac{2}{3}}{\left(\displaystyle \frac{{\dot{M}}_{{ \mathcal P }}}{{M}_{\oplus }{\mathrm{yr}}^{-1}}\right)}^{\tfrac{5}{6}}\hspace{0.2cm}\displaystyle \frac{{M}_{\oplus }}{\mathrm{yr}}.\end{eqnarray} \tag{ 34 }$$

If we substitute for the radial mass flux of the pebbles from Equation (29) and the pebble surface density from Equation (31), then the growth rate of a core is obtained as a function of time and the remaining model parameters, which can be used to examine the evolution of a core mass due to pebble accretion.

Equation (34) is in fact a first-order differential equation in terms of the mass of a core that can be easily integrated to obtain the evolution of the core mass. With some care we see that the time dependence of the right side of this equation appears only through the mass accretion rate ${\dot{M}}_{{ \mathcal P }}(t)$. We have already shown that the time dependence of the pebble radial mass flux can be approximated by a power-law relation, i.e., ${\dot{M}}_{{ \mathcal P }}(t)\propto {t}^{\xi }$, where the constant of the proportionality strongly depends on the parameter β₀, and the exponent ξ has a relatively weak dependence on the plasma parameter. Using Equation (34) we thus find the following relation:

$$\begin{eqnarray}&&{M}_{{\rm{c}}}\simeq 0.1{\left(\displaystyle \frac{{\beta }_{0}}{1000}\right)}^{\tfrac{11}{4}}{\left(\displaystyle \frac{r}{\mathrm{au}}\right)}^{-1}{\left(\displaystyle \frac{t}{{10}^{6}\mathrm{yr}}\right)}^{3+2.5\xi }\,{M}_{\oplus },\end{eqnarray} \tag{ 35 }$$

where the exponent (3 + 2.5ξ) ≃ 2.3 has a relatively weak dependence with the plasma parameter. However, as shown in Figure 6, the growth rate of a core mass is strongly dependent on the parameter β₀.

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The top panel of Figure 6 shows the mass of a growing core as a function of time for different values of plasma parameter, a fixed accretion rate, $\dot{M}={10}^{-8}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$, and various locations throughout the disk. The numbers labeling each curve are the radial distance considered in au. It is clearly seen that the mass of the core increases with time, although the growth rate depends on the wind strength. Lambrechts et al. (2014) showed that when the mass of a growing core reaches a critical mass M_cr, then the accretion of the pebbles is halted. They estimated the critical mass as a function of the radial distance: ${M}_{\mathrm{cr}}\simeq 20{(r/5\mathrm{au})}^{3/4}\,{M}_{\oplus }$ (Lambrechts et al. 2014). More disk parameters, however, have been incorporated in the most recent estimates of the critical mass (Ataiee et al. 2018; Bitsch et al. 2018). While Bitsch et al. (2018) generally confirmed the findings of Lambrechts et al. (2014), the results of Ataiee et al. (2018) suggest that the pebble critical mass depends on the disk turbulence strength. Our estimate of the critical mass, however, is based on the proposed relation by Lambrechts et al. (2014). In Figure 6, we identified the time at which the mass of a core reaches its critical mass by the blue squares, and the continuation of the growth process by ignoring this halt in accretion is extrapolated by the dashed curves (red). When the wind is strong, it can be seen that over the course of 10 million years, core growth due to pebble accretion occurs so slowly that it cannot reach its critical mass. But with decreasing wind power, the mass of a growing core reaches a critical mass over a shorter time period.

Although our nominal value for the accretion rate is $\dot{M}={10}^{-8}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$, we also displayed a gas surface density profile for $\dot{M}={10}^{-7}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$ in Figure 1. Since our gas density profile is directly proportional to the gas accretion rate, the disk surface density increases with the accretion rate for a fixed plasma parameter. The gas surface density dependence of the pebble dynamics has been studied by LJ2014, as illustrated in their Figure 5. They found that the pebble accretion rate strongly decreases with a slight reduction in the disk surface density. We can now explore whether a reduced pebble accretion rate in a strongly magnetized disk is a consequence of having a very low surface density. In Figure 4, we explored the role of the accretion rate in the pebble radial flux in a magnetized disk with a gas surface density comparable to that adopted in LJ2014.

We have shown that our disk model with ${\beta }_{0}={10}^{5}$ and $\dot{M}={10}^{-8}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$ is comparable to LJ2014's density profile; however, pebble growth in this weakly magnetized configuration, as discussed above, is not necessarily similar to that of LJ2014. This can be easily understood in terms of the coefficients α_eff and α_D, which are not comparable even in the weakly magnetized disks. This parameter study is generalized to investigate pebble growth in a disk with a stronger magnetic field but comparable gas surface density as in LJ2014. The surface density of a strongly magnetized disk with ${\beta }_{0}={10}^{3}$, however, becomes comparable to that of LJ2014 if the accretion rate is adopted around ${10}^{-6}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$, which is not consistent with observed values. We therefore consider a case with ${\beta }_{0}={10}^{4}$ and $\dot{M}={10}^{-7}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$, which leads to a gas surface density comparable to that of LJ2014. In the bottom panel of Figure 6, the plasma parameter is ${\beta }_{0}={10}^{4}$, but different accretion rates are considered. This figure shows that the growth of a core is enhanced when a higher accretion rate is adopted.

When a core reaches its pebble isolation mass, the gas is able to be accreted onto the core and form an extensive gaseous envelope. Considering our finding that in the presence of a strong wind, the mass of a core hardly reaches its critical value, it is unlikely that the formation of giant planets with extensive gas envelopes under these conditions is possible, at least within the pebble accretion scenario. The final fate of such cores will be ice planets that contain little quantities of gas in their envelopes. But in a PPD with a weak wind, a growing core is able to reach its isolation mass and eventually create giant planets. These theoretical speculations depend on other physical processes, such as the duration of the wind-launching and the radial extent of the regions with wind in a PPD. More sophisticated models, however, are needed to address these complicated issues.

We investigated pebble growth and its transport in a PPD with magnetically driven winds using a simplified steady-state model for the gas component. Using the results of previous numerical simulations about the behavior of the radial accretion stress and the vertical dust diffusion (Simon et al. 2013; Zhu et al. 2015), we found that the growth rate of micron-sized particles to pebbles strongly depends on the ratio α_D/α_eff, which in our model is a function of the midplane plasma parameter β₀ and the radial distance.

In our implemented magnetized gaseous disk model, even in a case with weak magnetic winds, contrary to most of the previous (semi)analytical works in this context, this ratio does not tend to unity. The turbulence in a PPD is highly anisotropic and it is unlikely that turbulence will play the same role in the dynamics of particles in the vertical and radial directions. Transport of the dust particles in a turbulent medium, like a PPD, is an interesting topic and has attracted considerable attention in recent years (e.g., Turner et al. 2010). Although the dust sub-layer can be subject to different kinds of instabilities, such as streaming instability (Youdin & Goodman 2005; Johansen & Youdin 2007), to maintain its thickness, the vertical transport of dust particles is commonly described using the diffusion approximation (Voelk et al. 1980; Cuzzi et al. 1993; Dullemond & Dominik 2004). We also checked that the dust layer in a disk with a high plasma parameter is not thinner than the streaming instability supported minimum of H_d/H_g ∼ 0.01 (Yang & Johansen 2014; Yang et al. 2017).

Under the steady-state approximation and some simplifying assumptions (Youdin & Lithwick 2007), the ratio of the thickness between a dusty disk and gaseous disk becomes equal to ${({\alpha }_{{\rm{D}}}/\mathrm{St})}^{1/2}$, where St < 1. Youdin & Lithwick (2007) pointed out that the diffusion coefficient α_D is not necessarily equal to the turbulent coefficient α. Numerous numerical simulations have been done over recent years to investigate dust dynamics in PPDs. For instance, Fromang & Nelson (2009) investigated dust settling in a disk with ideal MHD-driven turbulence and found that there is a disagreement between the results of their simulations and simple theoretical expectations, unless the diffusion coefficient varies vertically in proportion to the square of the local turbulent vertical velocity fluctuations. In another effort, Bai & Stone (2010) investigated the dynamics of dust particles in the midplane of a non-magnetized PPD with turbulence driven by the streaming instability. They also found that particle dynamics does not follow simple models that are obtained assuming the equality of the radial and vertical turbulent coefficients.

It is worth noting that our implemented relation for the relative velocity between dust particles Δv_t, i.e., Equation (13), is an essential part of the present model to calculate pebble growth rate and its evolution. This relation has been used by previous pebble production models in PPDs without winds, however, our physical arguments and recent developments in the non-ideal MHD disk simulations provided motivations to use this equation in PPDs with winds. In this case, we suggested that the viscosity coefficient in Equation (13) can be replaced by an effective viscosity coefficient, i.e., Equation (8), which includes a wind contribution as well. However, we are aware of the limitations of our treatment in estimating the relative velocity Δv_t. In our argument to support this replacement, we introduced the parameter ξ_T to denote the efficiency with which the released energy is dissipated by turbulence. Although this parameter should be less than unity, its exact value remains uncertain and it depends on the properties of the turbulence. Furthermore, available relations for the relative velocity between dust particles are presented as integrals where the turbulent energy spectrum is a model input function (e.g., see Equation (12) in Cuzzi & Hogan 2003). The Kolmogorov energy spectrum therefore has generally been adopted to obtain Equation (13) (Ormel & Cuzzi 2007). Non-ideal MHD-induced turbulence, on the other hand, exhibits a power spectrum with noticeable differences in comparison to the ideal MHD simulations, requiring a more realistic approach to compute the relative velocity Δv_t. In principle, the obtained power spectrum based on non-ideal MHD disk simulations (see Figure 9 in Zhu et al. 2015) could be implemented in the integral relations of Cuzzi & Hogan (2003) or Ormel & Cuzzi (2007) to calculate the relative velocity Δv_t. This approach could offer a potential avenue for improving the present model in the future.

Although we did not consider mass-loss due to winds, we found that the pebble surface is slightly enhanced compared to a case with weak winds. In fact, the role of the wind appeared only through changes in the radial and the vertical turbulent coefficients. However, during the period of growth and transport of the pebbles, a non-negligible fraction of the mass of the disk can be lost due to the presence of the winds. More sophisticated models are needed for modeling gaseous disks with magnetic-driven winds. For instance, Bai et al. (2016) presented a global disk model with magnetic winds in which both angular momentum and mass-loss by winds have been considered consistently. Suzuki et al. (2016) investigated the evolution of a PPD with magnetically driven winds within the framework of the standard accretion disk model (Shakura & Sunyaev 1973). The losses of mass and angular momentum by disk winds are incorporated using parameterized relations constrained by MHD simulations and the global energy budget of a disk. Suzuki et al. (2016) showed that the radial profiles of the disk quantities are quite variable, including an inner surface density with a positive radial slope. These theoretical findings that may affect the radial drift of dust particles deserve further studies. In addition, dust particles, especially smaller ones, may be uplifted to the surface layers of a PPD, and be removed from the disk through the wind. Although the details of these mechanisms are not clear (Miyake et al. 2016), it seems necessary to consider this key physical process in future work to improve the present study.

Based on the present study, in which enhancement of the dust-to-gas mass ratio due to the wind was not considered, the presence of the magnetically driven wind could reduce the propagation rate of the pebble production line and thus delay the onset of the core growth. Although it is not clear which parts of a disk are susceptible to the wind-launching, if the wind is assumed to be active over the entire disk, then the efficiency of planetary embryo growth by pebble accretion can be greatly reduced in the presence of winds. It is unlikely that there will be wind throughout the entire disk. Recently, Nolan et al. (2017) investigated steady-state disk-wind solutions by considering all non-ideal effects and found that disk winds occur over a limited radial extend. In addition, the location of the wind-launching region and its size were found to be strongly dependent on the mass accretion rate, magnetic field strength, and disk surface density profile. However, theoretical studies (e.g., Bai & Stone 2013) and observational evidence (e.g., Bjerkeli et al. 2016) have shown that especially in the inner regions of a PPD, such magnetically driven winds can be very active and even operate as the dominant angular momentum transport mechanism. Since the pebble production line spreads outward, at least in the inner region of a PPD, which is the most likely place for wind-launching, there would be a delay in production and transport of pebbles according to the results of our model. Although we implemented a simple model for pebble formation and its delivery, our results at least can show the expected trends of pebble dynamics under wind-launching conditions in a PPD.

As the wind becomes stronger, the pebble production line propagates with a slower rate and the pebble radial mass flux reduces. The rate of core growth reduces in the presence of a strong wind. The plasma parameter plays a vital role in our model, because the wind strength is quantified in terms of this parameter. We found a set of giant planets for ${\beta }_{0}={10}^{4}$ and $\dot{M}={10}^{-8}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$; however, cores formed at a larger orbital distance are expected to be more massive. Observational features of HL Tau, on the other hand, suggest that the preferred value is ${\beta }_{0}={10}^{4}$, as discussed in Hasegawa et al. (2017). Our analysis suggests that for the allowed range of the model parameters, pebble growth mostly leads to super-Earths. Current observational studies show that super-Earths are more common in comparison to gas giants, which are found rarely (at most around 10% of stars). This trend implies that gas-giant formation occurs under specific conditions that deserve further study.

The pebble production line in our magnetized disk model propagates to a large radial extend. For the allowed range of plasma parameter, therefore, the PPD size is required to be larger than 100 au, which is not consistent with ALMA observations of PPDs. In the old Ophiuchus star-forming region, for instance, Andrews et al. (2009) performed an extensive survey of PPDs and their characteristic size was found to be between 20 and 200 au. ALMA observations of 36 PPDs in the Lupus star-forming complex exhibited a similar range of disk sizes (Tazzari et al. 2017). Continuous dust replenishment in the PPDs that may alter the mass flux rate as an input of our model or loss of detection efficiency in observations of PPDs are possibilities that may help us to reconcile current observational features of PPDs with our disk-wind model. Further works are needed to address these issues.

We are grateful to the referee, Michiel Lambrechts, for a very constructive and thoughtful report that greatly helped us to improve the paper. M.S. and F.K. thank Golestan University for supporting their research, and Niels Bohr International Academy and the Niels Bohr Institute for their hospitality and support during part of this work. The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP/2007-2013) under ERC grant agreement 306614 (MEP).