Effect of lift force on the aerodynamics of dust grains in the protoplanetary disk

Sustainability of the spin of the dust grains

where R is the semi-major axis, R₀ is the typical radius of the disk, R₁ = R/R₀, f_d is the dust-to-gas mass ratio, $\overline{m}=2.35m_H$ is the mean particle mass of gas, and M_s = 1M_⊙ is the mass of the central star. We use the isothermal sound speed c_s and the mid-plane gas density ρ_g when estimating timescales. If we choose Σ₀ = 1.7 × 10³ g cm^-2, c_s,0 = 1.0 × 10⁵cm s^-1, Ω₀ = 2.0 × 10^-7s^-1, R₀ = 1 AU, f_d = 0.01, q = 3/2, and p = 1/4, the disk profile is similar to the minimum mass solar nebula (MMSN; Hayashi 1981).

Comparison of timescales

Now, we can obtain the size of dust grains that are able to keep spinning. We estimate these timescales just in the Stokes law regime because the lift force in other regimes is uncertain. There are two necessary conditions to realize Stokes law. One is that the gas can be regarded as a continuum medium, which is expressed as r_d≳ 9λ_mfp/4. The other is that the flow around the dust grains is laminar, which is represented as Re = 2u r_d/ν ≲ 20 (Shirayama 1992), where u is the relative velocity of the dust to the gas. Here, we should actually include the effect of turbulence in the expression of u as in Ormel and Cuzzi (2007) so that the physical situation is consistent with that of Equation (15). However, taking this effect into account causes complicated equations. Thus, as a first-step attempt, we assume that u is equal to the relative velocity between the orbital velocity of the gas and the Keplerian velocity, i.e., u = η v_K, where

$\eta \equiv \frac{2p+2q+3}{4}{\left(\frac{c_{\text{s}}}{v_{\mathrm{K}}}\right)}^2,$

(22)

which is given in (Adachi et al. 1976). By these conditions, we find that our estimation is valid in the following range:

$\begin{array}{ll}{r}_{\text{d,min}}& \equiv \frac{9\overline{m}{c}_{\text{s,0}}}{2{\sigma }_{\text{mol}}{\Sigma }_0{\Omega }_0}{R}_1^{q-p+3/2}\lesssim r_{\text{d}}\lesssim r_{\text{d,max}}\\ \equiv \frac{40\sqrt{2\pi }\overline{m}{R}_0}{3(2p+2q+3){\sigma }_{\text{mol}}{\Sigma }_0}{R}_1^{q+1}.\end{array}$

(23)

For MMSN, this condition is simply written as

$3.2R_1^{11/4}\lesssim r_{\text{d},1}\lesssim 89R_1^{5/2}.$

(24)

Figure 1 shows the two timescales t_col (solid lines) and t_spin-down (dashed lines) at R₁ = 1 for MMSN. We plotted t_col and t_spin-down in the range that satisfies the condition (24). From Figure 1, we can see that t_col is larger than t_spin-down, so that the spin of the dust would stop at R₁ = 1. The difference between the two timescales gets smaller as the dust grains become larger. Equations (18) and (21) show that large grains are likely to satisfy the condition t_col < t_spin-down. From Equation (24), the Stokes regime can be adopted for the larger dust grains at the outer region of the disk. Thus, we expect that the condition t_col < t_spin-down is satisfied at the outer region R₁ > 1. Figure 2 shows the parameter space where the dust grains keep spinning in the R₁ – r_d plane for MMSN. The Stokes regime is realized between the dashed green lines. The condition t_col < t_spin-down is satisfied above the solid red line. In the blue region, the condition is satisfied with the Stokes regime. There are dust grains that keep spinning with the Stokes regime in R₁≳ 1.3. The grains that can keep spinning have the size r_d ∼ r_d,max. The dotted magenta line shows the dust radius when St = 1, which is used in the ‘Discussion’ section.

Relative velocity between the dust grains

In this section, we investigate whether the mean relative velocity is comparable to or greater than the gas velocity in the Kepler rotational frame. Since this gas velocity is comparable to the typical relative velocity between a large grain and a small one compared to 1-m-sized dust, we take it as a reference value. First, we derive the equation of motion for a dust grain assuming that it moves at a terminal velocity. Next, we estimate the mean relative velocity by assuming an isotropic distribution for the spin angular momentum.

Here, for simplicity, we assume that the dust grains move on the mid-plane of the disk, which means that the z-component of the lift force is assumed to be zero, where the z-axis is taken as the disk axis, and we adopt the cylindrical coordinates described below. Since the direction of the spin angular momentum can be taken arbitrarily, the lift force can show the z-component. Nevertheless, we neglect the z-component of the velocity to simplify the calculation below.

where $\overrightarrow{e}$ with a subscript and μ represent a unit vector in the direction of the subscript and the cosine of the angle between ${\overrightarrow{\omega }}_{\text{d}}$ and the z-axis, respectively. We note here that the F_L values do not depend on the azimuth angle of the spin angular momentum.

As the second step, we transform this into coordinates rotating at the angular velocity of the Kepler rotation, that is, $v_{\theta }=v_{\text{K}}+v_{\theta }^{\prime }$. As the third step, we assume that the motion of the dust grain is stationary and that $\left|v_r\right|,\left|v_{\theta }^{\prime }\right|\ll v_{\text{K}}$.

This means that the dust grain stops spinning before moving at the terminal velocity independently of the dust size and the distance from the central star. Thus, as long as the lift force is exerted on the grain, the motion of the grain cannot reach a steady state. Alternatively, the grain motion can be considered to be determined by the merger of the parent grains (or scattering by the other grains). Nevertheless, we assume that the grain motion reaches the steady state for the first stage of this type of work.

These equations represent the balance between the Coriolis, drag, and lift forces.

Here, Equations (26) and (31) lead to g_L = g_L,maxμ, where $g_{\text{L,max}}\equiv \frac{A{\omega }_{\text{d}}}{{\Omega }_{\text{g}}}$, and we introduce the lift-drag ratio R_LD ≡ g_L,max/g_D to obtain g_L = g_DR_LDμ. Thus, x and y are expressed as functions of μ, g_D, and R_LD. We show x (μ) and y (μ) in Figure 3. Here, we assume g_D = 1 and R_LD = 1 as trial values.

Since x = - 1 when we neglect the lift force, the curve of x (μ) shows that the radial velocity of the dust grain can be a third or four times of that without the lift force. On the other hand, when μ < 0, y is almost constant and comparable to that without the lift force. When μ is larger than 0.5, y is smaller than -1, which means that the dust grain orbits more slowly than the gas. In addition, we see that x and y decrease as μ is close to unity. Therefore, the absolute value of the velocity tends to increase as μ increases.

Equation (38) means that the dust grains, on average, fall down to the star faster than without the lift force. Moreover, Equation (39) means that their average orbit is at almost the same velocity as the gas. We note that the standard deviation of the velocity represents the average of the relative velocity between the grains. Therefore, Equation (42) means that the average relative velocity exceeds the relative velocity between the gas and Kepler velocity, so that the collision rate is affected by the lift force when g_D = 1 and R_LD = 1.

We finally calculate the averaged relative velocity on the disk mid-plane < w_rel > for arbitrary values of g_D and R_LD. Figure 4 shows the contour lines of <w_rel > = 1 (the solid line) and 0.1 (the dashed line) on the g_D – R_LD plane. We see that the relative velocity is large when g_D is small and when R_LD is large, which corresponds to the situation where the lift force is efficiently exerted on the grains. The important fact is that there exists a region satisfying < w_rel > > 1, where the lift force non-negligibly affects the dynamics of the system of grains, compared to the case without the force.

Lift-drag ratio

where we take the MMSN disk parameters.

We see from Figure 2 that when r_d is approximately 10² cm at R₁ ∼ 1, which corresponds to g_D = St^-1 ∼ 0.1, then the conditions where t_col ∼ t_spin-down and the drag force is represented with Stokes law are marginally satisfied. In this case, we find R_LD of approximately 2.0 when α = 0.01 using Equation (49). Thus, from Figure 4, we obtain < w_rel > ∼ 0.1, which means that the averaged relative velocity due to the lift force is a tenth of η v_K for the 1-m-sized dust at 1 AU from the central star.

Dependence of the relative velocity on disk parameters

The situation stated in the previous subsection can be qualitatively or quantitatively changed by adopting disk parameters different from MMSN. If we take an f_d larger than 0.01, as proposed in Sekiya (1998) and Hasegawa and Tsuribe (2014), the averaged relative velocity < w_rel > can be approximately 1. The dust-to-gas ratio f_d is included just in the expression of t_col [Equation (16)], so that a larger f_d means a smaller t_col and thereby smaller r_d satisfying t_col = t_spin-down. This implies that the red line in Figure 2 moves down and that the blue region expands inside. When f_d is approximately 0.1, the blue region includes the dotted magenta line, which shows the grain size satisfying St = 1 at R₁ ∼ 0.5. Since R_LD is approximately 1 when α = 10^-4, we find from Figure 4 that the relative velocity < w_rel > is approximately 1. In this case, the relative velocity due to the lift force η v_K < w_rel > is approximately 60 m s ^-1, where we note that η v_K does not depend on R₁ when p = 1/4. This value exceeds the relative velocity due to the turbulence < v_d-d > _t ∼ 8 m s ^-1, which is computed with Equation (15). Therefore, the lift force is expected to efficiently affect the growth rate of the grain whose size is 10 cm at R₁ ∼ 0.5 if there is a grain-concentrated part.

Recently, a steeper density profile, which is denser at 1 AU, has been proposed (Desch 2007). When the density profile is steeper, a larger q is adopted. If q is larger, the radial profile of the minimum and maximum sizes of the grains is steeper [Equation (23) and dashed green lines in Figure 2]. Moreover, we can find that the red line in Figure 2 does not change as much as the green lines by comparing Equation (16) to (20). Thus, the blue region in Figure 2 slightly shifts to the inner region. Therefore, we expect that when the density profile is steeper with the normalization coefficient fixed, the innermost radius where the lift force continues to be exerted on the grains slightly decreases close to R₁ = 1.

Suppose that the disk has a larger surface density. Spin-down time [Equation (20)] and Stokes number [Equation (14)] are independent of the surface density Σ₀. Conversely, collision time is inversely proportional to Σ₀ [Equation (16)], so that r_d satisfying t_col = t_spin-down is proportional to ${\Sigma }_0^{-1/2}$ for St ≫1. Additionally, r_d,min and r_d,max in Equation (23) are inversely proportional to Σ₀. Therefore, the blue region moves outside and the minimum grain size in the blue region is nearly unchanged. The ratio R_LD becomes smaller with decreasing R₁, so that the relative velocity due to the lift force is smaller than that for the fiducial surface density.

Model refinement for other effects

while $t_{\text{spin-down}}\propto {\rho }_{\text{int}}^1$. Thus, when ρ_int decreases, t_spin-down decreases more rapidly than t_col, which means that the lift force is exerted on the grain over a shorter period of time. If the grain is lumpy, the coefficient of lift would become as large as a baseball or a golf ball.

It is also worth taking into account realistic collision processes between the grains, such as simple scattering (bouncing), minor merger, and destruction. Through the simple scattering, the grain may gain the spin angular momentum by means of surface friction, where the maximum surface velocity of the scattered grains is v_d-d, which is less than the mean surface velocity in the case of the major merger [Equation (47)]. In addition, in the case of a minor merger, which is realized when we consider the size distribution of grains (Windmark et al. 2012), the grains obtain less spin angular momentum compared to the case of the major merger. Thus, we expect less mean relative velocity when the grains undergo the scattering and minor merger. On the other hand, if the destruction (fragmentation) occurs when the grains collide, the grains may gain larger spin angular momentum. The experiments show that agglomerates (centimeter to decimeter size) are divided into many fragments that are of millimeter to centimeter size through low-velocity collisions (Deckers and Teisere 2012; Schräpler et al. 2012). If these fragments have a large spin angular momentum, they can be sufficiently affected by the lift force. How much spin angular momentum grains gain depends on many parameters, but we will defer this question to future work.

The z-component of the lift force, which is omitted in this paper, may affect the resultant relative velocity of the dust grain. The equation of motion in the z-direction includes the gravitational force by the central star, the drag force, and the lift force. Although the steady state cannot be realized because the gravitational force depends on the altitude from the disk mid-plane, the dust grains should gain a momentum in the z-direction. This causes an increase in the absolute value of the velocity, which may result in an increase in the relative velocity.