### Sustainability of the spin of the dust grains

In this section, we examine whether the dust grains keep spinning in the gas disk because the lift force does not act on non-spinning spherical objects. Here, we assume that the collisions between the dust grains induce the spin of the dust grains. The spinning dust is subjected to the torque due to the friction created by the background viscous fluid. After the spin-down timescale, the spin of the dust would stop. We can estimate the collision time *t*_{col} and the spin-down time *t*_{spin-down}. By comparing these timescales, we obtain the parameter space where the lift force can act on the spinning dust grains. These timescales depend on the disk structure. We adopt the parameters for the disk structure in this paper as follows:

\begin{array}{ll}{\Sigma}_{\text{g}}& ={\Sigma}_{0}{R}_{1}^{-q},\phantom{\rule{2em}{0ex}}\end{array}

(4)

\begin{array}{ll}{\Sigma}_{\text{d}}& ={f}_{\text{d}}{\Sigma}_{\text{g}}={f}_{\text{d}}{\Sigma}_{0}{R}_{1}^{-q},\phantom{\rule{2em}{0ex}}\end{array}

(5)

\begin{array}{ll}{c}_{\text{s}}& =\sqrt{\frac{\mathit{\text{kT}}}{\overline{m}}}={c}_{\text{s,0}}{R}_{1}^{-p},\phantom{\rule{2em}{0ex}}\end{array}

(6)

\begin{array}{ll}{\Omega}_{\text{K}}& =\sqrt{\frac{G{M}_{\text{s}}}{{R}^{3}}}={\Omega}_{0}{R}_{1}^{-3/2},\phantom{\rule{2em}{0ex}}\end{array}

(7)

\begin{array}{ll}{v}_{\mathrm{K}}& =R{\Omega}_{\text{K}}={\Omega}_{0}{R}_{0}{R}_{1}^{-1/2},\phantom{\rule{2em}{0ex}}\end{array}

(8)

\begin{array}{ll}{H}_{\text{g}}& =\frac{\sqrt{2}{c}_{\text{s}}}{{\Omega}_{\text{K}}}=\frac{\sqrt{2}{c}_{\text{s,0}}}{{\Omega}_{0}}{R}_{1}^{-p+3/2},\phantom{\rule{2em}{0ex}}\end{array}

(9)

\begin{array}{ll}{\rho}_{\text{g}}& =\frac{{\Sigma}_{\text{g}}}{\sqrt{\pi}{H}_{\text{g}}}=\frac{{\Sigma}_{0}{\Omega}_{0}}{\sqrt{2\pi}{c}_{\text{s,0}}}{R}_{1}^{p-q-3/2},\phantom{\rule{2em}{0ex}}\end{array}

(10)

where *R* is the semi-major axis, *R*_{0} is the typical radius of the disk, *R*_{1} = *R*/*R*_{0}, *f*_{d} is the dust-to-gas mass ratio, \overline{m}=2.35{m}_{H} is the mean particle mass of gas, and *M*_{s} = 1*M*_{⊙} 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 *Σ*_{0} = 1.7 × 10^{3} g cm^{-2}, *c*_{s,0} = 1.0 × 10^{5}cm s^{-1}, *Ω*_{0} = 2.0 × 10^{-7}s^{-1}, *R*_{0} = 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).

#### Collision timescale

The collision timescale is estimated as

{t}_{\text{col}}\sim {({n}_{\text{d}}\xb7\pi {r}_{\text{d}}^{2}\xb7<{v}_{\text{d-d}}>)}^{-1}\phantom{\rule{1em}{0ex}},\phantom{\rule{1em}{0ex}}

(11)

where *n*_{d} and *v*_{d-d} are the number density of the dust grains and the relative velocity between the dust grains, respectively. The parenthetic quantity <*Q*> represents the statistical average.

The dust number density is expressed as

{n}_{\text{d}}=\frac{{\Sigma}_{\text{d}}}{{H}_{\text{d}}{m}_{\text{d}}},

(12)

where *H*_{d} is the scale height of the dust layer and *m*_{d} is the mass of the dust grains. We approximate that the mass distribution function of the dust is the delta function because it is necessary for the dust grains to collide with similar scale grains so that the grains gain the angular momentum. Considering the equilibrium between turbulent diffusion and sedimentation (Birnstiel et al. 2010), *H*_{d} is obtained as

{H}_{\text{d}}={H}_{\text{g}}\xb7{\left(\frac{\alpha}{\text{St}}\frac{\text{1+2St}}{{\text{1+St}}^{2}}\right)}^{1/2},

(13)

where St≡*Ω*_{K}*t*_{s} is the Stokes number (*t*_{s} is the stopping time by drag force). We use alpha prescription *ν*_{turb} = *α* *c*_{s}*H*_{g} to describe the strength of the turbulence in the protoplanetary disk and assume *α* < St to avoid the situation *H*_{g} < *H*_{d}. At the Stokes drag law regime, the Stokes number is written as

\text{St}=\frac{{m}_{\text{d}}{\Omega}_{\text{K}}}{6\pi {r}_{\text{d}}{\rho}_{\text{g}}\nu}=\frac{2{\rho}_{\text{int}}{r}_{\text{d}}^{2}{\Omega}_{\text{K}}}{3{\rho}_{\text{g}}{c}_{\text{s}}{\lambda}_{\text{mfp}}}=\frac{2{\sigma}_{\text{mol}}{\rho}_{\text{int}}{\Omega}_{0}}{3\overline{m}{c}_{\text{s,0}}}{r}_{\text{d}}^{2}{R}_{1}^{p-3/2},

(14)

where *ρ*_{int} is the internal mass density of the dust grains, *ν* = *c*_{s}*λ*_{mfp}/3 is the kinematic viscosity, and *λ*_{mfp} is the mean free path of the gas particles. The mean free path is estimated as {\lambda}_{\text{mfp}}=\overline{m}/\left({\sigma}_{\text{mol}}{\rho}_{\text{g}}\right), where *σ*_{mol} is the cross section of collisions between H _{2} molecules. We adopt *ρ*_{int} ≃ 3 g cm^{-3} and *σ*_{mol} ≃ 2 × 10^{-15} cm^{2}. Equation (14) means that the Stokes number is independent of the normalization coefficient of the surface density *Σ*_{0}, which is canceled out because of the one in *λ*_{mfp}.

Since the gas was assumed to be in a turbulent state described by the alpha prescription, we set the mean relative velocity <*v*_{d-d} > = < *v*_{d-d} > _{
t
}, where < *v*_{d-d} > _{
t
} means the relative velocity between the grains in the turbulent gas. According to (Ormel and Cuzzi 2007), < *v*_{d-d} > _{
t
} with similar scale grains can be represented as

<{v}_{\text{d-d}}{>}_{t}={c}_{\text{s}}{\left(\frac{\alpha \text{St}}{\sqrt{1+\frac{1}{4}{\text{St}}^{2}{(1+\mathit{\text{St}})}^{2}}}\right)}^{1/2},

(15)

where we smoothly interpolate the two limiting solutions of St≫1 and St≪1. This expression is valid when the stopping time is larger than the turnover time of the Kolmogorov-scale eddy. The minimum size of the grain satisfying this condition is on the order of sub-millimeters for MMSN at 1 AU; hence, we focus on grains larger than approximately 1 mm in what follows.

Now we can express *t*_{col} as the function of *r*_{d} and *R*_{1} by using Equations (11) to (15) as

{t}_{\text{col}}=\frac{4\sqrt{2}{\rho}_{\text{int}}}{3{\Omega}_{0}{\Sigma}_{0}{f}_{\text{d}}}{r}_{\text{d}}{R}_{1}^{q+3/2}f\left(\text{St}\right),

(16)

where

f\left(\text{St}\right)=\frac{1}{\text{St}}{\left(\frac{1+2\text{St}}{1+{\text{St}}^{2}}\right)}^{1/2}{\left(1+\frac{1}{4}{\text{St}}^{2}{(1+\text{St})}^{2}\right)}^{1/4}.

(17)

We note that when *α* < St, *t*_{col} is independent of *α* because the effect of increasing <*v*_{d-d} > _{
t
} balances the one decreasing *n*_{d}. For the case with St≪1, the collision timescale is {t}_{\text{col}}\propto {\text{St}}^{-1}{r}_{\text{d}}^{1}{R}_{1}^{q+3/2}\propto {r}_{\text{d}}^{-1}{R}_{1}^{3+q-p}, while it is {t}_{\text{col}}\propto {\text{St}}^{-1/2}{r}_{\text{d}}^{1}{R}_{1}^{q+3/2}\propto {r}_{\text{d}}^{0}{R}_{1}^{2.25+q-0.5p} for the case with St ≫ 1. If we adopt the same parameters as MMSN, the collision time is

{t}_{\text{col}}=1.7\times 1{0}^{6}f\left(\text{St}\right){R}_{1}^{3}{r}_{\text{d},1}\phantom{\rule{1em}{0ex}}\mathrm{s},

(18)

where *r*_{d,1} = *r*_{d}/(1 cm).

#### Spin-down timescale

In the case of Stokes law, the angular momentum conservation around the spin axis of a spherical grain is given as

{I}_{\text{d}}\frac{d{\omega}_{\text{d}}}{\mathit{\text{dt}}}=-8\pi {\rho}_{\text{g}}\nu {r}_{\text{d}}^{3}{\omega}_{\text{d}},

(19)

where *I*_{d} is the moment of inertia of the grain. The torque acting onto a spherical body by viscous fluid is given in Rubinow and Keller (1961) and Takagi (1974). From this equation, the *t*_{spin-down} is estimated as

{t}_{\text{spin-down}}=\frac{{I}_{\text{d}}}{8\pi {\rho}_{\text{g}}\nu {r}_{\text{d}}^{3}}=\frac{{\rho}_{\text{int}}{r}_{\text{d}}^{2}}{5{\rho}_{\text{g}}{c}_{\text{s}}{\lambda}_{\text{mfp}}}=\frac{{\sigma}_{\text{mol}}{\rho}_{\text{int}}}{5\overline{m}{c}_{\text{s,0}}}{r}_{\text{d}}^{2}{R}_{1}^{p}.

(20)

In this second equation, we assume a spherical and uniform density grain whose moment of inertia is represented as {I}_{\text{d}}=2{m}_{\text{d}}{r}_{\text{d}}^{2}/5. The spin-down time becomes longer as the dust grain becomes larger. Here, we note that the spin-down time is independent of *Σ*_{0} for the same reason as the Stokes number [see Equation (14)]. For MMSN, *t*_{spin-down} is estimated as

{t}_{\text{spin-down}}=3.0\times 1{0}^{3}{r}_{\text{d},1}^{2}{R}_{1}^{1/4}\phantom{\rule{0.3em}{0ex}}\mathrm{s}.

(21)

#### 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}*≳* 9*λ*_{mfp}/4. The other is that the flow around the dust grains is laminar, which is represented as Re = 2*u* *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

3.2{R}_{1}^{11/4}\lesssim {r}_{\text{d},1}\lesssim 89{R}_{1}^{5/2}.

(24)

Figure 1 shows the two timescales *t*_{col} (solid lines) and *t*_{spin-down} (dashed lines) at *R*_{1} = 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 *R*_{1} = 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 *R*_{1} > 1. Figure 2 shows the parameter space where the dust grains keep spinning in the *R*_{1} – *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 *R*_{1}*≳* 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.

For preparation to derive the equation of motion, we express a projected vector of the lift force on the mid-plane in terms of the direction of the spin angular momentum of a dust grain. Since the direction of the lift force is perpendicular to the spin angular momentum and the velocity of the grain with respect to the gas, then

{\overrightarrow{F}}_{\text{L}}=A{\overrightarrow{\omega}}_{\text{d}}\times \overrightarrow{u},

(25)

where the coefficient satisfies A=\frac{\pi {\rho}_{\text{g}}{r}_{\text{d}}^{3}}{{m}_{\text{d}}} (see the ‘Background’ section). Since the *z*-component of \overrightarrow{u} is zero, the lift force vector projected on the mid-plane {\overrightarrow{F}}_{\text{L,mid}} is expressed as

{\overrightarrow{F}}_{\text{L,mid}}={\overrightarrow{F}}_{\text{L}}\xb7\left(\begin{array}{l}{\overrightarrow{e}}_{r}\\ {\overrightarrow{e}}_{\theta}\end{array}\right)=A{\omega}_{\text{d}}\mu \left(\begin{array}{l}-{u}_{\theta}\\ {u}_{r}\end{array}\right)\equiv {F}_{\text{L}}\left(\begin{array}{l}-{u}_{\theta}/u\\ {u}_{r}/u\end{array}\right),

(26)

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.

Next, we derive and reduce the equation of motion for a dust grain. Now, the forces exerted on the dust grain are the gravitational force of the central star, the drag force, and the lift force, so the equation of motion is expressed as

\frac{d\overrightarrow{v}}{\mathit{\text{dt}}}=-\frac{G{M}_{\odot}}{{r}^{2}}{\overrightarrow{e}}_{r}-{\overrightarrow{F}}_{\text{D}}+{\overrightarrow{F}}_{\text{L,mid}},

(27)

where we assume that the mass of the central star is the same as the solar one. As the first step for the reduction of Equation (27), we divide it into two equations for the *r* and *θ* components. Since the velocity of the disk gas does not have the radial component, the components of the velocity of the dust grain are represented as (*u*_{
r
},*u*_{
θ
}) = (*v*_{
r
},*v*_{
θ
}-*r* *Ω*_{g}), where *Ω*_{g} is the orbital angular velocity of the disk gas around the central star. Thus, Equation (27) is expressed as

\begin{array}{l}\frac{d{v}_{r}}{\mathit{\text{dt}}}-\frac{{v}_{\theta}^{2}}{r}=-\frac{G{M}_{\odot}}{{r}^{2}}-{F}_{\text{D}}\frac{{v}_{r}}{u}-{F}_{\text{L}}\frac{{v}_{\theta}-r{\Omega}_{\text{g}}}{u},\phantom{\rule{2em}{0ex}}\end{array}

(28)

\begin{array}{l}\frac{d{v}_{\theta}}{\mathit{\text{dt}}}+\frac{{v}_{r}{v}_{\theta}}{r}=-{F}_{\text{D}}\frac{{v}_{\theta}-r{\Omega}_{\text{g}}}{u}+{F}_{\text{L}}\frac{{v}_{r}}{u}.\phantom{\rule{2em}{0ex}}\end{array}

(29)

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 stationary assumption may be invalid when taking into account the timescales discussed in the ‘Sustainability of the spin of the dust grains’ section. The stopping time *t*_{s} is represented as

{t}_{\text{s}}=\frac{{m}_{\text{d}}}{6\pi {r}_{\text{d}}{\rho}_{\text{g}}\nu}\sim 1{0}^{4}\phantom{\rule{1em}{0ex}}{r}_{\text{d},1}^{2}{R}_{1}^{1/4}\sim 4\phantom{\rule{1em}{0ex}}{t}_{\text{spin-down}}.

(30)

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.

Finally, we non-dimensionalize the variables as

x=\frac{{v}_{r}}{\eta {v}_{\text{K}}},\phantom{\rule{1em}{0ex}}y=\frac{{v}_{\theta}^{\prime}}{\eta {v}_{\text{K}}},\phantom{\rule{1em}{0ex}}{g}_{\text{D}}=\frac{{F}_{\text{D}}}{u{\Omega}_{\text{K}}},\phantom{\rule{1em}{0ex}}{g}_{\text{L}}=\frac{{F}_{\text{L}}}{u{\Omega}_{\text{K}}},

(31)

where *η* is the constant satisfying the equation *r* *Ω*_{g} = *v*_{K}(1-*η*) and *Ω*_{K} is the angular velocity of the Kepler motion. Thus, we obtain two algebraic equations:

\begin{array}{l}2y={g}_{\text{D}}x+{g}_{\text{L}}(y+1),\end{array}

(32)

\begin{array}{l}\frac{1}{2}x=-{g}_{\text{D}}(y+1)+{g}_{\text{L}}x.\end{array}

(33)

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

The solution of the equations is

\begin{array}{l}x=\frac{-2{g}_{\text{D}}}{{g}_{\text{D}}^{2}+({g}_{\text{L}}-2)({g}_{\text{L}}-1/2)},\end{array}

(34)

\begin{array}{l}y=-1+\frac{-2({g}_{\text{L}}-1/2)}{{g}_{\text{D}}^{2}+({g}_{\text{L}}-2)({g}_{\text{L}}-1/2)}.\end{array}

(35)

Here, Equations (26) and (31) lead to *g*_{L} = *g*_{L,max}*μ*, 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*_{D}*R*_{LD}*μ*. 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.

Next, we calculate the average and dispersion of the velocity of the dust grain on the disk mid-plane, assuming that the spin angular momentum is isotropic, which is just for simplicity. Thus, the direction distribution satisfies f(\Omega )=\frac{1}{4\pi}, which is equivalent to f(\mu )=\frac{1}{2}, where *Ω* is a solid angle parameter. The average and dispersion of *x* are calculated by performing the integration below.

\begin{array}{ll}<x>\phantom{\rule{1em}{0ex}}& =\int \mathit{\text{xf}}\left(x\right)\mathit{\text{dx}}\\ =\frac{1}{2}\underset{-1}{\overset{1}{\int}}x(\mu )d\mu ,\\ <{x}^{2}>\phantom{\rule{1em}{0ex}}& =\int {x}^{2}f\left(x\right)\mathit{\text{dx}}\\ =\frac{1}{2}\underset{-1}{\overset{1}{\int}}{x}^{2}(\mu )d\mu ,\end{array}

(36)

where we transform the integration variable into *μ*. We also can derive the same expression for *y*,

\begin{array}{ll}<y>\phantom{\rule{1em}{0ex}}& =\frac{1}{2}\underset{-1}{\overset{1}{\int}}y(\mu )d\mu ,\\ <{y}^{2}>\phantom{\rule{1em}{0ex}}& =\frac{1}{2}\underset{-1}{\overset{1}{\int}}{y}^{2}(\mu )d\mu .\end{array}

(37)

By taking *g*_{D} = 1 and *R*_{LD} = 1, we can obtain the approximate value of the average and standard deviation of the velocity,

\begin{array}{l}<x>\phantom{\rule{1em}{0ex}}\simeq -1.4,\end{array}

(38)

\begin{array}{l}<y>\phantom{\rule{1em}{0ex}}\simeq -1.2,\end{array}

(39)

\begin{array}{l}\sqrt{<{x}^{2}>}\phantom{\rule{1em}{0ex}}\simeq 1.0,\end{array}

(40)

\begin{array}{l}\sqrt{<{y}^{2}>}\phantom{\rule{1em}{0ex}}\simeq 0.6,\end{array}

(41)

\begin{array}{l}<{w}_{\text{rel}}>\equiv \sqrt{<{x}^{2}>+<{y}^{2}>}\simeq 1.2.\end{array}

(42)

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.