FERROMAGNETISM AND PARTICLE COLLISIONS: APPLICATIONS TO PROTOPLANETARY DISKS AND THE METEORITICAL RECORD

We follow the formalism of Hubbard (2014), with m, m_i, and $f\equiv {m}_{i}/m$ denoting a dust grain's total mass, iron metal mass, and metal mass fraction, respectively. Dust grains are assumed to be spherical, with radius a, density ρ, and volume

$$\begin{eqnarray}&&\forall \,=\,\displaystyle \frac{4\pi }{3}{a}^{3}=\displaystyle \frac{m}{\rho },\end{eqnarray} \tag{ 1 }$$

of which

$$\begin{eqnarray}&&{\forall }_{i}=\displaystyle \frac{{m}_{i}}{{\rho }_{i}}\end{eqnarray} \tag{ 2 }$$

is taken up by metal with density ${\rho }_{i}$. These definitions allow us to calculate the metal volume fraction g:

$$\begin{eqnarray}&&g\equiv \displaystyle \frac{{\forall }_{i}}{\forall }=\displaystyle \frac{{m}_{i}\,\rho }{m\,{\rho }_{i}}=f\displaystyle \frac{\rho }{{\rho }_{i}}.\end{eqnarray} \tag{ 3 }$$

Note that collisionally aggregated dust grains in protoplanetary disks are expected to be porous. Accordingly, the densities of iron metal, fused solids, and porous solids are, respectively (Friedrich et al. 2015),

$$\begin{eqnarray}&&{\rho }_{i}\simeq 7.86\,{\rm{g}}\,{\mathrm{cm}}^{-3},\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&{\rho }_{s}\simeq 3\,{\rm{g}}\,{\mathrm{cm}}^{-3},\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&171{\rho }_{\phantom{s}}=\phi \,{\rho }_{s},\end{eqnarray} \tag{ 6 }$$

where ϕ is the volume filling fraction,

Outside limited regions of extremely strong magnetic fields (Hubbard 2014), the ambient magnetic field H in protoplanetary disks is generally too weak to magnetize iron metal near to saturation.¹ We therefore use the relationship

$$\begin{eqnarray}&&M=\displaystyle \frac{{\mu }_{r}H}{4\pi },\end{eqnarray} \tag{ 7 }$$

where M is the magnetization induced by the ambient field and ${\mu }_{r}$ is the magnetic relative permeability when considering magnetically soft materials, which are the main thrust of this paper. Further, we assume that the iron metal is evenly distributed throughout our dust grains. We parameterize the ambient magnetic field H in terms of the thermal pressure through the plasma β parameter:

$$\begin{eqnarray}&&\beta \equiv \displaystyle \frac{8\pi {{nk}}_{B}T}{{H}^{2}}=\displaystyle \frac{8\pi {\rho }_{g}{k}_{B}T}{\bar{m}{H}^{2}},\end{eqnarray} \tag{ 8 }$$

where n is the number density of the gas, ${\rho }_{g}$ its density, and $\bar{m}$ its the mean molecular mass. It follows that

$$\begin{eqnarray}&&{H}^{2}=\displaystyle \frac{8\pi {\rho }_{g}{k}_{B}T}{\bar{m}\beta }.\end{eqnarray} \tag{ 9 }$$

Given the strongly radially decreasing gas density expected in protoplanetary disks, Equation (9) implies that there should exist an outer radius outside of which induced magnetization is negligible.

We use the gas parameters of a Hayashi Minimum Mass Solar Nebula's (henceforth MMSN) midplane at $R=2.5\,{\rm{au}}$, appropriate for the asteroid belt (Hayashi 1981). Unfortunately, most of the magnetic parameters used in this paper are poorly constrained. The values we normalize to, denoted with overbars, are listed in Table 1. The relative permeability of iron metal, ${\mu }_{r}$, depends on its manufacture, composition, history and the amplitude of the ambient field, potentially varying from 0 up to tens of thousands for some nickel–iron alloys (Weast 1975). We normalize, possibly optimistically, to ${\bar{\mu }}_{r}=1000$, and hope that this work will inspire future research to shed light on this parameter.

Table 1.  Magnetic Interaction Parameters

Symbol	Normalization	Parameter
${\mu }_{r}$	1000	Metal relative permeability
f	0.2	Metal mass fraction
$g$	0.0076	Metal volume fraction
ϕ	0.1	Grain volume filling fraction
$\beta$	100	Plasma beta
${\rho }_{g}$	10−11 g cm−3	Gas density (MMSN at 2.5 au)
${{\rm{\Sigma }}}_{g}$	430 g cm−3	Gas surface density (MMSN at 2.5 au)
T	177 K	Gas temperature (MMSN at 2.5 au)
cs	$8\times {10}^{4}$ cm s−1	Gas sound speed (MMSN at 2.5 au)
$\bar{m}$	2.33 amu	Gas mean molecular mass
$\bar{C}$	0.2	Cross-section parameter (Equation (25))

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Similarly, the amplitude of the ambient magnetic field H is unclear, in no small part because we do not fully understand the behavior of the field in magnetically "dead" zones such as the midplane at $R=2.5\,{\rm{au}}$ (Gammie 1996). However, recent work has suggested that the Hall effect can allow significant field growth even there (Bai 2014), implying that the term "dead zone" may be a misnomer, leading us to normalize to $\bar{\beta }=100$. For the parameters in Table 1, and using Equations (7) and (9), we find

$$\begin{eqnarray}&&\bar{M}\simeq 10\,\mathrm{emu}/{\mathrm{cc}}^{3},\end{eqnarray} \tag{ 10 }$$

significantly less than the saturated magnetization of iron of Weast (1975):

$$\begin{eqnarray}&&{M}_{S}\simeq 1720\,\mathrm{emu}/{\mathrm{cc}}^{3}.\end{eqnarray} \tag{ 11 }$$

The appropriate iron metal fraction for dust in protoplanetary disks is unknown but, inspired by the low iron fraction of astrophysical silicates (Draine 2003), we adopt $\bar{f}\sim 0.2$, the overall chondritic iron mass fraction (Lodders 2003). To estimate the parameter g we also need an estimate for the overall grain density (or equivalently the overall grain volume filling factor). Collisionally grown grains are expected to be highly porous, but as the collisional speeds approach ones capable of rearranging the grains, the grains will be collisionally compacted and the volume filling fractions will rise (Seizinger et al. 2012). We adopt $\bar{\phi }=0.1$, which when inserted into Equations (3)–(6) implies that

$$\begin{eqnarray}&&\bar{g}=\bar{f}\,\displaystyle \frac{\bar{\phi }\,{\rho }_{s}}{{\rho }_{i}}=0.0076,\end{eqnarray} \tag{ 12 }$$

although we will be invoking grains both enriched in metallic iron (higher $f,g$), and depleted in metallic iron (lower $f,g$).

Consider two dust grains 1 and 2 close to each other, and embedded in an external magnetic field. It is easiest to operate in the limit of either identically sized grains or asymptotically different sized grains with ${a}_{1}\gg {a}_{2}$. The theory of turbulently stirred dust collisions is far simpler in the latter case, so we will henceforth require ${a}_{1}\gg {a}_{2}$. Accordingly, we place our larger grain 1 at the origin and consider it fixed, while following the motion of the smaller grain 2. We will assume that the dust grains are perfectly magnetically soft: i.e., that their magnetizations are controlled purely by a spatially uniform external magnetic field H according to Equation (7); and we adopt a spherical coordinate system with the polar coordinate θ and the axis aligned with the external field (and hence the magnetic dipole moments). For simplicity we assume that non-size dependent parameters other than f and g, i.e., ρ, ϕ, ${\mu }_{r}$, and M, are identical for both grains.

The magnetic energy of the dipole–dipole interaction is

$$\begin{eqnarray}&&{U}_{M}=-\displaystyle \frac{1}{{r}^{3}}[3({{\boldsymbol{M}}}_{1}\cdot \hat{r})({{\boldsymbol{M}}}_{2}\cdot \hat{r})-{{\boldsymbol{M}}}_{1}\cdot {{\boldsymbol{M}}}_{2}],\end{eqnarray} \tag{ 13 }$$

where $\hat{r}$ is the unit vector joining the two grains, and ${{\boldsymbol{M}}}_{\mathrm{1,2}}$ are the dipole moments. Because we have assumed that the dipole moments are aligned with the external field along the pole of our coordinate system, and we have assumed that the relative magnetic permeabilities of the grains are equal, Equation (13) reduces to

$$\begin{eqnarray}&&{U}_{M}=\displaystyle \frac{{\forall }_{i1}{\forall }_{i2}{M}^{2}}{{r}^{3}}[1-3{\cos }^{2}(\theta )],\end{eqnarray} \tag{ 14 }$$

where $(r,\theta )$ is the position of grain 2, and $\theta =0$ is the pole.

We can calculate the radial magnetic force using Equation (14):

$$\begin{eqnarray}&&{{\boldsymbol{F}}}_{M}=-{\partial }_{r}{U}_{M}\hat{r}.\end{eqnarray} \tag{ 15 }$$

Two point masses cannot interact in a way which would change their angular momentum. Further, magnetic dipoles embedded in a uniform external magnetic field feel no net force, but only a torque acting to align them with the external field. Accordingly, even though $-{\partial }_{\theta }{U}_{M}\ne 0$, our system conserves its orbital angular momentum. This can be understood by noting that while Equation (14) is written in terms of θ, the actual dependency in Equation (13) is on the alignment of the magnetic dipole moments; and the apparent force $-{\partial }_{\theta }{U}_{M}$ actually represents the torques the two grains exert on each other to align their dipole moments: the torque drives spin, rather than orbital angular momentum, and we are neglecting the spin of the particles. Because the particles would be torquing each other to stay magnetically aligned, we would expect any spin they would develop to be aligned with, and extracted from, their orbital angular momentum. Accordingly, our treatment of the motion of the grains as spin-free will underestimate the effect of their magnetic interactions.

As noted above, by assuming that the magnetization of the two grains is purely controlled by the external field, we are neglecting the fact that their magnetic dipole–dipole interactions will further work to align the dipole moments. Note that if the grains are close enough together, the dipole moment of one can induce a dipole moment in the second, which strongly alters the close-approach physics and further strengthens the attraction between the two particles (see, e.g., Mehdizadeh et al. 2010). This simplification therefore results in a further underestimate of the magnetic attraction for the two grains.

We can estimate the magnetic enhancement to the collisional cross-section by considering polar orbits (constant azimuth). Using Equations (14) and (15), the magnetic force on grain 2 is

$$\begin{eqnarray}&&{F}_{M}=\displaystyle \frac{3{\forall }_{i1}{\forall }_{i2}{M}^{2}}{{r}^{4}}[1-3{\cos }^{2}(\theta )].\end{eqnarray} \tag{ 16 }$$

If grain 2 has an initial velocity ${v}_{\theta }={v}_{0}$ at an initial position $({r}_{0},{\theta }_{0})$, then it has angular momentum per unit mass

$$\begin{eqnarray}&&L={r}_{0}{v}_{0},\end{eqnarray} \tag{ 17 }$$

and hence a polar velocity

$$\begin{eqnarray}&&{v}_{\theta }(r)=\displaystyle \frac{L}{r}.\end{eqnarray} \tag{ 18 }$$

We can use Equations (16) and (18) to write

$$\begin{eqnarray}\displaystyle \frac{{\partial }^{2}r}{\partial {t}^{2}} & = & \displaystyle \frac{{F}_{M}}{{m}_{2}}+\displaystyle \frac{{v}_{\theta }^{2}}{r}=\displaystyle \frac{3{\forall }_{i1}{\forall }_{i2}{M}^{2}}{{m}_{2}{r}^{4}}[1-3{\cos }^{2}(\theta )]\\ & & +{v}_{0}^{2}\displaystyle \frac{{r}_{0}^{2}}{{r}^{3}},\end{eqnarray} \tag{ 19 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial \theta }{\partial t}=\displaystyle \frac{{v}_{\theta }}{r}=\displaystyle \frac{{v}_{0}{r}_{0}}{{r}^{2}},\end{eqnarray} \tag{ 20 }$$

which define the motion of grain 2.

Note that for ${\theta }_{0}=0$, the centrifugal and magnetic forces are balanced at a critical radius defined through

$$\begin{eqnarray}&&{r}_{c}^{3}\equiv \displaystyle \frac{6{\forall }_{i1}{\forall }_{i2}{M}^{2}}{{m}_{2}{v}_{0}^{2}},\end{eqnarray} \tag{ 21 }$$

which also lets us define the dynamical timescale

$$\begin{eqnarray}&&{t}_{c}\equiv \displaystyle \frac{{r}_{c}}{{v}_{0}}.\end{eqnarray} \tag{ 22 }$$

Using Equations (21) and (22) we can rewrite Equations (19) and (20) as

$$\begin{eqnarray}&&\displaystyle \frac{{\partial }^{2}\tilde{r}}{\partial {\tilde{t}}^{2}}=\displaystyle \frac{1-3{\cos }^{2}(\theta )}{2{\tilde{r}}^{4}}+\displaystyle \frac{{({r}_{0}/{r}_{c})}^{2}}{{\tilde{r}}^{3}},\end{eqnarray} \tag{ 23 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial \theta }{\partial \tilde{t}}=\displaystyle \frac{({r}_{0}/{r}_{c})}{{\tilde{r}}^{2}},\end{eqnarray} \tag{ 24 }$$

where $\tilde{r}\equiv r/{r}_{c}$ and $\tilde{t}\equiv t/{t}_{c}$.

We can integrate Equations (23) and (24) forward in time for given values of ${r}_{0}/{r}_{c}$ and ${\theta }_{0}$, determining whether r will hit zero (i.e., a magnetic collision). However, the limiting value of ${r}_{0}/{r}_{c}$ for which collisions occur (when one exists) for any given value of ${\theta }_{0}$ is complicated. Nonetheless, the only control parameter for Equations (23) and (24) is ${r}_{0}/{r}_{c}$, so the collisional cross-section for magnetic interactions between the two dust grains must scale with r_c². We can therefore write

$$\begin{eqnarray}&&{\sigma }_{m}(v)=C\pi {r}_{c}^{2}\end{eqnarray} \tag{ 25 }$$

for the magnetic collisional cross-section, where C is a constant. For ${\theta }_{0}=0$, ${r}_{0}/{r}_{c}=0.49$ is the limiting value for which collisions occur, and we adopt $\bar{C}=0.2\lesssim .{0.49}^{2}$ as an approximation for C, noting that our analysis underestimates the strength of the magnetic interactions, and that grains only need to approach $r={a}_{1}+{a}_{2}\simeq {a}_{1}$ to collide.

This allows us to define the ratio of the magnetic collisional cross-section ${\sigma }_{m}$ to the geometrical cross-section ${\sigma }_{g}\equiv \pi {a}_{1}^{2}$ (recall that ${a}_{1}\gg {a}_{2}$):

$$\begin{eqnarray}&&\displaystyle \frac{{\sigma }_{m}(v)}{{\sigma }_{g}}=C{\left(\displaystyle \frac{{r}_{c}}{{a}_{1}}\right)}^{2}=C{\left[\displaystyle \frac{4{f}_{1}{f}_{2}\phi {\rho }_{s}{\mu }_{r}^{2}{\rho }_{g}{k}_{B}T}{\beta {\rho }_{i}^{2}\bar{m}{v}^{2}}\right]}^{2/3}.\end{eqnarray} \tag{ 26 }$$

For velocities where ${\sigma }_{m}(v)\gg {\sigma }_{g}$, the magnetic cross-section applies, and when ${\sigma }_{m}(v)\ll {\sigma }_{g}$, the velocities are too high for magnetic forces to play a role, and the geometric cross-section applies. We approximate the intermediate regime by assuming that $\sigma =\max ({\sigma }_{m},{\sigma }_{g})$.

Using Equation (26) we can define the critical magnetic velocity v_m such that ${\sigma }_{m}({v}_{m})={\sigma }_{g}$:

$$\begin{eqnarray}&&{v}_{m}=2{C}^{3/4}\sqrt{\displaystyle \frac{{f}_{1}{f}_{2}\phi {\rho }_{s}{\rho }_{g}{k}_{B}T}{\beta \bar{m}}}\displaystyle \frac{{\mu }_{r}}{{\rho }_{i}}.\end{eqnarray} \tag{ 27 }$$

To calculate how much magnetic interactions change collision rates we will operate in terms of v_m, in which case Equation (26) becomes simply

$$\begin{eqnarray}&&\displaystyle \frac{{\sigma }_{m}(v)}{{\sigma }_{g}}={\left(\displaystyle \frac{{v}_{m}}{v}\right)}^{4/3}.\end{eqnarray} \tag{ 28 }$$

Using the parameters in Table 1, we can write Equation (27) as

$$\begin{eqnarray}&&{v}_{m}\simeq {\left(\displaystyle \frac{C}{\bar{C}}\right)}^{4/3}\sqrt{\displaystyle \frac{{f}_{1}{f}_{2}\phi {\rho }_{g}T\bar{\beta }}{{\bar{f}}^{2}\bar{\phi }{\bar{\rho }}_{g}\bar{T}\beta }}\displaystyle \frac{{\mu }_{r}}{{\bar{\mu }}_{r}}\times 0.2\,\mathrm{cm}\ {{\rm{s}}}^{-1}.\end{eqnarray} \tag{ 29 }$$

This estimate for v_m is respectably high compared to estimates for the velocity at which dust grains bounced rather than stuck together in the solar nebula, and justifies the remainder of this paper (Güttler et al. 2010). Note that ${v}_{m}\propto$ ${\rho }_{g}^{1/2}$, so the strength of magnetic dipole interactions will generally decrease with orbital position.

In the case of magnetically hard magnetic dipoles not embedded in an external magnetic field, we can approximate that the magnetic torques act to keep the dipole moments optimally aligned. In that limit, Equation (14) simplifies to ${U}_{M}=-2{\forall }_{i1}{\forall }_{i2}{M}^{2}/{r}^{3}$. In this case, a grain with initial velocity ${v}_{\theta }={v}_{0}\,\mathrm{at}\,{r}_{0}={r}_{c}$ would be on an unstable circular orbit, and the corresponding estimate for $C\,\mathrm{is}\,C=1.$ This would be a reasonable approximation for dust grains, each with only a single magnetic domain, and for which Equation (29) would become

$$\begin{eqnarray}&&{v}_{m}\simeq 0.2{\left(\tfrac{C}{\bar{C}}\right)}^{4/3}\sqrt{\tfrac{{f}_{1}{f}_{2}\phi }{{\bar{f}}^{2}\bar{\phi }}}\tfrac{M}{\bar{M}}\tfrac{\mathrm{cm}}{{\rm{s}}}\simeq 171\sqrt{\tfrac{{f}_{1}{f}_{2}\phi }{{\bar{f}}^{2}\bar{\phi }}}\times \tfrac{\mathrm{cm}}{{\rm{s}}},\end{eqnarray} \tag{ 30 }$$

where we have used Equations (10) and (11). More extreme than Equation (29), this value for ${v}_{m}$ is large enough for magnetized interactions to dominate dust coagulation even at scales often associated with fragmentation (Güttler et al. 2010). Note that observations of interstellar dust, however, imply that only a small fraction of interstellar iron could be in the form of such single-domain magnetically saturated iron–nickel dust grains (Draine & Lazarian 1999).

Our current picture for the collisional coagulation phase of dust growth at and below the size associated with chondrule precursors is one where macroscopic grains are stirred by both turbulence and Brownian motion. The degree of gas–dust coupling is captured by the dust Stokes number

$$\begin{eqnarray}&&\mathrm{St}\equiv \tau {\rm{\Omega }},\end{eqnarray} \tag{ 31 }$$

where $\tau$ is the dust drag time and ${\rm{\Omega }}$ the local Keplerian frequency. Near the midplane of a vertically isothermal disk in vertical hydrostatic equilibrium we have

$$\begin{eqnarray}&&\mathrm{St}\simeq \tfrac{\sqrt{2\pi }a\rho }{{{\rm{\Sigma }}}_{g}}\end{eqnarray} \tag{ 32 }$$

where a is a dust grain's radius, $\rho$ its density, and ${{\rm{\Sigma }}}_{g}$ the gas surface density, and in this paper, where relevant, we assume that $\mathrm{St}\,\ll$ 1. In that limit, turbulence drives a relative velocity between dust grains which scales as (Voelk et al. 1980; Cuzzi & Hogan 2003):

$$\begin{eqnarray}&&{v}_{t}\propto \sqrt{\alpha \mathrm{St}}{c}_{s}\end{eqnarray} \tag{ 33 }$$

where the strength of the turbulence is measured through the Shakura–Sunyaev $\alpha$ parameter (Shakura & Sunyaev 1973). We can see from Equations (32) and (33) that larger dust grains collide more violently than smaller ones.

If the two dust grains are very different in size, as we assumed in Section 3, the turbulently driven velocity distribution is quasi-Maxwellian (Hubbard 2013), and we have

$$\begin{eqnarray}&&n(v,{v}_{0})={n}_{0}{\left(\tfrac{v}{{v}_{0}}\right)}^{2}{e}^{-{v}^{2}/2{v}_{0}^{2}},\end{eqnarray} \tag{ 34 }$$

where $n(v)$ is the number density of targets with relative velocity $| v|$ per unit relative velocity. The velocity scale ${v}_{0}$ is controlled by Equation (33). We assume Equation (34) henceforth, noting that it also applies in the Brownian motion limit appropriate for very small particles extremely well coupled to the gas.

The rate at which a dust grain will collide with other grains at velocities between ${v}_{1}\lt {v}_{2}$ is

$$\begin{eqnarray}&&R({v}_{1},{v}_{2})\equiv {\int }_{{v}_{1}}^{{v}_{2}}{dv}\,n(v)\sigma (v)v,\end{eqnarray} \tag{ 35 }$$

where $n(v)$ is the number density of targets with relative velocity $| v|$ per unit relative velocity, and $\sigma$ the effective collisional cross-section (which may depend on the relative velocity). In the absence of magnetic effects, we have only the geometric cross-section, i.e., $\sigma ={\sigma }_{g}$. Combining Equations (34) and (35) we find:

$$\begin{eqnarray}&&{R}_{g}\left(\tfrac{{v}_{1}}{{v}_{0}},\tfrac{{v}_{2}}{{v}_{0}}\right)={n}_{0}{\sigma }_{g}{v}_{0}^{2}{\int }_{\tfrac{{v}_{1}}{{v}_{0}}}^{\tfrac{{v}_{2}}{{v}_{0}}}{du}\,{u}^{3}{e}^{-{u}^{2}/2},\end{eqnarray} \tag{ 36 }$$

$$\begin{eqnarray}&&=\,-{n}_{0}{\sigma }_{g}{v}_{0}^{2}{[({u}^{2}+2){e}^{-{u}^{2}/2}]}_{{v}_{1}/{v}_{0}}^{{v}_{2}/{v}_{0}},\end{eqnarray} \tag{ 37 }$$

where the subscript $g$ (geometric) is used for non-magnetically mediated cases and we have performed the non-dimensionalising substitution $u\,=\,v/{v}_{0}$. Accordingly, the total geometric collision rate is given by

$$\begin{eqnarray}&&{R}_{g}={R}_{g}(0,\infty )=2{n}_{0}{\sigma }_{g}{v}_{0}^{2},\end{eqnarray} \tag{ 38 }$$

where one must recall that ${n}_{0}$ is a number density per unit velocity. Note also that $R$ (and hence all its variants in this paper) is a function of everything that the distribution $n$ depends on. This reduces to $R$ also depending on ${v}_{0}$ if Equation (34) is assumed.

Magnetic interactions only matter for velocities $v\lt {v}_{m}$. Requiring ${v}_{2}\lt {v}_{m}$ we can therefore write the rate of magnetically mediated collisions as:

$$\begin{eqnarray}{R}_{m}\left(\displaystyle \frac{{v}_{1}}{{v}_{0}},\displaystyle \frac{{v}_{2}}{{v}_{0}}\right) & = & {\displaystyle \int }_{{v}_{1}}^{{v}_{2}}{dv}\,n(v,{v}_{0}){\sigma }_{m}(v)v,\\ & = & {n}_{0}{\sigma }_{g}{v}_{0}^{2/3}{v}_{m}^{4/3}{\displaystyle \int }_{\tfrac{{v}_{1}}{{v}_{0}}}^{\tfrac{{v}_{2}}{{v}_{0}}}{du}\,{u}^{5/3}{e}^{-{u}^{2}/2}\\ & = & -{n}_{0}{\sigma }_{g}{v}_{0}^{2/3}{v}_{m}^{4/3}\left[\tfrac{{2}^{\tfrac{1}{3}}}{3}{\rm{\Gamma }}\left(\tfrac{1}{3},\tfrac{{u}^{2}}{2}\right)\right.\\ & & {\left.\phantom{{\rm{\Gamma }}\left(\tfrac{1}{3},\tfrac{{u}^{2}}{2}\right)}+{u}^{\tfrac{2}{3}}{e}^{-{u}^{2}/2}\right]}_{\tfrac{{v}_{1}}{{v}_{0}}}^{\tfrac{{v}_{2}}{{v}_{0}}},\end{eqnarray} \tag{ 39 }$$

where ${\rm{\Gamma }}$ is the incomplete Gamma function. The total increase in interactions due to magnetic effects is just

$$\begin{eqnarray}&&{\bar{R}}_{m}\equiv {R}_{m}(0,{v}_{m}/{v}_{0})-{R}_{g}(0,{v}_{m}/{v}_{0}),\end{eqnarray} \tag{ 40 }$$

and so the fractional increase is

$$\begin{eqnarray}\tfrac{{\bar{R}}_{m}}{{R}_{g}} & = & {\left(\tfrac{{v}_{m}}{{v}_{0}}\right)}^{\tfrac{4}{3}}\left[\tfrac{{2}^{-\tfrac{2}{3}}}{3}\left[{\rm{\Gamma }}\left(\tfrac{1}{3},0\right)-{\rm{\Gamma }}\left(\tfrac{1}{3},\tfrac{{v}_{m}^{2}}{2{v}_{0}^{2}}\right)\right]\right]\\ & & \phantom{{\rm{\Gamma }}\left(\tfrac{1}{3},\tfrac{{v}_{m}^{2}}{2{v}_{0}^{2}}\right)}+{e}^{-{v}_{m}^{2}/2{v}_{0}^{2}}-1.\end{eqnarray} \tag{ 41 }$$

The important control parameter is clearly the ratio ${v}_{m}/{v}_{0}$: if it is large, then magnetic effects are important for the bulk of the potential collisional partners, while if it is small most potential collisional partners are moving too fast for the magnetic effects to play a role.

While Equation (41) is somewhat complicated, many of its terms become negligible for large ${v}_{m}/{v}_{0}$, and we can approximate

$$\begin{eqnarray}\tfrac{{\bar{R}}_{m}}{{R}_{g}}\left(\tfrac{{v}_{m}}{{v}_{0}}\gtrsim 2\right) & \simeq & \tfrac{{2}^{-\tfrac{2}{3}}}{3}{\rm{\Gamma }}\left(\tfrac{1}{3},0\right){\left(\tfrac{{v}_{m}}{{v}_{0}}\right)}^{\tfrac{4}{3}}-1\\ & \simeq & 0.56{\left(\tfrac{{v}_{m}}{{v}_{0}}\right)}^{\tfrac{4}{3}}-1.\end{eqnarray} \tag{ 42 }$$

We plot both Equations (41) and (42) in Figure 1, and one can see that the approximation is extremely good for ${v}_{m}/{v}_{0}\gtrsim$ 2. Further, for ${v}_{m}/{v}_{0}\gt 2.56{\text{}},{\bar{R}}_{m}/{R}_{g}\gt 1$ and magnetic interactions play a dominant role, more than doubling the collision rate.

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If we start with a population of dust grains with varying iron metal mass fractions, then ${v}_{m}\propto \sqrt{{f}_{1}{f}_{2}}$ (see Equation (27)). For very small dust grains, which could even be too small to couple collisionally to turbulence and instead collide through Brownian motion (which also has a Maxwellian distribution), we expect ${v}_{m}\gg {v}_{0}$, and so ${\bar{R}}_{m}/{R}_{g}\simeq 0.56{({v}_{m}/{v}_{0})}^{4/3}$ (further simplifying Equation (42)). In this case, the collision rate between iron-rich grains is enhanced compared to that between iron-poor grains, which can help preserve an initial iron inhomogeneity. Take a scenario in which half of the iron is found in metal-poor grains making up $90 \%$ of the population with ${f}_{\mathrm{mp}}={f}_{0}$(associated with a magnetic velocity v_m), while the other half is found in the 10% of the grains that are metal-rich with ${f}_{\mathrm{mr}}=9{f}_{0}$. In that case, the ratio of the rates at which iron-rich grains will collide with each other compared to colliding with iron-poor grains is

$$\begin{eqnarray}&&\displaystyle \frac{0.1\times {({f}_{\mathrm{mr}}^{2})}^{2/3}}{0.9\times {({f}_{\mathrm{mp}}{f}_{\mathrm{mr}})}^{4/3}}\sim 0.48,\end{eqnarray} \tag{ 43 }$$

over four times the ratio that would occur in the absence of magnetic interactions. While collisional growth will still act to erase inhomogeneities, magnetic interactions dramatically reduce the rate at which it does so.

Collisional dust growth is expected to continue until the grains begin to collide rapidly enough that electrostatic sticking forces are inadequate to hold them together upon impact and they begin to bounce. This will occur for collisional velocities on the order of (Güttler et al. 2010):

$$\begin{eqnarray}&&{v}_{b}\sim {10}^{-2}-1\,\mathrm{cm}\,{{\rm{s}}}^{-1}.\end{eqnarray} \tag{ 44 }$$

These estimates are mostly below that of Equation (29), so it is interested to explore the effects of magnetic interactions on sticking rates. One must note that the release of magnetic potential energy means that magnetically mediated interactions occurring at $v={v}_{b}$ will lead to collisions at velocities $v\gt {v}_{b}$. However, this velocity increase is due to the magnetic potential energy well that the collisional participants must exit for bouncing to occur, and we expect some of the kinetic energy to be thermalized or go into rearranging the dust grains. Magnetic interactions should therefore increase the effective bouncing velocity, although confirmation will require future experimental and numerical work. If the magnetically mediated collisions are violent enough, they could even lead to fragmentation, although the iron components would remain bound (e.g., magnetic erosion; Hubbard 2014). For now, we assume that the critical velocities for bouncing are not changed by consideration of magnetic effects.

Using Equation (35), we can write the sticking rate as

$$\begin{eqnarray}&&S\equiv R(0,{v}_{b}/{v}_{0})={\int }_{0}^{{v}_{b}}{dv}\,n(v,{v}_{0}){\sigma }_{\mathrm{eff}}(v)v.\end{eqnarray} \tag{ 45 }$$

In the absence of magnetic interactions we can use Equation (37) to find:

$$\begin{eqnarray}&&{S}_{g}=2{n}_{0}{\sigma }_{g}{v}_{0}^{2}\left(1-\left[1+\displaystyle \frac{{v}_{b}^{2}}{2{v}_{0}^{2}}\right]{e}^{-{v}_{b}^{2}/2{v}_{0}^{2}}\right).\end{eqnarray} \tag{ 46 }$$

Defining ${v}_{c}\equiv \min ({v}_{b},{v}_{m})$ and using Equations (39) and (46), we can see that the increase in sticking rate due to magnetic interactions is given by

$$\begin{eqnarray}&&{\bar{S}}_{m}\equiv {R}_{m}(0,{v}_{c}/{v}_{0})-{R}_{g}(0,{v}_{c}/{v}_{0})\end{eqnarray} \tag{ 47 }$$

$$\begin{eqnarray} & = & n{\sigma }_{g}{v}_{0}^{2}\left(\displaystyle \frac{{2}^{\tfrac{1}{3}}}{3}{\left(\displaystyle \frac{{v}_{m}}{{v}_{0}}\right)}^{\tfrac{4}{3}}\left[{\rm{\Gamma }}\left(\displaystyle \frac{1}{3},0\right)-{\rm{\Gamma }}\left(\displaystyle \frac{1}{3},\displaystyle \frac{{v}_{c}^{2}}{2{v}_{0}^{2}}\right)\right]\right.\\ & & \left.+\displaystyle \frac{2{v}_{0}^{2}+{v}_{c}^{2}-{v}_{m}^{\tfrac{4}{3}}{v}_{c}^{\tfrac{2}{3}}}{{v}_{0}^{2}}{e}^{-{v}_{c}^{2}/2{v}_{0}^{2}}-2\right),\end{eqnarray} \tag{ 48 }$$

and the fractional increase is ${\bar{S}}_{m}/{S}_{g}$. We plot ${\bar{S}}_{m}/{S}_{g}$ in Figure 2. Note that for large ${v}_{b}/{v}_{0}\gtrsim 3$ the ratio is approximately independent of v_b: almost all interactions occur for velocities $v\sim {v}_{0}\ll {v}_{b}$, so including bouncing has little effect. Note also that even for small ${v}_{m}/{v}_{0}$, magnetic interactions can become arbitrarily important if ${v}_{m}\gt {v}_{b}$: all the interactions that do not bounce are magnetically mediated.

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5.1.1. Limiting Approximations

While the full form of ${\bar{S}}_{m}/{S}_{g}$ is lengthy, one is generally in the limit that ${v}_{c}\gg {v}_{0}$ or ${v}_{c}\ll {v}_{0}$, and both limits admit simplifying approximations. In the limit that ${v}_{c}\gg {v}_{0}$, we have (using Equation (42)):

$$\begin{eqnarray}&&\displaystyle \frac{{\bar{S}}_{m}}{{S}_{g}}\simeq \displaystyle \frac{{\bar{R}}_{m}}{R}\simeq 0.56{\left(\displaystyle \frac{{v}_{m}}{{v}_{0}}\right)}^{\tfrac{4}{3}}-1.\end{eqnarray} \tag{ 49 }$$

This occurs because only a negligible fraction of encounters are at a high enough velocity for bouncing to matter.

In the limit of ${v}_{c}\ll {v}_{0}$, we can approximate Equation (34) as

$$\begin{eqnarray}&&n\simeq {n}_{0}{\left(\displaystyle \frac{v}{{v}_{0}}\right)}^{2},\end{eqnarray} \tag{ 50 }$$

leading to

$$\begin{eqnarray}&&{S}_{g}\simeq {\int }_{0}^{{v}_{b}}{dv}\,{n}_{0}{\sigma }_{g}{\left(\displaystyle \frac{v}{{v}_{0}}\right)}^{2}v={n}_{0}{\sigma }_{g}{v}_{0}^{2}\times \displaystyle \frac{1}{4}{\left(\displaystyle \frac{{v}_{b}}{{v}_{0}}\right)}^{4},\end{eqnarray} \tag{ 51 }$$

$$\begin{eqnarray}&&{R}_{g}(0,{v}_{c}/{v}_{0})\simeq {n}_{0}{\sigma }_{g}{v}_{0}^{2}\times \displaystyle \frac{1}{4}{\left(\displaystyle \frac{{v}_{c}}{{v}_{0}}\right)}^{4}.\end{eqnarray} \tag{ 52 }$$

We also have

$$\begin{eqnarray}{R}_{m}(0,{v}_{c}/{v}_{0}) & \simeq & {\displaystyle \int }_{0}^{{v}_{c}}{dv}\,{n}_{0}{\sigma }_{m}{\left(\displaystyle \frac{v}{{v}_{0}}\right)}^{2}v\\ & \simeq & {n}_{0}{\sigma }_{g}{v}_{0}^{2}\times \displaystyle \frac{3}{8}{\left(\displaystyle \frac{{v}_{c}}{{v}_{0}}\right)}^{8/3}{\left(\displaystyle \frac{{v}_{m}}{{v}_{0}}\right)}^{4/3},\end{eqnarray} \tag{ 53 }$$

and hence

$$\begin{eqnarray}&&\displaystyle \frac{{\bar{S}}_{m}}{{S}_{g}}\simeq \displaystyle \frac{3}{2}\displaystyle \frac{{v}_{c}^{8/3}{v}_{m}^{4/3}}{{v}_{b}^{4}}-{\left(\displaystyle \frac{{v}_{c}}{{v}_{b}}\right)}^{4}.\end{eqnarray} \tag{ 54 }$$

We plot the ratio of these approximations for ${\bar{S}}_{m}/{S}_{g}$ to the actual value in Figure 3. We can see that Equation (49) is a good approximation as long as we have both ${v}_{m}/{v}_{0}\gt 2.5$ and ${v}_{b}/{v}_{0}\gt 3;$ and that Equation (54) is a good approximation for ${v}_{b}/{v}_{0}\lt 0.5$.

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In addition to bouncing, dust grains in protoplanetary disks colliding at velocities

$$\begin{eqnarray}&&v\gt {v}_{f}\simeq 100\,\mathrm{cm}\ {{\rm{s}}}^{-1}\end{eqnarray} \tag{ 55 }$$

will fragment (Güttler et al. 2010). This value for v_f lies well above our estimate for v_m for magnetically soft grains (Equations (29)), but is comparable to the limiting value for saturated magnetically hard grains (Equation (30)). Including magnetic forces could change this picture by allowing magnetic erosion (Hubbard 2014), but exploring that in detail is far beyond the scope of this paper and will require future experimental or numerical studies of high velocity encounters between aggregates of monomers, some of which are magnetized. Further, mass transfer is a possibility when colliding dust grains are very different in size, allowing for growth even at high collision speeds (Windmark et al. 2012). For simplicity, we nonetheless adopt the picture of Equation (55) for both the geometric and magnetized cases. Recall also that the collisional velocity scale v₀ is controlled by the dynamics of the disk. We have assumed turbulent stirring, with v₀ scaling with the square root of the Stokes numbers of the dust grains (see Equations (32) and (33)).

At some critical collisional velocity scale, dust grain collisions that result in sticking will balance those that result in fragmentation. We parameterize this balance through

$$\begin{eqnarray}&&R(0,{v}_{b}/{v}_{0})=\psi R({v}_{f}/{v}_{0},\infty ),\end{eqnarray} \tag{ 56 }$$

where ψ is the critical growth-neutral ratio of the sticking rate to the fragmentation rate. We also choose to define the ratio of the fragmentation and bouncing velocities as

$$\begin{eqnarray}&&\xi \equiv {v}_{f}/{v}_{b}.\end{eqnarray} \tag{ 57 }$$

Note that v_b, v_f, and ξ depend on the microphysics of the collisions, and we have assumed that they are independent of any magnetic interactions. Combining Equations (29), (44), and (55), we expect ${v}_{f}\gg {v}_{m}\gt {v}_{b}$ for magnetically soft grains. Accordingly, for a given dust grain size (and hence collisional velocity scale v₀), including magnetic interactions would not effect the fragmentation rate, but would increase the sticking rate. If ψ is constant, then magnetized grains would find a balance between sticking and fragmentation for larger collisional velocity scales, and hence at larger dust grain sizes. As is explored in Hubbard (2016), differences in dust grain sizes lead to aerodynamical sorting which could be correlated with the chondrule formation process. Such a correlation would provide an explanation for the observed partitioning of iron and of tungsten isotopes between chondrules and matrix (Grossman & Wasson 1982; Budde et al. 2016).

In the geometric case we can consider Equation (56) to be an equation for ${v}_{b}/{v}_{0}$(and hence ${v}_{f}/{v}_{0}$) as a function of ψ and ξ. In this case Equation (56) becomes

$$\begin{eqnarray}&&2-\left[{\left(\displaystyle \frac{{v}_{b}}{{v}_{0}}\right)}^{2}+2\right]{e}^{-\tfrac{{v}_{b}^{2}}{2{v}_{0}^{2}}}=\psi \left[{\left(\displaystyle \frac{\xi {v}_{b}}{{v}_{0}}\right)}^{2}+2\right]{e}^{-\tfrac{{\xi }^{2}{v}_{b}^{2}}{2{v}_{0}^{2}}},\end{eqnarray} \tag{ 58 }$$

noting that n₀ and σ have canceled, and v₀ only shows up in ratio with v_b. We write ${u}_{g}(\psi ,\xi )$ as the solution of Equation (58) for ${v}_{b}/{v}_{0}$. However, it is crucial to note that we have assumed that v_b is set by the microphysics, so u_g actually measures v₀. Accordingly, we define

$$\begin{eqnarray}&&{v}_{0,g}(\psi ,\xi ,{v}_{b})\equiv {v}_{b}/{u}_{g},\end{eqnarray} \tag{ 59 }$$

the collisional velocity scale for Equation (34) at which sticking and fragmentation are balanced for a set of parameters $(\psi ,\xi ,{v}_{b})$.

For the magnetic case we can similarly use

$$\begin{eqnarray}&&{R}_{m}(0,{v}_{b}/{v}_{0})=\psi {R}_{m}({v}_{f}/{v}_{0},\infty )\end{eqnarray} \tag{ 60 }$$

to define ${u}_{m}(\psi ,\xi ,{v}_{b},{v}_{m})$ as the solution of Equation (60) for ${v}_{b}/{v}_{0}$. We will use u_m to define

$$\begin{eqnarray}&&{v}_{0,m}(\psi ,\xi ,{v}_{b},{v}_{m})\equiv {v}_{b}/{u}_{m}.\end{eqnarray} \tag{ 61 }$$

The ratio ${v}_{0,m}/{v}_{0,g}$, which is also the ratio ${u}_{g}/{u}_{m}$ because v_b and v_f are not altered by magnetic interactions, describes how much more violently magnetized dust grains have to be stirred for their collisions to be growth neutral; and hence how much larger they can grow.

5.2.1. Exploring the Consequences of Magnetization on the Balance between Sticking and Fragmentation

The ratio ${v}_{0,m}/{v}_{0,g}$ depends on too many parameters to fully explore in this paper, but by restricting some of the parameters we can estimate the impact that magnetization has on the dust grain size at which sticking and fragmentation are balanced. Based on Equations (29), (44), and (55) we adopt

$$\begin{eqnarray}&&{v}_{b}=0.1\,\mathrm{cm}\ {{\rm{s}}}^{-1}\lt {v}_{m}\ll {v}_{f}=100\,\mathrm{cm}\ {{\rm{s}}}^{-1},\end{eqnarray} \tag{ 62 }$$

i.e., $\xi ={10}^{3}$. Given this ratio between v_b and v_f we can be certain that at least one of the limits ${v}_{b}\ll {v}_{0}$ or ${v}_{f}\gg {v}_{0}$ holds, and accordingly that at least one of the sticking or fragmentation rates is small. We will also assume that ψ, the growth-neutral ratio of the sticking to fragmentation rates, is neither very large nor very small. Combining Equation (62) with this condition on ψ implies that both the sticking and fragmentation rates must be small, and we have

$$\begin{eqnarray}&&{v}_{b}\ll {v}_{0}\ll {v}_{f},\end{eqnarray} \tag{ 63 }$$

which allows significant simplifications.

Because ${v}_{b}\ll {v}_{0}$ we can use Equation (51) to rewrite Equation (58) as

$$\begin{eqnarray}&&4\psi \left(\displaystyle \frac{{\xi }^{2}}{{u}_{g}^{2}}+\displaystyle \frac{2}{{u}_{g}^{4}}\right){e}^{-{\xi }^{2}{u}_{g}^{2}/2}-1\,=\,0.\end{eqnarray} \tag{ 64 }$$

We also have ${v}_{c}\equiv \min ({v}_{b},{v}_{m})={v}_{b}\ll {v}_{0}$ and ${v}_{m}\ll {v}_{f}$, so we can use Equation (53) to rewrite Equation (60) as

$$\begin{eqnarray}&&\displaystyle \frac{8}{3}\psi ({\xi }^{2}{u}_{m}^{-2/3}+2{u}_{m}^{-8/3}){e}^{-{\xi }^{2}{u}_{m}^{2}/2}-{\left(\displaystyle \frac{{v}_{m}}{{v}_{0}}\right)}^{4/3}=0.\end{eqnarray} \tag{ 65 }$$

We can parameterize the magnetic velocity v_m in terms of the bouncing velocity and u_m through:

$$\begin{eqnarray}&&\displaystyle \frac{{v}_{m}}{{v}_{0}}=\displaystyle \frac{{v}_{m}{v}_{0,m}{v}_{b}}{{v}_{0,m}{v}_{0}{v}_{b}}=\displaystyle \frac{{v}_{m}}{{v}_{b}}\displaystyle \frac{{v}_{0,m}}{{v}_{0}}{u}_{m}.\end{eqnarray} \tag{ 66 }$$

Because Equation (65) is an equation for ${v}_{0,m}$ (through the intermediary u_m), its solution for ${v}_{0,m}$ is unchanged by setting ${v}_{0}={v}_{0,m}$ in Equation (66). This reduces Equation (65) to

$$\begin{eqnarray}&&\displaystyle \frac{8}{3}\psi \left({\xi }^{2}{u}_{m}^{-\tfrac{2}{3}}+2{u}_{m}^{-\tfrac{8}{3}}\right){e}^{-\tfrac{{\xi }^{2}{u}_{m}^{2}}{2}}-{\left(\displaystyle \frac{{v}_{m}}{{v}_{b}}{u}_{m}\right)}^{\tfrac{4}{3}}=0.\end{eqnarray} \tag{ 67 }$$

For our fixed $\xi ={10}^{3}$, Equations (64) and (67) can be solved for u_g and u_m (and hence ${v}_{0,m}/{v}_{0,g}$) as functions of ψ and ${v}_{m}/{v}_{b}$. In Figure 4 we plot the ratio ${v}_{0,m}/{v}_{0,g}$, along with the implied difference in Stokes number, for growth-neutral grains. One immediate observation is that the equilibrium size is only moderately shifted even when ${v}_{m}\gg {v}_{b}$ and magnetization strongly increases the sticking rate. We can understand this by taking the derivative of the fragmentation rate (i.e., the right-hand side of Equation (58)) with respect to v₀:

$$\begin{eqnarray}&&{\partial }_{{v}_{0}}\left(\psi \left[{\left(\displaystyle \frac{{v}_{f}}{{v}_{0}}\right)}^{2}+2\right]{e}^{-\tfrac{{v}_{f}^{2}}{2{v}_{0}^{2}}}\right)\simeq \displaystyle \frac{\psi }{{v}_{0}}{\left(\displaystyle \frac{{v}_{f}}{{v}_{0}}\right)}^{4}{e}^{-\tfrac{{v}_{f}^{2}}{2{v}_{0}^{2}}},\end{eqnarray} \tag{ 68 }$$

where we have used $\xi {v}_{b}={v}_{f}\gg {v}_{0}$. Dividing the fragmentation rate by its derivative we can estimate how sensitive it is to small changes in v₀:

$$\begin{eqnarray}&&\displaystyle \frac{\mathrm{RHS}}{{\partial }_{{v}_{0}}\mathrm{RHS}}\simeq {v}_{0}{\left(\displaystyle \frac{{v}_{0}}{{v}_{f}}\right)}^{2}\ll {v}_{0}.\end{eqnarray} \tag{ 69 }$$

The exponential term in the fragmentation rate is an extremely sensitive function of velocity scale, so even significant changes in the sticking rate will be balanced by small changes in the collisional velocity scale.

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5.2.2. Saturated Magnetization Case

5.2.3. Varying f