Igneous Rim Accretion on Chondrules in Low-velocity Shock Waves

We assume a simple gas structure. In our model, the gas velocity with respect to the shock front, v _g , is expressed as a function of the distance from the shock front, x:

$$\begin{eqnarray}{v}_{g}=\left\{\begin{array}{ll}{v}_{0} & (x\lt 0),\\ {v}_{0}+({v}_{\mathrm{post}}-{v}_{0})\exp (-x/L) & (x\geqslant 0),\end{array}\right.\end{eqnarray} \tag{ 1 }$$

where v₀ is the preshock gas velocity with respect to the shock front, namely, shock velocity; v_post is the postshock gas velocity with respect to the shock front; and L is the spatial scale of the shock. The postshock gas velocity is given by the Rankine–Hugoniot relations as v_post = [(γ − 1)/(γ + 1)]v₀, where γ is the ratio of specific heats, γ = 1.4. We take L as a parameter, and its fiducial value is L = 10³ km (Miura & Nakamoto 2005). The temperature of the gas, T _g , is

$$\begin{eqnarray}{T}_{g}=\left\{\begin{array}{ll}{T}_{0} & (x\lt 0),\\ {T}_{0}+({T}_{\mathrm{post}}-{T}_{0})\exp (-x/L) & (x\geqslant 0),\end{array}\right.\end{eqnarray} \tag{ 2 }$$

where the preshock gas temperature is T₀ = 500 K and the postshock gas temperature is T_post = 2000 K. The postshock region is approximately isobaric (e.g., Susa et al. 1998; Iida et al. 2001). According to the Rankine–Hugoniot relation, the gas number density in the postshock region, n _g , can be expressed by

$$\begin{eqnarray}&&{n}_{g}={n}_{0}\frac{{T}_{0}}{{T}_{g}}\frac{4{s}_{0}^{2}-(\gamma -1)}{\gamma +1},\end{eqnarray} \tag{ 3 }$$

where n₀ is the preshock gas number density; ${s}_{0}\,={v}_{0}/{(2{k}_{{\rm{B}}}{T}_{0}/{m}_{{\rm{g}}})}^{1/2}$, with k_B the Boltzmann constant; and m_g = 3.34 × 10⁻²⁴ g is the gas molecule mass.

In the postshock region, the velocities of chondrules and dust particles with respect to the shock front are given by

$$\begin{eqnarray}&&m\frac{{dv}}{{dx}}=-\frac{{C}_{D}}{2}\pi {a}^{2}{m}_{g}{n}_{g}\frac{| v-{v}_{g}| }{v}(v-{v}_{g}),\end{eqnarray} \tag{ 4 }$$

where v, m, and a are the velocity, mass, and radius of a chondrule or particle, respectively. We consider the radii of chondrules, a_ch, and initial dust particles, a_d,init, as parameters, and their fiducial values are a_ch = 10³ μm and a_d,init = 20 μm. The densities of the chondrules and the dust particles are the same, ρ_ch = ρ_d = 3.3 g cm⁻³, whose values are taken from the typical values for chondrules (e.g., Hughes 1978; Friedrich et al. 2015). This density is close to that of forsterite, from which we take the other material properties in this paper. The drag coefficient, C _D , is given by (Arakawa et al. 2022)

$$\begin{eqnarray}\begin{array}{rcl}{C}_{D} & = & \frac{2}{3s}\sqrt{\frac{\pi T}{{T}_{g}}}\\ & & +\frac{2{s}^{2}+1}{\sqrt{\pi }{s}^{3}}\exp (-{s}^{2})+\frac{4{s}^{4}+2{s}^{2}-1}{2{s}^{4}}\mathrm{erf}(s)\\ & \simeq & \frac{2\sqrt{\pi }}{3s}+\frac{2{s}^{2}+1}{\sqrt{\pi }{s}^{3}}\exp (-{s}^{2})\\ & & +\frac{4{s}^{4}+2{s}^{2}-1}{2{s}^{4}}\mathrm{erf}(s),\end{array}\end{eqnarray} \tag{ 5 }$$

where $s=v/{(2{k}_{{\rm{B}}}{T}_{g}/{m}_{g})}^{1/2}$. This approximation works well when T ≃ T _g or s > 1. Either condition is satisfied behind the shock front when dust particles are molten.

The temperatures of chondrules and dust are derived by the equations of energy,

$$\begin{eqnarray}&&{{mc}}_{\mathrm{heat}}\displaystyle \frac{{dT}}{{dx}}=\displaystyle \frac{4\pi {a}^{2}}{v}({\rm{\Gamma }}-{{\rm{\Lambda }}}_{\mathrm{rad}}-{{\rm{\Lambda }}}_{\mathrm{evap}}),\end{eqnarray} \tag{ 6 }$$

where c_heat = 1.42 × 10⁷ erg g⁻¹ K⁻¹ is the specific heat, Γ is the heating rate via the energy transfer from gas, Λ_rad is the radiative cooling, and Λ_evap is the latent heat cooling by evaporation per unit area, respectively (e.g., Miura et al. 2002; Miura & Nakamoto 2005). The heating rate via the energy transfer from gas is given by

$$\begin{eqnarray}&&{\rm{\Gamma }}={m}_{g}{n}_{g}| v-{v}_{g}| ({T}_{\mathrm{rec}}-T){C}_{H},\end{eqnarray} \tag{ 7 }$$

where the recovery temperature, T_rec, and the heat transfer function, C _H , are (e.g., Gombosi et al. 1986)

$$\begin{eqnarray}\begin{array}{rcl}{T}_{\mathrm{rec}} & = & \frac{{T}_{g}}{\gamma +1}\left[2\gamma +2(\gamma -1){s}^{2}\right.\\ & & -\,\left.\frac{\gamma -1}{0.5+{s}^{2}+(s/\sqrt{\pi })\exp (-{s}^{2}){\mathrm{erf}}^{-1}(s)}\right]\end{array}\end{eqnarray} \tag{ 8 }$$

and

$$\begin{eqnarray}&&{C}_{H}=\frac{\gamma +1}{\gamma -1}\frac{{k}_{{\rm{B}}}}{8{m}_{g}{s}^{2}}\left[\frac{s}{\sqrt{\pi }}\exp (-{s}^{2})+\left(\frac{1}{2}+{s}^{2}\right)\mathrm{erf}(s)\right],\end{eqnarray} \tag{ 9 }$$

respectively. The radiative cooling rate is given by

$$\begin{eqnarray}&&{{\rm{\Lambda }}}_{\mathrm{rad}}={\epsilon }_{\mathrm{emit}}{\sigma }_{\mathrm{SB}}{T}^{4}-{\epsilon }_{\mathrm{emit}}{\sigma }_{\mathrm{SB}}{T}_{0}^{4},\end{eqnarray} \tag{ 10 }$$

where _emit is the emission coefficient and σ_SB = 5.67 × 10⁻⁵ erg cm⁻² K⁻⁴ s⁻¹ is the Stefan–Boltzmann constant. The emission coefficient is (Rizk et al. 1991; Miura & Nakamoto 2005)

$$\begin{eqnarray}&&{\epsilon }_{\mathrm{emit}}=\min \left[2.1543\times {10}^{-3}{\left(\displaystyle \frac{a}{1\,\mu {\rm{m}}}\right)}^{0.8253},1\right].\end{eqnarray} \tag{ 11 }$$

The latent heat cooling by evaporation is the product of the evaporation rate, J_evap, and the latent heat of evaporation, L_evap,

$$\begin{eqnarray}&&{{\rm{\Lambda }}}_{\mathrm{evap}}={J}_{\mathrm{evap}}{L}_{\mathrm{evap}},\end{eqnarray} \tag{ 12 }$$

where J_evap is

$$\begin{eqnarray}\begin{array}{rcl}{J}_{\mathrm{evap}} & = & 691{\left(\displaystyle \frac{{p}_{{{\rm{H}}}_{2}}}{100\,{\text{dyn cm}}^{-2}}\right)}^{1/2}{\left(\displaystyle \frac{T}{{T}_{\mathrm{melt}}}\right)}^{-1/2}\\ & & \times \,\exp \left(-\displaystyle \frac{3.17\times {10}^{4}\,{\rm{K}}}{T}\right)\,{\rm{g}}\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1},\end{array}\end{eqnarray} \tag{ 13 }$$

for forsterite (Miura et al. 2002; Miura & Nakamoto 2005), and L_evap = 1.12 × 10¹¹ erg g⁻¹ (Miura et al. 2002). ³ The ambient gas pressure, ${p}_{{{\rm{H}}}_{2}}$, is the summation of the hydrostatic pressure and the ram pressure,

$$\begin{eqnarray}&&{p}_{{{\rm{H}}}_{2}}={n}_{{\rm{g}}}{k}_{{\rm{B}}}{T}_{g}+\frac{1}{3}{m}_{g}{n}_{g}{(v-{v}_{g})}^{2},\end{eqnarray} \tag{ 14 }$$

and the melting temperature is T_melt = 2171 K. The latent heat cooling by evaporation becomes effective when the temperature of dust or a chondrule approaches the melting temperature.

Dust particles begin to melt when their temperature reaches the melting temperature. Following Miura & Nakamoto (2005), we consider the phase transition of chondrules and dust between the solid and liquid. We express the phase transition by the solid (unmelted) mass fraction, S,

$$\begin{eqnarray}&&{{mL}}_{\mathrm{melt}}\displaystyle \frac{{dS}}{{dx}}=\displaystyle \frac{4\pi {a}^{2}}{v}({\rm{\Gamma }}-{{\rm{\Lambda }}}_{\mathrm{rad}}-{{\rm{\Lambda }}}_{\mathrm{evap}}),\end{eqnarray} \tag{ 15 }$$

where L_melt = 4.47 × 10⁹ erg g⁻¹ is the latent heat of melting. During the phase transition (0 < S < 1), we assume that the temperature of a chondrule or dust remains constant and the net energy input is used for the phase transition. Molten dust particles crystallize according to the cooling. Droplets, which are dust particles with S = 0, do not solidify immediately at T_melt since they cause supercooling (e.g., Nagashima et al. 2006, 2008; Arakawa & Nakamoto 2016). In this study, we assume that the crystallization temperature of droplets is T _c = 1000 K. For simplicity, we do not consider collisions between droplets that cause crystallization.

We consider the size evolution of dust particles. The size change of dust particles, $\left({{da}}_{d}/{dx}\right)$, is given by the summation of its contribution from boiling and evaporation:

$$\begin{eqnarray}&&\displaystyle \frac{{{da}}_{d}}{{dx}}={\left(\displaystyle \frac{{{da}}_{d}}{{dx}}\right)}_{\mathrm{boil}}+{\left(\displaystyle \frac{{{da}}_{d}}{{dx}}\right)}_{\mathrm{evap}}.\end{eqnarray} \tag{ 16 }$$

Dust particles boil when their vapor pressure, p_eq, exceeds the gas pressure, ${p}_{{{\rm{H}}}_{2}}$ (Miura et al. 2002), where

$$\begin{eqnarray}&&{p}_{\mathrm{eq}}=3.20\times {10}^{8}\exp \left(-\displaystyle \frac{6.18\times {10}^{4}\,{\rm{K}}}{T}\right)\,{\rm{bar}}.\end{eqnarray} \tag{ 17 }$$

The boiling rate of a dust particle is

$$\begin{eqnarray}&&-4\pi {a}_{d}^{2}{\rho }_{d}{L}_{\mathrm{boil}}{\left(\displaystyle \frac{{{da}}_{d}}{{dx}}\right)}_{\mathrm{boil}}=\displaystyle \frac{4\pi {a}_{d}^{2}}{{v}_{d}}({\rm{\Gamma }}-{{\rm{\Lambda }}}_{\mathrm{rad}}-{{\rm{\Lambda }}}_{\mathrm{evap}}),\end{eqnarray} \tag{ 18 }$$

where L_boil = 1.64 × 10¹¹ erg g⁻¹ is the latent heat of boiling. The size of a dust particle also decreases owing to evaporation from the precursor surface, and its rate is

$$\begin{eqnarray}&&-4\pi {a}_{d}^{2}{\rho }_{d}{\left(\displaystyle \frac{{{da}}_{d}}{{dx}}\right)}_{\mathrm{evap}}=\displaystyle \frac{4\pi {a}_{d}^{2}}{{v}_{d}}{J}_{\mathrm{evap}}.\end{eqnarray} \tag{ 19 }$$

Miura & Nakamoto (2005) showed that evaporation is the main mechanism to shrink the dust size since the period of dust boiling is very short. We do not consider the size evolution of chondrules since evaporation works more for smaller grains.

Chondrules accrete droplets and/or crystallized them in the postshock region owing to their relative velocities. The accretion rate is

$$\begin{eqnarray}&&\displaystyle \frac{{{dm}}_{\mathrm{ch}}}{{dx}}={{Qm}}_{d}{n}_{d}\pi {a}_{\mathrm{ch}}^{2}\displaystyle \frac{{v}_{\mathrm{imp}}}{{v}_{\mathrm{ch}}},\end{eqnarray} \tag{ 20 }$$

where Q is the accretion efficiency, n_d is the number density of the dust particles, and v_imp = ∣v_d − v_ch∣ is the impact velocity. We simply consider that the number density of the dust particles is

$$\begin{eqnarray}&&{n}_{d}=\chi \frac{{m}_{g}{n}_{g}}{{m}_{d}},\end{eqnarray} \tag{ 21 }$$

where χ = 0.1. This dust-to-gas ratio is higher than 0.01 (the column dust-to-gas mass ratio in the minimum-mass solar nebula; Hayashi 1981), considering that chondrules and igneous rims are accreted in dusty environments (Rubin 2010; Schrader et al. 2013; Tenner et al. 2015). The final accreted rim thicknesses can be scaled by χ. We note that the realistic χ value is affected by dust velocity, accretion from chondrules, dust recondensation, dust evaporated from planetesimals (Tanaka et al. 2013; Arakawa et al. 2022), and disruption in collisions (e.g., Jacquet & Thompson 2014; Liffman 2019). The assumption of χ is discussed in Section 4. The growth rate of the rim thickness is given by

$$\begin{eqnarray}&&\displaystyle \frac{{{da}}_{\mathrm{ch}}}{{dx}}=\displaystyle \frac{1}{4\pi {\rho }_{\mathrm{ch}}{a}_{\mathrm{ch}}^{2}}\displaystyle \frac{{{dm}}_{\mathrm{ch}}}{{dx}},\end{eqnarray} \tag{ 22 }$$

under the assumption that the density of the accreted rim is the same as ρ_ch. The average thickness of the rims, Δ, is the difference between the chondrule size and the size of the chondrules with the accreted rims.

We consider the accretion criteria for molten dust particles and solid dust particles. For molten dust particles, we consider the accretion criteria of the molten (or partially molten) particles on the solid chondrules. The accretion condition between molten (or partially molten) particles is given by the Weber number, We, which is the ratio of the impact energy to the surface energy,

$$\begin{eqnarray}&&\mathrm{We}=\displaystyle \frac{2{\rho }_{d}{{av}}_{\mathrm{imp}}^{2}}{\sigma },\end{eqnarray} \tag{ 23 }$$

where σ = 400 erg cm⁻² is the surface tension (Murase & McBirney 1973). According to Mundo et al. (1995), we consider that molten dust particles are accreted when the Weber number is smaller than the critical Weber number for accretion,

$$\begin{eqnarray}&&{\mathrm{We}}_{\mathrm{cr}}=\displaystyle \frac{9}{2}{\beta }^{4}\mathrm{Ca}+3(1-\cos {\rm{\Theta }}){\beta }^{2}-12,\end{eqnarray} \tag{ 24 }$$

where β ≃ 2–3 is the ratio of the maximum spreading diameter of the droplet to its initial diameter (Chandra & Avedisian1991), Θ is the contact angle, and Ca is the capillary number,

$$\begin{eqnarray}&&\mathrm{Ca}=\displaystyle \frac{\eta {v}_{\mathrm{imp}}}{\sigma },\end{eqnarray} \tag{ 25 }$$

which is the function of the viscosity, η (Hubbard 2015),

$$\begin{eqnarray}&&{\mathrm{log}}_{10}\left(\displaystyle \frac{\eta }{1\,{\rm{P}}}\right)=-3.55+\displaystyle \frac{5084.9\,{\rm{K}}}{T-584.9\,{\rm{K}}}.\end{eqnarray} \tag{ 26 }$$

We ignore the dependence of the viscosity on the solid mass fraction (e.g., Roscoe 1952; Liu et al. 2018) since droplets are rapidly crystallized in the postshock region. We simply adopt the maximum We_cr values by taking β = 3 and $\cos {\rm{\Theta }}=-1$. These values do not affect results much since the critical Weber number is significantly affected by Ca, i.e., v_imp, and T. We assume Q = 1 when We < We_cr; otherwise, Q = 0, for molten dust particles.

The accretion efficiency of solid dust particles is

$$\begin{eqnarray}Q=\left\{\begin{array}{ll}{Q}_{\mathrm{ad}} & ({v}_{\min }\leqslant {v}_{\mathrm{imp}}\leqslant {v}_{\max }),\\ 0 & ({\rm{otherwise}}),\end{array}\right.\end{eqnarray} \tag{ 27 }$$

where Q_ad = 0.5, ${v}_{\min }=0.1\,\mathrm{km}\,{{\rm{s}}}^{-1}$, and ${v}_{\max }=1\,\mathrm{km}\,{{\rm{s}}}^{-1}$, taking from the fiducial case of Arakawa et al. (2022). The velocity range of the accretion is based on the experimental studies, which consider the adhesion of ceramic particles on a ceramic substrate (e.g., Hanft et al. 2015). Dust particles with impact velocities less than ${v}_{\min }$ bounce and do not stick. We ignore erosion in high-velocity collisions. We do not consider collisions between dust particles.

We explain how igneous rims are accreted behind the shock front in our fiducial case, with v_imp = 13 km s⁻¹, n₀ = 10^13.5 cm⁻³, L = 10³ km, a_ch = 10³ μm, and a_d,init = 20 μm. The temperature of the dust particles increases quickly and reaches the melting temperature (top left panel of Figure 2). The temperature of the dust particles is constant during melting, and it increases again after they become droplets (S = 0; top middle panel of Figure 2). The temperature of the droplets becomes peaked at x ≃ 4 km when the heating rate is balanced with the cooling rate, which becomes higher as the temperature increases. Droplets are cooled by evaporation, initially, and then radiation. Droplets are crystallized at T _c . Through this evolution, the dust size shrinks and the final size is 13 μm (top right panel of Figure 2).

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Chondrules are also heated, and they reach the peak temperature at x ≃ 110 km. The peak temperature is lower than the melting temperature. This is because chondrules have a higher radiative cooling rate than dust since the emission coefficient increases as the size of the particle increases (Equation (11)). Chondrules do not melt, and S_ch is always 1.

The impact velocity between chondrules and dust particles initially increases (bottom left panel of Figure 2). Due to the stopping length dependence on the size,

$$\begin{eqnarray}\begin{array}{rcl}{l}_{\mathrm{stop}} & \simeq & 3.3\times {10}^{3}\,\mathrm{km}\\ & & \times \,\left(\frac{a}{1\,\mathrm{mm}}\right){\left(\frac{v}{v-{v}_{g}}\right)}^{2}{\left(\frac{{n}_{0}}{{10}^{13.5}{\text{cm}}^{-3}}\right)}^{-1},\end{array}\end{eqnarray} \tag{ 28 }$$

the relative velocity of a dust particle to the gas decreases, and that of a chondrule does not change. The impact velocity becomes maximum, v_imp = 9.6 km s⁻¹, which is close to ∣v_post − v₀∣ = 10.8 km s⁻¹, at x ≃ 76 km. The impact velocity decreases as the relative velocity of a chondrule to gas decreases. The impact velocity once reaches 0 but increases again (Arakawa & Nakamoto 2019; Arakawa et al. 2022; see also the Appendix). The second-peak impact velocity is 1.2 km s⁻¹ at x ≃ 2.8 × 10³ km, and then the impact velocity decreases.

The Weber number and the critical Weber number are given by the temperature and impact velocity and are shown in the bottom middle panel of Figure 2. The Weber number is larger than the critical Weber number during T_d > T_melt. The Weber number peaks at x ≃ 74 km and begins to decrease, reflecting its v_imp dependence. The critical Weber number continues to increase owing to its temperature dependence. As a result, the Weber number becomes smaller than the critical Weber number at x ≃ 71 km, where dust is in the supercooling state (T_d ≃ 1.6 × 10³ K). The condition of the droplet accretion (We < We_cr) is satisfied in the supercooling state since

$$\begin{eqnarray}\begin{array}{rcl}T & \lt & 584.9\,{\rm{K}}\\ & & +\,\displaystyle \frac{5084.9\,{\rm{K}}}{5.07+\mathrm{log}\left[\left(\tfrac{{a}_{d}}{20\,\mu {\rm{m}}}\right)\left(\tfrac{{v}_{\mathrm{imp}}}{10\,\mathrm{km}\,{{\rm{s}}}^{-1}}\right)\right]},\end{array}\end{eqnarray} \tag{ 29 }$$

which is about 1600 K. The droplet accretion occurs in T _c < T < T_cr ≃ 1600 K. The droplet accretion stops when the supercooling droplets crystallize at x = 1.1 × 10³ km. Through the droplet accretion, chondrules acquire 7.9 μm igneous rims (bottom right panel of Figure 2).

The crystallized dust particles continue to be accreted onto chondrules. When the dust particles crystallize at x = 1.1 × 10³ km, the impact velocity is 1.0 km s⁻¹ and decreasing. Dust particles are accreted except around the second-peak impact velocity, where ${v}_{\mathrm{imp}}\gt {v}_{\max }$. Finally, the thickness of an igneous rim becomes 13.3 μm. ⁴

We find that there are two stages of the accretion of igneous rims: droplet accretion and subsequent crystallized dust accretion (see Figure 1). The component dust particles of both layers experience a melting event. The accretion of the crystallized dust particles occurs since chondrules still have the relative velocity to gas when the dust crystallizes. Considering T _g = T_d , which works well when the dust stopping length, l_stop,d, is shorter than the shock scale, L, the dust crystallizes when the dust temperature becomes T _c at x = x _c , which is given by

$$\begin{eqnarray}&&{T}_{g}={T}_{c}\iff {x}_{c}=L\mathrm{ln}\left(\frac{{T}_{\mathrm{post}}-{T}_{0}}{{T}_{c}-{T}_{0}}\right)\simeq L.\end{eqnarray} \tag{ 30 }$$

When the relative velocity of chondrules is not totally damped at L, i.e., l_stop,ch ≳ L, chondrules accrete crystallized dust particles. The effect of the relation between l_stop,ch and L is described in Section 3.2.2.

3.2.1. Shock Velocity, Preshock Gas Number Density, and Initial Dust Size

Igneous rims are accreted on chondrules in low-velocity shock waves. In this section, we present when the igneous rim accretion occurs and how thick the rims are. The accretion of igneous rims occurs when droplets are formed: the dust becomes droplets and does not evaporate completely. These conditions depend on the shock velocity, preshock gas number density, and initial dust size (Miura & Nakamoto 2005). We plot the condition for the formation of droplets in Figure 3. For example, the droplet formation occurs when v₀ is between 12 and 21 km s⁻¹ in the case in which n₀ = 10^13.5 cm⁻³, L = 10³ km, a_ch = 10³ μm, and a_d,init = 20 μm. Dust evaporates completely when the shock velocity is higher than this range, and dust does not melt when the shock velocity is lower. Both boundary velocities of the melting and evaporation of dust become lower as the preshock gas number density increases. This is because the heating rate via the energy transfer from gas is proportional to the preshock gas number density (Equation (7)), although the radiative cooling rate is independent of the preshock gas number density (Equation (10)). We note that chondrules also melt in high n₀ and/or v₀.

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The shock velocity range for the droplet formation becomes wider as the initial dust size increases since a higher shock velocity is needed to evaporate completely. In these shock velocity ranges, dust particles evaporate and their sizes shrink. The accreted dust sizes are different according to the shock velocity, preshock gas number density, and initial dust size. We plot the size ratios between the final and initial dust sizes, a_d,last/a_d,init in Figure 3. The dust sizes shrink more as the shock velocity and preshock gas number density are higher. This suggests that if we consider a certain dust size, igneous rims are composed of larger grains as the disk gas dissipates when the shock velocity is the same. The observed dust sizes of the components of the igneous rims range from ≤3 to 100 μm, and the average dust sizes are ∼10 μm in Allende and ∼4 μm in Semarkona (Rubin 1984). Such size distributions are reproduced in a single rim-forming shock event when the precursor dust particles have size distributions, where the precursor dust sizes are between the observed maximum size and the boundary size of the dust evaporation.

The thicknesses of the accreted rims are shown in Figure 4. This figure shows that the thickness increases as the shock velocity increases. This dependence is mainly because the dust number density increases as the shock velocity increases (Equations (3) and (21)). This makes the thicknesses of the rims from the accretions of both droplets and crystallized particles larger. However, the thicknesses of the rims from crystallized particles are not an increasing function of the shock velocity when the preshock gas number densities are n₀ = 10^13.5 and 10¹⁴ g cm⁻³. In these cases, the impact velocity of the crystallized particles exceeds ${v}_{\max }$ (Appendix). This makes the region where the crystallized particles can be accreted by chondrules shorter as the shock velocity increases.

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The preshock gas number density also affects the rim thickness through the dust number density. The rims from the accretion of the droplets become thicker as the preshock gas number density increases owing to the dust number density. In addition to the dust number density, the preshock gas number density affects the rim thickness of the crystallized particles through the length of the accretion region. We assume that the accretion region of the crystallized particles finishes when ${v}_{\mathrm{imp}}\lt {v}_{\min }$ (Equation (27)). When the gas number density is high, the stopping length becomes short (Equation (28)), and the impact velocity becomes lower than ${v}_{\min }$ quickly (Appendix).

The rim thicknesses are similar between the a_d,init = 40 μm cases and a_d,init = 20 μm cases. This is because the precursor dust size mainly changes the peak locations of the dust temperature and impact velocity, which are much shorter than the accretion ranges of the droplets and crystallized dust. This indicates that our results are good examples to consider the rim thicknesses that are composed of the observed coarse grains.

3.2.2. Shock Scale

The spatial scales of the shock waves are expected to depend on the shock-driving mechanisms, shock-driving bodies, and particle trajectories (e.g., Desch et al. 2012; Morris et al. 2012). We present how the shock scale affects the accretion condition and thickness of igneous rims. Figure 5 shows the size ratio between the final dust and initial dust (a_d,last/a_d,init) and the thickness of the accreted rims as a function of the shock scale in the a_d,init = 40 μm and a_d,init = 20 μm cases. The other parameters are v₀ = 13 km s⁻¹, n₀ = 10^13.5 cm⁻³, and a_ch = 10³ μm. As in the case of the shock velocity and preshock gas number, dust does not melt completely when the shock scale is short (L < 100 km for a_d,init = 40 μm dust and L < 30 km for a_d,init = 20 μm dust), and dust is evaporated when the shock scale is long (L > 5 × 10⁴ km for a_d,init = 40 μm dust and L > 10⁴ km for a_d,init = 20 μm dust). In the larger dust size case, a longer shock scale is needed for melting and evaporation.

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The total thicknesses increase as the shock scale increases. The droplets are accreted more as L increases since the accretion of the droplets takes place at x < x _c ∼ L (Equation (30)). The thickness of the accreted crystallized particles is also an increasing function of L when L < l_stop,ch. This is because the accretion range of crystallized dust particles increases as L increases. The accretion of crystallized dust particles occurs between x _c ∼ L and the location where the impact velocity becomes lower than ${v}_{\min }$ after the impact velocity reaches the second peak (Figure 1, Section 3.1). In the accretion of crystallized dust particles, the impact velocity is almost equal to the relative velocity of chondrules to gas in this range when L ≳ 100 km since crystallized dust particles are well coupled with gas. When L ≪ l_stop,ch, the chondrule velocity barely changes behind the shock front since the gas velocity recovers to the preshock velocity at l_stop,ch (Arakawa & Nakamoto 2019). As L increases, chondrules need longer x to recover their velocities, which makes the accretion range of crystallized dust particles longer. The thickness of the accreted crystallized particles is peaked around L ∼ l_stop,ch and begins to decrease. When L ≫ l_stop,ch, the chondrule velocity is close to the gas velocity at x _c , and the accretion rate of crystallized dust particles decreases (Appendix).

3.2.3. Chondrule Size

The observed igneous rims tend to increase as the chondrule size increases (Matsumoto et al. 2021). This tendency is reproduced in our results. Figure 6 shows the thicknesses of the rims in the case in which v₀ = 10 km s⁻¹, n₀ = 10^14.5 cm⁻³, L = 10³ km, and a_d,init = 20 μm. As the chondrule size increases, the stopping length of chondrules becomes longer. This makes high impact velocity accretion for droplets (Appendix). Small-size chondrules do not have the rims of the accreted crystallized particles since L > l_stop,ch (Section 3.2.2).

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Although the observed igneous rims have a range in thickness, the observed rims are ∼10 times thicker than the rims in our results. Multiple rim-forming shock events are needed to explain the thickness of the observed igneous rims, if the accretion behind the shock waves is the main source of the observed igneous rims. We consider that multiple rim-forming shock events are plausible when chondrules form by shock events. The rim-forming shock events are expected to be more frequent than the chondrule-forming shock events. If a planetesimal generates a chondrule-forming bow shock, dust particles in higher impact parameters experience lower peak temperatures (Boley et al. 2013). Moreover, the shock velocity depends on its longitude (Nagasawa et al. 2019). These indicate that when a chondrule-forming bow shock is generated in a part of an orbit and at a certain impact parameter, the rim-forming bow shock is also generated in a wider part of the orbit and higher impact parameter, which means more chance to form igneous rims. Such shock regions will help explain the depletion of volatile elements (Hubbard & Ebel 2014). Igneous rims are accreted in multiple melting events, as chondrules experience multiple (partial) melting events (e.g., Rubin & Wasson 1987; Baecker et al. 2017).