**1.2.1 Size distribution**

Fig. 3 shows the size distribution of chondrules compiled from measurement data in some literatures (Nelson & Rubin, 2002; Rubin, 1989; Rubin & Grossman, 1987; Rubin & Keil, 1984). The horizontal axis is the diameter *D* and the vertical axis is the cumulative fraction of 2 Will-be-set-by-IN-TECH

Fig. 1. Schematic of formation process of a chondrule. The precursor of chondrule is an aggregate of *μ*m-sized cosmic dusts. The precursor is heated and melted by some

standard model of the early solar gas disk (Iida et al., 2001).

**1.2 Physical properties of chondrules**

of time.

the shock front.

**1.2.1 Size distribution**

mechanism, becomes a sphere by the surface tension, then cools to solidify in a short period

precursors remain un-accelerated because of their inertia. Therefore, after passage of the shock front, the large relative velocity arises between the gas and dust particles (panel (b)). The relative velocity can be considered as fast as about 10 km s−<sup>1</sup> (Iida et al., 2001). When the gas molecule collides to the surface of chondrule precursors with such large velocity, its kinetic energy thermalizes at the surface and heats the chondrule precursors, as termed as a gas drag heating. The peak temperature of the precursor is determined by the balance between the gas drag heating and the radiative cooling at the precursor surface (Iida et al., 2001). The gas drag heating is capable to heat the chondrule precursors up to the melting point if we consider a

The chondrule formation models, including the shock-wave heating model, are required not only to heat the chondrule precursors up to the melting point but also to reproduce other physical and chemical properties of chondrules recognized by observations and experiments. These properties that should be reproduced are summarized as observational constraints (Jones et al., 2000). The reference listed 14 constraints for chondrule formation. To date, there

Here, we review two physical properties of chondrules; size distribution and three-dimensional shape. The latter was not listed as the observational constraints in the literature (Jones et al., 2000), however, we would like to include it as an important constraint for chondrule formation. As discussed in this chapter, these two properties strongly relate to the hydrodynamics of molten chondrule precursors in the gas flow behind

Fig. 3 shows the size distribution of chondrules compiled from measurement data in some literatures (Nelson & Rubin, 2002; Rubin, 1989; Rubin & Grossman, 1987; Rubin & Keil, 1984). The horizontal axis is the diameter *D* and the vertical axis is the cumulative fraction of

is no chondrule formation model that can account for all of these constraints.

Fig. 2. Schematic of the shock-wave heating model for chondrule formation. (a) The precursors of chondrules are in a gas disk around the proto-sun 4.6 billion years ago. The gas and precursors rotate around the proto-sun with almost the same angular velocity, so there is almost no relative velocity between the gas and precursors. (b) If a shock wave is generated in the gas disk by some mechanism, the gas behind the shock front is suddenly accelerated. In contrast, the precursor is not accelerated because of its large inertia. The difference of their behaviors against the shock front causes a large relative velocity between them. The precursors are heated by the gas friction in the high velocity gas flow.

chondrules smaller than *D* in diameter. Table 1 shows the mean diameter and the standard deviation of each measurement. It is found that the chondrule sizes vary according to chondrite type. The mean diameters of chondrules in ordinary chondrites (LL3 and L3) are from 600 *μ*m to 1000 *μ*m. In contrast, ones in enstatite chondrite (EH3) and carbonaceous chondrite (CO3) are from 100 *μ*m to 200 *μ*m.

It should be noted that the true chondrule diameters are slightly larger than the data shown in Fig. 3 and Table 1 because of the following reason. This data was obtained by observations on thin-sections of chondritic meteorites. The chondrule diameter on the thin-section is not necessarily the same as the true one because the thin-section does not always intersect the center of the chondrule. Statistically, the mean and median diameters measured on the thin section are, respectively, <sup>√</sup>2/3 and <sup>√</sup>3/4 of the true diameters (Hughes, 1978). However, we do not take care the difference between true and measured diameters because it is not a substantial issue in this chapter.

It is considered that in the early solar gas disk the dust aggregates have the size distribution from ≈ *μ*m (initial fine dust particles) to a few 1000 km (planets). In spite of the wide

chondrite meteorite chondrule number diam. *D* ref.

Table 1. Diameters of chondrules from various types of chondritic meteorites and the standard deviations. ∗BO = barred olivine, RP = radial pyroxene, C = cryptocrystalline. all =

and vertical axes are axial ratios of *b*/*a* and *c*/*b*, respectively. A point (*b*/*a*, *c*/*b*)=(1, 1) means a perfect sphere because all of three axes have the same length. As going downward from the point, the shape becomes oblate (disk-like shape) because *a* = *b* > *c*. On the other hand, the shape becomes prolate (rugby-ball-like shape) as going leftward because *a* > *b* = *c*. The chondrule shapes in the measurement are classified into two groups: spherical chondrules in group-A and prolate chondrules in group-B. Chondrules in group-A have axial ratios of *c*/*b* >∼ 0.9 and *b*/*a* >∼ 0.9. In contrast, chondrules in group-B have smaller values of *b*/*a* as

It is considered that the deviation from a perfect sphere results from the deformation of a molten chondrule before solidification. For example, if the molten chondrule rotates rapidly, the shape becomes oblate due to the centrifugal force (Chandrasekhar, 1965). However, the shapes of chondrules in group-B are prolate rather than oblate. Tsuchiyama et al. (Tsuchiyama et al., 2003) proposed that the prolate chondrules in group-B can be explained by spitted droplets due to the shape instability with high-speed rotation. However, it is not clear whether the transient process such as the shape instability accounts for the range of axial

If chondrules were melted behind the shock front, the molten droplet ought to be exposed to the high-velocity gas flow. The gas flow causes many hydrodynamics phenomena on the molten chondrule droplet as follows. (i) Deformation: the ram pressure deforms the droplet shape from a sphere. (ii) Internal flow: the shearing stress at the droplet surface causes fluid flow inside the droplet. (iii) Fragmentation: a strong gas flow will break the droplet into many tiny fragments. Hydrodynamics of the droplet in high-velocity gas flow strongly relates to the physical properties of chondrules. However, these hydrodynamics behaviors have not been investigated in the framework of the chondrule formation except of a few examples that neglected non-linear effects of hydrodynamics (Kato et al., 2006; Sekiya et al.,

To investigate the hydrodynamics of a molten chondrule droplet in the high-velocity gas flow, we performed computational fluid dynamics (CFD) simulations based on cubic-interpolated propagation/constrained interpolation profile (CIP) method. The CIP method is one of the high-accurate numerical methods for solving the advection equation (Yabe & Aoki, 1991;

L3 Inman BO 173 1038±937 (Rubin & Keil, 1984) L3 Inman RP+C 201 852±598 (Rubin & Keil, 1984) L3 ALHA77011 BO 163 680±625 (Rubin & Keil, 1984) L3 ALHA77011 RP+C 70 622±453 (Rubin & Keil, 1984)

Hydrodynamics of a Droplet in Space 385

<sup>−</sup><sup>237</sup> (Nelson & Rubin, 2002)

<sup>−</sup><sup>70</sup> (Rubin, 1989)

<sup>−</sup><sup>101</sup> (Rubin & Grossman, 1987)

type type type∗ [*μ*m]

LL3 total of 5 types all 719 574+<sup>405</sup>

EH3 total of 3 types all 689 219+<sup>189</sup>

CO3 total of 11 types all 2834 148+<sup>132</sup>

all types are included.

ratio of group-B chondrules or not.

2003; Uesugi et al., 2005; 2003).

**1.3 Hydrodynamics of molten chondrule precursors**

≈ 0.7 − 0.8.

Fig. 3. Size distributions of natural chondrules in various types of chondritic meteorites (LL3, L3, EH3, and CO3). The vertical axis is the normalized cumulative number of chondrules whose diameters are smaller than that of the horizontal axis. Each data was compiled from the following literatures; LL3 chondrites (Nelson & Rubin, 2002), L3 chondrites (Rubin & Keil, 1984), EH3 chondrites (Rubin & Grossman, 1987), and CO3 chondrites (Rubin, 1989), respectively. The total number of chondrules measured in each literature is 719 for LL3, 607 for L3, 689 for EH3, and 2834 for CO3, respectively.

size range of solid materials, sizes of chondrules distribute in a very narrow range of about 100 − 1000 *μ*m. Two possibilities for the origin of chondrule size distribution can be considered; (i) size-sorting prior to chondrule formation, and (ii) size selection during chondrule formation. In the case of (i), we need some mechanism of size-sorting in the early solar gas disk (Teitler et al., 2010, and references therein). In the case of (ii), the chondrule formation model must account for the chondrule size distribution. The latter possibility is what we investigate in this chapter.

#### **1.2.2 Deformation from a perfect sphere**

It is considered that spherical chondrule shapes were due to the surface tension when they melted. However, their shapes deviate from a perfect sphere and the deviation is an important clue to identify the formation mechanism. Tsuchiyama et al. (Tsuchiyama et al., 2003) measured the three-dimensional shapes of chondrules using X-ray microtomography. They selected 20 chondrules with perfect shapes and smooth surfaces from 47 ones for further analysis. Their external shapes were approximated as three-axial ellipsoids with axial radii of *a*, *b*, and *c* (*a* ≥ *b* ≥ *c*), respectively. Fig. 4 shows results of the measurement. The horizontal 4 Will-be-set-by-IN-TECH

Fig. 3. Size distributions of natural chondrules in various types of chondritic meteorites (LL3, L3, EH3, and CO3). The vertical axis is the normalized cumulative number of chondrules whose diameters are smaller than that of the horizontal axis. Each data was compiled from

(Rubin & Keil, 1984), EH3 chondrites (Rubin & Grossman, 1987), and CO3 chondrites (Rubin, 1989), respectively. The total number of chondrules measured in each literature is 719 for

size range of solid materials, sizes of chondrules distribute in a very narrow range of about 100 − 1000 *μ*m. Two possibilities for the origin of chondrule size distribution can be considered; (i) size-sorting prior to chondrule formation, and (ii) size selection during chondrule formation. In the case of (i), we need some mechanism of size-sorting in the early solar gas disk (Teitler et al., 2010, and references therein). In the case of (ii), the chondrule formation model must account for the chondrule size distribution. The latter possibility is

It is considered that spherical chondrule shapes were due to the surface tension when they melted. However, their shapes deviate from a perfect sphere and the deviation is an important clue to identify the formation mechanism. Tsuchiyama et al. (Tsuchiyama et al., 2003) measured the three-dimensional shapes of chondrules using X-ray microtomography. They selected 20 chondrules with perfect shapes and smooth surfaces from 47 ones for further analysis. Their external shapes were approximated as three-axial ellipsoids with axial radii of *a*, *b*, and *c* (*a* ≥ *b* ≥ *c*), respectively. Fig. 4 shows results of the measurement. The horizontal

the following literatures; LL3 chondrites (Nelson & Rubin, 2002), L3 chondrites

LL3, 607 for L3, 689 for EH3, and 2834 for CO3, respectively.

what we investigate in this chapter.

**1.2.2 Deformation from a perfect sphere**


Table 1. Diameters of chondrules from various types of chondritic meteorites and the standard deviations. ∗BO = barred olivine, RP = radial pyroxene, C = cryptocrystalline. all = all types are included.

and vertical axes are axial ratios of *b*/*a* and *c*/*b*, respectively. A point (*b*/*a*, *c*/*b*)=(1, 1) means a perfect sphere because all of three axes have the same length. As going downward from the point, the shape becomes oblate (disk-like shape) because *a* = *b* > *c*. On the other hand, the shape becomes prolate (rugby-ball-like shape) as going leftward because *a* > *b* = *c*. The chondrule shapes in the measurement are classified into two groups: spherical chondrules in group-A and prolate chondrules in group-B. Chondrules in group-A have axial ratios of *c*/*b* >∼ 0.9 and *b*/*a* >∼ 0.9. In contrast, chondrules in group-B have smaller values of *b*/*a* as ≈ 0.7 − 0.8.

It is considered that the deviation from a perfect sphere results from the deformation of a molten chondrule before solidification. For example, if the molten chondrule rotates rapidly, the shape becomes oblate due to the centrifugal force (Chandrasekhar, 1965). However, the shapes of chondrules in group-B are prolate rather than oblate. Tsuchiyama et al. (Tsuchiyama et al., 2003) proposed that the prolate chondrules in group-B can be explained by spitted droplets due to the shape instability with high-speed rotation. However, it is not clear whether the transient process such as the shape instability accounts for the range of axial ratio of group-B chondrules or not.
