**1. Introduction**

#### **1.1 Droplet in space**

It is considered that our solar system 4.6 billion years ago was composed of a proto-sun and the circum-sun gas disk. In the gas disk, originally micron-sized fine dust particles accumulated by mutual collisions to be 1000 km-sized objects like as planets. Therefore, to understand the planet formation, we have to know the evolution of the dust particles in the early solar gas disk. One of the key materials is a millimeter-sized and spherical-shaped grain termed as "chondrule" observed in chondritic meteorites.

Chondrules are considered to have been formed from molten droplets about 4.6 billion years ago in the solar gas disk (Amelin et al., 2002; Amelin & Krot, 2007). Fig. 1 is a schematic of the formation process of chondrules. In the early solar gas disk, aggregation of the micron-sized dust particles took place before planet formation (Nakagawa et al., 1986). When the dust aggregates grew up to about 1 mm in size (precursor), some astrophysical process heated them to the melting point of about 1600 − 2100 K (Hewins & Radomsky, 1990). The molten dust aggregate became a sphere by the surface tension (droplet), and then cooled again to solidify in a short period of time (chondrule). The formation conditions of chondrules, such as heating duration, maximum temperature, cooling rate, and so forth, have been investigated experimentally by many authors (Blander et al., 1976; Fredriksson & Ringwood, 1963; Harold C. Connolly & Hewins, 1995; Jones & Lofgren, 1993; Lofgren & Russell, 1986; Nagashima et al., 2006; Nelson et al., 1972; Radomsky & Hewins, 1990; Srivastava et al., 2010; Tsuchiyama & Nagahara, 1981-12; Tsuchiyama et al., 1980; 2004; Tsukamoto et al., 1999). However, it has been controversial what kind of astronomical event could have produced chondrules in early solar system. The chondrule formation is one of the most serious unsolved problems in planetary science.

The most plausible model for chondrule formation is a shock-wave heating model, which has been tested by many theoreticians (Ciesla & Hood, 2002; Ciesla et al., 2004; Desch & Jr., 2002; Hood, 1998; Hood & Horanyi, 1991; 1993; Iida et al., 2001; Miura & Nakamoto, 2006; Miura et al., 2002; Morris & Desch, 2010; Morris et al., 2009; Ruzmaikina & Ip, 1994; Wood, 1984). Fig. 2 is a schematic of dust heating mechanism by the shock-wave heating model. Initially, the chondrule precursors were floating in the gas disk without any large relative velocity against the ambient gas (panel (a)). When a shock wave was generated in the gas disk, the gas behind the shock front was accelerated suddenly. On the other hand, the chondrule

Fig. 2. Schematic of the shock-wave heating model for chondrule formation. (a) The

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.

chondrite (CO3) are from 100 *μ*m to 200 *μ*m.

substantial issue in this chapter.

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

Hydrodynamics of a Droplet in Space 383

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

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

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

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

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