1. Introduction

The presence of crystalline silicates in comets was first identified by infrared spectra of dust in the coma of comet 1P/Halley (Bregman et al., 1987). Since then, infrared features of crystalline silicate have been observed in a number of comae, but the origin of crystalline silicates is a matter of debate. Experimental studies on thermal annealing of amorphous magnesium silicate smokes show that a dust temperature in a coma of a typical comet is too low for amorphous silicateFootnote 1 to crystallize in situ (Hallenbeck et al., 1998, 2000). Therefore, thermal annealing of amorphous silicates in the hot inner solar nebula and incorporation of crystalline silicate into comet nuclei are suggested (see a review by Hanner, 1999). In contrast, the interstellar composition of gas in the comae indicates the preservation of interstellar volatiles in the cold outer nebula (Biermann et al., 1982; Mumma, 1996). To reconcile hot and cold materials, crystalline silicates are considered as being transported from the inner solar nebula to the outer comet-formation region (Bockel’ee-Morvan et al., 2002). Nuth et al. (2000) claim that the fractional abundance of crystalline silicate in the solar nebula increases with time and thus can be used to date a comet’s formation age. Harker and Desch (2002) considered in-situ annealing of silicate at elevated temperatures by nebular shocks that could proceed with crystallization up to 20 AU, depending on the model parameters (Nakamoto and Miura, 2005). Without the outward transport, however, the in-situ annealing mechanism predicts no crystalline silicate in the Kuiper Belt comets, contrary to observations (Watanabe, 2004). As noticed by Huebner (2002), none of the thermal mechanisms allow comets to retain the interstellar composition of ices in their nuclei.

We notice that a detection of crystalline silicates in a cometary coma does not necessarily mean the presence of the crystallites in the nucleus. Comet nuclei may consist of interstellar ices along with interstellar dust that is almost free from crystalline silicates. If low-temperature crystallization of amorphous silicate, as pointed out by Molster et al. (1999), takes place in a comet coma, then one could automatically reconcile observable signatures of both interstellar ices and crystalline silicates in the coma. Recently, Carrez et al. (2002) have found that MgSiO3 glasses irradiated by 300 keV electrons crystallize at room temperature. We propose yet another nonthermal mechanism to preserve interstellar composition of volatiles in comets.

2. Nonthermal Crystallization Model

It is considered that a comet is an agglomeration of interstellar dust consisting of an amorphous silicate core, an organic refractory (OR) inner mantle, and an icy outer mantle (e.g., Greenberg, 1982; Jessberger et al., 1988). Because of the fluffiness of cometary dust, crystallization does not make essential difference between the aggregates and their constituent interstellar grains. The radius of the amorphous silicate core of the interstellar grain is a = 0.07 μm and the thickness of the OR mantle is hr = 0.03μm (Li and Greenberg, 1997). We assume Mg-bearing glass as the main component of the amorphous silicate core (see Jessberger et al., 1988; Kimura et al., 2003). Figure 1 depicts our nonthermal mechanism of crystallization at moderate temperatures (Yamamoto and Chigai, 2005). A typical temperature of cometary dust is several hundred K at heliocentric distances around 1 AU from the sun. This is insufficient for the amorphous silicate core to crystallize but is sufficiently high for the icy mantle to sublime and produce the observed interstellar composition of gas in a coma. Such moderate heating also increases the diffusion rate of reactive molecules and triggers reactions among them in the OR mantle. In consequence, the energy released by the reactions increases the temperature and subsequently expedites further reactions. This is a positive feedback process resulting in chain reactions, a sudden rise in the temperature of the OR mantle, and heat flow into the amorphous silicate core. The heat flow brings about crystallization from the core surface towards the interior.

Fig. 1.
figure 1

Illustration of our nonthermal crystallization mechanism.

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The degree of crystallinity fc(r, t) at a distance r from the center of the core at a time t is given (Haruyama et al., 1993) by

((1))

where k is Boltzmann’s constant and Ec is the effective activation energy of crystallization. In Eq. (1), AeEc/kT indicates the timescale for amorphous-to-crystalline transition (Brucato et al., 1999; Fabian et al., 2000). Here, the vibration period A of diffusing atoms in silicate is 5 × 10-14 s (Duschl et al., 1996). Equation (1) would overestimate the crystallization degree because it allows crystallization above the melting point (see Section 4). The temperature T (r, t) is determined by the equation of heat conduction:

((2))

where χ is the thermal diffusivity. Heating due to deposition of latent heat of crystallization and cooling due to thermal emission and sublimation of organic molecules are negligible because the time scales of these processes are much longer than that of heat conduction.

The duration of heating by reactions in the OR mantle is almost instantaneous compared to the timescale of heat conduction into the silicate core. We model such heating by putting an instantaneous spherical surface heat source on the silicate core surface and take the temperature distribution of the silicate core at t = 0 to be T (r, 0) = const · δ(r - a). Here the proportional constant (i.e., the strength of the source) is determined from the total energy released by the reactions given by

((3))

where cp is the specific heat per unit cell, g = 1 + hr/a + (1/3)(hr/a)2 is a geometrical factor, nr is the number density of reactive molecules in the OR mantle, Er is the amount of energy released per reaction, and Ω is a volume of the unit cell of silicate. The initial temperature distribution in the core is thus given by

((4))

We ignore the temperature prior to heating by the reactions because no crystallization takes place in practice at temperatures before the reactions. The solution to Eq. (2) for the initial condition (4) is given (Carslaw and Jaeger, 1959, p. 259) by

((5))

for z ≪ a, z = a -r being the depth from the core surface. In most cases of interest, the crystallized region is confined to the silicate core surface as shown below. Inserting Eq. (5) into Eq. (1), one obtains the degree of crystallinity at depth ζ = z/(χ A)½ for t → ∞ to be

((6))

where

((7))

Note that the material dependence of ƒc(ζ, ∞) is put together into the dimensionless parameter θ. The θ degree of crystallinity on the surface is given from Eq. (6) by

((8))

This indicates that θ > 1 is the condition of substantial crystallization near the surface.

3. Degree of Crystallinity

The parameter θ is the key quantity that determines the degree of crystallinity. For forsterite whose density is ρ = 3.2 g cm-3 and molecular weight is μ = 140, we find Ω = μmH/ρ = 7.3 × 10-23 cm3 where mH is the hydrogen mass. The specific heat per unit cell is expressed as cp = ƒk at temperatures higher than the Debye temperature, where ƒf is freedom of motion of atoms (plus contributions from electrons) composing a unit cell. The specific heat measured for forsterite is 40–50 cal mol-1 K-1 at T = 900–2000K (Chase et al., 1985) which corresponds to ƒf = 21–26. Following Hofmeister (2001), we estimate the thermal diffusivity of amorphous silicate with the sound velocity cs ≃ 3 × 105 cm s-1 and phonon mean free path λ ≈ (3Ω/4π)1/3 t{ 2.6 × 10-8 cm to be χ = csλ/3 ∼ 2.6 × 10-3 cm2 s-1 and the length scale (χ A)½ to be 1 × 10-8 cm. We take nr = 1021 to 1022 cm-3, corresponding to concentrations of reactive molecules of roughly 1 to 10% in the OR mantle (Greenberg, 1976). The energy released per reaction, Er, is a few to 10 eV (Schutte and Greenberg, 1991; Sorrell, 2001). Using these values, θ is expressed as

((9))

The θ value being on the order of unity implies that amorphous silicates in comets do crystallize to a certain degree by the present mechanism. Figure 2 shows the degree of crystallinity ƒc(r,∞) calculated from Eq. (6) for θ = 0.9 to 9 (nr = 1021-1022 cm-3). Also plotted is the case for θ = 18 (e.g., nr = 1022 cm-3, Er = 10 eV) as a possible upper limit. The crystallized region is localized near the core surface and hence the assumption posed in deriving Eq. (6) is not violated.

We examine whether the range of crystallinity derived from the present model provides the strengths of crystalline silicate peaks as observed in the 10 μm band. The strength of a crystalline silicate peak is evaluated by the absorption cross section at a wavelength λ of the peak relative to that at λ = 8 μm, where the spectra show a continuum baseline arising from emission by the OR mantle. The absorption cross section is calculated for three-layered spherical grains consisting of an amorphous silicate core coated by crystalline olivine together with an outermost OR mantle (see Friedman, 1998). We use the complex refractive indices for organic refractory given by Li and Greenberg (1997), amorphous silicate by Scott and Duley (1996), and forsterite by Sogawa et al. (2006). The fraction ƒ̄ of crystalline silicate in the volume V of the silicate core is estimated as

((10))

which gives ƒ̄c=0.4-20% for nr = 1021-1022 cm-3(0.9 ≲ θ ≲ 9) from Fig. 2. We focus for simplicity on the strengths of the major peaks at 10.0 and 11.2μm, which are attributed to forsterite. Figure 3 shows that the silicate peak strengths increase with crystallinity ƒ̄c The strengths at 10.0 and 11.2μm are comparable and, for θ = 0.9–9, the strengths of both peaks could reach up to 2-3 times the strengths due to amorphous silicate (ƒ̄c=0). These characteristics of the peaks are in harmony with infrared observations of cometary comae.

Fig. 2.
figure 2

The degree of crystallinity ƒfc(r,∞) for θ = 1 to 18 as a function of distance r from the center of the silicate core having the radius a.

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4. Discussion

We have proposed a nonthermal crystallization model of cometary silicate grains. The degree of crystallinity depends only on θ, which measures the amount of energy de26 posited in the OR mantle. The volume fraction of crystallites as much as 20% is realized depending on the θ-values. As shown in Fig. 3, the crystallinity obtained in the present model may reproduce the observed strengths of crystalline silicate features in the 8–13μm spectra. See Kimura et al. (2008) for discussion of the detailed spectra. In spite of the success in reproducing the observed infrared features, the present mechanism does not require mixing of amorphous and crystalline silicates. A natural consequence of the present model is that volatiles of interstellar composition may be preserved composition in comets.

Fig. 3.
figure 3

Crystalline silicate strengths expected for observed major features as a function of crystallinity ƒ̄c

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The present mechanism would also work for grains in other objects such as in protoplanetary disks if (1) the condition θ > 1 to induce crystallization is fulfilled and (2) moderate heating is available to trigger exothermal reactions in OR mantles. Crystalline silicates found in debris disks might be brought in part from comet-like objects in the disks. It is also possible to make crystalline silicate by the present mechanism in the vicinity of protostars. Recent observations showed the existence of crystalline silicates around protostars at an early stage of 105 years (Ciardi et al., 2005). This mechanism may also be triggered by shock waves and work at high temperature regions near O or B stars, although radiation pressure by these stars prevent any dust from staying in the vicinity of the stars. An upper limit of the degree of silicate crystallinity in the interstellar medium is estimated to be about 5% based on a comparison between a model and infrared observations (Li et al., 2007). To assess the degree of silicate crystallinity in the interstellar medium, it would be necessary to investigate the balance between the nonthermal crystallization and amorphization by cosmic rays. More observations of crystalline silicate features in low-temperature environments will provide a clue to specify the temperature that triggers the nonthermal crystallization.

Our results show a possibility of efficient crystallization by nonthermal process. As stated in Section 2, our model permits amorphous silicate to crystallize above its melting temperature. If we exclude crystallization at higher temperatures than the melting temperature, we would have a lower volume fraction of crystalline silicate. On the other hand, our model neglects the finiteness of the particle size for the determination of the temperature. Actually the heat transmitted to the center of the particle cannot run away easily. As a result, the mean temperature of the particle would not decrease as we have modeled. We expect that this particlesize effect increases the volume fraction of crystal. Therefore, it is worth examining this nonthermal process more in detail by taking into account a more realistic crystallization model and the particle-size effect.

Crystallization is a process for atoms to overcome an activation energy of crystallization at an appreciable probability. In general, low-temperature crystallization is possible if (1) a significant amount of input energy is stored and (2) there is a trigger to release the stored energy to induce crystallization. d’Hendecourt et al. (1982) observed that the energy stored in their OR is released at ∼30 K. Recently, Kaito et al. (2006) prepared particles of amorphous Mg silicate coated with amorphous carbon. They found that their crystallization temperatures are lowered substantially due to the energy released by graphitization of amorphous carbon. Inclusion of CH4 in the amorphous carbon mantle further reduces the crystallization temperature to room temperature (Kaito et al., 2007). Although their experiments does not necessarily simulate crystallization of cometary silicate, they prove that a silicate core proceeds with forsterite crystallization upon releasing energy in its outer layer. The temperature to trigger the nonthermal crystallization could be further lowered, if the stored energy is released at much lower temperatures. Experiments on a search for mechanisms of low-temperature crystallization are encouraged to better understand the ubiquity of crystalline silicate in various objects.