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Astron. Astrophys. 338, 364-370 (1998)

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3. Modeling results and the possible dust properties

Following Greenberg & Hage (1990), we model the comet dust of P/Borrelly as fluffy aggregates of core-mantle interstellar particles. The optical constants [the complex indices of refraction, [FORMULA]] for interstellar grains used in the modeling of comet P/Borrelly are based on the determination of the composition of interstellar core-mantle grains by fitting both the interstellar extinction curve and interstellar polarization (Li & Greenberg 1997). For the 10 µm silicate emission feature, we employ amorphous olivine for the silicate core, using the [FORMULA] of Dorschner et al. (1995) for amorphous olivine MgFeSiO4 for wavelengths longward of 2 µm. For [FORMULA], we adopt the [FORMULA] of Draine & Lee (1984). For [FORMULA], we adopt the [FORMULA] of crystalline olivine from Huffman & Stapp (1973), because both crystalline olivine and amorphous olivine absorb through electronic transitions in the far ultraviolet. Finally, the real part of the optical constants [FORMULA] is calculated from [FORMULA] by using the Kramers-Kronig relation. For the grain mantle, we shall first employ the optical constants of H, C, O, N-rich organic refractory residues (Li & Greenberg 1997).

The porosity [FORMULA] is treated as a free parameter with a wide range of [FORMULA] from 0 (compact) to 0.975 being considered. As a starting point , we adopted the Halley dust size (mass) distribution obtained by spacecrafts (see Fig. 3a of Greenberg & Hage 1990) which can be expressed by a polynomial function (e.g., see Lamy et al. 1987). Adjustment of the Halley size distribution can be made by modifying one of the coefficients to, for example, enhance the smaller grains. For the mass ratio of the organic refractory mantle to the silicate core, according to the mass spectra of comet P/Halley dust as measured by the PUMA mass spectrometer on board the spacecraft Vega 1 (Kissel & Krueger 1987), we adopt m[FORMULA]/m[FORMULA] = 1 . The effects of a lower m[FORMULA]/m[FORMULA] ratio will be discussed later. The lower mass (size) limit was set at [FORMULA] which is equivalent to an individual tenth micron interstellar grain. Particles with radii smaller than tenth micron contribute very little to the thermal emission in comet P/Halley (Hanner et al. 1987). The upper mass limit was set at the maximum liftable mass [FORMULA], the mass of the largest dust grain which can be dragged away from the nucleus. Adopting the nucleus size (equivalent to an [FORMULA] radius sphere) estimated by Lamy et al. (1995), the gas production rate detected by A'Hearn et al. (1995) [scaled by an [FORMULA] heliocentric dependence (A'Hearn et al. 1995)], and assuming a nucleus density [FORMULA] (Rickman 1986), a grain mass density [FORMULA] (corresponding to a porosity [FORMULA]), we estimated [FORMULA] from Eq. 19 in Newburn & Spinrad (1985). For a larger dust mass density (corresponding to a lower porosity), [FORMULA] becomes smaller, but one can expect that even a significant variation in [FORMULA] will not affect the resulting near infrared (hereafter NIR) emission spectrum because those high mass particles are so cold that their contribution is negligible (as long as the grain size distribution is not too flat).

Our calculations show that, within the Halley size distribution, if the particles (with organic residue mantles) are compact they are then too cold to give excess emission at the silicate band. With the dust size distribution adjusted to be greatly weighted toward smaller grains, the silicate feature is enhanced as expected (Gehrz & Ney 1992) but is then far too narrow compared with the observation of comet P/Borrelly. We have tried to fit the observation by both varying the porosity and adjusting the dust size distribution. It turns out that none of these attempts provides a satisfactory match. In Fig. 1a, b, c we present the "best-fitting" (to the silicate emission) spectra using amorphous olivine cores and organic residue mantles, calculated from [FORMULA] respectively. The corresponding grain size distributions are plotted in Fig. 1d. It can be seen from these figures that the theoretical spectra are a bit sharper than the observation. In addition, the NIR spectrum (3 - 5 µm) is too low compared to the observation. We note that, lower porosity or enhancement of larger particles in the size distribution could broaden the silicate feature, but then, the whole feature becomes too shallow and the peak position shifts to longer wavelengths (the particles are too cold).

[FIGURE] Fig. 1a-d. The infrared thermal emission spectrum of comet P/Borrelly (in [FORMULA]). The observational data (points) are taken from Hanner et al. (1996). The predicted spectra of the porous aggregate comet dust model of silicate core-organic refractory mantle interstellar grains with a porosity [FORMULA] are plotted as solid lines in a , b and c , respectively. The chi-square, [FORMULA] (here [FORMULA], the number of observational data points; [FORMULA] is the number of free parameters: the porosity and the dust size distribution) which, to some extent, can describe the goodness of the fit, is about 18.4, 18.9, 19.0 for [FORMULA], respectively. The dotted line in a is a blackbody ([FORMULA]) emission (Hanner et al. 1996). The corresponding dust size distributions are shown in d : solid - the Halley dust size distribution; dotted - [FORMULA]; short dashed - [FORMULA]; long dashed - [FORMULA].

Suppose that the chemical composition of organics of SP comets could have been modified by solar irradiation or possibly as proposed by A'Hearn et al. (1996), that the original chemical composition for the solar nebula out of which the SP comets were formed is different from where LP comets were formed. This has led us to consider an extreme case - amorphous carbon, which may be characteristic of the most highly processed organics materials - highly depleted in H, O, N. The indices of refraction of amorphous carbon are adopted from those compiled by Rouleau & Martin (1991). As shown in Fig. 2a, a satisfactory fit to the 10 µm silicate feature and the NIR emission can be obtained by the amorphous carbon model with a porosity [FORMULA]. The dust size distribution is enhanced toward smaller particles. The fits provided by other porosities are not as good as that by [FORMULA]. It can be seen in Fig. 2b that, although the fit for [FORMULA] is not bad, the theoretical spectrum is a bit deficient in the NIR region. Fig. 2c shows that the predicted NIR spectrum from [FORMULA] is too low compared to the observation. If we increase the weight of smaller particles, the fit to the NIR spectrum improves; however, then the silicate feature becomes too sharp. We have also tried lower porosities ([FORMULA]), but then we found that the dust grains are too cold so that the calculated spectra shift the peak positions to longer wavelengths and are deficient in the NIR.

[FIGURE] Fig. 2a-d. The theoretical spectra calculated for the porous aggregate comet dust model of silicate core-amorphous carbon mantle grains with a porosity [FORMULA] (a , [FORMULA]), 0.90 (solid line in b , [FORMULA]), and 0.95 (c ). The dotted line in a is a blackbody ([FORMULA]) emission (Hanner et al. 1996). Also plotted in b (dashed line, [FORMULA]) is the spectrum produced by the [FORMULA] model with a thinner mantle ([FORMULA]) and with the same size distribution as for the [FORMULA] model (dotted line in d ). In c , both the solid line ([FORMULA]) and the dotted line ([FORMULA]) are model spectra for [FORMULA]. The corresponding dust size distributions are shown in d : solid - the Halley dust size distribution; dotted - [FORMULA]; short dashed - [FORMULA]; long dashed - [FORMULA] (corresponding to the dotted curve in c ); dotted - short dashed - [FORMULA] (corresponding to the solid curve in c ).

Therefore we conclude that the amorphous olivine core - amorphous carbon mantle model with a porosity [FORMULA] (Fig. 2a) provides the best fit to the observations. Integrating over the mass range of interest in this work, the total dust mass required by the model with [FORMULA] is [FORMULA] g . If we assume an average outflow velocity [FORMULA] for all the grains rather than taking into account the outflow velocity distribution as a function of grain size, following the formula given by Hanner et al. (1985), we derive the dust production rate to be [FORMULA]. If we adopt the water production rate [FORMULA] measured at [FORMULA] (A'Hearn et al. 1995) scaled by a heliocentric evolution [FORMULA] (A'Hearn et al. 1995) as the gas production rate, the ratio of dust to H2O production rate is then [FORMULA]. One should keep in mind that the dust production rate deduced from the infrared (IR) emission may not reflect the actual dust mass loss since large particles are too cold to be well constrained by the IR emission (Crifo 1987; Fulle 1998). As long as the size distribution for the large particles are not too flat, some degree of variation in the slope of the large particle size distribution will not affect the IR emission spectrum, but result in considerably differences in the dust mass lose rate. Actually, the IR emission spectrum in the wavelength range considered in this work is contributed only by grains smaller than [FORMULA] (within the size distribution as derived for [FORMULA]). If the upper mass limit is set at [FORMULA], the corresponding dust production rate would be [FORMULA].

Alternatively, we have also tried to model the IR emission spectrum in terms of a power law dust size distribution [FORMULA]. We found that a model with [FORMULA] and [FORMULA] provides a good match. Actually, the modified Halley size distribution (for [FORMULA], see Fig. 2d) can be approximated by two power law distributions (for [FORMULA], [FORMULA]; for [FORMULA], [FORMULA]).

It is possible that the carbonaceous mantle could have undergone partial evaporation in the coma. We have also taken this into account by considering a model with a thinner mantle, [FORMULA]. Intuitively, we expect that, for a lower [FORMULA] which leads to a lower dust temperature, a higher porosity and/or a steeper dust size distribution, which results in a higher temperature, are needed to account for the emission spectrum. In the case [FORMULA] which implies that half of the mantle has evaporated, the original porosity [FORMULA] then becomes [FORMULA]. Using the same size distribution as derived for [FORMULA] (see Fig. 2d), our calculations show that the fit by the model with [FORMULA] and [FORMULA] is reasonably good (plotted as dashed line in Fig. 2b), but the silicate feature is slightly too sharp and the NIR emission is a bit too low. Increasing the porosity or enhancing the small particles, the fit to the NIR emission gets better but the silicate feature becomes even sharper. Decreasing the porosity or enhancing the large particles, the silicate feature becomes broader but then the model fails to fit the NIR emission. For a mantle thickness [FORMULA], the match to the overall spectrum is even poorer.

The modeling results as presented above clearly indicate some differences between the dust properties of P/Borrelly and those of P/Halley. First of all, the dust aggregates of P/Borrelly are somewhat more compact compared to P/Halley. The best fit to the P/Borrelly observation is provided by [FORMULA], while Greenberg & Hage (1990) have shown that, a higher porosity, in the range of [FORMULA], fits the silicate emission of P/Halley well. Moreover, the dust size distribution of P/Borrelly is steeper (weighted toward smaller size grains) than that of P/Halley. Furthermore, the organic mantle materials of P/Borrelly, best fit by amorphous carbon, appear to have been strongly processed and are depleted in H, O, N compared to P/Halley.

These differences are not surprising. Actually, there is no reason to expect the dust properties of P/Borrelly to be identical to those of P/Halley. Since P/Borrelly has passed through the inner solar system many more times than P/Halley and therefore been subjected much more to the solar irradiation, the dust grains within the surface layer of the nucleus could have been significantly modified. In particular, the organic refractory materials formed in the interstellar medium and then incorporated into the protosolar nebula and finally aggregated into comets could have undergone further carbonization. Here the term "carbonization" means that the organic materials, subjected to the processing of solar ultraviolet photons, would partially lose their H, O, N atoms and thus become carbon-rich (Jenniskens et al. 1993). In other words, the elements H, O, N relative to C would be more depleted than in comet P/Halley organics. Observations do show that some SP (Jupiter family) comets are depleted in C2 and C3 (however, CN is approximately constant, see A'Hearn et al. 1995 for details). This can be explained by attributing the "missing carbon" to the carbonization of the original interstellar organics. The fact that some C2 and C3 come directly from the volatile nuclear ices (which are relatively depleted in SP comets) while CN is mostly produced from grains (A'Hearn et al. 1995) is consistent with the idea of carbonization. While this is supported by the results of the EURECA space experiments which have indicated the carbonization of the "first generation" organic refractory materials by solar irradiation (Greenberg et al. 1995) there may be other ways of explaining the C2 and C3 depletion. For example, it has also been suggested that the chemical abundance in the solar nebula out of which the SP comets formed was different from that of LP comets (A'Hearn et al. 1995). This is not easy to understand because SP comets are formed further out than LP comets (see e.g. Levison 1996) so are closer to interstellar medium composition. On the other hand, if it is the case that the crystalline silicates formed in the hot, inner region of the solar nebula, extensive radial mixing would have occurred so that these materials could have been transported to the outer region where the cometesimals were forming (Hanner et al. 1994a, 1994b), although where and how the crystalline silicates formed is still not known (see e.g. Greenberg et al. 1996). The solar irradiation can also lead to a lower porosity than that of Halley dust due to the packing effect (Mukai & Fechtig 1983; also see Smoluchowski et al. 1984). The dust size distribution could be weighted toward smaller grains; i.e., smaller grains are enhanced as a consequence of evaporation and subsequent fragmentation in the coma. There are both observational and theoretical indications of dust fragmentation in the coma of comet P/Halley. As the volatile ice sublimates from the nucleus, it leaves behind the refractory particles and loosens the aggregates. If the fragmentation indeed results from the sublimation of volatile materials which act as "glue", one may expect relatively more drastic and more complete fragmentation in the coma of SP comets since volatiles are relatively depleted in SP comets (Weissman & Campins 1993). A statistical study of the cometary dust size distribution indeed seems to suggest that the dust size distribution of short-period comets is somewhat steeper than that for long-period comets (Fulle 1998).

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© European Southern Observatory (ESO) 1998

Online publication: September 8, 1998
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