E
The Astrophysical Journal, 595:522–530, 2003 September 20 # 2003. The American Astronomical Society. All rights reserved. Printed in U.S.A.
IRREGULAR PARTICLES IN COMET C/1995 O1 HALE-BOPP INFERRED FROM ITS MID-INFRARED SPECTRUM F. Moreno and O. Mun˜oz Instituto de Astrofı´sica de Andalucı´a, CSIC, P.O. Box 3004, 18008 Granada, Spain;
[email protected]
R. Vilaplana Universidad Polite´cnica de Valencia, Valencia, Spain
and A. Molina Departamento de Fı´sica Aplicada, Universidad de Granada; and Instituto de Astrofı´sica de Andalucı´a, CSIC, P.O. Box 3004, 18008 Granada, Spain Received 2003 April 3; accepted 2003 June 4
ABSTRACT An analysis of the Infrared Space Observatory (ISO) infrared spectra of comet C/1995 O1 Hale-Bopp has been conducted. The particles in the coma are assumed to be irregular aggregates that are built by a diffusionlimited aggregation (DLA) procedure and are composed of olivine, both amorphous and crystalline, and glassy carbon. To simulate the morphological structure of the interstellar dust, the silicate component is placed in the inner layers of the aggregate, while the carbon dipoles occupy the outermost layers. The particle emissivities are calculated using the discrete dipole approximation (DDA) method of Draine & Flatau. The dust size distribution functions at different heliocentric distances (2.8, 2.9, and 3.9 AU) have been determined as the solution to three overdetermined systems of equations for the infrared flux at each wavelength in the 8–40 lm region. These size distributions can be well fitted by a power law, nðrÞ / r , having power indexes that decrease with heliocentric distance as ¼ 3:6 0:3 at 2.8 AU and 3:3 0:3 at 3.9 AU. Subject headings: comets: general — comets: individual (Hale-Bopp (C/1995 O1)) — dust, extinction — infrared: solar system On-line material: color figure
While it is possible to obtain the scattering parameters for nonspherical particles at size parameters X ¼ 2r=d15 with a variety of computer codes (see, e.g., Mishchenko et al. 2000 and references therein for a summary of the available numerical techniques), and for large nonabsorbing particles in the geometric optics limit (see, e.g., Liou, Takano, & Yang 2000), there is not a general method that allows one to obtain the scattering functions and the absorbing and scattering efficiencies for arbitrary sizes and refractive indexes for irregular particles, such as those one could find in a comet. All the previous works on the interpretation of cometary infrared spectra have relied on the assumption that the spectra can be assumed as a sum of several separate dust components. Thus, Brucato et al. (1999) assumed the presence in the cometary dust of amorphous olivine particles, amorphous carbon particles, and crystalline olivine particles. They employed laboratory-measured mass absorption coefficients for both the amorphous carbon and crystalline olivine particles. Similarly, Harker et al. (2002) also assumed the presence of crystalline silicate (pyroxene and olivine) particles as being separated from the other contributions. We propose here a new method, which we believe resembles in a closer way the actual grains that populate the comet dust. We assume that the cometary dust grains are irregular particles of a mixed composition, namely, amorphous olivine and carbon, plus crystalline olivine, in a percentage compatible with the findings of Greenberg (1998) for a comet nucleus composition. The irregular particles are assumed here to be aggregates that are built up by a diffusion-limited aggregation (DLA) procedure (Witten & Sander 1983), and
1. INTRODUCTION
The shape of the cometary spectra in the thermal region (8–45 lm) has revealed the presence of crystalline silicate particles, in particular Mg-rich crystalline olivine (Mg2SiO4, forsterite) (Campins & Ryan 1989; Hanner, Lynch, & Russell 1994; Colangeli et al. 1995). A careful interpretation of those spectra has also revealed the presence of amorphous carbon and amorphous silicates, as well as crystalline pyroxene (Brucato et al. 1999; Wooden et al. 1999; Harker et al. 2002). These findings are consistent with the composition of cometary dust particles as detected by Giotto and Vega (Kissel et al. 1986). The silicate emission features are clearly seen in the observations of comet Hale-Bopp from ISO (Crovisier et al. 2000). These spectra have been shown to have remarkable similarities to the spectra of dusty disks surrounding young stars, such as HD 100546, an intermediate star between Vega-like and Herbig Ae-Be stars, suggesting that the crystalline nature of the silicates in comet HaleBopp dates from the early evolution of the solar system (Waelkens, Malfait, & Waters 1999). The interpretation of the cometary thermal spectrum is difficult for a variety of reasons. First, the optical constants of crystalline olivine and pyroxene are not available in the full spectral range needed to perform the equilibrium temperature calculations (see x 2). In addition, the crystalline components are anisotropic, with different refractive indexes along the principal axes, so some kind of average must be performed in order to compute the absorption efficiencies. A second complication arises from the variety of shapes and sizes of the grains present in the comet comae. 522
IRREGULAR PARTICLES IN COMET HALE-BOPP the relevant scattering parameters are obtained by the discrete dipole approximation (DDA) code of Draine & Flatau (1994). A detailed description of the procedure is given in the forthcoming section.
energy emitted in the infrared, as Z a2 1 S ðÞQabs ð; aÞ d r2h 0 Z 1 2 ¼ 4a Qabs ð; aÞB ½T ðaÞ d
2. ANALYSIS OF THE INFRARED SPECTRA
0
The three spectra of comet C/1995 O1 Hale-Bopp under analysis were obtained by the Infrared Space Observatory (ISO) on 1996 September 26, 1996 October 7, and 1997 December 28, when the comet was at a heliocentric distance of 2.9, 2.8, and 3.9 AU, respectively (Crovisier et al. 2000). The analysis of the spectrum taken on 1996 September 26 by Brucato et al. (1999) involved the sum of thermal emissions from various contributors, namely, amorphous carbon, amorphous olivine, and crystalline olivine, plus a continuum blackbody contribution from large particles. Crovisier et al. (2000) also made an interpretation of the spectrum in terms of the sum of several components, but involving two blackbodies to fit both the short- and long-wavelength parts of the spectrum, besides the contribution from amorphous silicates and Mg-rich crystalline olivine and crystalline orthopyroxene. The most sophisticated interpretation to date is that by Harker et al. (2002), who consider fractal, porous spherical grains. They obtained the dust size distribution, but considering separate contributions from each component (amorphous carbon, pyroxene, and olivine and crystalline olivine and pyroxene). We return to these works in x 3, where we compare our results with those of previous works. Our approach considers the thermal spectrum as given by the contribution of irregular particles of variable size composed of various materials, specifically amorphous carbon and olivine and crystalline olivine, the basic ingredients of comet dust composition. Our aim is not to perform a detailed mineralogical analysis, but a simple compositional model in which only the most relevant compounds are taken into account. Thus, for simplicity, we have not included other possible components in the mixture. For instance, the 9.3 lm feature seen for the first time in comet Hale-Bopp marks the presence of pyroxene, probably amorphous enstatite (Hayward, Hanner, & Sekanina 2000), while crystalline pyroxene may be contributing to some spectral features at 9.3, 10, and 10.5 lm (see, e.g., Hayward et al. 2000; Harker et al. 2002). As mentioned in x 1, our approach is different from previous assumptions, in that single particles of mixed composition are responsible for the spectrum observed, instead of separate dust components of different compositions. We believe that our approach is more consistent with the actual dust grains that populate the cometary comae. The procedure we have envisaged can be described as follows. The thermal flux for a single grain can be obtained by the expression F ða; Þ ¼
a2 ð; aÞB ½T ðaÞ D2
ð1Þ
(Hanner et al. 1999), where ð; aÞ is the grain emissivity, a is the grain radius, D is the Earth-comet distance, and B ½T ðaÞ is the Planck function for grain temperature T ðaÞ. The temperature from a single grain is computed from the balance between the energy absorbed in the visual and the
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ð2Þ
(Hanner et al. 1999), where S ðÞ is the solar flux at 1 AU and rh is the heliocentric distance in AU. For a normalized size distribution nðaÞ, where nðaÞda is the number of particles of radii between a and a þ da, equation (1) can be written in integral form as Z amax nðaÞa2 ð; aÞB ½T ðaÞ da ; ð3Þ F ð Þ ¼ 2 D amin where amin and amax are the lower and upper limits of the size distribution function considered. If we consider a discrete number m of particle sizes (a1 ; . . . ; am ) that are contributing to the observed spectrum, with abundances n1 ; . . . ; nm at n wavelengths 1 ; . . . ; n , with n > m, equation (3) can be written as an overdetermined system of n equations as follows: i¼m X ni a2i i j Bi j Di ¼ Fj ; j ¼ 1; . . . ; n ; ð4Þ 2 D i¼1 where i j is the emissivity of a particle of radius ai at a wavelength j , Bi j is the Planck function at the equilibrium temperature appropriate for a particle of radius ai placed at an heliocentric distance rh at a wavelength j , and Di are the corresponding intervals in radius. The Fj are the observed flux vectors at wavelengths j . The system is to be solved for n ¼ 80 wavelengths between 8 and 40 lm and m ¼ 20 radii logarithmically distributed between 0.1 and 5 lm. The logarithmic distribution in the radius is performed because it is expected that there will be a larger number of particles in the bins corresponding to the smaller radii than at the larger radii. We seek a vector solution for this system of equations, whose components ni must all be greater than or equal to zero. To solve the system of equations, we need to compute the emissivity of the particles as a function of size and wavelength and the equilibrium temperatures as a function of size and heliocentric distance. The particles are assumed to be irregular in shape and are built by a DLA mechanism, as described by Witten & Sander (1983). The random aggregate has been built up by a computer code, in which a seed particle is located in the origin of a lattice and then another particle is allowed to walk at random from far away until it arrives at a position adjacent to the occupied site. The procedure is then repeated until a certain total number of particles is reached. We built in this way an aggregate particle consisting of 900 dipoles. Two views of the particle are shown in Figure 1. The composition of the aggregate is taken to be consistent with the composition of the refractory components in a comet nucleus, as described by Greenberg (1998). Following this author, 26% of the mass is silicates, 23% is complex organics dominated by carbon, and 9% is in the form of polycyclic aromatic hydrocarbon (PAH) particles. The rest is assumed to be mostly water, mixed with about 2%–3% of CO, CO2, CH3OH, and other simple molecules. Assuming a mean density for the silicates and carbon of 3.27 and 1.85 g
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Fig. 1.—Two views of the aggregate particle used in the modeling procedure. It has a total of 900 dipoles. White circles represent amorphous olivine, light gray circles correspond to crystalline olivine, and dark gray circles represent the carbon component. [See the electronic edition of the Journal for a color version of this figure.]
cm3, respectively, and neglecting the PAHs and volatile components, we find that 60% of the particle (by volume) is carbon and the remaining 40% is silicate. We further assume that all silicates are in the form of olivine, in a 1 : 1 ratio of crystalline to amorphous abundance; i.e., 20% of the particle volume is crystalline olivine, and the remaining 20% is amorphous olivine. The distribution of particles in the aggregate has been set so as to have the olivine particles occupying the inner parts of the aggregate and the carbon particles placed in the outermost layers (see Fig. 1). In this way we tried to simulate the morphological and chemical structure of interstellar dust, as suggested by Greenberg (1998). The optical constants were taken from different sources as follows. The glassy carbon optical constants were taken from Edoh (1983). The amorphous olivine optical constants were taken from Dorschner et al. (1995; their Mg2yFe22ySiO4 olivine, with y ¼ 0:5) and those for crystalline olivine from Steyer (1974). For the amorphous components, the optical constants are isotropic and are available in the whole wavelength region from the far-UV to the farIR. The crystalline olivine is an anisotropic, orthorhombic, biaxial mineral, so the index of refraction depends on the three principal directions, or propagation axes. In addition, their optical constants are not available at wavelengths smaller than 3 lm (Steyer 1974). This becomes an obvious problem when the equilibrium temperature has to be obtained, as described below. The emissivities are computed by applying Kirchoff’s law; i.e., the emissivity becomes equal to the absorption efficiency, Qabs (see, e.g., Hanner et al. 1999; Harker et al. 2002). The absorption efficiencies are computed by the DDA code of Draine & Flatau (1994), which is publicly available on the World Wide Web. To perform the calculations, each particle in the aggregate is assumed to be a dipole of a certain composition, as described above. The absorption efficiencies are computed at 80 wavelengths between 8 and 40 lm and 20 radii logarithmically spaced between 0.1 and 5 lm. Each value of the absorption efficiency
corresponds to an average of 36 orientations of the target (three around the angle , three around h, and four around ). The whole procedure took about 5.2 days CPU time on a Pentium 4 processor at 1.7 GHz. One important source of concern when obtaining the emissivity spectra is the large value of the refractive indexes at some wavelengths. The DDA code is known to produce very accurate results in the scattering functions and scattering efficiencies when the refractive index m satisfies the criterion kmk < 1 (Draine & Flatau 1994). As the materials involved in the computations of the emissivities have refractive indexes much larger than 1 at several wavelengths, we first give an indication of the magnitude of the errors obtained by comparing with results produced by Mie theory for spheres. As there is not an effective medium theory that produces exact results, we performed the estimates only for particles of homogeneous composition and spherical shape. Also, as the larger indexes are found at a number of wavelengths for the anisotropic crystalline olivine (at the strongest resonance features), we compare the results for the three mutually perpendicular axes independently, first using the DDA code, then with Mie theory. To be consistent with the number of dipoles used for the aggregate, we used the DDA for a homogeneous sphere composed of 912 dipoles. Figure 2 shows the comparison at four values of particle radii and at the three crystallographic axes. The agreement is generally good, with differences of a few percent, except at the largest values of the particle radii, r > 4 lm, and at a few limited frequency ranges, where we found the largest discrepancies. At those frequency ranges, it is generally found that the DDA efficiencies are larger than those given by Mie theory, by an amount that in some cases reaches 50%. We address this point later in this section. The reason for the choice of the particles to be irregular can be immediately understood by a quick view at Figure 3. In this graph we show the resulting absorption efficiency for spheres having the same composition as the aggregates shown in Figure 1, together with one of the Hale-Bopp spectra under analysis. It is clear not only that the spheres do
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Fig. 2.—Absorption efficiency computed for spheres having the refractive indexes of crystalline olivine (Steyer 1974). The rows of panels correspond to Eka (top), Ekb (middle), and Ekc (bottom). The sphere radii are indicated at the top. The dashed lines correspond to Mie calculations, while the solid lines correspond to DDA calculations for a sphere having 912 dipoles.
Fig. 3.—Absorption efficiency spectrum computed for spheres composed of 60% carbon, 20% crystalline olivine, and 20% amorphous olivine (by volume), in comparison with a scaled ISO spectrum of Hale-Bopp at a heliocentric distance of 2.9 AU (thick solid line). The absorption efficiencies have all been divided by their maximum value in the spectral range shown. The thin solid line corresponds to a particle radius of 0.1 lm, the dashed line to 0.78 lm, the dash-dotted line to 3.3 lm, and the dotted line to 5 lm.
not reproduce correctly the 10 lm silicate peak, as was first shown by Yanamandra-Fisher & Hanner (1999), but also, as the spheres become larger than about 1 lm, that the shape of the absorption efficiency spectra becomes poorly correlated with the observed ISO spectrum. This is direct evidence of the irregularity of the particles composing the comet dust. On the other hand, in Figure 4, we show the same absorption efficiency spectra as for spheres, but now for the irregular aggregates. Here it can be seen that the spectra show all the spectral features present in the cometary spectrum, at each size considered. The reason for the choice of the radii of the particles to be between 0.1 and 5 lm is related to the observed behavior of the emissivity spectrum with wavelength. Figure 4 shows how the 10 lm spectral feature is correlated to the observed cometary spectrum, but as the particle size increases, the width of the feature also increases, becoming much larger than that observed. The same happens for all the other features, although it is less evident at larger wavelengths than at the 10 lm feature. It is interesting to note that the width of the feature has been shown to change among comets (Hanner et al. 1994), so it could be indicative to some extent of differences in particle size distribution among different
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TABLE 1 Absorption Efficiencies Computed Using the Maxwell-Garnett (M-G) Mixing Rule and the DDA Code for Inhomogeneous Spheres (lm)
r (lm)
Qabs (M-G)
Qabs (DDA)
Dipoles (DDA)
0.1.........
0.10 0.20 0.10 0.45 0.80 1.00
1.263 1.097 1.306 1.315 1.151 1.103
1.282 1.113 1.323 1.333 1.170 1.116
47,833 47,833 33,552 33,552 33,552 47,833
0.5.........
Fig. 4.—Absorption efficiency spectrum computed for the aggregate shown in Fig. 1, which is composed of 60% carbon, 20% crystalline olivine, and 20% amorphous olivine (by volume), in comparison with a scaled ISO spectrum of Hale-Bopp at a heliocentric distance of 2.9 AU (thick solid line). The absorption efficiencies have all been scaled to 1 at 11.3 lm. The thin solid line corresponds to particle radius of 0.1 lm, the dashed line to 0.78 lm, the dash-dotted line to 3.3 lm, and the dotted line to 5 lm. The radius for a nonspherical particle is defined as that of a sphere having the same volume as the irregular particle.
comets. Another reason to place the upper limit at 5 lm is that most of previous analyses of Hale-Bopp’s infrared spectra have demonstrated that the 10 lm feature arises from predominantly submicron grains (see, e.g., Hanner et al. 1999; Williams et al. 1997; Hayward et al. 2000; Harker et al. 2002). Also, by fitting a blackbody continuum to one of the ISO Hale-Bopp spectra, Crovisier et al. (1997) conclude that the fitted blackbody temperature was 20% higher than the equilibrium blackbody, indicating that the thermal emission arises from particles that are small compared to wavelength. We should also recognize that we cannot extend the modeling to particles larger than about 5 lm because of the large size parameters that would be involved in the calculations, a technical problem impossible to solve with the available computers. Another quantity that should be computed before the system can be solved is the equilibrium temperature of the particles as a function of radius and the heliocentric distance. Here we face two problems. On the one hand, as stated above, for one of the (essential) components of the dust grains (crystalline olivine) there are no estimates of the refractive index at wavelengths smaller than 3 lm (Steyer 1974). An approach used by Harker et al. (2002) is to assume that at those wavelengths the refractive index becomes similar to that of the amorphous silicates. However, even assuming that, another complication arises: the absorption efficiency Qabs must be calculated for nonspherical particles at wavelengths for which the size parameter becomes very large, and at present it is impossible to handle absorbing, nonspherical particles having size parameters as large as X 300 (which would corresponds to the worst case, i.e., particles of 5 lm in radius at a wavelength of 0.1 lm) with any computer code. Therefore, in order to estimate the grain temperatures, we assumed composite spherical particles of amorphous olivine plus carbon compounds. We used the Maxwell-Garnett effective medium theory to compute the equivalent refractive index for spherical particles composed of 60% carbon plus 40% amorphous olivine
in volume, having the refractive indexes cited above for each compound. Hanner et al. (1996) also applied an effective medium theory to obtain grain temperatures to compute mid-IR spectra of several comets, for grain composites having 50% carbon and 50% bronzite by volume. In order to assess the possible inaccuracies in this particular application of the Maxwell-Garnett rule, we computed the absorption efficiencies for spheres of that composition using the DDA code and compared the results with Mie theory, although only in the range in which the size parameter is low enough to permit computation with the DDA code for a maximum of 47,833 dipoles. In order to simulate random olivine inclusions in the sphere, we built a spherical target in which we set, at random, 40% of the dipoles to be of amorphous olivine ‘‘ composition ’’ (refractive index) and 60% of the dipoles to have the glassy carbon refractive index. Table 1 shows the resulting Qabs for both Mie computations in combination with the Maxwell-Garnett rule and the DDA code at the smallest wavelengths and correspondingly largest size parameters. The agreement between the absorption efficiencies for the cases shown in the table is quite satisfactory, the discrepancies being always less than 2%. At wavelengths of 3 lm and larger, it is feasible to cover all the region of sizes r 0:1 lm. Figure 5 shows the absorption efficiency versus sphere radius at four wavelengths, with the DDA code and with Mie theory in combination with the Maxwell-Garnett rule. The agreement between the absorption efficiencies is generally good, the discrepancies being generally less than about 15%. This suggests that the Maxwell-Garnett rule may be used, at least as a first approximation, for this particular application, to compute the absorption efficiencies for these composite spheres. The grain temperatures are obtained from the numerical solution of equation (2), from min ¼ 0:03 lm to max ¼ 300 lm, a wavelength range that is wide enough to ensure the validity of the computed temperatures. The solar flux was approximated by a blackbody at an effective temperature of T ¼ 5900 K. The equation was solved iteratively by Brent’s method (Brent 1973), using the source code obtained from Press et al. (1992). The validity of the results obtained was checked by comparing with calculations done by Hanner et al. (1999) for glassy carbon spheres at various radii and heliocentric distances, for which we found the same results. The resulting grain temperatures for the inhomogeneous spheres as a function of the grain size are depicted in Figure 6, for the three different heliocentric distances at which the comet was observed with ISO. With those temperature data, we are now ready to solve the system of equations. Our approach consists in solving the system with the constraint
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Fig. 5.—Absorption efficiency as a function of the radius of the sphere composed of 60% carbon and 40% olivine (by volume), computed with the DDA (thin lines) and the Mie theory in combination with the Maxwell-Garnett mixing rule (thick lines). The computations are shown at various wavelengths, as indicated. The number of dipoles used to represent the sphere in the DDA code is also shown.
that the solution should be positive. We performed the computations by the simplex multidimensional minimization method, as described in Press et al. (1992). The best fits are shown in Figure 7, in which the best-fitted spectrum is shown together with the corresponding ISO spectrum at each heliocentric distance. The overall shapes of the spectra
Fig. 6.—Grain temperature at the three heliocentric distances as a function of grain radius for spheres composed of 60% carbon and 40% olivine (by volume).
are very well reproduced, the standard deviation of the fits being ¼ 7:2, 8.9, and 6.3, at 2.8, 2.9, and 3.9 AU, respectively. A perfect match to the data cannot be expected, as we have not included all the minerals that have been detected in the comet dust in our compositional model. The size distribution functions that gave the best fits are shown in Figure 8. As can be seen, the size distribution functions may be well fitted to a power law, which is defined as nðrÞ / r , where nðrÞdr is the number of particles per unit volume with radius between r and r þ dr and is the power index. The power index of the size distribution function shows a slight tendency to decrease with heliocentric distance, being 3:6 0:3 at 2.8 AU, 3:4 0:3 at 2.9 AU, and 3:3 0:3 at 3.9 AU. It is also interesting to note that the maxima of the size distribution, at the three heliocentric distances, all occur at the same particle radius of 0.12 lm, although the magnitude of the peak is not very significant with respect to the adjacent size distribution values at 0.1 and 0.15 lm. Concerning errors, we stated above that there may be an important source of error in the determination of Qabs for the largest (r 4 lm) particles. As it can seen in Figure 2, and because of some resonance frequencies in the olivine crystal, the DDA code tends to overestimate Qabs in a few spectral intervals (e.g., near 26 lm) by a factor of about 1.5 larger than that computed by Mie theory for spheres. The mean errors (summing over the whole wavelength range from 8 to 40 lm) are 7.1% on the a-axis, 8.7% on the b-axis,
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Fig. 8.—Size distribution functions corresponding to the best-fitted spectra of Fig. 7, at the three heliocentric distances. Also shown are the least-squares fits to the size distribution, whose slopes are 3:6 0:3, 3:4 0:3, and 3:3 0:3 at 2.8, 2.9, and 3.9 AU, respectively. Fig. 7.—Observed ISO spectra (thick lines; Crovisier et al. 2000) and the modeled spectra (thin lines), at the three heliocentric distances, as labeled. The modeled spectra refer to the corresponding fluxes at 80 wavelengths between 8 and 40 lm used to solve the system of eq. (4).
and 7.6% on the c-axis. In order to estimate the effect of these errors on the computed dust size distributions, we solved the system of equations (eq. [4]), keeping all the coefficients to their value, except for the bins corresponding to particle effective radius of r 4 lm, which we multiply by a factor of 1.3, a factor that is larger than the error corresponding to the sum of the errors for each axis. As could be expected from the lower abundance of these large particles with respect to the smaller ones, no essential modifications in the retrieved size distribution functions are found. Thus, the size distribution power indexes change to 3:7 0:4 at 2.8 AU, 3:3 0:3 at 2.9 AU, and 3:1 0:3 at 3.9 AU. Given the size distribution, the total number of particles (within the ISO aperture) and the total mass can be derived, for a mean density of the particles of 2.42 g cm3, which is computed from a mixture of carbon (60% by volume) and olivine (40%) at the densities given above. A summary of all the derived parameters appears in Table 2. One of the conclusions from these results is that, neglecting temporal changes in the dust composition and structural changes (e.g., fluffiness) in the particles, the size distribution function does change with the heliocentric distance. Such
changes have been predicted by other techniques, such as the numerical dust tail inversion models (see, e.g., Fulle, Cremonese, & Bo¨hm 1998). 3. COMPARISON WITH PREVIOUS WORK
The main difference between this work and all the previous investigations and analysis of cometary thermal spectra is that our approach involves the existence of composite, irregular, inhomogeneous particles contributing to the thermal flux instead of separate contributions to the spectrum from various homogeneous (and in most cases, spherical) particles of differing composition. Also, our model does not make use of any laboratory-measured emissivities. Instead, we computed the emissivities from the refractive indexes available in the literature. Thus, Brucato et al. (1999) fitted the ISO Hale-Bopp spectrum at 2.9 AU by considering separate contributions from laboratorymeasured crystalline olivine and amorphous carbon mass absorption coefficients, as well as Mie calculations for spheres for the amorphous olivine component, plus a contribution from unspecified large particles needed to fit the long-wavelength side of the spectrum. Therefore, no precise estimates of the actual particle size can be made with their approach. They conclude that the observed Hale-Bopp spectrum can be fitted by both a small-particle component (r < 1 lm), responsible for the emission features, and a
TABLE 2 Retrieved Parameters from the Fit to Hale-Bopp Infrared Spectra
Date
rh (AU)
Standard Deviation ()
Power Index
Total Number of Particles (1025)
Total Mass (1012 g)
1996 Oct 7 ................ 1996 Sep 26 .............. 1997 Dec 28..............
2.8 2.9 3.9
7.2 8.9 6.3
3.6 0.3 3.4 0.3 3.3 0.3
2.5 4.1 2.1
3.3 6.9 1.6
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large-particle component (r > 50 lm), which provides only a featureless continuum spectrum. Harker et al. (2002) used laboratory-measured optical extinction for Mg-rich and orthopyroxene powders to calculate the emission from the submicron crystals and used Mie theory to compute the grain temperatures. The thermal spectra were computed as the sum of several contributions, including amorphous carbon, pyroxene, and olivine and crystalline olivine and pyroxene. They used fractal, porous grains, in combination with an effective medium theory, to calculate the thermal spectra and made an interpretation of the extended spectral energy distribution combining ISO observations plus other ground-based observations. They used a modified powerlaw size distribution (Hanner 1983). They concluded that the peak grain size distribution function remained constant at each heliocentric distance, the peak being at r ¼ 0:2 lm. These authors also found that the slope of the size distribution steepens ( ¼ 3:4 to 3.7) as the comet approached perihelion. We also found the same tendency in power index, the size distribution having a larger slope when the comet was closer to perihelion ( ¼ 3:6 at 2.8 AU) than when the comet was farther from perihelion ( ¼ 3:3 at 3.9 AU). This is remarkable, taking into account the different approaches used in the modeling procedure. It is also interesting to note that Fulle (1999) has estimated the power index to be in the range 3:7 < < 3:0, from analysis of tens of comets. In particular, Fulle et al. (1998), from dust tail images analysis of comet Hale-Bopp at 4 AU, found a time-averaged size distribution power index of 3:6 0:1, which is consistent with our values and those of Harker et al. (2002). Regarding the Harker et al. (2002) conclusions on changes on composition with heliocentric distance, we just can state that they are in principle not needed to explain the observations, at least of the overall shape of the spectra, but we of course cannot rule out their existence, since we have not made a detailed compositional modeling. Wooden et al. (1999) also presented a detailed mineralogical analysis of the spectra of Hale-Bopp, deriving a peak grain size of 0.15 lm, very close to our small maximum at r ¼ 0:12 lm. Nevertheless, Wooden et al. (1999) gave another interpretation, based on the presence of porous interstellar aggregates of silicate core–organic refractory mantle particles having a peak size distribution at 24 lm, as derived by Li & Greenberg (1998). Hayward et al. (2000), from their analysis of 3– 13 lm imaging and spectra of comet Hale-Bopp, derived a peak grain size of 0.13 lm near the comet’s perihelion, which is still closer to the value derived by us. Regarding compositional differences, our models assume a crystalline-to-amorphous silicate ratio of 1 : 1. This value is between the ratio given by Brucato et al. (1999), 3.45 : 1 at 2.9 AU, and the ratio of 0.5 : 1 given by Harker et al. (2002). As explained by Harker et al. (2002), these discrepancies may be attributable to the temperature assigned to each grain species. The silicate-to-carbon mass ratio assumed here, consistent with the nucleus composition model by Greenberg (1998), is rather different from that found by those authors: Brucato et al. (1999) gave 8 : 1, and Harker et al. (2002) gave values between 3.8 : 1 and 4.5 : 1, depending on the heliocentric distance, while we assumed 1.1 : 1.
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4. CONCLUSIONS
We have developed a model of cometary grains, irregular in shape and inhomogeneous in composition. The grains are assumed to be composed of olivine (crystalline and amorphous) and carbon, with mass fractions of 0.54 and 0.46, respectively, and with an amorphous-tocrystalline olivine ratio of 1 : 1. The grains are assumed to have the silicates in the inner layers of the particles, to be consistent with the morphological structure of core-mantle interstellar dust grains. In this approach, the grain temperatures are obtained by Mie theory for spheres, but the absorption efficiencies are calculated by the DDA code, which can handle irregular and inhomogeneous particles. The grain size distribution is obtained as the vector solution (restricted to have all the components greater than or equal to zero) to an overdetermined system of equations in which the coefficients are proportional to the product of the Planck function at the grain temperature and the emissivity. In this way, we obtained a size distribution function having a small maximum at 0.12 lm, which, while it is not very significant compared to the adjacent values, does keep constant with the heliocentric distance, at least for the analyzed Hale-Bopp spectra at 2.8, 2.9, and 3.9 AU. The power index of the size distribution function tends to decrease slightly with heliocentric distance, being 3:6 0:3 at 2.8 AU, and 3:3 0:3 at 3.9 AU. These two facts agree very well with previous modeling by Harker et al. (2002), which is based on a rather different approach. Unfortunately, because of the need for a very large amount of CPU time and memory storage, we could not extend our analysis to published spectra acquired at wavelengths shorter than 8 lm, as the size parameter would become too large to prevent absorption efficiency calculations with the available numerical codes and computer facilities. For the same reason, the thermal equilibrium temperatures could not be obtained for the irregular shapes, but only for spheres (although inhomogeneous in composition). These problems can be overcome only if a significant improvement in the particle scattering codes and/or computer speed takes place. Similarly, regarding laboratory measurements, the determination of refractive indexes of cometary dust components, such as crystalline olivine, to fill in the wavelength gap from the far-UV to 3 lm should also be accomplished. Also, laboratory measurements of emissivities as a function of particle size for the relevant cometary dust compounds would be of greatest interest. We would like to acknowledge Bruce T. Draine and Piotr J. Flatau for making available their DDA code. Thanks also go to Jacques Crovisier for making available to us the ISO Hale-Bopp spectra and to Martha Hanner and Vladimir B. Il’in for supplying refractive indexes of olivine from the thesis work by T. R. Steyer. We are also grateful to an anonymous referee for his/her comments and suggestions. R. V. acknowledges support by the ‘‘ Programa Incentivo a la Investigacio´n de la UPV.’’ This work was supported by contracts AYA2001-1177 and PNE-001/2000-C-01.
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MORENO ET AL.
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