1. Introduction

The interstellar extinction law is one of the primary sources of information about the interstellar grain population, and one often obtains direct information on the composition of interstellar dust from spectral features in extinction (Draine, 2003). These spectral features also provide strong constraints on interstellar grain models. With the advent of ground-based and space borne infrared (IR) telescope facilities, the IR extinction continuum and absorption features have been receiving increasing attention and play an essential role in recovering the intrinsic energy distribution of celestial objects and inferring the characteristics of interstellar dust.

In the interstellar extinction curve, the 2175 Å bump is outstanding in the ultraviolet (UV), while in the IR, there are a number of prominent absorption features as well: (1) the ubiquitous 9.7μm and 18μm features respectively due to the Si-O stretching and O-Si-O bending modes of amorphous silicates; (2) the 3.4μm feature due to the CH stretching mode of aliphatic hydrocarbon dust, as ubiquitously present in the ISM of the Milky Way and external galaxies as the 9.7μm and 18μm silicate bands, except this feature is not seen in dense molecular clouds (see Pendleton, 2004 for a review); (3) the 3.3μm and 6.2μm weak features seen in both local sources and GC sources (Schutte et al., 1998; Chiar et al., 2000), respectively due to the C-H stretching and C-C stretching modes of polycyclic aromatic hydrocarbon (PAH) molecules; and (4) in dense clouds the 3.1μm feature due to the O-H stretching mode of water ice as well as a number of weaker features at 4.68μm (CO), 7.68μm (CH4), 4.28μm, 15.2μm (CO2), 3.54μm, 9.75μm (CH3OH).

The 9.7μm silicate extinction profile varies among different sightlines; in particular, its optical depth Δτ9.7 μm (relative to the visual extinction AV) shows considerable variations from the local diffuse ISM (LDISM) to the Galactic center (GC): AV/Δτ9.7 μm ≈ 18.2 for LDISM differs from that of the GC (AV/Δτ9.7 μm ≈ 8.4) by a factor of ∼2.2 (see Table 1; also see Draine, 2003).Footnote 1 Roche and Aitken (1985) argued that AV/Δτ9.7 μm varies because there are fewer carbon stars in the central regions of the Galaxy and the production of carbon-rich dust may be substantially reduced compared with the outer Galactic disk. However, as shown in Table 1, the 3.4μm C-H feature of aliphatic hydrocarbon dust also exhibits a similar behavior: Δτ3.4 μm ∼ 274 for LDISM is higher than that of the GC (AV/Δτ3.4 μm ∼ 146) by a factor of ∼1.9 (see Table 1). If the argument of Roche and Aitken (1985) was valid, one would expect a much smaller AV/Δτ3.4 μm ratio in the LDISM than that of the GC. Sandford et al. (1995) tried to quantitatively explain this phenomena by assuming that the abundance of the C-H carrier (relative to other dust components) gradually increases from the local ISM toward the GC.Footnote 2 However, this requires that amorphous silicate dust and aliphatic hydrocarbon dust should not be solely responsible for the visual extinction. If one has to invoke an additional dust component (most likely a population of carbon dust which does not show the characteristic 3.4μm feature, say, graphite) making an appreciable contribution to AV, one would encounter a severe carbon budget problem (see Snow and Witt, 1996).

Table 1.
figure Tab1

Observational values of AV/Δτ3.4 μm and AV/Δτ9.7 μm .

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Along the lines of sight toward the GC, there are dense molecular clouds.Footnote 3 In cold, dense molecular clouds, interstellar dust is expected to grow through coagulation (as well as accreting an ice mantle) and the dust is likely to be porous (Jura, 1980). In this work, we demonstrate that the observed variations of AV/Δτ9.7 μm and AV/Δτ3.4 μm from the LDISM to the GC could be explained in terms of composite porous dust.

2. Model

We consider a composite porous dust model consisting of amorphous silicate, carbon, and vacuum (in dense clouds silicate dust and carbon dust are coated with water ice). We take the optical constants of Draine and Lee (1984) for amorphous silicate, of Li and Greenberg (1997) for carbonaceous organic refractory (to represent the carbon dust component), of Li and Greenberg (1998) for water ice. The mass densities of silicate dust, organic refractory dust and ice are taken to be ρsil ≈ 3.5 g cm-3, ρcarb ≈ 1.8 g cm-3 and ρice ≈ 1.2 g cm-3, respectively. We take the mass ratio of organic refractory dust to silicate dust to be mcarb/msil = 0.7 and the mass ratio of water ice to organic refractory dust and silicate dust to be mice/(mcarb + msil) = 0.8, as inferred from the cosmic abundance constraints (see appendix A of Li and Lunine, 2003a).

For the dust in the local diffuse ISM, we assume the dust to be a solid compact mixture of amorphous silicate and organic refractory materials with mcarb/msil = 0.7. We take the dust size to be a = 0.1μm, the typical grain size for the dust in the diffuse ISM (see Draine, 1995). For the dust in the dense molecular clouds along the lines of sight toward the GC, we assume that silicate dust and organic refractory dust are equally coated with an ice layer and then form a porous aggregate (see Li and Lunine, 2003b). For porous dust, a key parameter is the porosity P (or fluffiness; the fractional volume of vacuum in a grain). We will consider a range of porosities. We assume all grains are spherical in shape; the porous grain size a is defined as the radius of the sphere encompassing the entire porous aggregate. In order to find suitable porosity P and dust size a for the dust in the dense molecular clouds to reproduce the observed AV/Δτ9.7 μm and AV/Δτ3.4 μm ratios toward the GC, we leave both P and a adjustable.

We use Mie theory in combination with the Maxwell- Garnett and Bruggeman effective medium theories (Bohren and Huffman, 1983; see eqs. 7-9 of Li and Lunine, 2003b and Kimura et al., 2008b) to calculate the optical properties of composite porous grains. This approach is valid for computing the integral scattering characteristics (e.g. extinction, scattering, absorption cross sections, albedo and asymmetry parameter; see Hage and Greenberg, 1990; Wolff et al., 1994).

For illustration, we plot in Fig. 1 the 1–100μm extinction cross sections Cext(λ) of compact dust (for the local diffuse ISM) and porous dust (for dense clouds toward the GC). For a given dust size, both AV and Δτ3.4 μm, Δτ9.7 μm decrease with the porosity P, although the degree of decrease is somewhat less significant for Δτ3.4 μm and Δτ9.7 μm than for AV (thus AV/Δτ3.4 μm and AV/Δτ9.7 μm decrease moderately with the increase of P; see Table 2). This is because with the same size, a porous grain contains less dust material than its solid counterpart. The effective dielectric functions of porous dust are reduced. In the IR the extinction for dust of ∼0.1μm is dominated by absorption (i.e. in the Rayleigh regime) and is roughly proportional to the imaginary parts of the dielectric functions (see Li, 2008). Therefore porous dust produces smaller Δτ3.4 μm and Δτ9.7 μm than its solid counterpart of the same size. In the optical, both absorption and scattering are important. The introduction of vaccum leads to a reduction of the dielectric functions of the dust which will decrease both the absorption and scattering.

Fig. 1.
figure 1

Extinction cross sections Cext(?) of different types of dust. The 3.1μm water ice O-H feature shows up in the extinction profiles of porous dust.

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For models consisting of single-sized dust, the ratio of AV to the optical depth of the 9.7μm silicate feature is simply AV/Δτ9.7 μm ≈ 1.086Cext(V)/ΔCext(9.7μm), where Cext(V) is the extinction cross section at V band (γ = 5500Å), ΔCext(9.7μm) is the excess extinction cross section of the 9.7μm feature above the continuum, and the factor “1.086” arises from the conversion of extinction (in magnitude) to optical depth. We obtain ΔCext(9.7μm) by integrating the 9.7μm model extinction profile (with the continuum subtracted) over wavelength (see Fig. 2 for illustration) and then dividing the integrated value with the width of the interstellar 9.7μm silicate absorption feature. The same procedure is applied to the 3.4μm feature to calculate ΔCext(3.4μm) so as to obtain AV/Δτ3.4 μm (see Fig. 3 for illustration).

Fig. 2.
figure 2

Schematic illustration of obtaining ΔCext(9.7μm), the excess 9.7μm extinction cross section above the continuum. It is obtained by (1) fitting the 9.7μm model profile with a Drude function and subtracting the continuum which is fitted with a six-order polynomial, (2) integrating the continuum-subtracted extinction profile (which is approximated by a Drude function) over wavelength, and finally (3) dividing the integrated value with the width of the interstellar 9.7μm silicate absorption feature.

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Fig. 3.
figure 3

Same as Fig. 2 but for the 3.4μm feature. In the right panel, the 3.1μm water ice O-H feature shows up in the extinction profile of porous dust.

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3. Results and Discussion

We calculate AV/Δτ3.4 μm and AV/Δτ9.7 μm for various dust models with a range of porosities and dust sizes. The results are tabulated in Table 2. For a given dust size a = 0.1μm, porous dust results in smaller AV/Δτ3.4 μm and AV/Δτ9.7 μm values than compact dust (see Section 2). This is encouraging that porous dust which is likely present in the dense molecular clouds along the sightlines toward the GC indeed is on the right track to account for the observed AV/Δτ3.4 μm and AV/Δτ9.7 μm variations from the local diffuse ISM to the GC. More specifically, from Table 2 we see that the dust with P ∼ 0.8–0.9 and a ∼ 0.5–1μm can approximately explain the observed variations of AV/Δτ3.4 μm and AV/Δτ9.7 μm (by a factor of ∼2) from the local diffuse ISM to the GC. Both high porosities (P ∼ 0.8–0.9) and large sizes (a ≥ 0.5μm) are required for the GC dust to account for the lower AV/Δτ3.4 μm and AV/Δτ9.7 μm ratios.

Table 2.
figure Tab2

AV/Δτ3.4 μm and AV/Δτ9.7 μm calculated for various dust models.

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In Fig. 4, we show AV/Δτ3.4 μm and AV/Δτ9.7 μm as a function of dust size for P = 0.8, 0.9, 0.95. It is clearly seen that for a given porosity, the variation of AV/Δτ3.4 μm with dust size exhibits a tendency similar to that of AV/Δτ9.7 μm. This suggests that with a dust size distribution taken into account, we would still maintain the variation tendency. For small, highly porous grains (a < 0.05μm), AV/Δτ3.4 μm and AV/Δτ9.7 μm are nearly independent of size since they are more or less in the Rayleigh regime in the optical-IR and therefore AV/Vdust, Δτ3.4 μm/Vdust and Δτ9.7 μm/Vdust are independent of grain size (where Vdust is the dust volume). The visual extinction AV reaches its maximum at a ∼ λ/[2π a (n - 1)] (where n is the real part of the refractive index of the porous dust at wavelength λ; see Li, 2008). At even large sizes, while AV reaches a constant (i.e. in the geometric optics regime) Δτ3.4 μm and Δτ9.7 μm increase with a (till they reach their respective peak values). This explains why AV/Δτ3.4 μm and AV/Δτ9.7 μm decrease with a after they reach their peak values.

Fig. 4.
figure 4

Variations of AV/Δτ3.4 μm and AV/Δτ9.7 μm with dust size.

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For AV/Δτ3.4 μm, our model with mcarb/msil = 0.7 (and P = 0.8, a = 0.5–1.5μm) is consistent with the observed factor-of-two variations in the local ISM and toward the GC (see Tables 1, 2). However, the model values of AV/Δτ9.7 μm for both the diffuse ISM dust (AV/Δτ9.7 μm ≈ 38.2) and the GC dust (AV/Δτ9.7 μm ≈ 16.3) are higher by a factor of ∼1.5–2 than that observed in the local diffuse ISM (AV/Δτ9.7 μm ≈ 18.2) and the GC (AV/Δτ9.7 μm ≈ 8.4). This discrepancy may result from the underestimation of the silicate mass fraction. With an increased silicate mass fraction, say, mcarb/msil = 0.5 which is consistent with the in situ measurements of comet Halley (Jessberger and Kissel, 1991) and widely adopted in cometary dust modeling (Greenberg, 1998; Greenberg and Li, 1999; Kolokolova et al., 2004; Kimura et al., 2006, 2008a; Mann et al., 2006; Kolokolova and Kimura, 2010), we obtain AV/Δτ3.4 μm ≈ 252 and AV/Δτ9.7 μm ≈ 27.1 for the local ISM (assuming compact dust), and AV/Δτ3.4 μm ≈ 154 and AV/Δτ9.7 μm ≈ 11.3 for the GC (assuming porous dust). These values are closer to that observed. It is expected that with a smaller mcarb/msil (i.e. a larger silicate mass fraction), one would obtain a smaller AV/Δτ9.7 μm while AV/Δτ3.4 μm does not change much. Thus the observed variations of AV/Δτ3.4 μm and AV/Δτ9.7 μm from the local ISM to the GC could be explained. It is worth noting that, based on a detailed analysis of the GC 5–8μm absorption spectra obtained from the Kuiper Airborne Observatory, Tielens et al. (1996) argued that silicate dust may contribute as much as 60% of the interstellar dust volume. This would translate to mcarb/msil ≈ 0.34 if we assume that the remaining 40% of the interstellar dust volume is all from the 3.4μm C-H feature carrier (which is indeed a very generous assumption).

Admittedly, the proposed explanation is oversimplifed. In the future we will consider more realistic models in which more dust species (e.g. hydrogenated amorphous carbon with a range of C/H ratios), the distribution of dust along the line of sight toward the GC (e.g. see Sandford et al., 1995), a distribution of dust sizes, and the possible porous nature of the diffuse ISM dust (e.g. see Mathis and Whiffen, 1989) will be considered.