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Astron. Astrophys. 348, 594-599 (1999)

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3. Results and discussion

In Paper I, we had presented results for the extinction by the porous silicate and graphite grains in the wavelength region from [FORMULA] to [FORMULA]. The enhancement in the extinction for porous grains was clearly indicated at certain wavelengths and for some wavelengths it deviated from that obtained for the solid grains. This motivated us to study the effect of porosity on the extinction for the silicate and graphite grains in the near infrared wavelengths between [FORMULA] and [FORMULA] as well as in the ultraviolet region between [FORMULA] and [FORMULA]. In this paper we present the results for the entire spectral range from [FORMULA] to [FORMULA]; i.e. 22 wavelengths. The prolate spheroidal grains are assumed to be randomly oriented.

Silicates: Fig. 2 shows the plots of the extinction efficiency [FORMULA] vs 1/[FORMULA] for the porous (N = 152,1184,4088) silicate grains in the wavelength range of [FORMULA]-[FORMULA] for the grain size (i)[FORMULA] (ii)[FORMULA] (iii)[FORMULA] and (iv)[FORMULA].

[FIGURE] Fig. 2. Extinction efficiencies ([FORMULA]) for porous Silicate grains.

These plots show that for small grain sizes (i.e. [FORMULA] and [FORMULA]) there is not much variation in the extinction with the porosity whereas for larger grain sizes ([FORMULA] and [FORMULA]) the extinction is modified considerably especially in the visible and the UV region. For these large grain sizes each porous grain model is valid upto a certain wavelength (see Table 1) beyond this wavelength the extinction curve would show irregularity; e.g. especially N = 152 in the UV spectral range; i.e. [FORMULA]-[FORMULA]. Fig. 2 also shows [FORMULA] curves for solid spherical grains obtained using Mie theory.

Graphites: Fig. 3 shows plots of [FORMULA] vs [FORMULA] for the porous graphite grains with (i)[FORMULA] (ii)[FORMULA] (iii)[FORMULA] and (iv)[FORMULA]. Except for the curves with a very small grain size, i.e. (i)[FORMULA], all the other curves i.e. for [FORMULA] and [FORMULA] show the variation in the extinction with the porosity. As mentioned earlier in the case of silicates, for large grain sizes; i.e. [FORMULA] and [FORMULA]; each porous grain model will be valid upto a certain wavelength (check Table 2) and beyond that wavelength the extinction curves for that model would start showing irregularities (e.g. N = 152 in the UV spectral range). These curves also show a shift in the central wavelength of the extinction peak, as well as the variation in the width of the peak, as the porosity increases.

[FIGURE] Fig. 3. Extinction efficiencies ([FORMULA]) for porous Graphite grains.

In order to emphasize these aspects we show in Fig. 4, the extinction efficiency [FORMULA] for the porous graphite grains for [FORMULA] in the wavelength region [FORMULA]-[FORMULA]. This plot shows the shift in the absorption peak; from [FORMULA] for solid Mie curve to [FORMULA] for porous grains with N = 152. It also shows the variation in the width of the `bump'; i.e. it broadens as the porosity increases. These results on the porous graphite grains show that the inhomogeneity within the grains can produce shift in the central wavelength of the extinction bump as well as the variation the width of the bump. For small graphite grains [FORMULA], Perrin & Sivan (1990) had found that the width of the extinction bump was modified by porosity but the maximum of the bump did not shift with porosity. Draine & Malhotra (1993) have found the shift in the central wavelength of the bump in the coagulated graphite grains, but they did not find any appreciable change in the profile width.

[FIGURE] Fig. 4. Extinction efficiencies ([FORMULA]) for porous Graphite grains in the UV region for grain size [FORMULA].

Interstellar Extinction Curve: The interstellar extinction curve (i.e. the variation of extinction with wavelength) is usually expressed by the ratio E([FORMULA]-V)/E(B-V) vs [FORMULA] and the ratio R of total to selective extinction is expressed as [FORMULA]. The average observed value of R is 3.1 (e.g. Savage & Mathis, 1979).

We use the extinction efficiencies of the porous silicate and graphite grains and the power law grain size distribution (i.e. [FORMULA], see e.g. Mathis et al., 1977) to reproduce the interstellar curve.

The average observed interstellar extinction curve (Savage & Mathis, 1979) is then compared with the model curve formed from a [FORMULA] minimized and best fit linear combination of porous silicate and graphite grains in the following way:

The two model interstellar extinction curves for silicate and graphite porous grains are linearly combined as p times silicate and q times graphite to render a net curve for comparison with the observed curve. By varying p and q individually from 0.1 to 1.0 in steps of 0.1, a set of 20 curves are generated and the comparison with the observed curve gives a set of reduced [FORMULA]values. The combination of p and q which gives a minimum [FORMULA] value in this set is chosen and such combinations and corresponding minimum [FORMULA] values are shown in the Table 4.


[TABLE]

Table 3. Best fit values of p, q and minimized [FORMULA] for porous grains of sizes [FORMULA], [FORMULA] and [FORMULA] and number of dipoles N = 152,1184 and 4088


The set of reduced [FORMULA] values is defined as (Bevington, 1969):

[EQUATION]

where pp is the degrees of freedom, [FORMULA] is the [FORMULA] model curve for the corresponding p and q linear combination of silicate and graphite porous grains and [FORMULA] is for the observed curve, [FORMULA] are the wavelength points with [FORMULA] for [FORMULA] points of the extinction curves.

Figs. 5, 6 and 7 show the best fit interstellar extinction curves for the porous grain models N = 152, N = 1184 and N = 4088 respectively. Table 4 shows the best fit parameters p, q and [FORMULA] for three size distributions that we have used for these curves. These values of p and q indicate the proportions of silicate and graphite components required to fit the observed extinction curve. It is seen that in most of the cases, the combination of 50% silicate and 40% graphite fit the observed extinction curve best.

[FIGURE] Fig. 5. Comparison of the observed interstellar extinction curve with the best fit model combination curve of porous silicate and graphite grains with the number of dipoles N = 152.

[FIGURE] Fig. 6. Comparison of the observed interstellar extinction curve with the best fit model combination curve of porous silicate and graphite grains with N = 1184.

[FIGURE] Fig. 7. Comparison of the observed interstellar extinction curve with the best fit model combination curve of porous silicate and graphite grains with N = 4088.

It is to be noted here that for the grain models with N = 152 (Fig. 5) we have considered small grain sizes that satisfy the DDA validity criteria. As a result models with N = 152 do not fit well in the ir and the visible spectral range but they fit reasonably well in the bump region (i.e. [FORMULA]-[FORMULA]). Grain Models with N = 1184 and N = 4088 with larger grain sizes fit better with the observed curve (Figs. 6 & 7).

Extinction in the UV and the `2175' feature: Despite continuing research over the last 30 years, the identification of the 2175 Å feature remains controversial. The most widely accepted explanation of the 2175 Å bump has been the extinction by small graphite interstellar grains (e.g. Hoyle and Wickramasinghe, 1962; Mathis et al., 1977 and Draine, 1988). However, as indicated earlier, the structure of the grains also plays an important role in the interstellar extinction. Using the extinction efficiencies for small ([FORMULA]-[FORMULA]) graphite and silicate porous grains in the wavelength region [FORMULA]-[FORMULA] we evaluate the interstellar extinction curve. Fig. 8 shows the interstellar extinction curve in the `bump' region i.e. [FORMULA]-[FORMULA] for the porous grains with N = 152.

[FIGURE] Fig. 8. Comparison of the observed interstellar extinction curve in the `bump' region with the best fit model combination curve of porous silicate and graphite grains with N = 152.

These results show that small porous grains with about 60%-70% porosity fit the observed extinction in the bump region reasonably well.

Albedo: In Fig. 9, we show the variation of the albedo [FORMULA] for the porous grain model with N = 1184 in the optical and the UV spectral range.

[FIGURE] Fig. 9. Curves showing the albedo variation in the optical and the UV range.

The visual albedo of the porous grain model is about 0.6, which is in agreement with the value suggested by Witt (1989). The albedo predicted by the composite fluffy dust (CFD) model is lower, about 0.5 (Mathis, 1996 and Dwek, 1997). The low albedo of the CFD model was found to be the cause of the excess far infrared emission of that observed with the COBE/DIRBE and FIRAS instruments from the diffuse interstellar medium (Dwek, 1997). This would mean that with the higher albedo the porous grain model which we have considered here would re-emit less radiation in the far infrared region than that is emitted by the CFD model and would fit the COBE data better.

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© European Southern Observatory (ESO) 1999

Online publication: July 26, 1999
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