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Astron. Astrophys. 339, 194-200 (1998)

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3. Analysis of dust bands

3.1. Analytical fit of the spectrum

Fig. 1 intuitively suggests that the NGC 7023 and Ophiuchus spectra are composed of overlapping Lorentzian profile, i.e. with relatively narrow peaks but wide wings, in evident contrast to the narrow profiles of the gaseous lines of Fig. 2. The Lorentz function is given by:

[EQUATION]

where

[EQUATION]

is the normalized Lorentz profile. Here [FORMULA] is the frequency, [FORMULA] the position of the band, [FORMULA] its width parameter. The peak amplitude of the band is given by y = [FORMULA]. It follows that its integrated intensity, A, is simply given by

[EQUATION]

where [FORMULA] is the full width at half maximum. If the Lorentzian shape of the bands has a physical signification, it must be expressed as a function of the frequency instead of the wavelength. This is why we chose to display the spectra as a function of the wave number (in cm-1), in the spectroscopic tradition. However, in the text we shall refer to the individual features by their wavelength in µm, which are probably more familiar to most astronomers in the field.

We started to fit the spectrum with a set of six Lorentz profiles at frequencies 1602, 1300, 1173, 886, 797 and 605 cm-1 (Eq. (1)) corresponding to the main bands at 6.2, 7.7, 8.6, 11.3, 12.7 and 16.5 µm. A linear baseline had also to be introduced. The 16.5 µm band is at the extreme limit of the observed spectral range and only half of it is seen in the Ophiuchus observation. For this last feature we fixed the central wavelength and width, and fitted only the intensity. Minor bands or lines at 1923, 1724, 1445, 1031, and 709 cm-1 (5.2, 5.8, 6.9, 9.7, 14.1 µm) are also present in the data. We fitted the minor features in the residuals of the first fit. The 6.9 and 9.7 µm lines coincide with the S(5) and S(3) H2 lines, which have been adjusted by Gaussian profiles. The three other lines which are probably dust features with an intensity a few percent of that of the main bands have been adjusted with Lorentz profiles. The complete fit with all the features of the NGC 7023 and Ophiuchus spectra are presented in Fig. 1. The automatic adjustment of the six main bands and the linear baseline provides a good fit to the spectrum. An even closer fit was obtained with the additional features. Intensities and widths of the main bands derived from this fit are given in Table 2.


[TABLE]

Table 2. Lorentz and Voigt fits to the Ophiuchus and NGC 7023 spectra
(1) Intensities of the bands([FORMULA]) obtained with the Lorentz and Voigt fitting procedures.
(2) Full width at half maximum for the Lorentzian fit.
(3) True (deconvolved) width from the Voigt fitting procedure.


Note that the wings completely account for more than half of the signal in the spectrum valley around 10 µm. The remaining signal (18% of the total power over the 5 to 16.5 µm wavelength range) is accounted for by the linear baseline which represents a true signal and is not instrumental. For Ophiuchus, at 12 µm the baseline intensity represents 50% of the zodiacal emission which has been subtracted from the ISOCAM spectrum (Sect. 2). It is thus significantly larger than our uncertainty on the zodiacal light subtraction estimated to be at most 20%. In NGC 7023 where the baseline intensity is about five times higher, this argument is even stronger. The linear baseline is not necessarily a true continuum emission. It could represent the sum of a set of numerous weak features which are not resolved with the CVF resolution. Analysis of high signal to noise spectra should allow to test this hypothesis at least in bright objects.

Gaussian and Lorentzian fits to the Ophiuchus spectrum are compared in Fig. 3; the small features have been removed to leave only the six main dust bands. The Gaussian fit is as good as the Lorentzian fit only if, in addition to the six UIR bands and the linear continuum, we allow for two broad components at about 840 and 1375 cm-1 (11.9 and 7.3 µm), thus six additional parameters. Without these broad components the gaussian fits fail in accounting for the signal at the edge of the bands. In this decomposition the broad Gaussian components and the linear baseline include 60% of the total power integrated from 5 to 16.5 µm while the baseline represents only 18% of this total power for the Lorentzian fit. Our preference for the Lorentz fit is also motivated by the physics of the emission from large molecules at high temperature (Sect. 4.2). In the following we focus on the interpretation of the Lorentz fit.

[FIGURE] Fig. 3. Comparison of the Lorentzian and Gaussian fits for the Ophiuchus spectrum. The symbols are the same as in Fig. 1. The small dust features and gas lines have been subtracted from the spectra of Fig. 1.

3.2. Effect of the CVF finite resolution

3.2.1. Voigt profiles

For the narrowest bands, the width derived from the Lorentz fits is not much larger than the spectral resolution. We assessed the effect of the finite resolution of the CVF (as interpolated from Table 1) on the bands profiles as follows. The convolution of a Gauss and a Lorentz profile is known as a Voigt profile (e.g. Lang 1980). The Voigt fitting algorithm may not properly converge for a large set of features. Thus, as in Fig. 3, the minor bands have been subtracted from the spectrum before fitting. The result is displayed in Fig. 3, with the same line styles as in Fig. 1.

As can be seen, the synthetic spectrum obtained with the Voigt procedure fits very properly the observations. It proves that the bands are intrinsically well fitted by Lorentz profiles and that the wings of the features are not produced by the finite resolution of the CVF. The intensities and deconvolved widths obtained with the Voigt fit are listed in Table 2. The intensities derived with the Lorentz and Voigt fitting procedures are very close to each other as it should be; the 10% difference observed for the 12.7 and 11.3 µm features is very probably due to the tight overlapping of the bands, which makes the separation of the respective contribution of the two bands somewhat uncertain. The fact that the amplitude differences are in opposite directions and average close to zero supports this interpretation.

For the three narrowest features (6.2, 8.6 and 11.3 µm) the intrinsic band widths derived from the Voigt fit are appreciably smaller than the values derived from the initial Lorentzian fit which shows that the instrumental convolution affects the spectral shape of the peak. Nevertheless, the fact that globally the fit is better with Lorentz than with Gauss profiles shows that, for all band wings, the intrinsic Lorentz shape dominates the instrumental broadening as well as any intrinsic broadening.

In conclusion, we propose that the Lorentz fit to the UIR bands provides an unequivocal method to properly separate the different components of the infrared emission, and should clarify the respective roles played by the underlying continuum and the features themselves. A first good approximation is already obtained with the six main features. Refinement is gained by adding the weaker bands. The problem of choosing appropriate baselines to compute the intensity of each band is also eliminated. An important result of our analysis is that a large fraction of the power emitted between the features belongs to the bands themselves. In other words, the quantities [FORMULA] listed in Table 2 represent the true intensity of the UIR bands, which would be grossly underestimated if measured with Gaussian fits. We note in passing that, although the physical processes involved are certainly different, this procedure is reminiscent of that propose by Fitzpatrick & Massa (1990) to quantify the UV extinction bump by a Drude profile, which is a modified Lorentz profile.

[FIGURE] Fig. 4. The main bands fitted by Voigt profiles (Lorentz profiles convolved by the instrument resolution, see Sect. 3.2.1 for details). The symbols are the same as in Fig. 1. The minor bands and the linear baseline of Fig. 1 have been subtracted from the data.

3.2.2. Comparison with SWS Observations

Higher resolution spectra have been obtained on a wide variety of objects with the Short Wavelength Spectrometer (SWS). Many published spectra such as those of the M17 H II - molecular cloud interface (Verstraete et al. 1996), compact H II regions (Roelfsema et al. 1996, 1998) and circumstellar envelopes (Molster et al. 1996) refer to objects with a very high radiation field. In such environments very small grains (VSGs) heated by the absorption of several photons contribute to the mid-IR emission. It is thus not straightforward to compare these data with the spectra, presented in this paper, obtained in regions where the VSGs are not hot enough to radiate in the ISOCAM wavelength range. An additional problem to bear in mind is that, in molecular clouds with very large column density like that associated with M17 and compact H II regions, the spectrum shape and the level of the signal around 10 µm may be significantly affected by extinction and in particular the silicate 9.6 µm feature. Thus, high radiation field objects are not best suited to study the emission properties of the smallest dust particles.

More recently, Moutou et al. (1998) published an SWS spectrum of NGC 7023. Fig. 1 of this paper shows that the feature pedestals and the 10 µm continuum are present in the SWS data as in the ISOCAM spectra. The NGC 7023 SWS spectrum, when displayed as a function of wavelength, shows some asymmetry of the 6.2 and 11.3 µm features, with larger wings on the red side. Note that a resonance curve is symmetrical as a function of the frequency (or wave number), but trails towards the red when displayed as a function of wavelength. This effect increases with the width of the line and is not completely negligible for the mid-infrared UIB features. In the Lorentz decomposition some asymmetry of the 6.2 and 11.3 µm is introduced by the underlying wings of the 7.7 and [FORMULA] features. However, these two effects do not entirely account for the asymmetry observed in the SWS spectra. By looking carefully at the Lorentz fits in Fig. 1, one can see that the blue side of the 11.3 µm band is slightly narrower than the fit. A combination of Lorentz functions would thus be necessary to account for the asymmetry.

It has also been known for some time that the "7.7 µm" feature is composed of at least two components, at wavelength of about 7.6 and 7.8 µm respectively, with varying relative intensities (Bregman 1989; Roelfsema et al. 1996). This appears clearly in the SWS spectra and can also be seen in the ISOCAM data. The single component (Lorentzian or Gaussian) fit of the 7.7 µm feature do not provide a good fit at the peak of the band. In Fig. 5, we show that a better fit is obtained with two bands at 7.6 and 7.8 µm. We stress that the fact that we replaced one broad component by two slightly shifted components does not affect the ability of the Lorentz profiles to account for the wings.

[FIGURE] Fig. 5. Fit of the 7.7 µm dust feature with two Lorentz profiles (solid line). The fit with one single 7.7 µm component is shown as a dash line. The 7.6 and 7.8 µm Lorentz profiles for the two components fit are shown with dotted lines. The minor features and gas lines have been subtracted from the data as in Fig. 3

The SWS data clearly show that the true band shape must be a combination of Lorentz profiles with different central frequencies and widths; this is particularly obvious for the 7.7 and 12.7 µm features the peaks of which are not well fitted by a single Lorentz profile (Fig. 3 and 5). The combination is important to fit the emission peaks but has little effect on the width and intensity of the band wings, which contain more than half of the total radiated power. Physically, it is not unexpected to observe a combination of Lorentz profiles since the width and position of the features may vary from particle to particle and with temperature (Sect. 4).

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© European Southern Observatory (ESO) 1998

Online publication: September 30, 1998
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