3 Results and discussion

   
3.1 IR emission by individual PAH-like species

The IR intensities emitted by the PAH-like species {C $_{{\rm 6p^2}}H_{{\rm 6p}}$} during their cooling in the environment of the object IRAS 21282+5050 are presented in Fig. 3 for four different sizes: {C $_{{\rm 24}}{\rm H}_{{\rm 12}}$}, {C $_{{\rm 54}}{\rm H}_{{\rm 18}}$}, {C $_{{\rm 96}}{\rm H}_{{\rm 24}}$} and {C $_{{\rm 150}} {\rm H}_{{\rm 30}}$}. For an easier comparison of the relative band intensities with the observations, we have chosen to plot each IR band as Lorentzian functions whose positions and widths have been derived from the observed spectrum (Table 5). Figure 3 shows that the emission in the 3.3 and 6.2 $\mu$m bands decreases with increasing the PAH size. This is particularly steep in the case of the 3.3 $\mu$m feature which is very weak in the spectrum of a large molecule such as {C $_{{\rm 150}} {\rm H}_{{\rm 30}}$}. We have also calculated the detailed band profiles for the four PAH-like species, taking into account the broadening mechanisms (Sects. 2.2 and 2.3.3). The profiles of the 11.3, 6.2 and 3.3 $\mu$m bands are presented in Figs. 4-6. They were obtained by integrating the emission at each temperature step during the cooling as described in Sect. 2.5. The contribution from the hot bands was constructed, using the anharmonic shifts listed in Table 2.

Figure 3: Calculated IR band intensities emitted by four PAH-like molecules in the environment of IRAS 21282+5050. The positions and widths of the bands are those of the observed AIBs (cf. Table 5).

Figure 4: Calculated 11.3 $\mu$m band profiles for different PAH-like species (solid line) including the contribution from the fundamental v=1 $\rightarrow$0 transition (dashed line) and from the hot bands (transitions 2 $\rightarrow$1 and 3 $\rightarrow$2; dot-dashed lines).

Figure 5: Calculated 6.2 $\mu$m band profiles for different PAH-like species (solid line) including the contribution from the fundamental v=1 $\rightarrow$0 transition (dashed line) and from the hot band (transition 2 $\rightarrow$1; dot-dashed line).

Figure 6: Calculated 3.3 $\mu$m band profiles for different PAH-like species (solid line) including the contribution from the fundamental v=1 $\rightarrow$0 transition (dashed line) and from the hot band (transition 2 $\rightarrow$1; dot-dashed line).

Figure 7: Dependence of the band width and asymmetry factor on the molecular size and on the values of the anharmonicity coefficients $\chi '$ and $\chi ''$. From top to bottom: the 11.3, 6.2 and 3.3 $\mu$m bands. The horizontal dotted lines represent the values observed in IRAS 21282+5050.

Each band can be described by a width and an asymmetry factor. The latter is defined as the ratio of the half width at half maximum (HWHM) measured on the red side to the HWHM measured on the blue side. The values of the asymmetry factor and of the band width (FHWM) calculated for different {C $_{{\rm 6p^2}}{\rm H}_{{\rm 6p}}$} species and different values of the anharmonicity coefficients $\chi '$ and $\chi ''$ are displayed in Fig. 7. We observe that the band widths and the asymmetry factors are larger for smaller molecules. The asymmetry is also larger at longer wavelengths (cf. the 11.3 $\mu$m profile compared to the 3.3 $\mu$m band profile). The asymmetry of the band profiles is partly due to the hot band contribution. Hot bands are in general limited to the 2 $\rightarrow$1 transition except in the case of the 11.3 $\mu$m band for which the 3 $\rightarrow$2 transition is still significant for the smallest species (cf. case of {C $_{{\rm 24}}{\rm H}_{{\rm 12}}$} in Fig. 4). The asymmetry of the 11.3 $\mu$m band is larger compared to the other bands even without including the contribution from hot bands. This implies that the dominant effect is the shift and broadening of the bands induced by the coupling between modes rather than the anharmonicity of the mode itself. It is therefore interesting to discuss the variations of the band profiles with the values of $\chi '$ and $\chi ''$ which quantify this coupling. The values adopted in the present work can be questioned on several points. First, the dependence of the band position and width with temperature is linear only at high temperatures. In the low temperature range, the regime differs. Defining a linear law over all the temperature range can therefore induce some errors. For the band width, this happens at higher temperature and we determined the linear law by including the value at 4 K to avoid negative values (cf. Fig. 2). The resulting error in the high-temperature range is at maximum 10% for the 6.2 $\mu$m band at 2000 K (Sect. 2.3.3) and differences of a few percents are excepted on the calculated profiles. For the band positions, the non-linear behaviour is effective only below $\sim$200 K (cf. Joblin et al. 1995) and has no or very little impact on the calculated profiles. A second and probably more important effect is that the values of $\chi '$ and $\chi ''$ were derived from the spectrum of a specific molecule: coronene and its methylated derivative. Different values for these coefficients would lead to a different band width and asymmetry. To search for a dependence of $\chi '$ and $\chi ''$ with molecular size, we looked more carefully at the available measurements on pyrene (C $_{{\rm 16}}{\rm H}_{{\rm 10}}$), coronene (C $_{{\rm 24}}{\rm H}_{{\rm 12}}$) and ovalene (C $_{{\rm 32}}{\rm H}_{{\rm 14}}$) (Joblin et al. 1995 and Joblin 1992). The comparison was only possible for the C-H bands which are the most intense in the IR spectra of PAH neutrals. All the $\chi ''$ coefficients were derived after extracting the rotational width (cf. 2.3.3). The values of $\chi '$ and $\chi ''$ are plotted in Fig. 8 as a function of size. Maximal variations of 20% were found between the three molecules. Although the considered sample is quite restrictive in size, the data suggest an increase of $\chi '$ and $\chi ''$ with the molecular size. Linearly extrapolated to a size of 100 carbon atoms, the value of $\chi '$ would be larger by a factor of 2 and 3 for the 3.3 and 11.3 $\mu$m bands respectively. This is likely to be an upper limit since a saturation in the increase is expected to occur at the solid-state limit. In the case of $\chi ''$, the factors are 2.5 and 4 respectively. The effect of increasing the values of $\chi '$ or $\chi ''$ on the band width and asymmetry is illustrated in Fig. 7. The band width is found to increase with $\chi '$ and $\chi ''$ with a dominant effect of $\chi '$. The asymmetry factor behaves differently. An increase of $\chi '$ leads to a significant decrease of the asymmetry in the case of the 3.3 and 6.2 $\mu$m bands. On the other hand, the asymmetry of the 11.3 $\mu$m band is slightly modified. This has to do with the difference in the emission regime at short wavelengths compared with that at long wavelengths. Indeed, the variations of the emitted intensity with temperature is slow at 11.3 $\mu$m but very steep at 3.3 and 6.2 $\mu$m.

Figure 8: Evolution of the anharmonicity coefficients $\chi '$ and $\chi ''$ with molecular size for the C-H stretch band (stars) and for the C-H out-of-plane bending band (extracted from the laboratory measurements of Joblin et al. 1995).

   
3.2 IR emission by a distribution of PAH-like species

   

Table 3: Calculated values for the 3.3 $\mu$m band width, $\Delta\nu_{3.3~\mu{\rm m}}$, and for the band intensity ratios, $I_{3.3~\mu{\rm m}}$/ $I_{11.3~\mu{\rm m}}$ and $I_{1.7~\mu{\rm m}}$/ $I_{3.3~\mu{\rm m}}$, using different size distributions of PAH-like molecules. Comparison with the observations in IRAS 21282+5050.

Distribution	$\beta$	$N_{{\rm Cmin}}$- $N_{{\rm Cmax}}$	$\Delta\nu_{3.3~\mu{\rm m}}$ (cm-1)	$I_{3.3~\mu{\rm m}}$/ $I_{11.3~\mu{\rm m}}$	$I_{1.68~\mu {\rm m}}$/ $I_{3.3~\mu{\rm m}}$
A	2	[24, 200]	37	0.18	1.1 $\times$ 10-2
B		[24, 1000]	37	0.14	1.1 $\times$ 10-2
C		[40, 200]	32	0.09	4.9 $\times$ 10-3
D	3.5	[30, 200]	38	0.20	9.8 $\times$ 10-3
E		[44, 200]	29	0.11	4.8 $\times$ 10-3
IRAS 21282+5050			40	0.17	4.9 $\times$ 10-3

A size distribution characterized by three parameters: $N_{{\rm Cmin}}$, $N_{{\rm Cmax}}$ and $\beta$ was then considered. These parameters can be constrained by using three independent spectral features. As shown by Schutte et al. (1993) and also observed in our calculations, the smallest sizes dominate the emission at short wavelengths. To characterize the minimum size $N_{{\rm Cmin}}$, we have therefore considered the width of the 3.3 $\mu$m band as well as its intensity relative to its overtone v = 2 $\rightarrow$0, detected at 1.68 $\mu$m in IRAS 21282+5050 by Geballe et al. (1994). Finally, the $I_{3.3~\mu{\rm m}}$/ $I_{11.3~\mu{\rm m}}$ ratio was used as a constraint on the average size of the distribution since this ratio strongly depends on the size (cf. Fig. 9).

Figure 9: The band intensity ratio $I_{3.3~\mu{\rm m}}$/ $I_{11.3~\mu{\rm m}}$ as a function of the minimal size ( $N_{{\rm Cmin}}$) for two size distributions of PAH-like molecules $N_{{\rm C}}$ $^{-\beta }$ with $\beta =$ 3.5 (solid line) and $\beta =$ 2 (dashed line).

The values of the three selected spectral features: $\Delta\nu_{3.3~\mu{\rm m}}$, $I_{1.68~\mu {\rm m}}$/ $I_{3.3~\mu{\rm m}}$, and $I_{3.3~\mu{\rm m}}$/ $I_{11.3~\mu{\rm m}}$ are presented in Table 3 for different size distributions. The first distribution with $\beta =$ 2, $N_{{\rm Cmin}}=$ 24 and $N_{{\rm Cmax}}=$ 200 (A in Table 3), which is similar to the one considered by Désert et al. (1990), provides correct values for $\Delta\nu_{3.3~\mu{\rm m}}$ and $I_{3.3~\mu{\rm m}}$/ $I_{11.3~\mu{\rm m}}$. As expected, increasing $N_{{\rm Cmax}}$ up to 1000 (B) does not affect the characteristics of the 3.3 $\mu$m band but simply leads to a decrease of $I_{3.3~\mu{\rm m}}$ relative to $I_{11.3~\mu{\rm m}}$. In both cases, the calculated $I_{1.68~\mu {\rm m}}$/ $I_{3.3~\mu{\rm m}}$ ratio is too high by a factor of 2. The observed value is matched by increasing $N_{{\rm Cmin}}$ to 40 (distribution C), but this leads to a 3.3 $\mu$m band which is too narrow and too weak relative to the 11.3 $\mu$m one.

Another type of size distribution has been considered with $\beta =$ 3.5. Such a steep distribution strongly favours the smallest sizes compared to the previous distribution with $\beta =$ 2 (cf. Fig. 10). A larger value of the minimum size ( $N_{{\rm Cmin}}=$ 30, distribution D) had therefore to be taken in order to fit $\Delta\nu_{3.3~\mu{\rm m}}$ and $I_{3.3~\mu{\rm m}}$/ $I_{11.3~\mu{\rm m}}$. Here, again, a larger value $N_{{\rm Cmin}}=$ 44 (E in Table 3) is required to account for the 1.68 $\mu$m intensity. It appears then that none of the considered size distributions can fit simultaneously the three selected spectral characteristics. This point is further discussed in the next section. Releasing the $I_{1.68~\mu {\rm m}}$/ $I_{3.3~\mu{\rm m}}$ constraint, we have retained in the following the set of parameters D ( $N_{{\rm Cmin}}=$ 30, $\beta =$ 3.5, $N_{{\rm Cmax}}=$ 200), which provides results as good as distribution A without involving too large sizes. Table 4 lists the calculated band intensity ratios relative to the 11.3 $\mu$m band for the selected size distribution D in the environment of IRAS 21282+5050. As can be seen, these ratios match well the observed values. The agreement is especially good for the 6.2 and 3.3 $\mu$m bands. The largest discrepancy is for the 8.6 $\mu$m band. This is at least partly due to the difficulty to extract the 8.6 $\mu$m band from the wing of the strong "7.7'' $\mu$m feature.

Figure 10: Normalised abundance for a PAH-like population following a power-law size distribution $N_{{\rm C}}^{-\beta }$ with $\beta =$ 2 and $\beta$ = 3.5.

Figure 11: Calculated 11.3 $\mu$m band profile (solid line) for the PAH-like population of size distribution D (Table 3) compared with the observations in IRAS 21282+5050 (crosses). The calculated band includes the fundamental v=1 $\rightarrow$0 transition (dashed line) and the hot band (transition 2 $\rightarrow$1; dot-dashed line).

   

Table 4: The relative IR band intensity ratios calculated for the PAH-like population of size distribution D (Table 3) compared with the values measured in IRAS 21282+5050.

	$I_{3.3~\mu{\rm m}}$/ $I_{11.3~\mu{\rm m}}$	$I_{6.2~\mu{\rm m}}$/ $I_{11.3~\mu{\rm m}}$	$I_{7.7~\mu{\rm m}}$/ $I_{11.3~\mu{\rm m}}$	$I_{8.6~\mu{\rm m}}$/ $I_{11.3~\mu{\rm m}}$
Calculated spectrum	0.20	1.01	3.82	0.54
Observed spectrum	0.16	0.98	2.62	0.79

Figure 12: Calculated 6.2 $\mu$m band profile (solid line) for the PAH-like population of size distribution D (Table 3) compared with the observations in IRAS 21282+5050 (crosses). The calculated band includes the fundamental v=1 $\rightarrow$0 transition (dashed line) and the hot band (transition 2 $\rightarrow$1; dot-dashed line).

The calculated band profiles for the distribution D are displayed in Figs. 11-13 for the 11.3, 6.2 and 3.3 $\mu$m bands respectively. The agreement between the calculated and observed bands appears to be nice as can be seen from the figures and from the values of the widths and asymmetry factors (Table 5). In the case of the 11.3 $\mu$m band, the calculated band falls exactly at the position of the observed band. The observed profile with a steep rise on the blue side and extended wing on the red side, is also very well reproduced by the calculations. This is also the case for the profile of the 6.2 $\mu$m feature, although the positions of the calculated and observed bands differ by 7 cm^-1. Finally, the calculated profile at 3.3 $\mu$m appears to be the less satisfactory with an asymmetry factor larger than the observed one (Fig. 13).

As previously discussed, the asymmetry of the profiles in the case of individual molecules results from the anharmonicity effects. In the case of a distribution of PAH-like species, it also reflects the variety of sizes and therefore of emission temperatures (Verstraete et al. 2001). The profiles obtained for the distribution D can be compared to the profiles of the average size of the distribution, $<N_{{\rm C}}>$ = 48. As can be seen in Fig. 14, the red wing is more extended when a distribution of molecules is considered. The effect increases for the bands at shorter wavelengths. At 6.2 $\mu$m, an increase of the width and asymmetry of 3 cm^-1 and 12% respectively is observed. The increase is up to 10 cm^-1 and 20% in the case of the 3.3 $\mu$m band. The increase of asymmetry provided by the distribution leads to a better fit of the AIBs except in the case of the 3.3 $\mu$m band. This is due to the contribution of the smallest sizes ( $N_{{\rm C}}$ < 48) that is up to 83% for the 3.3 $\mu$m band and 60% for the 11.3 $\mu$m band (cf. Fig. 15). Finally, calculations have also been performed for the "7.7'' and 8.6 $\mu$m bands. The "7.7'' $\mu$m AIB is known to consist of several components (Joblin et al. 2000 and Verstraete et al. 2001). Widths between 20 and 30 cm^-1 were extracted for the individual bands. Our calculations provide a value of 21 cm^-1which is consistent with these studies. The 8.6 $\mu$m band appears to be singular since the calculated band does not account for the observed intensity nor for the observed profile.

Figure 13: Calculated 3.3 $\mu$m band profile (solid line) for the PAH-like population of size distribution D (Table 3) compared with the observations in IRAS 21282+5050 (crosses). The calculated emission bands in the fundamental v=1 $\rightarrow$0 transition (dashed line) and in the v=2 $\rightarrow$1 hot band (dot-dashed line) have been superposed on an underlying continuum represented by linked crosses.

   

Table 5: Peak positions and full widths at half maximum of the emission bands calculated for the PAH-like population of size distribution D (Table 3) compared with the values observed in IRAS 21282+5050.

	Calculations			Observations
Band	Band position	Band	Band width	Band position	Band	Band width
	(cm-1)	asymmetry	(cm-1)	(cm-1)	asymmetry	(cm-1)
3.3 $\mu$m	3045	1.42	38	3040	0.93	41
6.2 $\mu$m	1601	1.46	42	1608	1.44	43
7.7 $\mu$m	1323	1.01	21	"$\sim$1283''	-	"98''
8.6 $\mu$m	1135	0.88	10	"1164''	-	"45''
11.3 $\mu$m	889	3.3	19	889	3.75	22

Figure 14: Comparison of the profiles calculated for the distribution D (solid line) and for a single molecule representing the average size of the distribution $< N_{{\rm C}}$ > = 48. From top to bottom: the 3.3, 6.2, 11.3 $\mu$m bands.

Figure 15: Contribution of the smallest sizes ( $30 \le N_{{\rm C}} \le 48$; dotted lines ) and of the largest sizes ( $50 \le N_{{\rm C}} \le 200$; dash-dotted lines) to the total profiles calculated for distribution D (solid line). From top to bottom: the 3.3, 6.2, 11.3 $\mu$m bands.

Figure 16: Summary of calculated spectra from this work compared to the AIB spectrum in IRAS 21282+5050 (top spectrum; ISO-SWS observations by M. Jourdain de Muizon, L. B. d'Hendecourt, A. Heras and collaborators). The spectra are shifted for clarity with the bottom spectrum corresponding to the distribution D of PAH-like species and the medium spectrum to the average size of the distribution ( $< N_{{\rm C}} >~ =$ 48 C). The solid lines correspond to calculated profiles and the others are lorentzian profiles at the observed positions and widths (Table 5).

The results of our calculations are summarized in Fig. 16. The calculated profiles are displayed for the 3.3, 6.2 and 11.3 $\mu$m bands. For the other bands (dashed lines), the observed positions and widths (Table 5) were used in order to favour the comparison between the observed and calculated spectra. A remarkable agreement is also found between the calculated and observed IR flux. Absolute values were determined for the calculated spectrum by assuming a total column density of 1.8 $\times$ 10²¹ cm^-2 (A_V= 1) and 10% of interstellar carbon in PAH species (using [C/H] $_{{\rm ISM}}=$ 2.6 10^-4, Snow & Witt 1996). Finally, we have assumed that the PAH emission is spread over a 3.6'' aperture according to the high spatial resolution images of IRAS 21282+5050 obtained by Meixner et al. (1993).

   
3.3 Discussion

Generic photophysical properties for the population of interstellar PAHs were defined using laboratory data or quantum chemical calculations on small molecules and extrapolated to larger sizes. The Einstein coefficients A_i are from ionized species whereas the anharmonicity coefficients $\chi '$ and $\chi ''$ are known only for neutrals. Another major assumption we made is that the IR frequencies and their temperature dependence are the same for all the PAHs species. From these hypotheses, the parameters of a distribution ( $N_{{\rm Cmin}}=$ 30; $\beta =$ 3.5; $N_{{\rm Cmax}}=$ 200) have been adjusted to match the ratio $I_{3.3~\mu{\rm m}}$/ $I_{11.3~\mu{\rm m}}$ and the width of the 3.3 $\mu$m AIB that are observed in the object IRAS 21282+5050. The calculated spectrum appears to provide a good match of the relative intensities of the AIBs as well as to account for the profiles of the 6.2 and 11.3 $\mu$m AIBs. The shape of these bands appears to be characteristic of the anharmonicity of molecular modes. Restricted inhomonegeous broadening is caused by the distribution of molecular sizes and therefore of temperatures. No spectral diversity (i.e. change of the IR spectrum from one species to the others) was included. The very good fit obtained for the 6.2 and 11.3 $\mu$m profiles leaves indeed very little room for such a spectral diversity. Including a dispersion of the central frequencies with the size and the specific geometry of the molecules would lead to a change of the band shape with difficulties to fit the observed band. In our model, the observed band profile is naturally explained. The case of the 3.3 $\mu$m band appears to be different with an observed profile more symmetric than the calculated one. Several explanations might be tentatively given to account for this discrepancy. Inhomonegeous effects might be more important for this band. As seen previously, the 3.3 $\mu$m band is dominated by the smallest sizes which are likely to have enhanced spectral diversity compared to larger sizes. Another reason, which might be invoked is the fact that we used values of the anharmonicity coefficients $\chi '$ and $\chi ''$from neutrals. From the point of view of the Einstein coefficients the 3.3 $\mu$m band is strongly perturbed by ionisation as shown by quantum chemical calculations (Langhoff 1996 for instance). The dependence of the band shape with temperature might then also be perturbed. As opposed to the other AIBs, the "7.7'' $\mu$m band is very broad and shows some sub-structures. A decomposition into Lorentzian profiles leads to at least 4 components in this spectral region (Joblin et al. 2000; Verstraete et al. 2001). This might give evidence for spectral diversity. Still, this work underlines the fact that the AIB spectrum does not contain much spectral diversity. In particular, our model was able to match the profile of the 6.2 and 11.3 $\mu$m bands by assuming that all PAHs in the distribution emit at the same frequencies $\nu_{i}$(T) when heated at the same temperature T. This could mean that the IR spectrum of the emitting species reaches a solid-like limit although these species behave as molecules from a photophysical point of view. An interesting question concerns the size at which such a limit is attained. Another possibility is that interstellar PAHs have similar spectra because they have similar structures. In this model, we have used the formula C $_{{\rm 6p^{2}}}$H $_{{\rm 6p}}$ which is characteristic of compact PAHs. May be photodissociation gives rise to natural selection among interstellar PAHs, only leaving species with special characteristics, for instance compact species. As an illustration Joblin et al. (1997) have shown that the condensed form of the tricoronene (C₇₂H₂₄ compared with the more linear form C₇₂H₂₈) was more stable upon UV laser radiation.