4 Probing the PAH model

To first order, the AIB profiles essentially remain identical while the exciting radiation hardens considerably: $T_{\rm eff} = 23$000 K in NGC 2023 to 45000 K in M17-SW corresponding to the mean energy of the photons absorbed by a PAH of 6 and 9 eV respectively. Such contrasted internal energies imply differences in the emission temperature of a few 100 K for a given molecular size. With this temperature change, band position shifts and band broadening of at least 10 cm_1 are expected (Joblin et al. 1995). How can then the interstellar AIB profiles be so stable? One possibility is that some process keeps the temperature distribution of interstellar PAHs essentially the same.

To investigate this issue, we have computed the temperature distribution and associated infrared (IR) emission of an interstellar PAH population using a modified version of the model described in Pech et al. (2000 hereafter PJB) where we study in detail the temperature fluctuations and the temperature distribution of PAHs. Improving on former work (Léger et al. 1989b; Schutte et al. 1993; Cook & Saykally 1998), this model takes into account the full spectral distribution of the exciting radiation and implements the recent results of Joblin et al. (1995) on the temperature dependence of the band profiles of neutral PAHs. In addition, it must be emphasized that the PAH IR emission cross-section of the PJB model was derived from studies on PAH cations: indeed, we show in Sect. 4.4.1 that singly ionized PAHs reproduce better the astronomical AIB spectra confirming previous work (Langhoff 1996; Cook & Saykally 1998; Allamandola et al. 1999; Hudgins & Allamandola 1999). This observational requirement is supported by theoretical studies on the charge state of interstellar PAHs for physical conditions comparable to those of our sample of objects (Bakes & Tielens 1994; Dartois & d'Hendecourt 1997). We first briefly describe the equations giving the PAH cooling curve and emission, compute the temperature distribution of interstellar PAHs and then compare our model PAH emission spectrum to the data. For the sake of clarity, we limit the model-to-data comparison to NGC 2023 and M17-SW which are the extreme cases of our data sample and we focus on the well-defined 3.3, 6.2, 8.6 and 11.3 $\mu$m-bands. In order to emphasize the temperature fluctuations of PAHs and the resulting temperature distribution, we have written the model equations in terms of cross-sections rather than Einstein A-coefficients: this formulation is however strictly equivalent to that of PJB (see, e.g. , Schutte et al. 1993, Eq. (15)).

4.1 Emission by PAHs

In the interstellar medium, a PAH containing $N_{\rm C}$ carbon atoms absorbs radiation from the surrounding stars at a rate:

$$R_{\rm abs}\ ({\rm photons\;s}^{-1})=\int^\infty_0 \sigma^{a}_{\nu}\; \frac{F_{\nu}}{h\nu}\; {\rm d}\nu$$ (1)

with $\sigma^{a}_{\nu}$ the visible-ultraviolet (UV) absorption cross-section of PAHs (Verstraete & Léger 1992) and $F_{\nu}$ the incident stellar flux assumed to be a diluted blackbody with the dilution factor and effective temperature given in Table 1. The absorption cross-section is proportional to $N_{\rm C}$; it also has a cut-off (due to the electronic transitions) in the visible to near-IR range whose wavelength increases with $N_{\rm C}$ (Verstraete & Léger 1992). The mean photon energy absorbed by a molecule is given by:

$$E_{\rm abs} = \frac{P_{\rm abs}}{R_{\rm abs}} \quad{\rm with}... ...\rm abs}=\int^\infty_0 \sigma^{a}_{\nu}\;F_{\nu}\; {\rm d}\nu.$$ (2)

$P_{\rm abs}$ is the power absorbed per molecule. In the standard interstellar radiation field of Mathis et al. (1983), we find that a PAH with $N_{\rm C} = 25$ or 500 carbon atoms absorbs $2.4\times 10^{-27}$ or $4\times 10^{-27}$ Watt per C-atom with $E_{\rm abs} = 4.8$ or 2.3 eV, respectively. The values for $N_{\rm C} = 25$ are in good agreement with the laboratory results of Joblin et al. (1992) on small PAHs.

As shown by Léger et al. (1989a) and Schutte et al. (1993), the IR emission of a PAH is well treated within the thermal approximation and using Kirchhoff's law. In the following, we will discuss the complete spectrum emitted by PAHs in terms of the spectral energy distribution (SED) noted $S(\nu)$ which is equal to $\nu I_{\nu}$. The SED (in Watt) emitted by a single molecule containing $N_{\rm C}$ carbon atoms is thus:

$$S(\nu',N_{\rm C}) = 4\pi\,R_{\rm abs} \int^\infty_0 {\cal P}_{\nu... ...u'}\;B_{\nu'}\left( T_{\rm c}(t,\nu,N_{\rm C}) \right) \; {\rm d}t \;{\rm d}\nu$$ (3)

where

$${\cal P}_{\nu} = \frac{\sigma^{a}_{\nu}} {R_{\rm abs}} \times \frac{F_{\nu}}{h\nu}$$ (4)

is the probability density for absorption of a photon of energy $h\nu$, $\sigma^{\rm e}_{\nu'}$ the emission cross-section in the IR, $t_{\rm e}$ is the time at the end of cooling (see Sect. 4.2) and $T_{\rm c}(t,\nu,N_{\rm C})$ is the cooling curve of the molecule. In the following, the frequency of photons emitted in the IR will be labelled $\nu'$ and that of stellar photons absorbed in the visible-UV range will be written $\nu$.

The PAH IR emission cross-section is defined as in PJB. The band profiles have a Lorentzian shape and at 3.3, 6.2, 8.6 and 11.3 $\mu$m, the band position and width follow temperature dependent laws. Each Lorentz profile is normalized to the peak value of the band cross-section, $\sigma_{\nu}$ and we assume that the integrated cross-section (or Einstein A-coefficient, see Schutte et al. 1993) of each band profile is conserved with temperature. As already noted above, the resulting IR cross-section describes the properties of singly-ionized PAHs. To match the observed AIB, the 8.6 $\mu$m-band had to be multiplied by 3. Because the available laboratory results do not account for the complex shape of the observed 7.7 $\mu$m-band (see Sect. 3), we define an empirical 7.7 $\mu$m-band with a shape derived from the observations (profile centre at 1300 cm_1 and ${\it FWHM} = 113$ cm_1 from the present data) and with the (laboratory determined) Einstein A-coefficient of PJB. This profile of the 7.7 $\mu$m-band is furthermore assumed temperature independent. We also added the 16.4 $\mu$m-AIB detected by ISO towards many sightlines (Moutou et al. 2000; van Kerckhoven et al. 2000). This band is often associated with the "classical'' 3 to 13 $\mu$m-AIBs and its intensity is not related to the hot dust continuum appearing in more excited objects (see Fig. 1), i.e. , as for the other AIBs. We represent the 16.4 $\mu$m-band with a Lorentz profile centered at 609 cm_1, with ${\it FWHM}=6$ cm_1 (both being observational values, see Moutou et al. 2000) and an Einstein A-coefficient of $3.1\times 10^{-2}$ s^-1 per carbon atom which corresponds to the average value of the laboratory measurements given in Moutou et al. (1996, Table 3). As we will see below, the intensity of the 16.4 $\mu$m-band can constrain the power-law index of the PAH size distribution. At longer wavelengths, we adopted the far-IR ( $\lambda\geq 20$ $\mu$m) cross-section of Schutte et al. (1993).

The size distribution of PAHs is defined by d $N=B\,N_{\rm C}^\beta\,{\rm d}N_{\rm C}$, i.e., the number of PAHs with a number of C-atoms between $N_{\rm C}$ and $N_{\rm C}+{\rm d}N_{\rm C}$. Writing the size distribution with respect to $N_{\rm C}$ frees us of any assumptions on the PAH geometrical shape. The number of H-atoms per molecule is assumed to be $N_{\rm H}=f_{\rm H}\times\sqrt{6N_{\rm C}}$ (Omont 1986, in the case of symmetric PAHs) with $f_{\rm H}$ the hydrogenation fraction of a molecule: if $f_{\rm H}=0$ the molecule is completely dehydrogenated, if $f_{\rm H}=1$ the molecule is fully hydrogenated. Unless otherwise stated, we will assume $f_{\rm H}=1$. The mass distribution is then straightforwardly found as d $N\times (m_{\rm C}\;+\;m_{\rm H}\times f_{\rm H}\sqrt{6/N_{\rm C}})$ where $m_{\rm C}$ is the carbon atomic mass and $m_{\rm H}$ the atomic mass of hydrogen. The normalization factor, B, is determined from the PAH abundance in terms of the fraction of interstellar carbon ([C/H] $_{\rm ISM}=2.6\times 10^{-4}$, Snow & Witt 1996) locked up in PAHs. Finally, the emergent SED emitted by PAHs is

$$S^{\rm e}(\nu')=\int^{N_{\max}}_{N_{\min}} S(\nu',N_{\rm C})\; {\rm d}N(N_{\rm C})$$ (5)

with $N_{\min}$ and $N_{\max}$the smallest and largest number of C-atoms per molecule respectively. We also define the total luminosity emitted by PAHs as $L_{\rm e} = \int^{\infty}_0 S^{\rm e}_{\nu'} \; {\rm d}\,{\rm ln}\,\nu'$.

4.2 The infrared cooling of PAHs

Upon absorption of a UV-photon, the PAH temperature is raised to a peak value, $T_{\rm p}$. The molecule then cools rapidly by the emission of IR vibrational mode photons. The peak temperature $T_{\rm p}$ is found from first principles:

$$U_{\rm f}-U_{\rm i} = \int^{T_{\rm p}}_{T_{\rm i}} C(T)\,{\rm d}T$$ (6)

where C(T) is the heat capacity of PAHs as given by Dwek et al. (1998, Eq. (A4) of their Appendix), $U_{\rm i,f}$ are the initial and final molecular internal energies and ${T_{\rm i}}$ the PAH temperature prior to photon absorption or the mean molecular temperature between two photon absorptions. We assumed that $U_{\rm f}>>U_{\rm i}$ with $U_{\rm f}=h\nu$ the energy of the absorbed photon. Furthermore, we took ${T_{\rm i}}$ to be the PAH temperature at the end of cooling (see the definition below). The heat capacity, as a function of temperature, is approximated by a polynomial in T; it is also proportional to $N_{\rm C}$but independent of $N_{\rm H}$, i.e. , the contribution of the hydrogen atoms to C(T) is assumed to be always the same whatever the size of the molecule (Dwek et al. 1998). In this case and for a given energy per C-atom in the molecule ( $h\nu/N_{\rm C}$), $T_{\rm p}$ is fixed.

The cooling curve $T_{\rm c}(t,\nu,N_{\rm C})$ is obtained by numerical integration (vs. temperature) of the energy conservation relationship during cooling,

$$P_{\rm e}\left(T_{\rm c}\right)\,{\rm d}t = - C(T_{\rm c})\,{\rm d}T$$ (7)

where $P_{\rm e}$ is the total power emitted at all wavelengths by the molecule and dT the temperature drop during the time dt. This simple differential equation determines a family of cooling curves, $T_{\rm c}$, as a solution, each cooling curve being completely defined by the knowledge of $N_{\rm C}$ and of the initial condition, i.e. , $T=T_{\rm p}$ at t=0.

The power emitted by a PAH at temperature T is:

$$P_{\rm e}(T)=\int^\infty_0 \sigma^{\rm e}_{\nu'} \times 4\pi B_{\nu'}(T)\, {\rm d}\nu'.$$ (8)

Cooling curves were computed on a grid of 50 points with constant temperature intervals. Under these conditions, we found that the energy conservation requirement:

$$\int^{\infty}_0 P_{\rm e}(T_{\rm c}(t)) \; {\rm d}t = h\nu$$ (9)

is always met within a few percent.

Each molecule sees its cooling interrupted by the absorption of the next photon which is a stochastic process (Désert et al. 1986). Moreover, gas-grain exchanges must be taken into account (Rouan et al. 1992; Draine & Lazarian 1998) to reliably estimate the temperature of a PAH in the low temperature part of the cooling curve. In fact, the emission from the cool (but long-lasting) tail of the cooling curves will peak in the far-IR/submillimeter, far away from the 3 to 20 $\mu$m-range on which we focus here. The detailed treatment of this problem is outside the scope of this paper and we adopt the following simple criterion to truncate our cooling curves. The mean time elapsed between two photon absorptions is $1/R_{\rm abs}$: it is for instance 6.9 and 2.6 hours in the radiation field of NGC 2023 and M17-SW respectively and for a molecule with $N_{\rm C}=20$. Consequently, all cooling curves have been truncated at $t_{\rm e}=1/R_{\rm abs}$ to account for the repeated absorptions of UV photons by a molecule; $t_{\rm e}$ decreases as $N_{\rm C}^{-1}$ because $R_{\rm abs}\sim N_{\rm C}$. We thus define the lowest temperature of a PAH heated by a photon of energy $h\nu$ as $T_{\rm i}=T_{\rm c}(t_{\rm e},\nu,N_{\rm C})$, i.e. , equal to the temperature at the end of the cooling. For instance, in the radiation field of NGC 2023 small PAHs ( $N_{\rm C}=20$) eventually reach temperature as low as 15 K at time $t_{\rm e}$ whereas bigger PAHs ( $N_{\rm C}=500$) remain at around 35 K.

Figure 4: Cooling curves for various energy per carbon atom (the zero time point has been excluded). The numbers in parenthesis give the value of $E_{\rm C}=h\nu /N_{\rm C}$ in meV/C. The corresponding $(h\nu ,N_{\rm C})$ combinations were as follows: a)  $E_{\rm C}=680$ meV/C, solid line is ($h\nu =13.6$ eV, $N_{\rm C}=20$), b) 453 meV/C, solid = (13.6, 30) and dot-dash = (9.06, 20), c) 272 meV/C, solid = (13.6, 50) and dot-dash = (5.44, 20), d) 136 meV/C, solid = (13.6, 100) and dot-dash = (2.72, 20), e) 27.2 meV/C, solid = (13.6, 500) and dot-dash = (0.54, 20).

In Fig. 4, we show the cooling curves obtained for various energies per C-atom $E_{\rm C}=h\nu /N_{\rm C}$. As expected, the peak temperatures are identical for a given $E_{\rm C}$. We note that the $T_{\rm c}$-curves do not change much for different $N_{\rm C}$-values while $E_{\rm C}$remains fixed: in fact if the emission cross-section were only proportional to $N_{\rm C}$ all the cooling curves with ($h\nu$, $N_{\rm C}$) combinations yielding the same $E_{\rm C}$ would merge. Cooling curves with larger $N_{\rm C}$ have higher temperatures because emission in the C-H modes is less important (in a fully hydrogenated PAH we have $N_{\rm H}/N_{\rm C}=\sqrt{6/N_{\rm C}}$). The small variation in the $T_{\rm c}$-curves at fixed $E_{\rm C}$ then reflects the decrease of the emission cross-section per carbon atom with increasing size.

4.3 The temperature distribution of interstellar PAHs

As noted above, the observed similarity of the AIB profiles in our sample means that the range of PAH emission temperatures does not vary much: to test this idea, we determine here the temperature distribution of PAHs. Moreover, as we show below, the effect of the size distribution and exciting radiation field parameters are illustrated in a synthetic fashion in the temperature distribution.

The observed AIB spectrum results principally from the superposition of many blackbodies at different temperatures (Eqs. (3) and (5)). Following Eq. (3) it is clear that a given blackbody will have a significant contribution to the emergent spectrum if its temperature T is high and/or if it remains a long time at that temperature: a PAH with a temperature between T and $T+{\rm d}T$ (corresponding to the time interval $[t,t+{\rm d}t]$) will thus contribute to the total emitted luminosity $L_{\rm e}$ (see the end of Sect. 4.1) with a weight proportional to $P_{\rm e}(T)\times {\rm d}t$ or the fraction of the total available energy, $h\nu$ (the energy of the absorbed photon), that is dissipated between t and $t+{\rm d}t$. This definition of the temperature weight is different from previous work (Draine & Anderson 1985; Désert et al. 1986) which only considered the time that a given grain spends in the temperature interval $[T,T+{\rm d}T]$, namely dt/dT: the present definition which takes into account the emitted power actually reflects the contribution of a given temperature to the final spectrum.

From Eq. (7), we see that the temperature weight in the total luminosity is simply $C(T)\times {\rm d}T$. Taking into account the probability distribution of the exciting photons and the fraction of PAHs containing $N_{\rm C}$ carbon atoms, we define the temperature weight $w_{\rm T}$ as:

$$w_T = \frac{C(T_{\rm c})\;\delta T} {h\nu} \times {\cal P}_\n... ..._{\rm C}^{\beta+1}\;\delta\,{\rm ln}\,N_{\rm C}} {N_{\rm PAH}}$$ (10)

where $N_{\rm PAH}$ the total number of PAHs and $\delta T$ is the temperature bin in $T_{\rm c}$, $\delta\nu$ the frequency bin of the exciting radiation field and $\delta\,ln\,N_{\rm C}$ the logarithmic bin of the size distribution. All these bins have been taken as fixed. With this definition we have $\sum_{(T,N_{\rm C},\nu)} w_{T} = 1$.

To obtain the PAH temperature distribution in a given exciting radiation field, we build the histogram of all cooling curves for all molecular sizes in temperature bins of constant size ( $\Delta T = 100$ K), each temperature T being given the weight w_T.

Figure 5: The distribution of PAH temperatures $T\times p(T)$ normalized to 1 at peak value. In the panels a), b) and c), we used the parameters for the radiation field of NGC 2023 (see Table 1). The fixed parameters of each plot are given in brackets. In a), we vary $N_{\max}$ and the triple-dot dash line shows the effect of assuming that all molecules absorb the same photon ( $E_{\rm abs} = 6.2$ eV). Variations of $N_{\min}$ are explored in b). The power-law index, $\beta$, is changed in c). In d), we explore the effect of $T_{\rm eff}$ while keeping the dilution factor of NGC 2023 (see text) and $N_{\min} = 20$. The solid line corresponds to the case of the NGC 2023 radiation field. The dot-dash line in d) shows the case of the M17-SW radiation field with $N_{\min} = 30$, the value required to fit the observed AIB profiles (see Sects. 4.4.1 and 4.4.2).

The density value in the histogram is then simply $\sum^{T+\Delta T}_{T'=T} w_{T'}$. The distribution per temperature interval, p(T) is then found by normalizing the histogram so that $\int_0^{\infty} p(T){\rm d}T = 1$.

Figure 6: The spectral energy densities corresponding to the temperature distributions shown in Fig. 5. All the curves have been normalized to 1 at the peak value of the 7.7 $\mu$m-band (see text). In panels a) to c), the exciting radiation field is that of NGC 2023 and we vary the parameters of the PAH size distribution with respect to some reference values, namely, $N_{\min} = 20$, $N_{\max} = 200$ and $\beta = -2.25$. In d), we look at the effect of changing the effective temperature of the exciting radiation field while the dilution factor is always equal to that of NGC 2023. In each panel, the parameters kept constant are given in brackets.

From our weight definition, p(T)dT is the fractional contribution of all PAHs with temperatures in the range $[T,T+{\rm d}T]$ to the total emitted luminosity $L_{\rm e}$.

The PAH temperature distributions, normalized to 1 at their peak values to facilitate the comparison, are shown in Fig. 5: they all present a maximum between 300 and 800 K. The shape of these distributions can be understood as follows. Assuming that all molecules absorb a photon of energy $h\nu$, the temperature weights are proportional to $C(T)\times N_{\rm C}^{\beta+1}$. At low T, all the C(T)-curves corresponding to the different $N_{\rm C}$'s sum up to give w_T. As the temperature increases, the largest molecules do not contribute anymore to w_T because they are too cold: this fade out more than compensates for the rise of C(T) with the temperature and produces a maximum in the temperature distribution.

Figure 5 illustrates the sensitivity of the PAH temperature distribution to the size distribution parameters, $N_{\min}$, $N_{\max}$ and $\beta$ (Figs. 5a to 5c) and to the hardness of the exciting radiation field, parameterized as $T_{\rm eff}$ (Fig. 5d). The associated changes in the AIB spectrum are shown in Fig. 6. In all these figures (except the M17-SW case in Fig. 5d), we used the same dilution factor for the radiation field ( $W_{\rm dil}=2.66\times 10^{-13}$, corresponding to NGC 2023): indeed, as long as one is in the regime of temperature fluctuations, the spectral distribution of the PAH emission spectrum does not depend on the radiation field intensity (Sellgren 1984) while its absolute level scales with it (see Eq. (3)). Moreover, for the purpose of comparison, all the spectra of Fig. 6 have been normalized to 1 at the peak value of the 7.7 $\mu$m-band because its profile is independent of the PAH temperature (see Sect. 4.1).

In Fig. 5a, we consider the effect of taking the full spectral distribution of exciting photons instead of the same mean absorbed photon energy, $E_{\rm abs}$ (i.e. , ${\cal P}_{\nu}=\delta[h\nu-E_{\rm abs}]$). In the following, we will refer to the $E_{\rm abs}$-approximation whenever it is assumed that each molecule absorbs the same photon of energy $h\nu=E_{\rm abs}$ at the rate $R_{\rm abs}$ in the given radiation field. The $E_{\rm abs}$-approximation only affects the high temperature tail of the distribution. As we will see in Sect. 4.4.2, this tail matters for the width of the 3.3 $\mu$m-band. Nevertheless, the rest of the AIB spectrum is little affected by the $E_{\rm abs}$-approximation. $N_{\rm max}$ has a weak influence on the temperature distribution (Fig. 5b) and its impact, which is most noticeable at long wavelengths ( $\lambda \geq$ 10 $\mu$m, see Fig. 6a), is easily drowned by small changes of $\beta$, the index of the power-law size distribution of PAHs (see Figs. 5c and 6c). When $\beta$ decreases, the contribution of the smallest PAHs becomes more important and the cold tail of the temperature distribution is somewhat reduced (Fig. 5c): this leads to an enhanced 3.3/11.3 $\mu$m-band ratio and to somewhat broader bandwidths (Fig. 6c and Figs. 8 to 11). Note that, due to this broadening, the band peak value diminishes (see the 6.2 $\mu$m-band in Fig. 6c) because the band integrated cross-section is conserved with temperature. Both $N_{\min}$ and $T_{\rm eff}$ (which determines $E_{\rm abs}$, here $T_{\rm eff} = 10\,000$ to 45000 K corresponds to $E_{\rm abs} = 4$ to 9 eV) affect the hot tail of p(T) (Figs. 5b and 5d) which merely reflects the hottest part of the cooling curve (Fig. 4). This behaviour of p(T) follows from the fact that the PAH peak temperature only depends on the energy content per carbon atom in a molecule, $E_{\rm C}$ (see Sect. 4.2) and that a typical value for $E_{\rm C}$ is $E_{\rm abs}/N_{\min}$ ($N_{\min}$ is close to the mean value of $N_{\rm C}$ because of the steep power-law size distribution). In fact, the temperature distribution and the shape of the emission spectrum are similar for different $(E_{\rm abs},N_{\min})$-couples yielding the same energy per carbon atom. As expected, an increase of $N_{\min}$ leads to colder PAHs (Fig. 5b) which contribute more to the long-wavelength bands (Fig. 6b) while an increase of $T_{\rm eff}$ has the reverse effect (see Figs. 5d and 6d). We pointed out earlier that the PAH temperature distribution must remain unchanged in order to explain the similarity of the observed AIB profiles: we show in Fig. 5d that this condition is fulfilled if $N_{\min}$ is raised along with $T_{\rm eff}$. Specifically, an increase of $T_{\rm eff}$ from 23000 K (the case of NGC 2023) to 45000 K (the case of M17-SW) requires $N_{\min} = 20$ to 30, respectively. In the next sections, we show that this requirement allows to consistently reproduce the overall AIB spectrum as well as the individual band profiles.

4.4 Modelling the AIB spectrum

We now present the PAH IR-emission as computed from the formalism described above in the case of NGC 2023 and M17-SW. We selected these two objects because they present the largest contrast in the effective temperature of the exciting radiation field: we therefore expect these spectra to provide the strongest constraints on our model.

4.4.1 The complete spectra

Figure 7: The spectral energy density in Watt per hydrogen for our sample of objects from the SWS data (dotted curve) and assuming a column density of $1.8\times 10^{21}$ cm-2. The resolving power ( $\lambda /\Delta \lambda$) is 200 for NGC 2023 and 500 for M17-SW. The model PAH emission is the dot-dash line. The solid line shows the model complete spectrum which includes the broadband continuum described in Sect. 3. We took $N_{\max} = 200$ and $\beta =-3.5$ for both objects. For NGC 2023, we take $N_{\min} = 20$, $f_{\rm H} = 0.8$ and 10% of the interstellar carbon in PAHs while we require $N_{\min} = 30$, $f_{\rm H} = 0.5$ and 8% of the interstellar carbon in PAHs for M17-SW (see text). In the upper panel, the dashed line shows the case of neutral PAHs from our model with $N_{\min} = 20$, $N_{\max} = 200$, $\beta =-3.5$, $f_{\rm H}=1$. For clarity, the SED of neutral PAHs has been normalized to the peak value of the 11.3 $\mu$m-band in the SED of ionized PAHs (dot-dash line). This normalization amounts to a PAH abundance representing 6% of the interstellar carbon.

Having determined the temperature distribution of a population of PAHs, we can now compute the corresponding IR emission using Eqs. (3) and (5). The PAH size distribution parameters were constrained as follows. We note that the overall shape of the 3-13 $\mu$m AIB spectrum which is dominated by emission from molecules at $T\geq 400$ K is not much affected by changes in $N_{\max}$ or $\beta$ (see Figs. 6a and 6c). For simplicity, we fix from now on $N_{\max} = 200$. First, $N_{\min}$ is fixed so as to reproduce the observed 3.3/11.3 $\mu$m-band ratio in a given exciting radiation field (described by $T_{\rm eff}$): we require $N_{\min} = 20$ in NGC 2023 while $N_{\min} = 30$ in M17-SW. The hydrogenation fraction, $f_{\rm H}$, is constrained from the 7.7/11.3 $\mu$m-band ratio: a good match to the observed band ratios is obtained with $f_{\rm H} = 0.8$ for NGC 2023 and $f_{\rm H} = 0.5$ for M17-SW. The index of the power law size distribution, $\beta$, can be constrained from the requirement that the predicted 16.4 $\mu$m-band (on top of the broadband continuum) matches the observed band: we find that $\beta =-3.5$ reproduces the 16.4 $\mu$m-band well in both objects. Unfortunately, individual PAH species show a large spread in the integrated cross-section of the 16.4 $\mu$m-band (see Table 3 of Moutou et al. 1996) which does not constrain $\beta$: specifically, to match the observed 16.4 $\mu$m-band we require $\beta = -2.25$ for the lowest cross-section value while $\beta={-6}$ for the largest cross-section value. More laboratory work on PAHs is warranted in order to understand the trend of the 16.4 $\mu$m-band profile with respect to $N_{\rm C}$ and to the various isomer states. As discussed by PJB, an additional constraint on $\beta$ is provided by the intensity and width of the AIB profiles (see the next section). The $\beta$-value can then be adjusted in order to match the peak position and the FWHM of the 3.3 $\mu$m-band (see Sect. 4.4.2): in fact, the value of -3.5 found from the 16.4 $\mu$m-band gives a good fit in the case of NGC 2023 and M17-SW (see Fig. 8).

At this point, it is interesting to note that that the 6-to-13 $\mu$m part of the spectrum is produced by molecules with emission temperatures between 400 and 1000 K. During temperature fluctuations, most PAHs go through these temperatures: this is why the maximum of the temperature distribution (Fig. 5) always falls in this range. In fact, the emission of any PAH with $N_{\rm C}\leq 160$ shows ratios of band peak values which are quite comparable to the observed 6-13 $\mu$m AIB spectrum (see PJB and Draine & Li 2001). On the other hand, the match of the observed band position and width in the present data sample and in CAM-CVF spectra of similar interstellar regions requires small molecules (see next section). However, taking into account the full size distribution of PAHs will add emission mostly at 3.3 $\mu$m (from the smaller species, see Fig. 6b) and in the far-infrared (from the larger species, see Fig. 6a) without changing much the 6-13 $\mu$m region. Thus, in the framework of the present model, the observed invariance of the AIB spectrum in CAM-CVF data (6-16 $\mu$m) implies that the AIB carriers are small enough to undergo large temperature fluctuations (at least 400 K) during which they emit in this wavelength range.

Our best fit SED's are displayed in Fig. 7: we assumed a total column density of $1.8\times 10^{21}$ cm^-2 (i.e. , 1 visible magnitude of extinction) for both NGC 2023 and M17-SW with 10% and 8%, respectively of the interstellar carbon in PAH cations (using [C/H] $_{\rm ISM}=2.6\times 10^{-4}$, Snow & Witt 1996). The hydrogenation fraction of PAHs was taken to be 80% for NGC 2023 and 50% for M17-SW. To compare in a meaningful fashion the predicted PAH emission to the observed spectrum, the broadband continuum described in Sect. 3 has been added to our model spectrum. This continuum consists of the modified blackbody along with the 1000 and 1450 cm_1 broad profiles (in M17-SW an additional broad profile at 600 cm_1 is required, see Fig. 2). All the main AIBs (at 3.3, 6.2, 7.7, 8.6 and 11.3 $\mu$m) are matched within 20%. We note that the model fails to reproduce the 12.7 $\mu$m-band in M17-SW: this may be due to an ill-definition of the underlying broadband continuum in this complex spectral region (see Fig. 2). The AIB spectrum corresponding to the neutral PAHs (we took the IR cross-sections from Léger et al. 1989b) is shown in the case of NGC 2023 with a PAH abundance corresponding to 6% of the interstellar carbon, with $N_{\min} = 20$, $N_{\max} = 200$ and $\beta =-3.5$. The $N_{\min}$-value was fixed so as to reproduce the position and width of the 3.3 $\mu$m-band (see next section) and it is the same as for the cations. In the case of neutral PAHs, the C-C/C-H band ratio is reduced by about one order of magnitude with respect to the cations. Matching the observed AIB with neutral PAHs would require a very low hydrogen coverage ( $f_{\rm H} \leq 0.1$) which would produce a strongly discrepant 8.6 $\mu$m-band and also would be at odds with theoretical predictions on the hydrogen coverage of PAHs (Allain et al. 1996).

In the following, we compare in detail the profiles of the observed 3.3, 6.2, 8.6 and 11.3$\mu$m-AIBs to our model results: we restrict our comparison to these bands because they are well defined in the data and involve a small number of Lorentz fit components.

4.4.2 Individual AIB profiles

To do a detailed model-to-data comparison, we now study the individual AIB profiles. Due to anharmonic couplings, the line profiles of PAH vibrational transitions are redshifted and broadened as the temperature of the molecule increases and both follow a linear law (Joblin et al. 1995). The temperature dependent broadening laws of PJB used in this modelling do not include the rotational width. The centroid and width of each band profile are given by $\nu(T)=\nu_0+\chi_{\rm c}\times T$and $\Delta\nu(T)=\Delta\nu_0 + \chi_w\times T$ respectively. The coefficients of the linear T-laws have been measured in the laboratory on a restricted sample of small ( $N_{\rm C}\leq 32$), neutral PAHs: we assume that the temperature dependence of the vibrational band profile are the same for all PAH sizes as well as for cations and neutrals. To match the position of the observed AIB profiles, we had to slightly change the band centers at 0 K: namely, for the 3.3, 6.2, 8.6 and 11.3 $\mu$m-bands we took $\nu_0= 3079$ (3076), 1636 (1627), 1169 (1141) and 899 (896) cm_1 where the laboratory values are in parenthesis. Such shifts are reasonable in view of the laboratory results on the spread in band positions for different species.

In order to highlight the match in band width and position, our model profiles are compared to the observed AIBs in a normalized fashion.

Figure 8: The normalized profile of the 3.3 $\mu$m-AIB as seen by ISO-SWS compared to our model emission profile also normalized to one (see text). The contribution of the continuum and of all bands other than the 3.3 $\mu$m have been subtracted. The resolving power of the data ( $\lambda /\Delta \lambda$) is 200 for NGC 2023 and 500 for M17-SW. For all the model profiles, we took $N_{\max} = 200$. In a), the SWS spectrum is the dotted curve, the result of our model best fit with $N_{\min} = 20$ and $\beta =-3.5$ is the solid line and the dashed line represents the case where $\beta = -2.25$ (while $N_{\min} = 20$). In b), we show the contribution of the small ( $20 \leq N_{\rm C} \leq 35$, dashed line) and large ( $35 \leq N_{\rm C} \leq 200$, dot-dashed line) PAHs to the model best fit of NGC 2023. In c), the observed 3.3 $\mu$m-band profile of M17-SW (dotted curve) is shown as well as our model best fit (solid line) with $N_{\min} = 30$ and $\beta =-3.5$. The model profile for $N_{\min} = 20$ (while $\beta =-3.5$) is shown as the dashed line. The dot-dashed line shows a model using the $E_{\rm abs}$-approximation ( $E_{\rm abs}=9.1$ eV, see Sect. 4.3) with $N_{\min} = 30$ and $\beta =-3.5$.

For all the results of best fit models represented, the peak absolute intensities do not deviate by more than 20% from the observations (Fig. 7). We use the $N_{\min}$-values derived from the complete spectra, i.e. , $N_{\min} = 20$ and 30 for NGC 2023 and M17-SW respectively. Noting that $N_{\min}$ has a strong effect on the PAH emission (Figs. 5c and 6c and Figs. 8c to 11c), it is remarkable that the same value allows to match both the absolute peak intensity and the profile shape of the 3.3 $\mu$m-band. The value of $\beta$ is then chosen so as to obtain a good match of the width of all bands: we find that $\beta =-3.5$ is a good compromise. It must be noted, however, that the values of $N_{\min}$ and $\beta$ are strongly dependent on the adopted profile temperature laws.

Figure 9: Same as Fig. 8 for the 6.2 $\mu$m-band. In the case of M17-SW, the profile for the $E_{\rm abs}$-approximation has not been represented.

Figures 8 to 11 show our model representation of the AIB profiles. The contribution of vibrational hot bands are not included in our model profiles: in the case of the 6.2 and 11.3 $\mu$m-bands, hot bands provide an additional broadening of less than 5 cm_1 (PJB). In Fig. 8c, we note the effect of making the $E_{\rm abs}$-approximation (i.e., all molecules are heated by photons of the same energy $E_{\rm abs}$, see Sect. 4.3): the lack of the hot tail in the PAH temperature distribution yields a narrower band profile. Hence, using this approximation would lead to an underestimate of $N_{\min}$: e.g., in the case of NGC 2023, $N_{\min}=15$ is required to match the observed 3.3 $\mu$m-band if one makes the $E_{\rm abs}$-approximation. This approximation has, however, negligible consequences for the other bands. The effect of a larger $\beta$, -2.25 (corresponding to the classical power-law exponent of -3.5 when the size distribution is expressed in terms of the grain radius, Mathis et al. 1977) is shown as the dashed line in Figs. 8a to 11a: a low $\beta$-value is clearly required to match the observed band profiles. Decreasing $\beta$, however, has a moderate impact on the band profiles because the then enhanced contribution of small molecules is compensated by a strong increase of the band width leading to a decrease in intensity (the integrated cross-sections of the IR bands, $\sigma\times\Delta\nu$, are assumed to be conserved with temperature).

Figure 10: Model results for the 8.6 $\mu$m-band. The data is displayed as in Fig. 8 with the following changes. In a), the solid line shows the model profile if the bandwidth temperature dependence derived from laboratory results is used. The dot-dash line shows the model profile for a hypothetical broadening 5.5 times faster (see text) and the dashed line corresponds to $\beta = -2.25$ case for the fast broadening law. In b), we show the contribution of small and large PAHs in the case of the fast broadening law. Finally, the display in c) is the same as in a for the M17-SW case except for the dash line where $N_{\min} = 20$ and $\beta =-3.5$ with the fast broadening law. The unresolved lines at 1035, 1110 and 1245 cm_1 correspond to 0-0 S(3) of H2, a forbidden line of [ArIII] and 0-0 S(2) of H2.

The harder radiation field of M17-SW results in PAHs at higher temperatures (see Fig. 5d), consequently the model AIB-profiles are clearly too broad and redshifted (see the dashed line in the bottom panel of Figs. 8 to 11) if we use the same size distribution as for NGC 2023. For the profile width and position, $N_{\min} = 30$ is also required in order to match the observed AIBs of M17-SW. This change of $N_{\min}$ has a physical interpretation: the photo-dissociation and fragmentation rates rise steeply with the molecular temperature (Léger et al. 1989a). The $N_{\min}$values we find are smaller than the theoretical predictions of Allain et al. (1996).

Figure 11: Same as Fig. 9 for the 11.3 $\mu$m-band. Note that the 11 $\mu$m-AIB has not been withdrawn from the data.

Except for the 8.6 $\mu$m-band, all AIB profiles are well explained by the PAH emission model. The hot (small) molecules dominate the emission profiles, in particular in the AIB wings; they also produce the observed red asymmetry (6.2 and 11.3$\mu$m-bands). The cool (large) molecules contribute the blue wing of the AIB. We want to stress that this match of the AIBs is obtained by using the cross-sections and profile temperature dependence measured on small molecules: no additional broadening, reflecting the decrease of the level lifetime with increasing molecular size (see Sect. 5), has been taken into account here.

The observed width of the 8.6 $\mu$m-band is not at all reproduced by the temperature broadening law derived from laboratory work. In fact, a broadening 5.5 times faster ( $\chi_w=6.0\times10^{-2}$ cm_1/K) with temperature would match the observations (see Fig. 10). In this latter case, we note that (i) the symmetrical shape of the 8.6 $\mu$m-band is well explained by the slow redshift of the band centroid with temperature ( $\chi_{\rm c}=0.84\times 10^{-2}$ cm_1/K whereas the other AIBs have 2 to $3 \times 10^{-2}$ cm_1/K) and (ii) the emission of large molecules dominates the core of the 8.6 $\mu$m-band (conversely to the other AIBs).

We have not modelled the 7.7 $\mu$m-band because it includes several components whose assignment to features in laboratory spectra of PAHs is not straightforward. However, it is interesting to note that the observed width (23 to 29 cm_1) of the 7.6 $\mu$m-band (see Sect. 3) is comparable to the width predicted by PJB.

We emphasize here that the model profile of a PAH emission band is the result of the superposition of many Lorentz profiles conversely to the implicit assumption behind the decomposition performed in Sect. 3. Consider for instance a PAH containing $N_{\rm C}$ carbon atoms which has been heated by a photon of energy $h\nu$: the emerging band profile of this molecule will be the sum of all the Lorentz profiles corresponding to the temperatures of the cooling curve, $T_{\rm c}(t,\nu,N_{\rm C})$ (i.e. , the time integral in Eq. (3)). The total band profile of a PAH interstellar population is eventually obtained from the weighted sum of the emerging band profiles from all molecules, for all energies of the exciting photon (see Eqs. (3) and (5)).