2 Modelling the IR emission spectrum of ``interstellar PAHs''

The photophysics of interstellar PAHs have been described for example by Léger et al. (1989). After the absorption of an UV photon, the system follows a rapid evolution towards the ground electronic state (internal conversion). The energy is redistributed in the vibrational modes by the Intramolecular Vibrational Redistribution (IVR) mechanism, and the isolated excited species then cool down by emitting in their IR active modes. Some of the energy can also be lost through electronic fluorescence (Poincaré fluorescence; Léger et al. 1988). The contribution of the electronic fluorescence to the cooling process was found to be about 10% for the small cation coronene C $_{{\rm 24}}$H $_{{\rm 12}}^{+}$ and to become negligible for larger PAH species. This mechanism was therfore not considered in the calculations.

   
2.1 Infrared vibrational emission and cooling

Assuming a statistical distribution of the energy among the vibrational modes, the infrared emission rate $k^{i}_{{\rm IR,v}}$ of the mode i in the transition $v \rightarrow v-1$ is given by:

$$k_{{\rm IR},v}^{i}=A^{v,v-1}_{i}\times\frac{\rho^{*}(U-vh\nu_{i})}{\rho(U)}$$ (1)

A_i^v,v-1 being the Einstein coefficient of spontaneous emission in the $v \rightarrow v-1$ transition, $\rho(U)$ the density of states and $\rho^{*}(U-vh\nu_{i}$) the density of states for all modes except the i emitting mode. It has been shown (Léger et al. 1989; Schutte et al. 1993) that the thermal approximation can be used to describe the infrared cooling of PAHs. In this approximation, the emission in a given vibrational mode from a PAH of internal energy U is calculated as the average emission of an oscillator connected to a thermal bath. This model assumes that, the energy content of the other modes being large enough, the PAH itself can be considered as a heat bath at the temperature T which corresponds to an average energy U of the molecule. Expressed in terms of specific heat, this can be written as:

$$U(T)=\int_{0}^{T}C(T'){\rm d}T'.$$ (2)

In the case of an assembly of harmonic (or nearly harmonic) oscillators, the integrated form is readily obtained:

$$U(T)=\sum_{i=1}^{s}\frac{h\nu_{i}}{\exp(h\nu_{i}/kT)-1}$$ (3)

with s being the total number of vibrational modes in the molecule. The emission rate for the transition $v \rightarrow v-1$ of the mode i is then given by:

$$k_{{\rm IR},v}^{i}=v\times A^{1,0}_{i} \times \exp(-{v}h\nu_{i}/kT) \times (1-\exp(-h\nu_{i}/kT)) .$$ (4)

The summation of Eq. (4) over all the $v \rightarrow v-1$ transitions gives the expression of the total IR emission rate of the mode i as:

$$k_{{\rm IR}}^{i}=\sum_{v}k_{{\rm IR},v}^{i}=A^{1,0}_{i}\times\left[\exp(h\nu_{i}/kT)-1\right]^{-1}.$$ (5)

   
2.2 Line profiles

Several mechanisms can contribute to the broadening of the IR bands emitted by PAHs during their cooling.

2.2.1 Homogeneous band width

The homogeneous IR band width of a vibrationnally excited PAH is governed by IVR. Driven by the coupling between states, IVR is effective at vibrational energies larger than $\sim$2000 cm^-1 in the case of PAHs (Felker & Zewail 1984). It leads to a lifetime of the levels much shorter than the radiative lifetime (10^-11 s compared to 10^-7 s for the electronic fluorescence and 10^-1 s for the infrared emission). Therefore this mechanism dominates the band width associated with a Lorentzian profile. As the vibrational energy increases, this band width becomes larger (Ionov et al. 1988; Joblin et al. 1995). It is found to increase linearly with temperature, a behaviour which is well-explained by theory.

2.2.2 Anharmonic shifts and hot bands

Measurements on gas-phase PAHs have also revealed the dependence on temperature of the IR band positions (Joblin et al. 1995). The positions are shifted towards lower frequencies as the temperature increases, following a linear law. This effect has a similar origin as the homogeneous band width described above. Due to the coupling between modes (intermode anharmonicity), the IR band associated with the mode i is shifted relative to the frequency $\nu_{i}$(0), that can only be attained when all the other modes are not populated. This temperature shift induces a consequent broadening of the total band emitted during the cooling of PAHs, as was shown in the case of the 3.3 $\mu$m feature by Joblin et al. (1995).

Another cause of broadening is due to the anharmonicity of the modes themselves (intramode anharmonicity; hot bands as described by Barker et al. 1987). Indeed the transitions $v \rightarrow v-1$ with v $\geq$ 2 are usually shifted relative to the fundamental $1 \rightarrow 0$ transition. As a consequence, if the molecule is sufficiently heated to populate the levels v $\ge$ 2, the hot bands can create an additional asymmetry in the total emitted feature.

2.2.3 Rotational broadening

The values of the rotational constants of PAHs are very small. For instance, the value of B for coronene is 0.011123 cm^-1 (Cossart-Magos & Leach 1990). It is smaller for larger PAHs since the rotational constant is expected to scale as the inverse of the square of the carbon number (Le Coupanec et al. 1998). As a result, the rotational sub-structure is blurred by the other broadening mechanisms. Only a global rotational envelope can be observed. This is illustrated for instance by the gas-phase spectrum of coronene at 770 K (Joblin et al. 1995). The width of the rotational envelope can be quantified by the separation of the maxima of the P and R branches.

$$% \Delta\nu_{{\rm rot}} = 4\sqrt{BkT_{{\rm rot}}}.$$ (6)

Calculations of the rotational temperature of interstellar PAHs led to an average value of 100 K (Le Coupanec et al. 1998). According to Eq. (6), the total rotational width is therefore 2.4 cm^-1 for C₂₄H₁₂ and becomes inferior to 1 cm $^{{\rm -1}}$ for large molecules ($N_{{\rm C}}$ $\geq$ 50). The contribution of the rotational width was therefore neglected in the calculations.

2.2.4 Molecular diversity

In this model, we have assumed that all PAHs have exactly the same IR active modes. However, the exact positions of the modes are expected to change from one PAH to the other and therefore to provide an additional broadening to the observed spectra. This is well-known for small molecules ( $N_{{\rm C}}\le 32$) but has still to be studied for larger systems $N_{{\rm C}}$ up to a few hundreds. The possible influence of molecular diversity on the calculated spectrum is discussed in Sect. 3.3.

   
2.3 Laboratory and observational inputs

In the calculations, we aim to use as much as possible the photophysical properties of PAHs derived from laboratory measurements. As mentioned earlier, the relative intensities of the AIBs are more consistent with the carriers being PAH cations rather than neutrals (Szczepanski & Vala 1993; DeFrees et al. 1993; Schutte et al. 1993; Hudgins & Allamandola 1995; Langhoff 1996; Cook & Saykally 1998). Laboratory data on PAH cations were therefore used for the IR band strengths. For the band profiles, only data on neutral PAHs are available. In all cases, the data concern 24-32 carbon PAHs and we had to assume that the derived properties can be extrapolated to larger systems ($\sim$100 C).

2.3.1 Specific heat

The exact calculation of the specific heat C(T) or the internal energy U(T) requires for each molecule the whole set of vibrational frequencies. To simplify, three typical frequencies have been associated with the C-C and C-H modes. The frequencies involving H atoms were readily determined as those of the three IR active C-H modes at 3050, 1150 and 885 cm^-1. For the C-C bonds, three representative frequencies at 1499, 840 and 301 cm^-1 were derived from the set of modes of dehydrogenated coronene (Boissel et al. 1997). Considering that the total number of modes 3N-6 can be split into 3 $N_{{\rm H}}$ C-H modes, and 3( $N_{{\rm C}}$-2) C-C modes, Eq. (3) can then be written as:

$$\begin{array}{rl} U(T) &= N_{{\rm H}}\times\sum_{i=1}^{3}{\di... ...u_{i{\rm C}}} {\exp(h\nu_{i{\rm C}}/kT)-1}}\cdot \end{array}$$ (7)

Figure 1: Absorption rate by PAHs in IRAS 21282+5050 (see Sect. 2.3.2).

   
2.3.2 UV absorption

In the interstellar medium, PAHs are excited by UV photons from stars. In this paper, we have considered the particular case of the planetary nebula IRAS 21282+5050. According to Cohen & Jones (1987), the nebula is powered by a central star of spectral type O7(f)-[WC11]. The flux $F_{\nu}$ from the star was taken from Kurucz (1979) assuming a stellar temperature of 28 000 K (Leuenhagen & Hamann 1998). A data compilation on protoplanetary nebulae (Bujarrabal et al. 2001) gives the distance (D=3 kpc) and the total luminosity ( $L=5.3 \times 10^{3}~L_{\odot}$) of this object. The maximum PAH emission is located at about 1'' from the central source (Meixner et al. 1993), e.g. at a projected distance of the central source d_*= 0.0145 pc. This corresponds to a dilution factor of $W_{{\rm dil}}=2.44 \times 10^{-11}$ and an integrated UV photon flux of $G=\int_{6 {\rm eV}}^{13.6 {\rm eV}}F_{\nu}$ ${\rm d}\nu =$ 3.7$\times$10⁵ $G_{{\rm o}}$, where $G_{{\rm o}}$ is the average value for the interstellar medium ( $G_{{\rm o}}=$ 1.6$\times$10^-10 W cm^-2; Habing 1968).
The rate of photons absorbed by PAHs is given by:

$$R_{{\rm abs}}(N_{{\rm C}}^{-1} {\rm s}^{-1})= \int_{}^{}\sigma_{\nu}\frac{F_{\nu}}{h\nu}{\rm d}\nu$$ (8)

where $\sigma_{\nu}$ is the UV absorption cross-section of PAHs.

We have used the UV absorption cross-section $\sigma_{\nu}$ measured by Joblin et al. (1992) on mixtures of neutral PAHs. This cross section consists of a far-UV rise, a broad feature centred at about 210 nm and a tail extending down to 400 nm. There is not much data on the UV cross-section of PAH cations. One study performed in boron oxide matrices (Robinson et al. 1997) shows that PAH cations are likely to have the same far-UV rise as neutral PAHs. In addition, PAH cations have also relatively sharp absorption bands in the visible part of their spectrum whose intensities and positions depend on the considered species (Salama et al. 1995; Salama et al. 1999). However, in the environment of a hot star, as considered here, the absorption in the UV is largely dominant. The visible bands of PAH cations were therefore not included and the UV-visible cross section of a mixture of neutral PAHs was used (Fig. 1c from Joblin et al. 1992). As shown by the authors, this UV cross section is proportional to the number of carbon atoms. Figure 1 displays the absorption rate of photons by PAHs in the environment of IRAS 21282+5050 as a function of wavelengths. The total absorption rate is $R_{{\rm abs}}$ = 2.65$\times$10^-4 $N_{{\rm C}}^{-1}$ s^-1.

   
2.3.3 The vibrational spectrum

To calculate the IR spectrum emitted by a population of PAHs, it is necessary to know their vibrational spectra including the band strengths (in terms of Einstein coefficients) as well as the band positions and widths and their evolution with temperature.

Band strengths

The IR vibrational spectra of many PAHs$^{\rm +}$ of sizes up to 32 carbon atoms are known. We have therefore considered the two largest compact cations: coronene C $_{{\rm 24}}$H $_{{\rm 12}}^{\rm +}$ and ovalene C $_{{\rm 32}}$H $_{{\rm 14}}^{\rm +}$ which have been studied experimentally (Szczepanski & Vala 1993; Hudgins & Allamandola 1995) and theoretically (Langhoff 1996). The different IR modes for these species have been gathered together into the main astrophysical features in order to derive Einstein A_i coefficients for bands corresponding to the 3.3, 6.2, "7.7'', 8.6, 11.3 and 12.7 $\mu$m AIBs. Classically, the vibrations at 6.2 and "7.7'' $\mu$m are attributed to C-C stretching modes, and the others to C-H modes. We note that the spectral range 11.6-13 $\mu$m includes the major feature at 12.7 $\mu$m and a plateau in which minor bands can be observed (Hony et al. 2000). The classical interpretation is that the 11.3 $\mu$m band is due to solo H whereas bands at longer wavelengths are attributed to other types of H (duo, trio, quatro; Allamandola et al. 1989). We did not consider the 11.6-13 $\mu$m plateau and put all the intensity into the 12.7 $\mu$m band. In order to extrapolate the spectra to larger sizes, we assumed that the Einstein coefficients A_i are proportional to the number of C and H atoms for the C-C and C-H modes respectively. The spectra of the PAH-like species considered in our model consist therefore of bands at the positions of the interstellar bands with A_i coefficients deduced from the C $_{{\rm 24}}$H $_{{\rm 12}}^{\rm +}$ and C $_{{\rm 32}}$H $_{{\rm 14}}^{\rm +}$ spectra and scaling with the number of atoms (N $_{{\rm C}}$ or N $_{{\rm H}}$; cf. Table 1). These values are quite comparable to those deduced by Schutte et al. (1993) in their standard model. In particular, the cross sections of the 6.2, 7.7 and 8.6 $\mu$m bands are significantly enhanced compared to those of neutral PAHs.

   

Table 1: Einstein coefficients for the IR active modes of "interstellar'' PAHs. ^a Generic spectrum based on the spectra of C $_{{\rm 24}}$H $_{{\rm 12}}^{\rm +}$ and C $_{{\rm 32}}$H $_{{\rm 14}}^{\rm +}$ (Hudgins & Allamandola 1995; Szczepanski & Vala 1993 and Langhoff 1996). ^b Standard model by Schutte et al. (1993).

Molecule	$A_{{\rm 3.3~\mu m}}$	$A_{{\rm 6.2~\mu m}}$	$A_{{\rm ''7.7~\mu m''}}$	$A_{{\rm 8.6~\mu m}}$	$A_{{\rm 11.3~\mu m}}$	$A_{{\rm 12.7~\mu m}}$
	(s $^{{\rm -1}}$/$N_{{\rm H}}$)	(s $^{{\rm -1}}$/$N_{{\rm C}}$)	(s $^{{\rm -1}}$/$N_{{\rm C}}$)	(s $^{{\rm -1}}$/$N_{{\rm H}}$)	(s $^{{\rm -1}}$/ $N_{{\rm Hsolo}}$)	(s $^{{\rm -1}}$/ $N_{{\rm Hother}}$)
Generic spectrum a	6.59	2.14	5.4	0.82	5.49	0.57
Standard b	8.95	0.93	2.68	0.82	1.87	0.58

In addition to the main AIBs, we have also considered two types of bands measured on neutral PAHs but which have not been studied yet on PAH cations. First, the overtone of the C-H stretch which has been measured in the laboratory by Joblin (1992) and detected at 1.68 $\mu$m in the object IRAS 21282+5050 (Geballe et al. 1994). The Einstein coefficient A₂₀ measured for this band is 1/6 of A₁₀. Second, PAHs have modes in the far-infrared domain that have to be included in the calculation of the cooling process. Moutou et al. (1996) have reported the far-IR spectra of many neutral PAHs. The positions of these bands are quite variable from one molecule to the other. However, for compact PAHs, accumulation points were found at 16, 18.2, 26 and 50 $\mu$m. We have included these bands in the calculations with Einstein coefficients corresponding to the average values measured by Moutou et al. (1996) on compact molecules: 0.057, 1.4$\times$10^-2, 1.5$\times$10^-3 and 1.5$\times$10^-2 (s^-1/ $N_{{\rm C}}$) for the $A_{{\rm 16~\mu m}}$, $A_{{\rm 18.2~\mu m}}$, $A_{{\rm 26~\mu m}}$ and $A_{{\rm 50~\mu m}}$ coefficients respectively.

Widths and positions of the IR bands

The temperature dependence of the band positions and widths was studied on a few gas-phase neutral molecules by Joblin et al. (1995). Due to the coupling between modes, the positions of the IR bands shift towards lower frequencies as the temperature increases and the band widths increase. Both effects appear to be linear at least in the high-temperature range. They are expected to be general, independent of the PAH charge state. Temperature laws for the bands' positions and widths were derived mostly from measurements on neutral coronene (Joblin et al. 1995). Compared to the authors' work, only the laws for the band widths were refined by subtracting the contribution of rotation which was not negligible in these experiments at thermal equilibrium ( $T_{{\rm rot}}$ = $T_{{\rm vib}}$). Also, the band widths measured in neon matrices at 4 K (Joblin et al. 1994) were included before fitting the experimental data points, in order to avoid negative widths at low temperatures. In the case of the 3.3 $\mu$m band, a third-order polynomial had to be included to fit the data at temperatures lower than 850 K. In all other cases, a linear fit was found to be satisfactory with a maximum error of 10% for the 6.2 $\mu$m band at 2000 K (cf. Fig. 2).

Figure 2: Widths of the 3.3, 6.2 and 11.3 $\mu$m bands as a function of temperature. The points at high temperatures are the measurements in gas-phase by Joblin et al. (1995) from which a rotational width was subtracted (except for the 11.3 $\mu$m band which has a prominent Q branch). The solid line shows the linear fit of the gas-phase data. To describe the width at low temperatures, the point in Ne matrices at 4 K (Joblin et al. 1995) was included and new fitting curves were derived (dotted lines).

The temperature dependence of the band positions and widths is summarized in Table 2. The values are those of coronene and derivatives except for the position $\nu_{{\rm L}}$(0) of the 11.3 $\mu$m band. Indeed, the position of the C-H out-of-plane bending mode is very sensitive to the number of adjacent H. Coronene has only duo H, whereas the 11.3 $\mu$m band corresponds to solo H. We therefore used the value of $\nu_{{\rm L}}$(0) derived from measurements on ovalene C $_{{\rm 32}}{\rm H}_{{\rm 14}}$ which contains 2 hydrogens solo (Joblin 1992; Joblin et al. 1994). The 6.2 $\mu$m band width of coronene was measured to be rather constant with temperature, a behaviour which was peculiar in all the measurements reported by Joblin et al. (1995) and could be due to the high symmetry of the molecule. We therefore used for this band the width and position measured for methyl-coronene CH $_{{\rm 3}}$-C $_{{\rm 24}}$H $_{{\rm 12}}$, a closely related molecule with lower symmetry (Joblin 1992). The implication of the choice of the coefficients $\nu_{{\rm L}}$(0), $\chi '$, $\triangle\nu(0)$, $\chi ''$ on the calculated IR spectrum is further discussed in Sect. 3.1. Table 2 also includes the anharmonic shifts used to calculate the positions of the hot bands (transitions $v \rightarrow v-1$, with $v \ge 2$) according to:

$$\nu_{i,v}(T)=\nu_{i,1}(T)-(v-1)(\nu-\nu_{{\rm HB}}).$$ (9)

For the 3.3 $\mu$m band, an anharmonicity of 120 cm^-1 has been measured by Geballe et al. (1994). No measurements are available for the 6.2 and 11.3 $\mu$m bands. A value of 5 cm^-1 was assumed according to Barker et al. (1987). Concerning the widths, we used the same total width for the hot band of the 3.3 $\mu$m band as that measured for pyrene C $_{{\rm 16}}$H $_{{\rm 10}}$ at 1.68 $\mu$m on the 2-0 overtone transition ( $\triangle\nu=$ 74 cm^-1; Joblin 1992) and also observed in IRAS 21282+5050 (Geballe et al. 1994). In the case of the 6.2 $\mu$m and 11.3 $\mu$m hot bands, there is no experimental data and we assumed that the hot bands follow the same evolution with temperature as the fundamental vibrations. Hot bands have not been considered for the 7.7 and 8.6 $\mu$m bands since several bands fall in the same region (Verstraete et al. 2001) which makes a detailed analysis of individual profiles very difficult.

   

Table 2: Empirical laws for the IR band positions and widths as a function of temperature: $\nu (T)=\nu _{{\rm L}}(0)+\chi 'T$ and $\Delta \nu (T)=\Delta \nu (0)+\chi ''T$, derived from measurements on neutral PAHs (a) C $_{{\rm 24}}$H $_{{\rm 12}}$, (b) C $_{{\rm 24}}$H $_{{\rm 11}}$-CH $_{{\rm 3}}$ and (c) C $_{{\rm 32}}$H $_{{\rm 14}}$ (Joblin 1992; Joblin et al. 1995 and Fig. 2 in this work). (*) In the case of the 3.3 $\mu$m band, the $\Delta \nu$(0) value represents the value at the origin of the linear domain at high temperature (T > 850 K). For T= [0, 850 K], we used a third-order polynomial 6.48+10 ^-5T²+8 $\times$ 10 ^-9T³. The hot band (transition $v \rightarrow v-1$ with $v \ge 2$) positions are derived according to Eq. (9) using the shift $\nu$ - $\nu _{{\rm HB}}$ ; (d) Geballe et al. (1994), (e) Barker et al. (1987).

	Band position		Band width		Hot band shift
	(cm-1)		(cm-1)		(cm-1)
Band	$\nu_{{\rm L}}$(0)	$\chi '$	$\Delta \nu$(0)	$\chi ''$	$\nu-\nu_{{\rm HB}}$
3.3 $\mu$m	3076 (a)	-3.2 $\times$ 10-2(a)	-14.69 (*)	3.8 $\times$ 10-2 (*)	120 (d)
6.2 $\mu$m	1627 (b)	-3.8 $\times$ 10-2 (b)	1.00 (b)	2.30 $\times$ 10-2(b)	5 (e)
7.7 $\mu$m	1326 (a)	-2.4 $\times$ 10-2 (a)	0.56 (a)	8.3 $\times$ 10-3(a)	-
8.6 $\mu$m	1141 (a)	-8.4 $\times$ 10-3 (a)	0.66 (a)	1.14 $\times$ 10-2(a)	-
11.3 $\mu$m	896 (c)	-2.3 $\times$ 10-2(a)	0.54 (a)	1.24 $\times$ 10-2(a)	5 (e)

   
2.4 Model of a PAH-like species population

The IR emission of a PAH-like species population is considered here. This population is defined by the generic formula C $_{{\rm 6p^2}}{\rm H}_{{\rm 6p}}$ and a continuous size distribution given by a power law $N_{{\rm C}}^{-\beta }$ between $N_{{\rm Cmin}}$ and $N_{{\rm Cmax}}$. The formula C $_{{\rm 6p^2}}{\rm H}_{{\rm 6p}}$ corresponds to compact species when p is an integer (Omont 1986). We used steps of 2 carbons which correspond in the calculations to a non-integer (non-physical) value of $N_{{\rm H}}$.

In the following, the notation {C $_{{\rm 6p^2}}{\rm H}_{{\rm 6p}}$} is given for the PAH-like species of formula C $_{{\rm 6p^2}}{\rm H}_{{\rm 6p}}$ and whose photophysical properties are described above. When considering a size distribution, the three parameters: $N_{{\rm Cmin}}$, $N_{{\rm Cmax}}$ and $\beta$ are reported. For instance, values of $N_{{\rm Cmin}}=$ 20, $N_{{\rm Cmax}}=$ 178 and $\beta =$ 2 were used by Désert et al. (1990) in their model. Finally, we assumed that 1/3 of hydrogen atoms are solo following Schutte et al. (1993).

   
2.5 Summary of calculations

The internal energy of a given PAH is initialized at the value $U_{{\rm in}}=h\nu_{{\rm UV}}$ corresponding to the absorption of one UV photon. The peak molecular temperature $T_{{\rm p}}$ is calculated from Eq. (3). The molecule then cools down in its different i modes from different v levels according to the emission rates given by Eq. (4). At each step $\delta T=1$ K, the internal energy changes by $\delta U$ and the fraction of energy emitted in the transition $v \rightarrow v-1$ of the i mode is given by:

$$\delta E_{i,v}(T)=\frac{k^{i}_{{\rm IR},v}}{\sum_{i,v}k^{i}_{{\rm IR},v}} \delta U.$$ (10)

This energy is associated with a band of Lorentzian profile centered at $\nu_{i,v}$(T) and of width $\Delta\nu_{i,v}$(T). The total energy emitted in the i $^{{\rm th}}$ band E_i( $U_{{\rm in}}$) as well as the total band profile $\psi_{i}$( $U_{{\rm in}}$) are obtained by adding all the individual contributions during the cooling of the molecule from $T_{{\rm p}}$ to a final temperature $T_{{\rm f}}$. In the calculations $T_{{\rm f}}$ has been taken at 0 K but the energy emitted in the mid-IR for T below $\sim$50 K is negligible. In a second step, the integration over the absorbed photon distribution is performed (see Sect. 2.3.2 and Fig. 1). The last step consists in integrating over the size distribution of PAHs as described in Sect. 2.4.