1 Introduction

Thanks to the European satellite Infrared Space Observatory (ISO), the well-known "Unidentified'' Infrared Bands (UIBs) at 3.3, 6.2, 7.7, 8.6, 11.3 and 12.7 $\mu$m have been observed in many regions of the interstellar medium (ISM) in our galaxy, such as high-latitude cirrus, reflection nebulae, planetary nebulae and HII regions as well as in external galaxies. These bands are the signatures of CC and CH bonds of hydrogenated aromatic structures, which has lead to a more recent re-naming as Aromatic IR Bands (AIBs). Several candidates have been proposed: polycyclic aromatic hydrocarbon molecules (PAHs) (Léger & Puget 1984; Allamandola et al. 1989) and various carbonaceous grains such as coals (Papoular et al. 1989), hydrogenated amorphous carbons (Borghesi et al. 1987) and quenched carbonaceous composites (Sakata et al. 1987). The AIBs were observed in different environments with UV flux ranging from 1 to 10⁵ times the average value in the solar neighbourhood (Boulanger 1999). The overall spectral shape was found to be constant and the band intensities were seen to scale with the intensity of the UV radiation field. This clearly rules out emission from grains at thermal equilibrium but is fully consistent with a transient heating following the absorption of a single UV photon. This excitation mechanism, first invoked by Andriesse (1978) and Sellgren (1984), requires the carriers to be very small. Recently, Cook et al. (1998) have shown, in the laboratory, that UV-excited PAHs emit in the 3-15 $\mu$m range and that the observed band widths are comparable to those of the AIBs. The IR emission spectrum of a PAH size distribution exposed to the radiation field of stars has been calculated by several authors. In particular, Schutte et al. (1993) have studied the influence on the emergent spectrum of the PAH photophysical properties such as the oscillator strengths of the IR active modes. They derived a generic spectrum for interstellar PAHs (standard model). This spectrum consists of bands at frequencies corresponding to the AIBs and oscillator strengths that have been adjusted to fit the observed spectra. These oscillator strengths differ from those of neutral PAHs but seem to be consistent with interstellar PAHs being ionized. Using a different approach, Cook & Saykally (1998) calculated the spectrum from a collection of PAHs whose IR properties were measured in the laboratory. The simulated spectrum was obtained by using the spectral characteristics (frequencies and oscillator strengths) of each species in the mixture. A single emission temperature was considered to simplify the calculations. Mean band widths derived by Cook et al. (1998) from measurements on UV-excited perylene C₂₀H₁₂ and coronene C₂₄H₁₂ were used. The authors came to the same conclusion that PAH cations are better candidates than neutrals to account for the AIBs. However, they showed that the small sizes that were considered in the calculations (larger PAH in the mixture: C₂₄H₁₂) have larger band widths than the AIBs. The generic spectrum of Schutte et al. (1993) is therefore likely to be dominated by larger PAHs, although the spectral characteristics of such species are presently not known.

In this paper, we combine the approach presented by Schutte et al. (1993) and that of Cook & Saykally (1998) to calculate the emission spectrum of a PAH population containing sizes of up to two hundred carbon atoms and subjected to the radiation field of a hot star. Photophysical properties measured on small PAHs were extrapolated to larger species (cf. Sect. 2.3). They include the IR band oscillator strengths, the band frequencies and widths and their evolution with temperature (cf. Sect. 2.3.3 (b)). Emission spectra were calculated by adding up, for a given initial internal energy (e.g. energy of the absorbed UV photon), the various contributions during the cooling of a given isolated PAH. Integration is then performed over the distribution of PAH sizes and the distribution of absorbed stellar photons. In this paper, we present the particular case of the planetary nebula IRAS 21282+5050 for which a nice mid-IR spectrum was measured with the ISO Short Wavelength Spectrometer (observations by M. Jourdain de Muizon, L. d'Hendecourt, A. Heras and collaborators; cf. Joblin et al. 2000). This object was also selected because of the detection by Geballe et al. (1994) of the 1.68 $\mu$m band, the overtone of the 3.3 $\mu$m band. As discussed in Sect. 3.2, the intensity of this band can be used to constrain the minimum size of PAHs. Our calculations based on the thermal model (Léger et al. 1989) are similar to those performed by Schutte et al. but they also consider the detailed profiles of the emitted bands which further constrain the model. Barker et al. (1987) first suggested that the asymmetry of the AIBs and in particular that of the 6.2 and 11.3 $\mu$m bands is related to the anharmonicity of the modes. Anharmonic effects have been carefully described and included in the calculations. The calculated profiles are presented in Sect. 3 and compared to the observations.

The molecular nature of the AIB spectrum is then discussed. Implications for the nature of the carriers of the bands are then given (Sect. 3.3).