Issue 
A&A
Volume 509, January 2010



Article Number  A12  
Number of page(s)  13  
Section  Interstellar and circumstellar matter  
DOI  https://doi.org/10.1051/00046361/200912708  
Published online  12 January 2010 
The longwavelength emission of interstellar PAHs: characterizing the spinning dust contribution
N. Ysard  L. Verstraete
Institut d'Astrophysique Spatiale, UMR 8617, Université ParisSud, 91405 Orsay, France
Received 17 June 2009 / Accepted 24 September 2009
Abstract
Context. The emission of cold dust grains at long
wavelengths will soon be observed by the Planck and Herschel satellites
and will provide new constraints on the nature of interstellar dust. In
particular, the microwave galactic anomalous foreground detected
between 10 to 90 GHz, proposed as coming from small spinning
grains (PAHs), should help to define these species better. Moreover,
understanding the fluctuations of the anomalous foreground
quantitatively over the sky is crucial for CMB studies.
Aims. We focus on the longwavelength emission of
interstellar PAHs in their vibrational and rotational transitions. We
present here the first model that coherently describes the
PAH emission from the nearIR to microwave range.
Methods. We take quantum effects into account to
describe the rotation of PAHs and compare our results to current models
of spinning dust to assess the validity of the classical treatment
used. Between absorptions of stellar photons, we followed the
rovibrational radiative cascade of PAHs. We used the exactstatistical
method of Draine & Li to derive the distribution of
PAH internal energy and followed a quantum approach for the
rotational excitation induced by vibrational (IR) transitions. We also
examined the influence of the vibrational relaxation scheme and of the
lowenergy crosssection on the PAH emission. We study the emissivity
of spinning PAHs in a variety of physical conditions (radiation field
intensity and gas density), search for specific signatures in this
emission that can be looked for observationally, and discuss how the
anomalous foreground may constrain the PAH size distribution.
Results. Simultaneously predicting the vibrational
and rotational emission of PAHs, our model can explain the observed
emission of the Perseus molecular cloud from the IR to the microwave
range with plausible PAH properties. We show that for
mm
the PAH vibrational emission no longer scales with the radiation field
intensity (G_{0}), unlike the
midIR part of the spectrum (which scales with G_{0}).
This emission represents less than 10% of the total dust
emission at 100 GHz. Similarly, we find the broadband
emissivity of spinning PAHs per carbon atom to be rather constant for
and for proton densities cm^{3}.
In the diffuse ISM, photon exchange and gasgrain interactions play
comparable roles in exciting the rotation of PAHs, and the emissivity
of spinning PAHs is dominated by the contribution of small species
(bearing less than 100 C atoms). We show that the
classical description of rotation used in previous works is a good
approximation and that unknowns in the vibrational relaxation scheme
and lowenergy crosssection affect the PAH rotational emissivity
around 30 GHz by less than 15%.
Conclusions. The contrasted behaviour of the PAH
vibrational and rotational emissivities with G_{0}
provides a clear prediction that can be tested against observations of
anomalous and dust midIR emissions: this is the subject of a companion
paper. Comparison of these emissions complemented with radio
observations (21 cm or continuum) will provide constraints on
the fraction of small species and the electric dipole moment of
interstellar PAHs.
Key words: ISM: general  dust, extinction  radiation mechanisms: general
1 Introduction
The midIR spectrum of the interstellar medium (ISM) shows prominent bands from 3 to 17 m, which account for one third of the energy emitted by interstellar dust. Such bands are emitted by very small (subnanometricsized) dust particles during internal energy fluctuations triggered by the absorption of a stellar photon (Sellgren 1984). The positions of these bands suggest there are aromatic, hydrogenated cycles in these grains. Léger & Puget (1984) and Allamandola et al. (1985) proposed polycyclic aromatic hydrocarbons (PAHs) as the carriers of these bands. Despite two decades of experimental and theoretical efforts, the match (band positions and intensities) between data only available on small species and observations still remains elusive, as illustrated recently by the work of Peeters et al. (2002) and Kim & Saykally (2002). Given their important role in the ISM (e.g., gas heating, Habart et al. 2001; UV extinction, Joblin et al. 1992), it is necessary to find other ways to constrain the properties of interstellar PAHs.
The Planck and Herschel data will soon trace the emission of cold interstellar grains in the interstellar medium. Due to their small size, PAHs are heated sporadically (every few months in the diffuse ISM) by absorption of stellar photons and have a high probability of being in lowenergy states (Draine & Li 2001). Interstellar PAHs may thus contribute significantly to the emission at long wavelength ( mm). In this context, an unexpected emission excess called anomalous foreground, correlated to dust emission, has been discovered between 10 and 90 GHz (de OliveiraCosta et al. 2002; Leitch et al. 1997). In this spectral range, several galactic emission components (synchrotron, freefree, and thermal dust) contribute with a comparable magnitude, and only recently has the anomalous foreground been separated in WMAP data (MivilleDeschênes et al. 2008). Spinning, small dust grains were first proposed by Draine & Lazarian (1998) (hereafter DL98) as a possible origin to this anomalous component. Since then, analysis of observations has suggested that the anomalous foreground is correlated to small grain emission (Casassus et al. 2006; Lagache 2003).
In this paper, we study the emission of PAHs with particular emphasis on the longwavelength part of the spectrum. In this spectral range, the emission is dominated by species in lowenergy states for which each photon exchange represents a large energy fluctuation. To derive the internal energy distribution of PAHs, we use here the exact statistical method described in Draine & Li (2001). We include lowfrequency bands to the vibrational mode spectrum of interstellar PAHs and examine the influence of the internal vibrational redistribution hypothesis (see Sect. 3) often used to describe their vibrational relaxation. We describe the rotation of PAHs with a quantum approach where specific processes in the rovibrational relaxation are naturally included (molecular recoil after photon emission; rovibrational transitions that leave the angular momentum of the molecule unchanged or Q bands). We present in a variety of interstellar phases the rovibrational (IR) and broadband rotational (spinning) emission of PAHs in a consistent fashion. Our model results are compared to observations of the interstellar emission from the IR to the microwave range and show how the size distribution and electric dipole moments of interstellar PAHs may be constrained.
The paper is organized as follows. Section 2 describes the properties of PAHs adopted in this work. Section 3 discusses the internal energy distribution of PAHs and associated rovibrational emission. Sections 4 and 5 discuss the rotational excitation and emission of PAHs. In Sect. 6, we apply our model to the case of a molecular cloud in the Perseus arm. Finally, conclusions and observational perspectives are given in Sect. 7.
2 The properties of interstellar PAHs
In the ISM, the stablest PAHs are found to be compact species (Le Page et al. 2003; Léger et al. 1989). Small PAHs are planar, but above some poorly known size threshold (40 to 100 C atoms), interstellar formation routes may favour threedimensional species (bowl or cageshaped) containing pentagonal cycles (see Moutou et al. 2000, and references therein). As we see later, small PAHs (containing less than 100 C atoms) dominate the rotational emission, so we assume that interstellar PAHs are planar with a hexagonal ( ) symmetry. The PAH radius is thus , where is the number of carbon atoms in the grain (Omont 1986). The formula for such molecules is C , and their hydrogentocarbon ratio is , where is the hydrogenation fraction of PAHs. In this work, we assume that PAHs are fully hydrogenated ( ).
2.1 Absorption crosssection
The excitation, cooling, and emission of PAHs depend on their absorption crosssection (Sect. 3), which we describe now. We took the visibleUV crosssection from Verstraete & Léger (1992) and applied their sizedependent cutoff for electronic transitions in the visibleNIR range. The resulting crosssection compares well to the available data (Joblin et al. 1992). In Appendix A, we discuss the midIR bands considered in this work. Each vibrational mode is assumed to be harmonic, and the corresponding band profile has a Drude shape (Draine & Li 2001). The width is inferred from astronomical spectra, and the peak value is chosen so that the integrated crosssection is equal to the value measured in the laboratory. By adopting the observed bandwidth, we empirically account for the complex molecular relaxation and band broadening (Mulas et al. 2006; Pech et al. 2002) in interstellar PAHs. We adopt here the band strengths given in Pech et al. (2002) that were inferred from laboratory data. Different definitions of the PAH IR crosssections have been proposed by Rapacioli et al. (2005), Flagey et al. (2006), and Draine & Li (2007).
We used the database of Malloci et al. (2007) to define an average broadband crosssection of the farIR vibrations of PAHs. At frequencies below 500 cm^{1}, each species features many modes. However, for compact species, modes accumulate in three definite frequency ranges: modes with a frequency of less than 100 cm^{1}, modes between 100 and 200 cm^{1}, and modes between 200 and 500 cm^{1}. We therefore model the farIR crosssection of compact PAHs with 3 modes (Table 1). The frequency of each mode is the average of all modes falling within the given energy range, weighted by their corresponding integrated crosssections. We find that the frequency of the lowest energy mode depends on the molecular size as (see Fig. 1). Conversely, for the two other modes, the average energy is instead independent of the size. The integrated crosssection for these 3 modes was estimated as follows. From the Mallocci database, we first derived the fraction of for each of the 3 modes. Then, we assumed that the total below 500 cm^{1} is given by the integral of the absorption crosssection of Schutte et al. (1993): cm^{2} per Catom. The integrated crosssections of each of the 3 modes was finally obtained by multiplying the former value by the fractions inferred from the database.
The parameters of the farIR bands adopted here are given in Table 1, and the full crosssection is displayed in Fig. 2. We show in Sect. 3 that this set of IR bands provides a good match to observed interstellar spectra. Finally, we note that the inplane or outofplane character of each vibrational band is important because of the different associated weights in the rotational excitation (Sect. 4.1). This character is indicated in Tables A.1 for the midIR bands (Socrates 2001). In the case of the farIR bands, this character is not as well known and we assume that 1/3 (2/3) of the oscillator strength come from outofplane (inplane) transitions respectively (Table 1).
Table 1: FarIR rovibrational bands of PAH cations adopted in this work, with the percentage of the total oscillator strength in each band.
Figure 1: Frequencies of the first (lowest energy) vibrational mode versus for PAH cations (black diamonds) and PAH neutrals (grey circles) from the Mallocci database^{} (see text). The solid line is the relationship we adopt between the band position of cations and , and the dashed line shows the case of neutrals. The dotted lines show extreme cases for this relationship. 

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Figure 2: Absorption crosssection of PAH cations per carbon atom (1 Mb = 10^{18} cm^{2}). The solid, dotted, dashed, and dotdashed lines show the cases for = 24, 54, 96, and 216, respectively. 

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2.2 The rigid rotor model
While describing the rotation of a molecule, the relevant operator is
the total angular momentum ,
which includes the electrons and nuclei contributions without the spin.
We note that ,
,
and
are the principal axis of inertia. We assumed that PAHs are oblate
symmetric top molecules with the axis
perpendicular to the plane of the molecule and parallel to Oz.
We called
the inertia moment with respect to
and ,
the inertia moments with respect to
and ,
which were taken to be parallel to Ox and Oy.
The rotational Hamiltonian is then
(1) 
where J_{x}, J_{y}, and J_{z} are the projections of along the three inertia axis. Given the large number of carbon atoms in interstellar PAHs, we assume here that they are uniform disks with (symmetric tops), where M is the molecular mass. With , we get
(2) 
and the rotational energy is
where cm^{1} (neglecting the contributions of H to the molecular mass) and C = B/2 are the rotational constants associated to and . The quantum number K is the absolute value of the J_{z}eigenvalues. For a symmetric top molecule, the selection rules for rovibrational electric dipole transitions are and . By defining the dipole moment as , with along and in the molecular plane along (Townes & Schawlow 1975), two kinds of transitions can be distinguished. First, the parallel transitions with for which the change of dipole moment in the transition is parallel to the top axis () of the molecule. Second, the perpendicular transitions with , for which the change of dipole moment is perpendicular to . Parallel transitions thus correspond to outofplane vibrations, whereas perpendicular transitions correspond to inplane vibrations.
Since the available microwave data are broadband observations ( of the order of a few), we make several simplifying assumptions in the description of the rotational motion of PAHs. We thus assume that the rotational constant B is the same in all vibrational levels, and within the framework of a rigid rotor model, we neglect the centrifugal distortion terms in the energy equation that are usually small for large molecules (Lovas et al. 2005; Herzberg 1968ab).
Figure 3: Symmetric top molecule: , , and are the principal axis of inertia and J the total angular momentum of the molecule with K its projection along . 

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2.3 Electric dipole moment
The rotational emission of PAHs depends on their permanent electric
dipole moment, .
Symmetric (
), neutral, and fully
hydrogenated PAHs have .
Spectroscopic analysis of their IR emission bands suggests
that interstellar PAHs can hqve a cationic form that is partially
hydrogenated (Le Page et al.
2003), and maybe also substituted (Peeters et al. 2002,2004a).
For instance, a PAH having lost one H atom has
to 1 D, and a PAH cation where a C atom has been substituted
by N would also have
to 1.5 D depending on its size (Pino, private communication).
Moreover, it has been proposed that nonplanar PAHs containing
pentagonal rings may exist in the ISM (see Moutou et al. 2000,
and references therein): such species are known to have large dipole
moments as recently measured on coranulene, C
,
D (Lovas et al. 2005). In this
work, we express the electric dipole moment of interstellar PAHs as
in DL98:
where is the total atoms number in the molecule, Z is its charge, and m a constant. Unless otherwise stated, we will use m=0.4 D.
3 Internal energy distribution and rovibrational IR emission of isolated interstellar PAHs
Figure 4: Internal energy distribution of PAHs with (solid line and box for P(0)) and (dashed line and triangle) in the case of a) an MC ( G_{0}=10^{2}), b) the diffuse ISM (G_{0} = 1) and c) the Orion Bar (the radiation field is the sum of the CMB, the ISRF and a blackbody at 37 000 K corresponding to G_{0} = 14 000). To illustrate the effect of the radiation field hardness, we show in b) the case of an Orion Bar type radiation field scaled down to G_{0}=1 (grey lines). 

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After the absorption of a visibleUV photon, a PAH cools off by emitting IR rovibrational photons. These photons reduce its angular momentum and also may increase it by recoil (a purely quantum effect, see Sect. 4.1). Previous studies have mostly used a thermal description of molecular cooling. In fact, since PAHs spend a large fraction of their time at low internal energies (see Fig. 4), their emission at long wavelengths cannot be considered as a negligible fluctuation compared to their total internal energy. While estimating the emission of PAHs at low temperatures, the validity of a thermal approach is questionable. Because of the rapid energy redistribution between interactions (photon exchange or collision with gas phase species), PAHs rapidly reach thermodynamical equilibrium while isolated (Léger et al. 1989). To describe this situation, we used the exactstatistical method of Draine & Li (2001) to derive the stationary internal energy distribution of PAHs.
The energy distribution, P(E), depends on (a) the energy density of the exciting radiation field (where is the brightness); (b) the absorption crosssection, , of interstellar PAHs; and (c) their rovibrational density of states, . For the stellar contribution, we used the local interstellar radiation field (ISRF, Mathis et al. 1983) or a blackbody. We sometimes scale this stellar radiation field with a factor, G_{0}, equal to 1 in the case of the Mathis field^{}. Throughout this work, the contribution of the CMB is included in the radiation field. The vibrational absorption crosssection used has been described previously, and for simplicity, we did not include rotational bands. Unless otherwise stated, we took the crosssection of PAH cations. The density of states was obtained by first deriving the vibrational mode spectrum from a Debye model and then applying the algorithm of Beyer & Swinehart (1973) for each molecular size (see Appendix B).
Figure 5: Left: dust IR emission of the DIM (solid black lines and symbols) with cm^{2} (Boulanger 2000). The model dust emission is overlaid in bold grey. MRN type size distributions are used for all dust populations. The PAH vibrational contribution (dashed line) is from the present model with , a 1:3 mixture of neutrals and cations, full hydrogen coverage, and 43 ppm of carbon. The contributions of larger grains are from the model described in Compiègne et al. (2008). The dotted line is the contribution of graphitic very small grains of radii a=0.9 to 4 nm containing 39 ppm of C. The dotdashed line is the contribution of silicate big grains with a=0.4 to 250 nm using 37 ppm of Si. Right: dust emission from the Orion Bar (the noisy black line is the ISOSWS spectrum) with cm^{2}. The model dust parameters are as above but for PAHs less abundant (18 ppm of C) and more ionized (82% are cations). 

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After each absorption of a stellar photon, we assume that the
excitation energy of the PAH is rapidly redistributed among all the
vibrational modes (Mulas
et al. 2006). This radiationless and isoenergetic
process is called internal vibrational redistribution (IVR).
The energy distribution is then computed according to the
exactstatistical formalism described in Draine
& Li (2001). The energy scale (referenced to the zero
point energy) is divided into bins of energy E_{i}
and width
with i
= 0, 1, ..., M (M = 500) with
where cm^{1}
is the Lyman limit. When its energy is below that of the first excited
vibrational state, the molecule is in a rotational state. Letting P_{i}
be the probability of having a PAH in the energy bin i
we have
(5) 
where is the transition rate from the state j to the state i for a molecule. We solve this equation in the stationary case. More details can be found in Draine & Li (2001). Figure 4 shows the energy distribution for three different radiation fields, representative of a molecular cloud, the diffuse interstellar medium (DIM), and the Orion Bar. We see that the most probable energy increases when the radiation field intensity G_{0} increases, as expected for PAHs whose cooling by IR emission is interrupted more frequently by absorption events. The sharp cutoff at cm^{1} stems from the Lyman limit of photon energies in neutral regions. We also observe a tail at higher energies that comes from to multiphoton events, i.e., absorption of a second photon while the PAH has not completely cooled off. This tail becomes more significant as absorption events are more frequent, i.e., for large PAHs or intense radiation fields. Conversely, we show in Fig. 4b that the radiation field hardness has little influence on P(E). To estimate the effect of the rotational absorption that we neglected, we included a band centred at 1 cm^{1} of width 1 cm^{1} and corresponding to J=150 ( cm^{2}/C, see Sects. 4.2 and 4.3): we found P(E) to be twice as large between 1 and 10 cm^{1} and unchanged otherwise. This would change the emission around 1 cm at very low flux levels. The rotational excitation rates (Sect. 4.2), which depend on the populations of excited vibrational levels, are unaffected by this hypothesis.
Knowing the internal energy distribution of PAHs, we can
deduce their IR emission from the upper state u
by summing the contributions of all lower vibrational modes l
(Draine & Li 2001):
(6) 
(7) 
is the number of rovibrational photons emitted at energy . The G_{ul} functions are defined in Draine & Li (2001). The degeneracies g_{u} and g_{l} are the numbers of energy states in bins u and l, respectively (see Appendix B). Figure 5 shows a comparison between midIR observations and our model results with a power law size distribution (as in Mathis et al. (1977) hereafter MRN) that provides a simple representation of the actual n_{s}(a) (Kim et al. 1994). We see that the IR emission from the DIM is explained well with a standard abundance of carbon in PAHs and as large fraction of cations as in Flagey et al. (2006). A similarly good match is obtained for the Orion Bar spectrum with strong PAH depletion probably reflecting an efficient photodestruction in this excited environment. Figure 6a shows the behaviour of the PAH vibrational emission with the radiation field intensity G_{0}: whereas the IR part scales with G_{0}, the mmrange ( mm) does not. The emission in this spectral range is produced mostly by PAHs in lowenergy states for which the dominant heating source is absorption of CMB photons, and this is why this part of the spectrum does not vary with G_{0}. Finally, we emphasize that at low energies ( m), our model shows the broadband behaviour of the PAH vibrational emission. When a detailed vibrational mode spectrum is used, the PAH emission at m is a superposition of numerous narrow bands that may be detectable with some of the instruments onboard Herschel (Mulas et al. 2006).
Figure 6: PAH rovibrational emissivity for a MRN size distribution with . In a) we show the case of radiation field intensities corresponding to G_{0}=1 and 100 and in b) we illustrate the effect of IVR breakdown in case 1 (dotted line) and case 2 (dotdashed line) (see Sect. 3.1). 

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3.1 Decoupling of vibrational modes
Our derivation of P(E) assumes an efficient energy redistribution between vibrational modes (IVR) during the PAH relaxation. The IVR thus involves a coupling between vibrational modes via intramolecular transitions. However, it is known that IVR is no longer efficient when the excitation energy of the molecule is below some threshold (Mulas et al. 2006), and we will then speak of IVR breakdown. When , the excitation of energetically accessible vibrational modes is frozen according to microcanonical statistics at energy . The molecule then continues to cool via emission of the excited vibrational modes. To study the influence of decoupling on the longwavelength emission, we consider two extreme cases for the relaxation at :
 
 case 1: no IVR, cooling by IR forbidden modes. We assume these modes to be defined like those of Table 1 but with an oscillator strength 10^{4} lower and;
 
 case 2: no IVR, cooling only by the first vibrational mode at (cm^{1}) as defined in Table 1,
4 Angular momentum distribution
Building the angular momentum distribution of interstellar PAHs is driven by photon exchanges and gasgrain interactions (DL98). In the DIM, pervaded by the ISRF, and for a PAH bearing 50 carbon atoms, the mean time between absorptions of visibleUV photons (0.2 yr) is comparable to the mean time between emissions of rotational photons, as well as to the mean time between PAHhydrogen collisions (for a gas density of 100 cm^{3} and a temperature of 100 K). Photons absorbed in the visibleUV have a weak effect on the total angular momentum; indeed, each photon exchanged carries a unit angular momentum, and the numerous^{} IR photons emitted overwhelm the influence of the photon absorbed. To estimate the angular momentum distribution, we take the following processes into account:
 
 IR rovibrational photon emission;
 
 purely rotational photon emission;
 
 H_{2} formation;
 
 collisions with gas (neutral atoms and ions);
 
 plasma drag;
 
 photoelectric effect.
with the change of J produced by the event number i, and the mean time between two events i. As in Rouan et al. (1992), we assume an efficient intramolecular vibration to rotation energy transfer (IVRET). We take n(J) to be the same for all vibrational levels, equal to a Maxwell distribution , where n_{0} is a normalization factor. Indeed, Mulas (1998) and AliHaïmoud et al. (2009) show that this form of n(J) is a good approximation^{}. In the following, we establish the rate of Jchange due to the rovibrational emission of isolated PAHs modelled in Sect. 3. We also present the rate of purely rotational photon emission, while the contribution of the gasgrain interactions is described in Appendix D.
4.1 Rovibrational transitions
The rovibrational IR emission can be both an exciting and a damping
process for the rotation of a PAH. Assuming that the interstellar PAHs
are symmetric top molecules, the selection rules for the emission of an
IR rovibrational photon are:
,
and .
Transitions corresponding to
,
and 0 are called the P, Q,
and Rbands, respectively. As seen in
Sect. 2.2,
two types of transitions are possible: parallel ones (outofplane
vibrational motion) with
and perpendicular ones (inplane vibrational motion) with
.
The type of each transition is given in Tables A.1 and 1. For a
rovibrational transition,
,
the rate is proportional to the spontaneous emission coefficient and to
the probability for the grain to lose the corresponding transition
energy. The spontaneous emission rate is proportional to the A_{KJ}
factors, which represent the angular part of the transition
probability. Formulae for these factors are given in Appendix C. Expressed in terms
of crosssection, the transition rate is (with
and
in cm^{1}):
where the give the transition rate for , and for the vibrational mode i at a frequency . This frequency depends on (the frequency of the vibrational transition for J=0) and on J and K (see Appendix C). Since the J and Kterms, in , are always much smaller than , we take . We note that the inplane transitions provide higher (by a factor 2) rates than the outofplane ones, as a consequence of the A_{KJ} sum values. The total transition rates are obtained by summing over all bands: . Finally, the rate of change in J due to rovibrational transitions is . Rotational excitation dominates at low J, whereas damping is the dominant process for high Jvalues. For vibrational modes at higher frequency, W^{+}W^{} is smaller because the rotational terms are smaller in (see Appendix C): is therefore dominated by the contribution of vibrational modes at lowfrequencies.
4.2 Rotational emission
We consider here the spontaneous emission of purely rotational photons.
The selection rules for such transitions are
and
(Townes & Schawlow 1975).
No change in K occurs because, for
rotational transitions, the dipole moment of a symmetric top molecule
necessarily lies along its symmetry axis. The transition rate is then
simply related to the spontaneous emission coefficient
A_{J,
J1}:
where is the electric dipole moment of the molecule and the factor A_{KJ} has been used (Appendix C). In the highJ limit, tends to the classical expression of Larmor. Finally, the damping rate due to rotational emission is^{}
4.3 Equilibrium angular momentum J_{0}
Table 2: Physical parameters for the typical interstellar phases considered.
We estimate damping and excitation rotational rates for the interstellar phases described in Table 2. When solving Eq. (8), we obtain the equilibrium angular momentum J_{0}. Figure 7 shows J_{0} for the DIM and the Orion Bar. In all cases, J_{0} increases with because there are more steps in the random walk; indeed, the crosssection and absorption rate scale with (see Fig. 2), so more IR photons are emitted for larger . Similarly, J_{0} reaches higher values when the radiation field intensity or the gas density increases (see Fig. 8).
In the DIM, the excitation comes from gasgrain collisions, whereas the damping is dominated by the emission of rotational photons for and by IR photon emission for larger PAHs ( , whereas ). We note similar J_{0}values for the CNM, WNM and WIM (Fig. 7a) in spite of the large differences in gas density and temperature (see Table 2). In the Orion Bar, because of the intense radiation field, the excitation is driven by the photoelectric effect, whereas the damping is dominated by the emission of IR photons. For the case of MCs (not shown), J_{0} also rises and reaches high values (J_{0}=1300 for ). Indeed, the damping by IR photon emission is no longer efficient ( ), while the gasgrain rates are strong because of higher density. We show in Fig. 7b the influence of different choices of the vibrational mode spectrum (or crosssection) on J_{0}: changes are below 20% and affect only the large sizes ( ), which make a minor contribution to the rotational emissivity (see Sect. 5). We found comparable variations of J_{0} for the vibrational relaxation (IVR) cases 1 and 2, discussed in Sect. 3.1.
Figure 7: J_{0} versus for PAH cations. In a) we show the case of the diffuse medium. Solid line shows the CNM, dashed line the WNM and dotted line the WIM. In b) we illustrate the influence of the absorption crosssection in the CNM: cations (solid line), neutrals (dashed line) and extreme behaviours (grey lines) for the first vibrational mode versus (see Fig. 1) are shown. In c), we present the case of the Orion Bar. 

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Figure 8: a) Effect of varying G_{0} on J_{0} for PAH cations with , 30, 40, and 100 ( from bottom to top). The values for (triangles) have been divided by 2. The gas density is 30 cm^{3} and the other physical parameters for the gas are determined using CLOUDY (Ferland et al. 1998). b) Same as a) but for varying at G_{0} = 1. In both cases the radiation field is a blackbody with K and the gas parameters have been obtained with CLOUDY at thermal equilibrium (see text). 

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The processes contributing to the rotation of PAHs depend on the intensity of the radiation field (G_{0}) and on the gas density (). We now examine the influence of these parameters on J_{0}. Other important quantities (gas temperature; electron, proton, and C^{+} abundances) are obtained at thermal equilibrium with CLOUDY (Ferland et al. 1998)^{}. To study the influence of G_{0}, we took the radiation field to be a blackbody of effective temperature 22 000 K and scaled it in order to vary G_{0} between 10^{2} and 10^{5}. The shape of the radiation field is then always the same^{}. Figure 8a shows J_{0} as a function of G_{0} for different PAH sizes. As discussed before, J_{0} increases with G_{0} because the excitation rate from IR emission (which scales with G_{0}) is dominant. This rise becomes steeper for larger PAHs because the frequency of the first vibrational mode decreases (as ) and requires lower values of G_{0} to be excited (the maximum of P(E) moves to higher energies as G_{0} increases, see Fig. 4c). However, we note that J_{0} is affected little by variations of G_{0} over the range 0.01100. The result of varying is shown in Fig. 8b. In all cases, the incident radiation field is a blackbody with 000 K and G_{0} =1. The gasgrain processes become dominant for cm^{3} and . At lower gas densities, radiative processes prevail (excitation by IR emission and damping by rotational emission), and J_{0} does not change. We therefore expect the rotational excitation of small PAHs to hardly be variable in the DIM as found by Davies et al. (2006).
4.4 Modelling the rotational excitation by IR emission
Following the absorption of a stellar visibleUV photon, the rovibrational cascade of PAHs is a complex process that involves many states. Molecular, statetostate models require a detailed database, use MonteCarlo simulations (see Mulas et al. 2006, for references), and so far do not include the gasgrain interactions that are important. Given our incomplete knowledge of interstellar PAHs and the fact that their rotational emission may so far be seen in broadband data, former models of spinning dust (DL98 and AliHaïmoud et al. 2009) made simplifying assumptions to describe radiative processes and performed a classical treatment of the gasgrain interactions^{}. We assess here the impact of these assumptions on the rotational emissivity of PAHs.
First, the internal energy distribution of PAHs was derived in the thermal approximation. As discussed in Sect. 4, this is a questionable assumption for describing the longwavelength emission of PAHs and the change in angular momentum it induces. Moreover, the excitation rate by IR emission (the recoil due to emission of individual photons), which is a purely quantum effect, has been described as a random walk of the angular momentum starting with a nonrotating grain^{}. Finally, an efficient vibrational redistribution (IVR, see Sect. 3.1) was assumed throughout the energy cascade following a photon absorption. In this work, we improve on the first aspect by deriving the internal energy distribution of isolated PAHs using a microcanonical formalism and including the rotational density of states. Next, we follow a quantum approach to treat the rovibrational emission where the recoil due to photon emission and selection rules are naturally included. In Fig. 9a, we compare the absolute value of our rate of angular momentum change by rovibrational IR emission (see Sect. 4.1) to former works. For a PAH containing 24 carbon atoms, all rates decrease with the angular momentum and cross zero for J between 100 and 150 (singular points in our logarithmic representation). The main difference with previous models is that the excitationtodamping transition (the zero value) occurs at higher Jvalues in our case. This discrepancy diminishes with increasing sizes, and identical IR rates are found for species with more than 100 C atoms.
Figure 9: Rate of Jchange due to rovibrational emission for a PAH with and heated by the ISRF. a) Comparison with former models: our rate (solid line), the rate of DL98 corrected as in AliHaïmoud et al. (2009) (dashed line) and the rate of AliHaïmoud et al. (2009) (dotted line). b) Effect of IVR breakdown, case 1 and 2 (see Sect. 3.1). 

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Figure 10: Rotational emissivity of PAH cations. We take 430 ppm of carbon in PAHs, and assume m = 0.4 Debye. Panel a): effect of changing the fraction of large PAHs (decreasing with to 6). The case of a lognormal distribution centered around and width 0.4 is also shown. Panel b): rotational emission spectrum in different environments with a power law size distribution and . 

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We show in Fig. 9b the effect of relaxation schemes other than IVR. Between extreme cases 1 and 2 (see Sect. 3.1), the damping IR rate is multiplied by a factor 2, which leads to a balance at a lower Jvalue. Variations in are in fact quite similar to those induced by the other approximations discussed above. All these variations affect the rotational emissivity of PAHs, around 30 GHz, by at most 15% (in the CNM and including the gasgrain interactions)^{}. We conclude that, in spite of the assumptions made, former spinning dust models (which are fast computationally) provide a sufficiently accurate rotational emissivity.
5 Spinning dust emission
The power emitted by a PAH containing
carbon atoms in a rotational transition from state J
to J1 is equal to
(13) 
when taking the spontaneous emission rate A_{J,J1} from Eq. (10) and with the transition energy ( for rotational transitions). With and , we find that : small PAHs will therefore have a dominant contribution to the rotational emission. For instance, the peak value of the rotational emissivity of a PAH with is 10 times lower than for a species with (see also AliHaïmoud et al. 2009). With n_{s}, the size distribution of PAHs (the number of PAHs of a given size a or per proton), and the angular momentum distribution , we get
Since we are interested in a broadband spectrum, we take the rotational bandwidth to be 2Bc. As discussed before, we assume that is described correctly by a Maxwell distribution. Given the relationship (Fig. 7) and the size distribution, the emission of spinning PAHs can be calculated from Eq. (14). It scales with . We illustrate the relative significance of small species in Fig. 10a by changing the index of the power law size distribution, . Defining small PAHs by , they represent 50% (97%) of the total abundance for (6). As expected, the rotational emissivity for is highest. It is also blueshifted by 5 GHz and broadened by about the same amount with respect to the case. Figure 10b shows the case of different interstellar environments and illustrate the influence of the gasgrain processes, which become strong in the CNM (while subdominant in the WIM/WNM).
We emphasize here that the level, peak position and width of the rotational emission spectrum depend on (a) the fraction of small PAHs ( ), (b) m the scaling factor for the dipole moment, and (c) the physical parameters of the gas along the line of sight (temperature, density, and ionization). Microwave observations alone cannot constrain all these unknowns. A quantitative description of the spinning dust emission will require nearIR data (to constrain the fraction of small PAHs) and radio maps (21 cm and continuum) to derive the physical state of the gas.
6 The case of the molecular cloud G159.618.5
We present here a comparison of our model results with observations of the anomalous microwave foreground and IR emission. We show how these data can be explained with a coherent description of the IR and rotational emission of PAHs, and also discuss how the size distribution of PAHs can be constrained.
The COSMOSOMAS experiment has delivered maps of large sky fractions with an angular resolution of 1 degree in four frequency bands: 10.9, 12.7, 14.7, and 16.3 GHz. In this survey, Watson et al. (2005) studied the region G159.618.5 located in the Perseus molecular complex. Figure 11 shows the spectrum of this region from 3 to to 4000 GHz (COSMOSOMAS, WMAP, and DIRBE data). From mid to farIR, the Perseus molecular complex is dominated by a molecular ring (G159.618.5) surrounding an HII region. Unfortunately, the COSMOSOMAS resolution does not allow these two phases to be separated. The molecular ring is centred on the star HD 278942 (Andersson et al. 2000; Ridge et al. 2006). Its stellar wind is responsible for the HII region dug into the parent molecular cloud. To describe the emission spectrum of G15918.5, we assume it includes an MC component for the ring and a WIM component for the HII region. We discuss below the physical parameters of both phases.
According to Ridge et al. (2006), HD278942 is a B0 V star: its radiation field is modelled by a blackbody with K, scaled to get at the radius of the ring (G_{0} is obtained from IRAS data). The radius of the HII region is approximately 2.75 pc (Andersson et al. 2000) and implies a WIM density cm^{3} (Ridge et al. 2006). We obtained the WIM temperature from CLOUDY: 7500 K. Andersson et al. (2000) estimated the density and temperature of the gas in the molecular ring to be cm^{3} and K. For the WIM and MC phases, we derived the abundances of electrons, protons and C^{+}, and the gas temperature with CLOUDY. We took the total column density (MC+WIM) to be cm^{2} (Watson et al. 2005). From 12 m IRAS data and our PAH emission model, we estimated the total PAH abundance to represent ppm of carbon.
Following Watson et al. (2005), we took the thermal dust emission to be a grey body at 19 K with an emissivity index . Our bestfit parameters for the spinning PAH contribution (assuming a size distribution ) are then (model A) and cm^{2} for the MC, and cm^{2} for the WIM. In both media, we took m=0.6 D (Eq. (4)). Larger PAHs are required in the MC, possibly as a consequence of graingrain coagulation. However, the results of Dupac et al. (2003) indicate that emission of dust around 20 K requires a higher emissivity index, close to 2. When adopting and a dust temperature of 18 K (model B), we find an equally good fit of this region in the farIR (Fig. 11). In this case, the flux around 100 GHz is not explained. We speculate here that it may come from an additional and less abundant population of small PAHs in the MC component with: , m=0.1 D and cm^{2} (for the larger PAHs in the MC phase, cm^{2}). The corresponding spinning emission is shown in Fig. 11 (model B). The PAH size distribution would thus be bimodal, as already suggested by Le Page et al. (2003). As can be seen in Fig. 11, both models provide a good fit to the data. Discrimiting tests of these scenarios will be soon possible with the Planck polarized data. Small grains or PAHs are expected to be poorly aligned (Lazarian & Draine 2000; Martin 2007), whereas big grains are fairly wellaligned with a polarization fraction of 5 to 10%.
Figure 11: Comparison to observations (symbols) of the Perseus molecular cloud G159.618.5. The dashed lines show the freefree (low frequency) and big grains (highfrequency) emissions. The PAH rovibrational emission is represented by the dotdashed line. Dotted lines show the PAH rotational emission. The black model is model A, and the grey model is model B. See text for details. 

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7 Summary
The Planck and Herschel data will soon reveal the emission of interstellar dust at long wavelengths. Thanks to their small size, interstellar PAHs spend most of their time at low internal energy and can spin at frequencies of a few tens of GHz. The emission of PAHs is therefore expected to make a significant contribution at long wavelength. Recent observations have shown the existence of a 10100 GHz emission component (the anomalous foreground), related to the smallest dust grains. As suggested by DL98, the anomalous foreground may trace the emission of spinning PAHs.
In this work, we built the first model that coherently describes the emission of interstellar PAHs from the nearIR to the centimetric range and focused on the long wavelength part of this emission. To do so, we derived the internal energy of isolated PAHs down to low energies. We included lowfrequency vibrational bands ( m), through which PAHs cool at intermediate to low energies. They are important for their rotational emission. In the cooling cascade that follows the absorption of a stellar photon by a PAH, we treated rovibrational transitions in a quantum approach and examined the possibility that the hypothesis of vibrational redistribution (IVR) is not always fulfilled. Purely quantum effects (recoil due to photon emission; transitions that do not change the angular momentum or Qbands) are thus naturally included in our description of the rovibrational cascade. We obtained the rotational emission of PAHs from a randomwalk formalism, including all processes participating in excitation and damping, namely, rovibrational and pure rotational transitions and gasgrain interactions.
We have shown that the rovibrational emission of PAHs above 3 mm does not depend on the intensity of the radiation field (represented by G_{0}), unlike the midIR part of the spectrum that scales linearly with it. In the diffuse interstellar medium, PAHs may contribute up to 10% of the dust emission around 100 GHz. We also found the rotational emissivity of PAHs is dominated by small species (bearing less than 100 C atoms) and is hardly sensitive to G_{0} over the range 0.1100. Using plausible PAH properties, our model can explain both the IR and microwave emissions of a molecular cloud in the Perseus arm where the anomalous foreground is conspicuous. The level, peak position and width of the rotational emission spectrum depend on the fraction of small PAHs, the dipole moment distribution (mfactor), and the physical parameters of the gas phases present along the line of sight. A quantitative description of the emission of spinning PAHs will therefore involve observations at IR and radio wavelengths (21 cm and continuum). Comparing the rotational excitation rate obtained from our quantum treatment of the rovibrational cascade to former works, we showed that the classical approximation used so far has little effect on the rotational emissivity (the peak value varies by at most 15%). Similarly, departures from the IVR hypothesis lead to similar emissivity changes. We therefore conclude that a classical description of rovibrational transitions and the IVR hypothesis are good approximations for describing the rotational emission of PAHs.
Our results on the influence of the radiation field intensity led to a specific prediction that can be tested observationally. If the anomalous microwave foreground comes from spinning PAHs, it is expected to be correlated with the dust emission in the 12 mIRAS band, mostly carried by PAHs. In regions where G_{0} varies significantly, this correlation should improve when the 12 m flux is divided by G_{0}, indeed, the IR emission of PAHs scales with G_{0}, whereas their rotational emission is independent of G_{0}. This prediction was tested in a companion paper with WMAP and IRAS data (Ysard et al. 2010).
AcknowledgementsWe thank an anonymous referee whose comments helped us to significantly improve the content of this paper. We gratefully acknowledge stimulating discussions with B. Draine, C. Joblin, E. Dartois, T. Pino, and O. Pirali. We are grateful to M. Compiègne for his help in the dust SED modelling.
Appendix A: MidIR absorption crosssection
In Table A.1 we give the parameters defining the midIR vibrational bands considered in this work. In all cases we assume their profile to have a Drude shape. The bands at 3.3, 6.2, 7.7, 8.6, 11.3, and 12.7 m are defined as follows. For the PAH cations, we start from the integrated crosssections, , of Pech et al. (2002) that have been derived from laboratory data. The corresponding band profiles, however, do not provide a detailed match of observations. We therefore use band positions and widths as deduced from fits of ISOSWS spectra of a number of interstellar regions (Verstraete et al. 2001). As indicated by these observations and others (Peeters et al. 2002), we include a broad band at 6.9 m and split the 7.7 m into three subbands at 7.5, 7.6, and 7.8 m, where we use the observed of each subband as weights in defining their integrated crosssections. In addition, we introduce a band at 8.3 m to fill the gap between the 7.8 and 8.6 m bands, and multiply the 8.6 m band by a factor 3 to match observations. For neutral PAHs, we use the laboratory integrated crosssection of Joblin et al. (1995), and assume the same band profiles as for the cations. Furthermore, spectroscopic data (ISOSWS, Spitzer IRS and UKIRT) reveal other bands at 5.25 and 5.75 m, which have been ascribed to combinations of PAH vibrational modes involving the 11.3 m band and IRforbidden modes at 9.8 and 11.7 m, respectively (Tripathi et al. 2001; Roche et al. 1996). For the 5.25 m, we use the width and intensity ratio to the 11.3 m band given in Roche et al. (1996). The 5.75 m band has been derived from the observed spectrum of the Orion Bar (Verstraete et al. 2001). We also add the 17.1 m band recently seen in Spitzer data (Werner et al. 2004; Smith et al. 2004) and recognized as arising from PAHs (Peeters et al. 2004b; Smith et al. 2007).
Table A.1: MidIR bands of interstellar PAHs adopted in this work for cations and neutrals.
Appendix B: Vibrational modes and density of states of interstellar PAHs
B.1 Vibrational modes
Symmetric top ( symmetry) type PAHs, with carbon atoms and hydrogen atoms, have vibrational modes that can be divided into the following types: outofplane (op) CC modes, inplane (ip) CC modes, outofplane CH bending modes, inplane CH bending modes, and CH stretching (st) modes. Following Draine & Li (2001), we approximate the mode spectrum of each type of vibration with a twodimensional Debye model of maximum energy , where is the Debye temperature. We derive the mode spectra from the following expressions:
 
 for the CC modes:
(B.1)
where is the type of mode and N_{t} is the number of CC modes of a given type with K and K with:
= = (B.2)
 
 for the CH modes:
(B.3)
where , and N_{t} is the number of CH modes of a given type with K, K and K.
B.2 Density of states and degeneracies
We estimate the PAH density of harmonic vibrational states from direct
counts, a modetomode convolution method proposed by Beyer & Swinehart (1973). We
start from the rotational density of states, classically given by
with E
and B in cm^{1} and for a
symmetric top (Baer & Hase 1996).
We then obtain the rovibrational density of states
with the convolution method. The molecule's zeropoint energy has been
chosen as the zero of the energy scale, and the calculation was made
for bins with finite width of 1 cm^{1}.
To calculate the internal energy distribution of PAHs P(E),
we grouped this very large number of points into broader energy bins
with i=1 to 500. Each energy bin i
thus contains many states and its degeneracy, g_{i},
is estimated as (with g_{1} =
1)
(B.4) 
Figure B.1: Cumulative distribution of vibrational modes for coronene ( ), ovalene ( ) and circumovalene ( ), from the Malloci et al. (2007) database (solid lines) and with our Debye model (dotted lines). 

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Appendix C: Lines intensities in the rotational bands
In the case of symmetric top molecules, we give below the HönlLondon factors for the angular part of rovibrational transition rates, as well as the corresponding transition energies (Herzberg 1968ab). The frequency of the vibrational mode i is noted .
 
 for
and
(parallel transitions or out of plane transitions):
A_{KJ+} = A_{KJ} = A_{KJ0} =
 
 for
and
(perpendicular or in plane transitions):
Appendix D: Rate of angular momentum change for gasgrain interactions
We describe below the gasgrain interactions considered in our model of PAH rotation and the rate of change of Jthey induce, .
D.1 Collisions with gas atoms and plasma drag
For the collisions with gas neutrals and ions, as well as the plasma
drag, we apply the results of DL98 to planar PAHs.
In the case where ,
we use the correspondence principle to write
and obtain the following rates:
(D.1) 
for the damping contribution, and
(D.2) 
for the exciting contribution, where F and G are normalized rates defined in the Appendix B of DL98. We use the formalism of Bakes & Tielens (1994) to estimate the average charge of PAHs of a given size.
D.2 Rocket effect
Ejection of H or H_{2} from the edges of PAHs may
yield a significant rotational excitation if it occurs asymmetrically,
thus generating a systematic torque by the rocket effect (Rouan et al. 1992). We
note below
the kinetic energy of the
ejected fragment. In the case of H_{2},
this will happen if this molecule forms on preferential sites by
chemisorption, and if the distribution of these sites on the PAH is
asymmetric as a result of dehydrogenation. We calculate the change
of J assuming that H_{2}
molecules are ejected from the edge of the PAH with a cosine law:
=  
(D.3) 
The above numerical values are based on the following assumptions: (i) all molecular hydrogen is formed on PAHs with , a H_{2} formation rate cm^{3} s^{1} (Jura 1975) and eV ; (ii) the distribution of formation sites has an asymmetry of 1 site and we assume that the site in excess is always at the same location on the molecule; (iii) we neglect the influence of crossover events that may reduce the angular momentum (Lazarian & Draine 1999). With all these assumptions, the spin rate due to H_{2} formation estimated here is an upper limit. This rate, however, remains small compared to the other gasgrains processes.
D.3 Photoelectric effect
Stellar UV photons can pull out electrons from grains. These
photoelectrons carry away a significant kinetic energy (1 eV)
that heats the interstellar gas and impulses grain rotation. If we
assume that the photoelectrons leave the grain surface with a cosine
law distribution, we have
(D.4) 
where is the rate of photoelectrons ejections calculated with the formalism of Bakes & Tielens (1994) ( s^{1} for in the DIM).
Appendix E: Rotational temperature
A rotational temperature T_{J}
can be defined from the Maxwell distribution of angular momentum,
:
(E.1) 
Figure E.1 shows this temperature as a function of PAH size for several interstellar environments. We see that T_{J} is subthermal in the case of the DIM, and suprathermal for MCs and the Orion Bar. These results are in good agreement with Rouan et al. (1997). The rotational temperature is a relevant parameter in the study of the width of PAH vibrational or electronic transitions. The latter have been proposed as the origin of some unidentified diffuse interstellar bands (DIBs).
Figure E.1: Rotational temperature for PAH cations as a function of their size for several insterstellar environments: CNM (black line), WNM (gray dashed line), WIM (gray line), MC (black dotted line) and the Orion Bar (black dashed line). 

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Footnotes

... database^{}
 The molecules considered are perylene (C_{20}H_{12}), benzo[g, h, i]perylene (C_{22}H_{12}), coronene (C_{24}H_{12}), bisanthene (C_{28}H_{14}), dibenzo[bc, ef]coronene (C_{30}H_{14}), ovalene (C_{32}H_{14}), circumbiphenyl (C_{38}H_{16}), circumanthracene (C_{40}H_{16}), cirumpyrene (C_{42}H_{16}), and circumovalene (C_{66}H_{20}).
 ... field^{}
 G_{0} scales the radiation field intensity integrated between 6 and 13.6 eV. The Mathis radiation field, G_{0}=1, corresponds to an intensity of erg/s/cm^{2}.
 ... numerous^{}
 Energy conservation implies that 40 IR photons are emitted after each absorption.
 ... approximation^{}
 With this form of n(J) it is possible to define a rotational temperature (see Appendix E).
 ... is^{}
 In the limit , we estimated that the stimulated emission and absorption of CMB photons represent less than 20% of the spontaneous emission rate.
 ...(Ferland et al. 1998)^{}
 We assumed 130 (320) ppm of carbon (oxygen) to be in the gas phase. Parameters were taken from the optically thin zone of isochoric simulations with CLOUDY.
 ... same^{}
 We also varied the shape of the blackbody keeping G_{0} constant (as well as ). Similar J_{0} values were found for between 10^{4} and K.
 ... interactions^{}
 These models can also be applied to describe the rotational emission of other grain types.
 ... grain^{}
 Conversely, the damping part is correctly described by a classical model when (AliHaïmoud et al. 2009).
 ... interactions)^{}
 A correction of the same order was found by AliHaïmoud et al. (2009), who used the FokkerPlanck equation to derive n(J) instead of a Maxwell distribution.
All Tables
Table 1: FarIR rovibrational bands of PAH cations adopted in this work, with the percentage of the total oscillator strength in each band.
Table 2: Physical parameters for the typical interstellar phases considered.
Table A.1: MidIR bands of interstellar PAHs adopted in this work for cations and neutrals.
All Figures
Figure 1: Frequencies of the first (lowest energy) vibrational mode versus for PAH cations (black diamonds) and PAH neutrals (grey circles) from the Mallocci database^{} (see text). The solid line is the relationship we adopt between the band position of cations and , and the dashed line shows the case of neutrals. The dotted lines show extreme cases for this relationship. 

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In the text 
Figure 2: Absorption crosssection of PAH cations per carbon atom (1 Mb = 10^{18} cm^{2}). The solid, dotted, dashed, and dotdashed lines show the cases for = 24, 54, 96, and 216, respectively. 

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In the text 
Figure 3: Symmetric top molecule: , , and are the principal axis of inertia and J the total angular momentum of the molecule with K its projection along . 

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In the text 
Figure 4: Internal energy distribution of PAHs with (solid line and box for P(0)) and (dashed line and triangle) in the case of a) an MC ( G_{0}=10^{2}), b) the diffuse ISM (G_{0} = 1) and c) the Orion Bar (the radiation field is the sum of the CMB, the ISRF and a blackbody at 37 000 K corresponding to G_{0} = 14 000). To illustrate the effect of the radiation field hardness, we show in b) the case of an Orion Bar type radiation field scaled down to G_{0}=1 (grey lines). 

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In the text 
Figure 5: Left: dust IR emission of the DIM (solid black lines and symbols) with cm^{2} (Boulanger 2000). The model dust emission is overlaid in bold grey. MRN type size distributions are used for all dust populations. The PAH vibrational contribution (dashed line) is from the present model with , a 1:3 mixture of neutrals and cations, full hydrogen coverage, and 43 ppm of carbon. The contributions of larger grains are from the model described in Compiègne et al. (2008). The dotted line is the contribution of graphitic very small grains of radii a=0.9 to 4 nm containing 39 ppm of C. The dotdashed line is the contribution of silicate big grains with a=0.4 to 250 nm using 37 ppm of Si. Right: dust emission from the Orion Bar (the noisy black line is the ISOSWS spectrum) with cm^{2}. The model dust parameters are as above but for PAHs less abundant (18 ppm of C) and more ionized (82% are cations). 

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In the text 
Figure 6: PAH rovibrational emissivity for a MRN size distribution with . In a) we show the case of radiation field intensities corresponding to G_{0}=1 and 100 and in b) we illustrate the effect of IVR breakdown in case 1 (dotted line) and case 2 (dotdashed line) (see Sect. 3.1). 

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In the text 
Figure 7: J_{0} versus for PAH cations. In a) we show the case of the diffuse medium. Solid line shows the CNM, dashed line the WNM and dotted line the WIM. In b) we illustrate the influence of the absorption crosssection in the CNM: cations (solid line), neutrals (dashed line) and extreme behaviours (grey lines) for the first vibrational mode versus (see Fig. 1) are shown. In c), we present the case of the Orion Bar. 

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In the text 
Figure 8: a) Effect of varying G_{0} on J_{0} for PAH cations with , 30, 40, and 100 ( from bottom to top). The values for (triangles) have been divided by 2. The gas density is 30 cm^{3} and the other physical parameters for the gas are determined using CLOUDY (Ferland et al. 1998). b) Same as a) but for varying at G_{0} = 1. In both cases the radiation field is a blackbody with K and the gas parameters have been obtained with CLOUDY at thermal equilibrium (see text). 

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In the text 
Figure 9: Rate of Jchange due to rovibrational emission for a PAH with and heated by the ISRF. a) Comparison with former models: our rate (solid line), the rate of DL98 corrected as in AliHaïmoud et al. (2009) (dashed line) and the rate of AliHaïmoud et al. (2009) (dotted line). b) Effect of IVR breakdown, case 1 and 2 (see Sect. 3.1). 

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In the text 
Figure 10: Rotational emissivity of PAH cations. We take 430 ppm of carbon in PAHs, and assume m = 0.4 Debye. Panel a): effect of changing the fraction of large PAHs (decreasing with to 6). The case of a lognormal distribution centered around and width 0.4 is also shown. Panel b): rotational emission spectrum in different environments with a power law size distribution and . 

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In the text 
Figure 11: Comparison to observations (symbols) of the Perseus molecular cloud G159.618.5. The dashed lines show the freefree (low frequency) and big grains (highfrequency) emissions. The PAH rovibrational emission is represented by the dotdashed line. Dotted lines show the PAH rotational emission. The black model is model A, and the grey model is model B. See text for details. 

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In the text 
Figure B.1: Cumulative distribution of vibrational modes for coronene ( ), ovalene ( ) and circumovalene ( ), from the Malloci et al. (2007) database (solid lines) and with our Debye model (dotted lines). 

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In the text 
Figure E.1: Rotational temperature for PAH cations as a function of their size for several insterstellar environments: CNM (black line), WNM (gray dashed line), WIM (gray line), MC (black dotted line) and the Orion Bar (black dashed line). 

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In the text 
Copyright ESO 2010
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