Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://www.mrao.cam.ac.uk/~bn204/mk2/publications/2008/dustgal1.pdf Äàòà èçìåíåíèÿ: Wed Nov 25 23:11:21 2009 Äàòà èíäåêñèðîâàíèÿ: Tue Oct 2 19:09:24 2012 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: m 8

Mon. Not. R. Astron. Soc. 000, 1­16; (2008)

Printed 12 May 2008

A (MN L TEX style file v2.2)

A Semi-Empirical Model of the Infrared Emission from Galaxies
D. C. Ford, B. Nikolic and P. Alexander
Astrophysics Group, Cavendish Laboratory, J J Thomson Avenue, Cambridge CB3 0HE, UK

arXiv:0805.1732v1 [astro-ph] 12 May 2008

Accepted 2008 April 17. Received 2008 March 19; in original form 2007 July 14

ABSTRACT

We present a semi-empirical model for the infrared emission of dust around star-forming sites in galaxies. Our approach combines a simple model of radiative transfer in dust clouds with a state-of-the-art model of the microscopic optical properties of dust grains pioneered by Draine & Li. In combination with the S TA R B U R S T 9 9 stellar spectral synthesis package, this framework is able to produce synthetic spectra for galaxies which extend from the Lyman limit through to the far-infrared. We use it to probe how model galaxy spectra depend upon the physical characteristics of their dust grain populations, and upon the energy sources which heat that dust. We compare the predictions of our model with the 8- and 24-µm luminosities of sources in the Spitzer First Look Survey, and conclude by using the models to analyse the relative merits of various colour diagnostics in distinguishing systems out to a redshift of 2 with ongoing star formation from those with only old stellar populations. Key words: infrared: galaxies ­ infrared: ISM ­ dust, extinction ­ radiative transfer ­ galaxies: starburst ­ galaxies: high-redshift

1 INTRODUCTION It is increasingly clear that the infrared spectra of galaxies hold vital clues concerning galaxy energetics. Observations of our nearest neighbours tell us that around 60 per cent of their star formation is obscured by dust at visible wavelengths (Takeuchi et al. 2006), and the cosmic infrared background (CIB) indicates by its brightness that this is also true of sources at cosmological redshifts (Hauser & Dwek 2001). Furthermore, deep sub-millimetre surveys (Smail et al. 1997; Hughes et al. 1998; Smail et al. 1998; Blain et al. 1999) have revealed a large population of ultra-luminous infrared galaxies (ULIRGs) at z 2 (Blain et al. 2002) ­ clearly very dusty systems, and, if not harbouring active nuclei, very actively star-forming also (Smail et al. 1997). Given the degree of optical extinction in these systems (Swinbank et al. 2004), it is apparent that UV/optical studies of the cosmic star formation history are subject to substantial incompleteness. The advent of the Spitzer Space Telescope has allowed a much greater understanding of the sources which comprise the CIB. Its resolution and sensitivity has allowed more than two million infrared galaxies to be resolved in 49 deg2 of sky by the Spitzer WideArea Infrared Extragalactic (SWIRE) Survey (Lonsdale et al. 2003, 2004). Moreover, Spitzer's wavelength coverage, 3.6­160 µm, encompasses three emission regimes in the spectra of normal galaxies, each yielding information complementary to the others. In the rest-frame far-infrared (FIR) ­ here taken to extend from around 30 to 300 µm ­ thermal emission from large dust grains dominates. In the rest-frame mid-infrared (MIR) ­ here taken to extend from around 4 to 30 µm ­ emission from transiently-heated dust grains


dominates, marked by a series of broad emission features (see, e.g., Draine 2003, and references therein), the catalogue of which has recently been greatly expanded by Spitzer (Draine & Li 2007; Smith et al. 2007). These are attributed to polycyclic aromatic hydrocarbon (PAH) molecules, and so we shall refer to them as `PAH features'1 . Finally, in the rest-frame near-infrared (NIR) ­ here taken to refer to 4 µm ­ stellar emission dominates. This wealth of available information has motivated many studies which have sought to provide a framework in which this emission can be interpreted. Some of these take an empirical approach, matching unresolved sources to template spectra derived from a variety of local galaxies (Rowan-Robinson & Crawford 1989; Xu et al. 2001; Rowan-Robinson et al. 2004, 2005). These yield fast diagnostics which are readily applicable to large numbers of sources. But they provide little information about the physical processes which fundamentally shape the spectra. Others have sought to develop semi-empirical models of infrared spectra, considering the propagation of radiation through dusty media (e.g. Silva et al. 1998; Efstathiou et al. 2000; Takagi et al. 2003; Efstathiou & Rowan-Robinson 1995). This task can be split into two parts: modelling the optical properties of individual dust grains, and modelling the large-scale transport of radiation through some realistic dust geometry. Some authors (e.g. Fritz et al. 2006; Piovan et al. 2006a,b) have incorporated a detailed consideration of the radiative transport, including a treatment not only of absorption, but also of elastic photon scattering, which becomes the principal source of complexity. Others (e.g. Li & Draine

Email: dcf21@mrao.cam.ac.uk

In the literature, they are also commonly referred to as `Aromatic Infrared Bands' (AIBs), or, historically, as `Unidentified Infrared Bands' (UIBs)

1


2
2001; Draine & Li 2007) have developed highly sophisticated models of the optical properties of individual dust grains, but simplify radiative transfer. Essentially a computational trade-off exists: studies which include a detailed treatment of radiative transfer involve many evaluations of dust emissivities in differing environments. This is prohibitively time consuming with state-of-the-art models of the microscopic optical properties of individual dust grains and so simpler alternatives must be sought. In this paper, we use a state-of-the-art model for the individual dust grains ­ that of Draine & Li (2001) and Li & Draine (2001) ­ and then seek to make a minimal set of simplifications in our treatment of the radiative transfer problem such that it becomes computationally viable. In Section 2 we outline the dust geometries which we consider. In Section 3 we go on to present our modelling of the radiative transfer, and in Section 4 we construct UV­visible radiation fields appropriate for dust-heating by star-forming galaxies. In Sections 5 and 6 we draw upon models of dust grain populations and their microscopic optical properties from the literature. In Section 7 we conclude the development of our model with a simple framework for modelling the evolution of the metallicity and mass of gas in passively-evolving star-forming gas clouds. In Sections 8 and 9 we present the basic predictions of our model. In Section 10 we use the model to predict 8- and 24-µm luminosities for a sample of galaxies and compare to observations. Finally, in Section 11, we develop a simple model for the evolving colours of early- and latetype galaxies. Wherever required, we assume a flat CDM cosmology with H0 = 72 km s-1 Mpc-1 and = 0.7.
Dust Shell H number density nH (r) Grain number density ngr (r)

Ray Y

s Heat Source

r

0

X r

r

1

Figure 1. Dust in a spherical shell of dust around a single isotropic heating source.

region, essentially devoid of dust due to the sublimation of grains by energetic photons.

3 RADIATIVE TRANSFER 2 THE MODEL GEOMETRY In this paper we consider two basic geometries for the spatial distribution of dust and the source of illumination. The first geometry is a shell of dust grains ­ which we shall term a `circumnuclear' grain population ­ surrounding a point-like heating source, representing dust around a star-forming region. The second is a uniform distribution of dust grains ­ which we shall term a `diffuse' grain population ­ spread throughout a diffuse inter-stellar medium (ISM) within which the radiation field is assumed spatially uniform. In a future paper, we shall use them as components of a composite model for star-forming galaxies. The geometry adopted for our diffuse populations is the simpler. We assume the dust-bearing ISM to be optically thin at infrared wavelengths, such that the re-absorption of dust emission can be neglected. We further assume the UV­visible heating radiation field within it to be spatially uniform, hence all grains of any given size and composition have the same emissivity. Under these conditions, total dust emission is directly proportional to the number of grains present, and we therefore scale all quantities per unit volume of ISM. The circumnuclear geometry is illustrated in Figure 1. A point-like heating source lies at the centre of a spherical shell of dust, of inner radius r0 and outer radius r1 . Within the shell, we trace the density of the medium via the number density of hydrogen nuclei, assumed to be spherically symmetric and denoted nH (r), such that the column density Nc of hydrogen nuclei along a line of sight passing through a dust shell to its nucleus is given by:
r

We adopt a highly-simplified treatment of the radiative transfer which we argue is sufficiently accurate for a wide range of problems.

3.1 The circumnuclear geometry The evolution of the surface brightness I along ray Y in Figure 1 is governed by the time-independent equation of radiative transfer: dI ds (r)I + (r)nH (r) + nH (r) 4
I C

(2) where the first term on the right-hand side describes the absorption of radiation by dust, the second dust emission, and the third the scattering of photons into the ray. The integral in the third term is over solid angle at X ; is the scattering angle between d and the direction of Y . As mentioned above, we use nH (r) to parameterise the varying spatial density of material. C ,abs and C ,sca are the cross sections to absorption and scattering respectively, expressed per hydrogen atom, and averaged as described below over the compositions and sizes of particles within the grain population. (r) is the emissivity of the grain population ­ the power emitted per unit frequency into unit solid angle, normalised in the same way as for the cross sections. s measures distance along the ray. The extinction cross section, C ,ext , is the sum of the absorption and scattering cross sections: C
,ext

X

= -C

,ext nH

,sca

( ) d ,

Nc =

1

r

nH (r) d r.

(1)
II

0

= C

,abs

+ C

,sca

.

(3)

Astrophysically, the cavity at r < r0 might correspond to an H

The averaging of the quantities above over grains of varying com-


Infrared Spectra of Galaxies
positions and sizes is performed as follows: C
,j

3

= =


i

a

i C , j (a) i (a, r)

1 dni r (a) g nH da 1 nH dni r g (a) da

through the sphere of constant radius passing through X is related to H (r) via: (4) F (r) = 4 H (r). Equation (6) can thus be re-written: 1 r2 F (r) = nH (r) 4 (r) - C r2 r
,abs F

da, da,

(10)

(r)


i

(5)

a

where j {ext, abs, sca}, i denotes a population of grains of given composition, and ni r (a) denotes the spatial number density of g grains of each composition with radius smaller than a. The first simplification that we make is to neglect scattering, such that C ,sca = 0. We discuss the validity of this assumption in the following section. Integrating the remaining terms of Equation (2) over all rays passing through the point X in Figure 1, we obtain (Chandresekhar 1960; Rowan-Robinson 1980): 1 r2 H (r) = nH (r) - C r2 r where: J (r) = and: H (r) = 1 2
1 -1 ,abs J

(r) .

(11)

This equation is integrated numerically from the inner to the outer radius.

3.2 Assumptions made in the circumnuclear geometry The assumptions made in the previous section ­ i.e. the neglect of scattering and the radial beaming of dust emission ­ will have negligible effect upon the predictions of our model for column densities of dust which are optically thin at all wavelengths, i.e. for Nc 1023 H m-2 . For column densities of dust which are optically thick in the UV, but not in the infrared, our neglect of scattering will lead us to under-estimate the path lengths of UV/optical photons through the dust shell by a factor of 1­2, and so to under-estimate the absorption of UV/optical radiation by a similar factor. Since this effect is essentially the same as that of reducing the column density of dust, the effect when using these models to fit the spectral shape of real sources will be that we will over-estimate the dust masses of these objects. Our assumption that dust emission is beamed radially outwards only begins to fail for dust shells with higher column densities still, when they become optically thick even at infrared wavelengths. As the dust emission is assumed to take the shortest path out of the dust shell, we will under-estimate its re-absorption in these optically thick cases. In practice, this effect becomes significant for dust column densities 1026 H m-2 , as will be shown in Figure 11. Rowan-Robinson (1980) studied the effects of a similar set of assumptions in their calculation of radiative transport in hot-centred star-forming clouds, and for the range of systems they consider, they report errors of around 10 per cent. In addition to the two assumptions just discussed, it is also apparent that the adopted geometry is simplistic, having only one single heat source. We note, however, that this geometry is observationally indistinguishable from an ensemble of N smaller circumnuclear geometries, each heated by its own central heat source with luminosity scaled by a factor 1/N with respect to the single shell, and each co aining a dust shell with inner and outer radii scaled nt by factor 1/ N and dust density distribution n (r) scaled according to: n (r) = N n(r N ) (12) where n(r) is the density distribution of the single shell. The column density of dust around each heat source in this latter ensemble is the same as that in the former single circumnuclear shell; the radiation field incident upon grains on the inner edge of each dust shell is the same; the total mass of dust in the two cases is the same; and to a remote observer, the total solid angle subtended by the dust in the two cases is the same. In summary, although our circumnuclear model is nominally of dust around a single heat source, it is also a good model of systems where that luminosity production is distributed between several discrete sources.

(r) ,

(6)

1 2

1 -1

I dµ ,

µ = cos ,

(7)

I µ dµ ,

(8)

with denoting the angle made between each ray and the radial direction, as shown in Figure 1. Geometrically, J (r) may be visualised as the average surface brightness along all rays passing through X , averaged over 4 steradians. H (r) may similarly be visualised as the average projected onto the radial direction. Rowan-Robinson (1980) introduced what has become a widely-used decomposition of the surface brightness I along each ray into three components, depending upon where photons last interacted with matter: I = I + I + I ,
(1) I (1) (2) (3)

(9)

is radiation from the central heat source which has not where (2) (3) been absorbed by dust, I is radiation emitted by dust and I is scattered radiation ­ which we have already neglected. While we do not use this in our mathematical treatment of Equation (2), it is useful in our discussion presently. The relationship between J (r) and H (r) encodes the angular distribution of the radiation flux passing through X . In the limiting case of a radiation field propagating exclusively in the radial direction, H (r) = J (r). In the opposite limit of an isotropic radiation field, H (r) = 1/2J (r). Given a point heat source, as in Figure 1, (1) the former limit is applicable to the component I ; the radiation (2) field emanating from the heat source is purely radial. For I , however, H (r) < J (r). Our second, and final, assumption, is that J (r) = H (r) in Equation (6). For UV­visible wavelengths, this assumption holds because the heating radiation field is expected to dominate over (1) dust emission at these wavelengths, and so I I . In the infrared, (2) where I is expected to dominate I , J (r) is under-predicted, but the assumption continues to hold if (r) C ,abs J (r), that is to say, if the dust emission from the shell is not appreciably reabsorbed. Geometrically, this assumption is equivalent to assuming that dust emission is beamed along the outward radial direction. Finally, we note that the net outward flux F (r) of radiation


4
3.3 The diffuse geometry Our treatment of radiative transfer in diffuse dust grain populations is simpler than the above. Instead of having a heat source of luminosity L , we have an interstellar radiation field (ISRF), whose (1) energy density E we normalise to 0 , where 0 is that of the solar neighbourhood interstellar radiation field less the cosmic microwave background (CMB)2 , and is a free parameter. We adopt 0 = 7.46 â 10-14 J m-3 , derived from integration of the ISRF of Mathis et al. (1983) and Mezger et al. (1982). The total luminosity emerging from the model can then be written: L = 4 nHV + E
(1)

stellar population of age t with our star formation history:
t

L (t ) =

0

(t ) 106 M

L

SSP

(t - t ) dt .

(14)

5 THE DUST MODEL Whilst there exist fairly tight observational constraints on the composition and size distribution of dust grains in the Milky Way, relatively little is known about those in other galaxies (Draine 2003). In this paper, we therefore base our dust grain population upon that inferred from observation of our own galaxy. Following Li & Draine (2001, hereafter, LD01), we consider binary populations of `carbonaceous' and silicate grains. The former sub-population includes both PAH molecules and larger graphitic grains; the optical properties of these grains exhibit a smooth transition with grain radius, centred around a radius of a . The absorption cross sections of carbonaceous grains of radius a is taken to be:
cr Caabs (a) = , PAH P (a)CAHs (a) + [1 - ,ab PAH PAH

cA

4

(13)

where V is the volume of the dust-bearing ISM, and A its surface area through which the interstellar radiation field leaves the galaxy.

4 THE HEATING RADIATION FIELD In this paper, we consider models for the heating radiation field due to star formation. We generate these using Version 5.1 of the S TA R B U R S T 9 9 stellar spectral synthesis package (Leitherer et al. 1999; Vazquez & Leitherer 2005). This package offers the facil´ ity to model stellar populations with two classes of star formation histories (SFHs). The first is ongoing star formation, proceeding at constant star-formation rate (SFR) , which began at some time t previously; we hereafter term these models `continuous' SFHs. The second is a delta-function SFH, representing an instantaneous burst of star formation at some time t previously, of total mass m; we term this the `instantaneous' SFH. The stellar populations modelled for the instantaneous SFHs may be referred to as single stellar populations (SSPs), which is to say that all of the stars within them are coeval. S TA R B U R S T 9 9 models both stellar emission and also nebular continuum. For the former it uses the stellar evolution tracks of the Padova group (Fagotto et al. 1994), with the addition of tracks for thermally pulsing asymptotic giant branch (TP-AGB) stars to improve the accuracy of the modelling of low and intermediate mass stars.3 For the nebular continuum, the emission coefficients of Ferland (1980) are used. This is the source of a problem in S TA R B U R S T 9 94 : the data of Ferland (1980) do not extend beyond 4.5 µm, however they are extrapolated to 160 µm, yielding large, unphysical, infrared luminosities. For 4 µm, we use a more physical extrapolation, taking L -0.1 . In addition to the continuous and instantaneous SFHs modelled by S TA R B U R S T 9 9, we have also considered arbitrary SFHs, modelled by convolving the luminosity, LSSP (t ), of a 106 M single

(a)]C

gra ,abs

(a),

(15)

where the weighting parameter

(a) is given by: (16)



PAH

(a) = 1 - qgra â min 1, (a /a)3 ,

and qgra = 0.01 sets even the smallest PAH molecules to exhibit 1 per cent of the continuum absorption of graphitic grains. The tran° sition radius, a , is set by default to 50 A. For the size-distribution of grains in each of these populations, we follow the parametric forms used by Weingartner & Draine (2001); for the carbonaceous grains, we use: 1 nH Z Z 1, exp - a - at,g /ac
,g 3

dnca gr da

r

= D(a) +

Cg a

a at,g ° 3.5 A



g

F (a, g , at,g ) (17) a a < at,g

â

< <

,

at,g

where D(a) represents two log-normal peaks:

D(a) =

i=1



2

Bi 1 ln a/a0,i exp - a 2

,

(18)

which were introduced by LD01 to reproduce the mid-infrared luminosities observed by ISO, the Infrared Telescope in Space (IRTS) and by IRAS at 60 µm. The term F (a, g , at,g ) provides curvature: F (a, g , at ) = 1 + a/at (1 - a/at )-1

0 < 0,

(19)

We neglect the CMB in this normalisation because, in contrast to the starlight component of the ISRF, it would make no sense to enhance it by a factor . It should be noted that the CMB is also absent from all models presented in this paper. 3 It should be noted that this is a significant departure from previous versions of S TA R B U R S T 9 9, which used the stellar evolution tracks of the Geneva group, and did not model low mass stars, introducing serious errors in the modelling of old stellar populations. 4 See notice by Hunt, October 30, 2006, in the Knowledge Base of the S TA R B U R S T 9 9 website.

2

Z is the mass ratio of metals, Z = 0.02 is the solar mass ratio of metals, and all other symbols are as defined in Weingartner & Draine (2001). The normalisation constants Bi are given by: 3 exp -4.5 a3,i (2 )3/2 0
2

Bi =

bC,i m 1 + erf
3 2

C

+

° ln a0,i /3.5 A 2

(20)

where mC = 1.99 â 10-26 kg is the mass of a carbon atom, = ° 2.24 â 103 kg m-3 is the density of graphite, a0,1 = 3.5 A and


Infrared Spectra of Galaxies
Parameter Value -1.54 -0.165 0.0107 µm 0.428 µm 9.99 â 10- -2.21 0.300 0.164 µm 1.00 â 10- 0.75bC 0.25bC 6.0 â 10-5

5

100 Carbonaceous grains 10
12
3

1029

g g at,g ac,g Cg s s at,s Cs bC,1 bC,2 bC

13

-1 H

a4 / cm n

1

100 Silicate grains 10

Table 1. The parameters of the default size distribution which we adopt ­ the Weingartner & Draine (2001) preferred distribution for RV = 3.1 Milky Way sight lines.

dngr da

1

° a0,2 = 30 A are the wavelengths of the centres of the two log-normal peaks, and = 0.4. For the silicate grains, we use: 1 nH 1, exp - [(a - at,s ) /ac,s ]
3

10-9

10-8 a/m

10

-7

10

-6

Z Z

dnsirl g da

Cs = a

a at,s



s

F ( ; s , at,s )

(21)

â

° 3.5 A , at,s

Figure 2. The adopted size distributions for silicate (top) and carbonaceous (bottom) grains, assuming a solar metallicity environment. Unit areas under each distribution represent unit masses of grain material.

< <

a a

<

at,s of neutral and ionised PAH molecules with a weighting parameter f describing the ionisation fraction. We take this to have a default value of 80 per cent, matching that which Draine & Li (2001) find in their model fits to Galactic photo-dissociation regions. The recr sulting absorption cross section Caabs (a) is shown for a range of , s grain radii in Figure 3(c), and the absorption cross section Ci,labs (a) of the silicate grain population Figure 3(d).

By default, we set the parameters of these size distributions to those preferred by Weingartner & Draine (2001) for RV = 3.1 Milky Way sight lines, as given in Table 1. The effect of using instead the size distributions preferred by those authors for RV = 4.0 and RV = 5.5 Milky Way sight lines will be discussed in Section 8. The appearance of Z in Equation (17) is a departure from Weingartner & Draine (2001), who only considered Galactic environments. Thus, their size distributions are normalised to a dust-togas-mass ratio appropriate for solar metallicity environments. As the variation in this ratio with Z is quite poorly understood, we make the assumption that it is linearly proportional to Z , which is implicit in our renormalisation above. These distributions are plotted in Figure 2. To calculate the absorption cross sections of silicate and graphitic grains, we follow LD01 and use dielectric functions for these species (Draine & Lee 1984) and Mie theory (see, e.g., Bohren & Huffman 1998) to estimate the absorption cross sections of spherical particles of radius a. The treatment of graphitic grains is slightly complicated by the anisotropy of graphite's dielectric function. We follow LD01 in calculating an averaged absorption cross section using the `1/3-2/3 approximation' (Draine & Malhotra 1993). For the PAH molecules, LD01 give algebraic fits to terrestrial P laboratory measurements of CAHs (a) for neutral and ionised sam,ab ples. Draine & Li (2007, hereafter, DL07) revise these in the light of new near-infrared data (Mattioda et al. 2005b), and in order to fit the high-fidelity spectra of nearby star-forming galaxies observed by the Spitzer Infrared Nearby Galaxies Survey (SINGS; Kennicutt et al. 2003) project. We implement the cross sections given by both LD01 and DL07, which are shown in Figures 3(a) and 3(b) for ° neutral and ionised grains respectively, both of radius 5 A. In the remainder of this paper, we use the DL07 cross sections throughout, except in Section 10. In both cases, we average the cross sections

6 MODELLING THE EMISSIVITY OF THE DUST
i In this section, we outline how we model the emissivities (a, r) of dust grains as a function of the energy density E (r) of radiation to which they are subjected.

6.1 Transiently-heated grains Modelling emission from transiently-heated grains requires the calculation of the time-averaged probability distributions P(E ) for their internal energies. Exact treatment of this problem would require knowledge of all of their vibrational energy levels and transition probabilities. Our simplified analysis follows Draine & Li (2001). We use Debye models for the normal modes of the C/Si skeletons of PAH and silicate particles, with Debye temperatures as used by Draine & Li (2001). We use Einstein models for the stretching, in-plane bending, and out-of-plane bending modes of the peripheral C­H bonds of PAH molecules: we assume the modes of each bond to be quantum harmonic oscillators with the same fundamental frequencies, as given in Draine & Li (2001). To reduce the resulting mode spectra to a computationally tractable number of energy states, we follow Guhathakurta & Draine (1989) and Draine & Li (2001) in dividing them into Nbin bins (for our choice of bins, see Appendix A), with mean energies


6
(a) 0.01 (b)

Cabs / a2

10-4

10-6 Neutral PAHs; LD01 model Neutral PAHs; DL07 model C ; a = 5° A Ionised PAHs; LD01 model Ionised PAHs; DL07 model C ; a = 5° A

1

(c)

(d)

0.01 Cabs / a2

10-4

10-6

a = 5 â 10-10 m a = 1 â 10-8 m C -6 a = 1 â 10 m 0.1 1 /µm 10 100 0.1

a = 5 â 10 a = 1 â 10 a = 1 â 10

-10 -8 -6

m m Si m 1 /µm 10 100

Figure 3. The adopted absorption cross sections. Panels (a) and (b) show those for neutral and ionised PAH grains respectively, as given by LD01 and DL07 ° for grains of radius 5 A. A discussion of the near-infrared feature introduced into the cross sections of ionised PAHs by DL07 at 1.05 µm , and the negative feature at 1.905 µm , can be found in Mattioda et al. (2005a). Panels (c) and (d) show those for carbonaceous and silicate grains respectively for a variety of grain radii a, assuming a PAH ionisation fraction f = 0.8. Each trace is normalised with respect to the classical grain cross section of a2 .

U i , widths U i , and time-averaged occupation probabilities Pi . We denote as T ji the transition rate between bins i and j. The time evolution of Pi is then given by: dPi = dt

constraint:


i

Pi = 1,

(25)



j=i

Ti j P j -



T j i Pi .

(22)

j=i

The time-averaged steady-state probability distribution which we seek is that to which the above converges over time, and for which dPi /dt = 0. The elements of T ji with j > i describe the upward transitions of grains that result from photon absorption; we model these using equations (15­25) of Draine & Li (2001); in Appendix B we reproduce these relations and describe a numerical optimisation that we use in their calculation. The elements with j < i describe the radiative cooling of grains; here we use the `thermal continuous' approximation (equation 41 of Draine & Li 2001, reproduced here as Equation B4), which models the cooling of grains as a continuous process, where each state i only makes downward transitions to the adjacent state i - 1. This allows much faster solution of Equation 22 to find Pi . The diagonal terms are chosen (Draine & Li 2001) so that: Tii = -

using the method of Guhathakurta & Draine (1989). Given the vector Pi , we calculate the time-averaged emissivity of each grain usi ing the thermal approximation, under which (a, r) can be calculated using equation (56) of Draine & Li (2001):

=

2h c2

3


i

Pi , exp(h /k i ) - 1

(26)

where i is the characteristic temperature of bin i, as defined in Draine & Li (2001), the sum is over all bins i whose central energies are greater than h , and we have neglected the factor (1 + 3 uE /8 ) shown by those authors; this represents stimulated emission and may straightforwardly be shown to be negligible in all of the models presented in this paper.

6.2 Large grains For sufficiently large grains, the approach outlined above becomes inefficient. Their internal energies become much larger than the energies of the photons they absorb, and so their temperature fluctuations are not significant. Their internal energy probability distributions P(E ) tend towards delta functions (Li & Draine 2001). In this limit, we can model the energetics and emission of these grains by numerically solving the equation of radiative balance to find the