High-Redshift Galaxies: The Far-IR and Sub-Mm View

4. GENERALIZED SPECTRO-PHOTOMETRIC MODELS OF GALAXIES

Twenty years after the first serious models of stellar population synthesis (Tinsley 1977; Bruzual 1983), the most relevant recent progresses have been the attempts to provide a self-consistent description of the effects of dust (and gas) in galaxy spectra and spectral evolution. We review in this Section some recent efforts of generalized spectral synthesis of galaxies from the UV to the sub-mm, including dust effects (as for both the extinction of the primary optical spectrum, and dust re-radiation at longer $\lambda$ ) in the various galactic environments.

Dust plays an important role in all relevant galactic sites: (1) the neutral interstellar medium, whose associated dust is heated by the general radiation field (infrared cirrus, prominent in the 100µm IRAS band); (2) the dense cores of molecular clouds, where dust optical-depth is very high and prevents light from very young stars to be observed; (3) dust in the external layers of molecular clouds (PRD regions), heated by the interstellar radiation field and OB associations formed in the clouds; (4) dust around protostars; (5) dust around evolved giants and young planetary nebulae; (6) hot dust associated with HII regions.

The inclusion of dust means a dramatic complication of spectro-photometric models: the usual assumption of population-synthesis codes - that the global emission of a whichever complex stellar system is simply the addition of the integrated flux of all components independently on the geometry of the system - is no more valid: not only the extinction process depends in a complex way on the relative distributions of stars and dust, but also dust emission itself, at high dust column densities and according to geometry, may be optically thick.

In principle, accounting for dust effects in detail may require a very complex description of: (1) the physical-chemical-geometrical properties of grains, determining their interactions with the radiation field (e.g. amorphous, porous low-albedo grains vs. highly reflective grains); (2) the chemical composition of the ISM where grains have condensed (which affects the dust composition), given by the integrated contribution of all previously active stellar populations in the galaxy; (3) the modifications that grains and molecules undergo during the course of evolution, i.e. sublimation in strong UV radiation fields, sputtering, etc. (see Sect. 2.2).

These complications of the classical purely stellar evolutionary codes cannot be avoided, if we want a complete and reliable description of physical processes inside galaxies. As we will discuss in later Sections, this turns out to be particularly critical when describing what we called the active phases during galaxy evolution: neglecting dust effects in such cases would bring to entirely wrong conclusions.

On the other hand, the uncertainties introduced by the large number of new parameters are largely reduced by adopting a multi-wavelength (UV through mm) approach, which balances the unknowns with the number of constraints coming from a wide-band observed spectrum.

4.1 Semi-empirical approaches

A phenomenological approach to a global spectrophotometric description of galaxy evolution was recently discussed by Devriend, Guiderdoni & Sadat (1999). This paper elaborates separately the code for stellar population synthesis from that of dust emission. The former is treated with the most recent prescriptions. The dust emission is schematically represented as the contribution of four different components: the PAH emission features, very small grains, big grains illuminated by the general galactic radiation field (cold dust), and big grains illuminated by young stars in starburst regions. These four components are modelled using typical parameter values for the dust composition, radiation field intensity, mass, etc. Relative normalizations of the four components are finally calibrated using the observed relationship between the IRAS colours of galaxies and the bolometric luminosity.

This approach is quite fast as for computation time (in particular it overcomes the problem of solving the radiative transfer equation), and is particularly useful for statistical analyses of large galaxy databases.

4.2. Detailed self-consistent spectro-photometric models

More physically detailed descriptions of the galactic dust emission are discussed by several teams. These models interface two logical procedures:

(1) the first is to describe, given a prescription for the IMF, the history of star-formation in the galaxy as a function $\psi$ (t) detailing the mass in stars formed per unit time t, the actual gas metallicity Z(t), the abundances of various elements produced by stars during evolution, and the residual gas fraction g(t) as a function of time;
(2) the second step is to sum up, at any galactic age t, the fluxes from all populations of stars, by solving the radiative transfer equation taking into account how stars and the residual gas and dust are geometrically distributed.

4.2.1. Chemical evolution of the ISM

While point (1) above is addressed in detail by other contributions to these Book (Bruzual), we remind here a few basic concepts.

A galaxy is usually modelled from the chemical point of view as a single environment where primordial gas flows in according to an exponential law

(4.11)

The SFR follows a general Schmidt law

(4.12)

with the addition of one or more bursts of star-formation to describe starburst episodes possibly triggered by galaxy interactions or mergers. The typically adopted value for k is 1. For the initial mass function (IMF) the usual assumption is a Salpeter law

(4.13)

with typically M_min = 0.1 M (but higher values may apply for example in the case of starbursts). The observed photometric properties of galaxies of various types and morphologies are reproduced by varying in particular t_inf and $\nu$ .

Given the above parameters, the solution of the equations of chemical evolution allow to compute at any given galactic time all basic quantities, in particular the functions g(t) and Z(t), and then, after eq. (4.12), the number of stars generated at that time with metallicity Z(t). The integrated spectrum of each stellar generation (Single Stellar Population, SSP) then evolves according to the prescriptions of stellar evolution, defining a 2D sequence (spectral intensity L[ $\nu$ , t] vs. frequency $\nu$ as a function of time, t).

4.2.2. Geometrical distributions of gas and stars

In the model by Silva et al. (1998) three different stellar and ISM components are considered in the generic galaxy: (a) star-forming regions, comprising molecular clouds (MC), with young stars, gas and dust in a dense phase, and HII regions; (b) young stars escaped from the MC complexes; (c) diffuse dust ("cirrus") illuminated by the general interstellar radiation field.

For disk galaxies the adopted geometry is a flattened system with azimuthal simmetry and a density distribution for the 3 above components described by double exponentials: $\rho$ = $\rho_{0}^{}$ exp(- r/r_d) exp(- |z|/z_d).

For spheroidal galaxies, spherical symmetry is adopted with King profiles $\rho$ = $\rho_{0}^{}$ ((1 + [r / r_c]²)^- - (1 + [r_t/r_c]²)^-), with $\gamma$ = 3/2,[r_t/r_c] $\sim$ 200, r_c $\sim$ 300pc as typical values.

4.2.3. Models of the molecular clouds (MC)

High-resolution CO and radio observations show that MCs are highly structured objects containing very dense cores where stars are actually formed. Typical values for the MCs are: size $\sim$ 10 pc, mass $\sim$ 10⁶ M.

All star-formation in the Galaxy happens in dusty MCs, the early evolution phases of young star clusters occurr inside such dusty regions, hence are optically hidden. Later, on the lifetime of OB stars (10⁶ - 10⁷ yrs), the radiation power of young stars, stellar winds and the first SNs destroy the parent MCs and allow the young stellar population to appear in the optical.

Note that, bacause of the clumpiness of MCs, this is in any case a statistical process: in some clouds even the emission of the youngest OB stars is already visible, while in others all young stars are completely embedded in dust. Silva et al. (1998) describe schematically this transition of the MC from a dust-embedded phase to the optically dominated phase, as a process in which the fraction f of the light from the SSP generated within the cloud still embedded into dust decreases linearly with time as f (t) = 2 - t / t₀, t₀ being the time interval during which the SSP is entirely extinguished.

The spectrum emitted by the MC and filtered by dust is computed by solving the transfer equation, e.g. by assuming that the primary SSP spectrum comes from a point source in the center of the cloud (this rather crude assumption allows substantial semplifications in the numerical code, see above).

A more detailed description of molecular cloud structure and emission is provided by Jimenez et al. (2000). Their model is based on fully three-dimensional simulations of the density and velocity fields obtained by solving 3D compressible magneto-hydrodynamical (MHD) equations in supersonic turbolent flows, as typical of the motions in Galactic molecular clouds (Padoan et al. 1998). The MHD turbolence generates a large density contrast, with the density field spanning a range of 4 to 5 orders of magnitude. This brings to a highly filamentary and clumpy morphology. All this is consistent with observed properties of the clouds.

Young stars with M > 15 - 20 M in this model are heavily extinguished for virtually all their live. A detailed Monte Carlo approach is required to solve the radiative transfer equation. The simultaneous knowledge of the density and velocity fields allows also to estimate in great detail the molecular emission lines (CO).

4.2.4. Models of diffuse dust (cirrus)

Diffuse dust in the galaxy is responsible for a general attenuation of the light emitted by all stars and MC complexes. In this case the dust column density is not so high to require a detailed solution of the transfer eq. ( $\tau_{\nu}^{}$ for IR photons is small). One can express an effective optical depth to account for combined absorption and scattering (Rybicky and Lightman 1979): $\tau_{eff}^{}$ = $\tau_{a}^{}$ ( $\tau_{a}^{}$ + $\tau_{s}^{}$ ). The galaxy is divided into small volume elements V_i, such that the local radiation field in this elementary volume is

r²_{i, k} being the distance between the i-th and k-th volumes. This determines the temperature of the local diffuse dust, whose integrated flux seen by an observer in a direction $\theta$ is a simple sum over all volume elements of the diffuse dust emissivity:

$\tau$ being the optical depth from the V-element to the outskirts in that direction and j( $\lambda$ )_k = j( $\lambda$ )_k^mc + j( $\lambda$ )_k^star + j( $\lambda$ )_k^cirrus.

4.2.5. Modelling the SEDs of normal and starburst galaxies

Figure 2 shows the broad-band (UV through radio) spectrum of the prototype starburst galaxy M82, a closeby well studied object at 3.2 Mpc. The lines in the figure come from the fit obtained by Silva et al. (1998). The thin (cyan colored) continuous line peaking at 0.1 µm corresponds to the unextinguished integrated spectrum of all stellar population, while the long-dashed line is the reddened stellar continuum. The dot-dashed line is the contribution of dust in molecular clouds, while the dotted line comes from diffuse dust in the "cirrus". In this model, the optical-NIR spectrum of the galaxy is contributed mostly by old stellar populations unrelated to the ongoing starburst, whereas the starburst emission is mostly observable at $\lambda$ > 4 µm in the form of dust re-radiation, radio SN and free-free emissions.

Figure 2. The broad-band (UV through radio) data on the prototype nearby starburst galaxy M82. The ordinate axis is normalized to 10³⁰ erg/sec. [Courtesy of G.L. Granato].

Equal areas in the $\lambda$ L( $\lambda$ ) plot of Fig. (2) subtend equal amounts of radiant energy: it is then clear from the figure that in this moderate starburst $\sim$ 80% of the bolometric flux emerges as dust re-radiation above 5 µm. In higher luminosity starbursts and in Ultra-Luminous IR Galaxies (ULIRGs, e.g. Arp 220) this fraction gets close to 100%. On the contrary, for local normal galaxies the average fraction is only $\sim$ 30%, as found from comparison of the far-IR with the optical luminosity functions of galaxies (Saunders et al. 1990).