Free Access
Volume 527, March 2011
Article Number A109
Number of page(s) 39
Section Extragalactic astronomy
Published online 04 February 2011

Online material

Appendix A: Calculation of the extinction cross-section and scattering phase function

A prerequisite for the calculation of radiation fields and infrared emission is knowledge of the extinction cross-section of the population of grains, and, since we consider anisotropic scattering in the radiation transfer calculations, the scattering phase-function needs to be known as well.

The absorption cross-section of grains of composition i =  { Si, Gra, PAH0, PAH +  } , Cabs,   i, is obtained by integrating the absorption efficiencies Qabs,   i over the grain size distribution n(a):

(A.1)where Cabs,   i is given in units of  [cm2   H-1] , amin is the minimum grain size and amax is the maximum grain size.

thumbnail Fig. A.1

The extinction curve for the dust model used in this paper (solid line), which is a Milky Way dust model. With dashed-dotted line we plotted the observed mean extinction curve of our galaxy (Fitzpatrick 1999). Also plotted are the two components of extinction, the model absorption curve (dotted line) and the model scattering curve (dashed line).

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Similarly, the scattering cross-section of grains of composition i, Csca,   i, is obtained by integrating the scattering efficiencies Qsca,   i over the grain size distribution n(a): (A.2)Then, by summing over the grain composition i we obtain the total absorption and scattering cross-sections, Cabs and Csca: The extinction cross-section Cext is the sum of the absorption and scattering cross-sections: (A.5)We note that the extinction cross section Cext is defined as per unit H. In some applications it is useful to define a cross section per unit dust mass, which we denote here : (A.6)where ρg is the density of the grain material and is in units of cm2/g.

In turn, is related to the extinction coefficient κext, as used in the mathematical prescription of the dust distribution from Eqs. (4) and (6) using: (A.7)where ρdust(R,z) is the dust mass density at position (R,z) in the galaxy in units of g   cm-3 and κext(λ,R,z) is in units of cm-1.

Figures A.1 and A.2 show the resulting extinction curve of the dust model adopted here, together with the absorption and scattering components (Fig. A.1) and the components given by the different grain composition (Fig. A.2). As expected, the figures confirm that the model extinction curve fits well the observed mean extinction curve of our galaxy.

The averaged anisotropy of the scattering phase function g needed in the radiative transfer calculation is obtained in a similar manner to Eqs. (A.1)–(A.4). (A.8)(A.9)where Qphase,   i is the anisotropy efficiency.

thumbnail Fig. A.2

The extinction curve for the dust model used in this paper (solid line) together with the contributions to the extinction from the different dust compositions used in the model: Si (dotted line), Gra (dashed line), PAH (dashed-three-dotted line). As in Fig. A.1, the observed mean extinction curve of our galaxy is plotted with dashed-dotted line.

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Appendix B: The library of attenuations of stellar light for the diffuse stellar components

thumbnail Fig. B.1

The attenuation-inclination relation for the disk (top), bulge (middle) and thin disk (bottom) in the B band. In each panel the solid curves represent the attenuations calculated with the dust model used in this paper, incorporating a mixture of silicate, graphite and PAH molecules. The dotted curves represent the corresponding attenuations calculated with the dust model used in Paper III, incorporating only a mixture of silicates and graphites. In each panel the 7 different curves represent the attenuation-inclination relations for different : 0.1, 0.3, 0.5, 1.0, 2.0, 4.0, 8.0, with the values increasing from bottom curves to top curves.

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The second set of simulated data needed to fit the panchromatic SEDs is the library of attenuations in the UV/optical/NIR as a function of and i. As mentioned in Sect. 2.4, the revision of the dust model required a recalculation of the database for the attenuation of stellar light, as presented in Paper III. The overall concept and characteristics of the calculations are the same as in Paper III, but a small change, mainly in the zero point of the calculations, was apparent due to the change in the relative contribution of absorption and scattering to the total extinction. Here we only give an example of a comparison between the attenuation-inclination curves obtained using the new and the old dust model (Fig. B.1).

The attenuation-inclination relations for disks show a systematic change with inclination when changing the dust model. Thus the attenuation for the low inclinations is decreased more than for the high inclinations, with a tendency for the curves to converge at the edge-on inclinations. This means that the shape of the attenuation-inclination is steepened for the present dust model. The curves for bulges show the biggest offset when changing the dust model, but in most of the cases there is no change in the shape of the curves. As one can see the curves run almost parallel, except perhaps for the lowest values of opacity. The smaller change is seen for the thin disk component, where neither the shape nor the zero point are changed significantly.

We also did some tests to quantify the effect of the change in the dust model to the overall energy balance. By integrating the attenuation over all angles we obtained an estimate of the total energy absorbed by dust in a galaxy. This absorbed energy was found to be on average 10% smaller for the attenuations calculated using the new dust model than for those from Paper III.

Appendix C: Formulation of composite attenuation of stellar light

In Sect. 5.1 of Paper III a generalised formula was given, showing how the composite attenuation (that is the overall attenuation of an arbitrary combination of luminosity components from stellar populations in the young stellar disk, the old stellar disk and the bulge) can be derived from the library of attenuation of stellar light from the diffuse component and the attenuation of the clumpy component. At any wavelength, the composite attenuation depends on the relative luminosity of the three stellar components, which we have described in this paper in terms of the parameters SFR, F, old and B/D, which we used to describe the dust emission. Here we re-write the generalised expression for the composite attenuation (Eq. (16) from Paper III) for the specific parameterisation adopted in this paper.

At a given wavelength λ, the composite attenuation Δmλ in a galaxy is given by:

(C.1)where and Lλ are the intrinsic and the apparent luminosity densities. The quantities and Lλ can be further expressed as a summation of the corresponding quantities for the disk, thin disk and bulge: The apparent and intrinsic luminosity densities for the disk, thin disk and bulge are related as follows:

where , and are the attenuation values expressed in magnitudes for the diffuse component in the disk, thin disk and bulge and , and are the corresponding attenuations expressed in linear form. Using Eqs. (13), (15), (19) and Eqs. (C.4) − (C.6) we can rewrite Eqs. (C.2) and (C.3) as: By making the notation: (C.9)the composite attenuation from Eq. (C.1) becomes: (C.10)where (C.11)For the case that old = 0 in the optical and in the UV Eq. (C.10) becomes: (C.12)The use of the calibration factor Fcal in our procedure means that in practice the equations describing the attenuation due to the diffuse dust illuminated by the young stellar disk need to be rescaled to accommodate different values of F than those used in the calibration. For this we need to use the correction factor for the diffuse component corrd(F) as defined in Eq. (29), and rescale Eq. (C.5) in a similar way to the formulation of the radiation fields in Sect. 2.5.2: (C.13)In this case Eq. (C.11) becomes: (C.14)and Eq. (C.12) becomes: (C.15)Equation (C.14), together with Eqs. (C.10) and (C.11) are the analog of the original expression for the composite attenuation from Eq. (16) in Paper III. Equation (C.15) together with Eq. (C.12) are the analog of Eq. (17) from Paper III.

Appendix D: How to use the model to fit UV/optical to infrared/submm SEDs

The six physical parameters determining our model prediction of the SED are:

  • , SFR, F, old, B/D and i14


  • , SFR, F, old, and B/D

determine , the model prediction for the dust luminosity density in the IR/submm as given by Eq. (43) of this paper, whereas

  • , SFR, F, old, B/D and i

determine the attenuation in the measured UV/optical/NIR emission (whereby the dependence on SFR and old is a weaker dependence due to the dependence of the composite attenuation on the relative amplitudes of the young and old stellar populations, as described in Appendix C). This attenuation is given in magnitude form, Δmλ, by Eqs. (C.10), (C.11) and (C.14). In the following it is convenient to express it in the linear form: (D.1)We note that two of the six physical parameters, SFR and old, are extrinsic (that is, the quantity scales with the amount of material in an object). This is a consequence of our model galaxy having a fixed size, expressed in terms of the fixed reference scale length of the old stellar population in B-band of  kpc.

The dust emission SED of the diffuse component of a galaxy with a value15 for differing from will be:(D.2)\arraycolsep1.75ptwhere (D.3)where SFRmodel is the star-formation rate of the model galaxy having the reference size , oldmodel is the normalised luminosity of the old stellar disk population of the model galaxy having the reference size , SFR is the real star formation rate of the galaxy that we want to model, old is the real normalised luminosity of the old stellar disk population of the galaxy that we want to model, and (D.4)Equation (D.2) expresses the fact that radiation field energy density, and hence the colours of the dust emission, varies according to surface density of luminosity. In cases where a galaxy is unresolved ζ is unconstrained by the data, and becomes a further free parameter of the model. Since we may also not know the distance D to the galaxy, it is convenient to express the dust emission SED of the diffuse component from Eq. (D.2) as a flux density by dividing throughout by 4πD2, to obtain: (D.5)where θgal is the half angle subtended at the Earth by the actual B-band scalelength of the galaxy:

(D.6)We note that, provided the galaxy is sufficiently resolved for θgal to be measured, as expressed by Eq. (D.5) depends on the value of ζ (via SFR and old – cf. Eq. (D.3)), but is independent of the distance D.

Our model is constructed such that the total emitted luminosity Lλ,   star of UV and optical light powering the dust emission can be directly constrained from available measured apparent UV/optical spatially integrated fluxes, , corrected for attenuation (expressed here in linear form by Eq. (D.1)) as a function of , SFR, F, old, B/D, i and distance D:(D.7) where, as outlined in Sect. 5 of Paper III and in Appendix C of this paper the dependence on SFR, old and B/D is for the optical range only. Depending on the exact range of wavelengths for which is available, Lλ,   star can be integrated over wavelength to obtain constraints on the parameters SFR = SFRcon and/or old = oldcon as a function of and i. Specifically, if UV data is available we can require that (D.8)Combining Eq. (17) from Sect. 2.3.3 and Eqs. (D.7) and (D.8), this leads to: (D.9)Correspondingly, if optical data is available, we can require that(D.10) After manipulation of Eqs. (12), (13), (15) and (19) from Sects. 2.3.2, 2.3.2 and 2.3.3 and Eq. (D.7), this leads to: (D.11)where (D.12)(D.13)(D.14)Note that oldcon is a weak function of SFR as well as , B/D and i due to the need to take into account the contribution of optical photons by the young stellar population, as described in Sect. 2.3.1.

We are now in a position to determine the physical model parameters from a combined set of measured UV/optical flux densities , measured at wavelengths λiobs,   star, and IR/submm flux densities of the pure dust component (corrected for contamination by direct stellar light at short infrared wavelengths) , measured at wavelengths λiobs,   dust. To do so, we minimise the function(D.15) subject, if UV data is available, to the constraint from Eq. (D.9) and, if optical data is available, from Eq. (D.11). σiobs,   dust are the 1σ uncertainties in the measurements and Sλ,   dust is given by (D.16)where and are given by Eqs. (D.5) and (41), respectively. In the case that the distance, the optical structure and orientation parameters D, θgal, ζ, B/D and i are known the optimization problem posed by Eq. (D.15) is reduced from the parameter set (SFR, old, , F, B/D, ζ) to the parameter set (SFR, old, , F). With just 4 parameters, this might potentially be solvable purely considering the dust emission data, bearing in mind the orthogonal effect of these parameters on the amplitude and colour of the dust emission SEDs, (as described in Sect. 6) if at least 4 data points are available well sampling the whole MIR/FIR/submm range. However, this will often be the exception rather than the rule, and in any case the fit is primarily and more robustly constrained through the optical and UV measurements. This is firstly because, if both UV and optical measurements are available the number of the primary search parameters would be reduced from four to just two – and F. Secondly, and perhaps more significantly, the model would no longer have any degree of freedom in terms of scaling parameters, due to the fact that, as noted above, SFR and old are the only extrinsic parameters in the full parameter set.

Below we illustrate how to use the optical and UV constraints by giving a possible processing path for a galaxy with a known (spectroscopically determined) redshift, for which integrated UV and optical photometry were available, and for which B/D, i and θgal are known from optical imaging. The parameter set to be determined is thus , SFR, old, F:

  • step i): choose a trial value for each of and F

  • step ii): set SFR to SFRcon and old to oldcon from Eqs. (D.9) and (D.11) for the trial values of and F.

  • step iii): find SFRmodel and oldmodel from Eq. (D.3), substituting for ζ as defined in Eq. (D.4). The value of used in Eq. (D.4) should be derived from the optically measured value using the correction factors tabulated as a function of i and in Tables 1−5 of Paper IV.

  • step iv): using the trial values of and F, together with SFRmodel and oldmodel and ζ from step iii) find .

  • step v): substitute from step iv) into Eq. (D.5) to compute . Compute (note the use of the extrinsic value of SFR = SFRcon here). Substitute the values for and determined at the wavelengths of the IR/submm observations in Eq. (D.15) to determine χ2 for the trial combination of and F.

  • step vi): repeat steps i) to v), until the combination of and F that minimises χ2 is found. The values of SFR and old from steps ii) to vi) found for the pair of and F that minimises χ2 are then best fit parameters in SFR and old.

We note that in the case of edge-on galaxies UV data should not be used to constrain SFR, since typically only a few percent of the total UV disk luminosity will be seen, and the solution can be subjected to stochastic variations, because the received photons may only be emitted by a small numbers of star-forming regions. In this case the SFR is better constrained from the FIR emission, as modelled for NGC 891 (Paper I and Sect. 3).

Having found the best fit parameter set, the following two further steps can be made:

  • step vii: deredden the UV/optical/NIR spectrum using the fittedvalues of , F, and the measured B/D and i.

  • step viii: apply a population synthesis modelling fit to the dereddened UV/optical spectrum from vii to find the star formation history.

Appendix E: Tables describing the fixed parameters of our model

Table E.1

Geometrical parameters of the model, as defined in Paper III and Sect. 2.1. All length parameters are normalised to the B-band scalelength of the disk of the standard modal galaxy, as derived for NGC 891:  pc .

Table E.2

Intrinsic spectral luminosity densities of the old and young stellar populations, as defined in Sect. 2.3, for old = 1 and SFR = 1   M   yr-1, respectively, and tabulated at the sampling wavelengths of the radiation fields.

Table E.3

Parameters of the dust model considered in this paper (see Sect. 2.4). For parameter definition see Weingartner & Draine (2001).

Table E.4

Wavelength dependence of the fraction of photons escaping from the clumpy component into the diffuse medium as described in Sect. 2.2, for F = Fcal = 0.35.

© ESO, 2011

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