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Astron. Astrophys. 338, 807-812 (1998)

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2. Dust mass evaluation from FIR observations

The dust mass can be derived both from optical and from FIR observations. The value of the mass depends on the physical-chemical properties of the solid particles (i.e. grain radius, grain density and emissivity). The usual approach consists of assuming some values for the grain properties (Hildebrand 1983), while a color temperature is derived from the FIR emission. Different authors, however, present slightly different formulae for the evaluation of the dust mass (Thronson & Telesco 1986, Greenhouse et al. 1988, Young et al. 1989, Roberts et al. 1991 and Thuan & Sauvage 1992). The differences between these relations are only due to different assumptions on the grain parameters and to different derivations of the color temperature. It should be noticed that, within a flux uncertainty of 10[FORMULA] (which is typical of the high quality IRAS data), all the dust mass values obtained by using the different formulae are in agreement.

A single temperature model is a rough approximation in describing a galactic environment. The dust is in fact heated by the radiation field, which in turn depends on the sources of luminosity and on their spatial distribution in the galaxy. The total FIR emission is thus likely due to the contribution of dust at different temperatures. Moreover, the IRAS FIR measurements are not adequate to detect the emission coming from cold dust (10-20 K) which peaks at wavelengths between 200 and 300 µm.

I adopt the dust temperature distribution given by Kwan & Xie (1992):

[EQUATION]

[EQUATION]

[FORMULA] and [FORMULA] are the lower and upper limits of the temperature T; [FORMULA] and [FORMULA] are free parameters that determine the shape of the distribution. The equations relating the temperature distribution [FORMULA] to the luminosity emitted by the dust and to the dust mass are detailed in Kwan & Xie (1992).

Due to the observed spectral range, I adopt [FORMULA]=7 K and [FORMULA]=60 K, taking into account only temperature distributions peaking at a value intermediate between [FORMULA] and [FORMULA] and excluding those pairs of [FORMULA] and [FORMULA] which produce unrealistic distributions (as, for instance, monotonically increasing or decreasing functions).

The main problem in the present approach is to select the proper values for the parameters [FORMULA] and [FORMULA], the choice being constrained by the ratio of the flux densities at two different wavelengths.

I first identify a range of [FORMULA] and [FORMULA] pairs which produce the observed flux ratio. Since the same flux ratio can be obtained by functions having quite different shapes, a further constraint is needed. Unfortunately, for none of the galaxies in the sample submillimeter observations are available. Nevertheless, their color temperature may be derived from the flux density ratio (Henning et al. 1990). Taking into account the uncertainty in the computed color temperature, I select those distributions [FORMULA] that produce the observed flux density ratio and whose peak temperature is comparable to the color temperature. The selected family of [FORMULA] functions obviously satisfies two conditions which are not independent (being both related to the flux ratio). This fact could in principle affect the reliability of the results. It turns out, however, that pairs of [FORMULA] and [FORMULA] which satisfy the same constraints give the same dust mass within the flux uncertainty.

In order to check the method a simple numerical simulation has been performed. I considered an artificial galaxy which is not resolved by IRAS both at 60 and 100 µm with a given dust mass and, then, I evaluated the dust content following the present method. The galaxy dust mass [FORMULA] may be roughly estimated by using the equation

[EQUATION]

where r is the dust grain radius, [FORMULA] is the dust grain density and [FORMULA] is the number of dust grains. Eq. 3 implies the following approximations. The dust grain radius r characterizes the dust grains which contribute to the thermal emission at FIR wavelengths. Actually, the uncertainties in the total dust mass introduced by using an average radius instead of a size distribution, are much smaller than those arising from uncertainties in the dust emissivity spectral trend and in the dust temperature distribution (Kwan & Xie 1992). [FORMULA] is the total number of the dust grains within the observing beam. Since the galaxy is not resolved by IRAS, as it is often the case, Eq. 3 gives the total dust mass of the point source. A dust radius of 0.1 µm and a dust grain density of 3 g cm-3 are currently adopted (Hildebrand 1983) and also used in the present article.

Concerning the dust emissivity, the power-law approximation [FORMULA] is used in the computations, with [FORMULA] and [FORMULA] (Hildebrand 1983).

By assuming a value for [FORMULA] and the temperature distribution [FORMULA], and accounting for the spectral responses of the IRAS detectors at 60 and 100 µm, the luminosity emitted by the [FORMULA] dust grains, as observed within the filter bandpass, can be estimated. The flux ratio and the color temperature of the galaxy are then computed, while the dust mass of the source is derived from [FORMULA] by Eq. 3.

A galaxy with [FORMULA]=[FORMULA] [FORMULA] turns out to contain about [FORMULA]=1.6[FORMULA] dust grains. The galaxy distance is assumed to be 10 Mpc. By assuming a [FORMULA] with [FORMULA] and [FORMULA] the flux ratio turns out to be about 0.5. I use the flux ratio as a constraint to select the pairs [FORMULA]-[FORMULA] producing a [FORMULA] whose peak temperature is comparable to the galaxy color temperature of 36 K. Among these pairs also the pair [FORMULA] and [FORMULA] is found (i.e. exactly the values adopted for the input [FORMULA]). For the selected pairs of parameters I compute the dust mass which ranges between [FORMULA] [FORMULA]. Therefore, the derived values are in agreement with the dust mass of the galaxy within the flux uncertainties, thus confirming the reliability of the method. Futhermore, by using the single temperature model and taking into account the different formulae available (see Sect. 2) a dust mass of [FORMULA] [FORMULA] is derived. This result shows that the single temperature model underestimates the dust content.

The same test was performed for different "artificial galaxies" with different temperature distributions obtaining always consistent results.

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© European Southern Observatory (ESO) 1998

Online publication: September 17, 1998
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