4 Results

   
4.1 Spectral energy distributions

The fluxes from our observations are listed in Table 1. The 60 and 90 $\mu$m values agree within 20% with the IRAS 60 and 100 $\mu$m ones. For many sources, where IRAS provided only upper limits at 12 or 25 $\mu$m, now the 10-25 $\mu$m fluxes could be measured. Also, for 16 sources out of 41, submm/mm fluxes and good upper limits are provided, as well as NIR 1.2 and 2.2 $\mu$m fluxes for seven sources.

The spectral energy distributions (SEDs, as measured, not corrected for redshift) are shown in Fig. 1, supplemented by literature data. The remarkable features of the SEDs are:

1)

For each galaxy the maximum of the SED can now be clearly determined. It lies between 60 and 100 $\mu$m. The ISOPHOT long wavelength filters beyond 100 $\mu$m clearly outline the beginning of the Rayleigh-Jeans branch. In some cases (e.g. NGC 6240, 17208-0014) a somewhat plateau-like broad maximum is revealed, suggesting a high opacity even in the FIR and/or the presence of various cool to cold dust components (see also Fig. 3);

2)

For 20 sources the shape of the Rayleigh-Jeans branch can now be determined by the mm and submm data points. As discussed in Sects. 3.1 and 3.2, within the error budget the effects of different instrumental beams and a possible contribution of extended flux appear to be negligible. In particular for those cases (e.g. NGC 6240, 17208-0014), where ISOPHOT already indicated a broad maximum, the submm measurements confirm the high flux prediction from the 100-200 $\mu$m measurements. Also, where ISOPHOT indicated a steep Rayleigh-Jeans branch (e.g. 16090-0139, 23365+3604, 23389-6139) the mm and submm fluxes and upper limits are low, demonstrating the consistency between the instruments;

3)

Shortward of the maximum at around 60-100 $\mu$m the SEDs exhibit two basic shapes:

i)

a flat NIR plateau followed by a jump-like flux increase at about 10 $\mu$m (e.g. Arp 220 and 12112+0305). In some cases also indications of PAH emission around 7.7 $\mu$m and silicate 9.7 $\mu$m absorption features (e.g. Arp 220 and NGC 6240) are recognized;

ii)

a power-law-like flux increase from the NIR to the MIR (e.g. Mrk 463) or FIR (e.g. Mrk 231). Spectral PAH emission or silicate absorption features may be present, e.g. for Mrk 463, compare also with Fig. 3 in Rigopoulou et al. (1999), but due to the high continuum level they are diluted in the broad band photometry.

   
4.2 CO line and synchrotron emission contamination of submm/mm fluxes

The determination of the Rayleigh Jeans branch allows a detailed analysis with respect to the dust emissivity $\lambda^{-\beta}$ and the opacity $\tau_{\rm 100~ \mu m}$, as carried out in the next section, under the condition that the emission is of thermal nature. Therefore, beforehand one has to check possible contamination of the submm and mm fluxes by CO lines and/or synchrotron emission:

1)

For all our sources observed at 1300 $\mu$m with SEST the CO(2-1) line at 230 GHz (1304 $\mu$m) moves out of the filter band pass due to their redshifts of ;

2)

The CO(3-2) line at 345 GHz (869 $\mu$m) may contribute to the SCUBA 850 $\mu$m fluxes for those sources with . Direct CO(3-2) observations are only available for Arp220 (Mauersberger et al. 2000). Therefore, we estimated CO(3-2) line strengths from the CO(1-0) fluxes (Solomon 1997; Downes & Solomon 1998; Gao & Solomon 1999), adopting an intensity ratio $R = I_{\rm CO\,(3-2)}$/ $I_{\rm CO\,(1-0)} = 0.9$ as found for Arp 220. The strength of the redshifted CO(3-2) line, however, is reduced according to the transmission of the SCUBA 850 $\mu$m filter which lies between 55% and 30% for our sources with . It turns out that only four sources have a CO(3-2) line contribution to the 850 $\mu$m flux which exceeds 5%: Arp 220 and Mrk273 (both 11%), Mrk231 (22%), and NGC6240 (35%). These contributions are smaller or of the order of our adopted photometric uncertainties. Therefore we decided not to correct for the CO(3-2) line contribution;

3)

In 20 cases with available radio fluxes (Condon et al. 1990; Crawford et al. 1996) extrapolations with spectral indices between 0.5 and 1.0 from the cm range towards shorter wavelengths are far below (<10%) our measured submm and mm fluxes (many of the extrapolations lie even below the range plotted in Fig. 1). The only exception is Mrk463 (with two Seyfert nuclei) where the 1.3 mm flux is variable, thus dominated by synchrotron emission (Chini et al. 1989a; Marx et al. 1994).

Our conclusion is therefore, that, like the FIR emission, the bulk of the submm/mm flux is of thermal nature being emitted by dust.

   
4.3 Dust parameters

In order to characterise the dust emission, the SEDs are fitted with modified blackbodies. Such fits, however, are not unique. They rely largely on the mass absorption coefficient $\kappa$ and its wavelength dependence $\beta$, both still being a matter of debate. Values of $\beta$ between 1 and 2 are commonly used (e.g. Hildebrand 1983). In case of a flat Rayleigh-Jeans tail the SEDs can also be modelled by several dust components. Since the interpretation of the dust emission as well as the derivation of the dust mass depend on the blackbody models used, we investigate the two main cases. They represent simplified formalisms, each relying on implicit assumptions, and a realistic description probably lies between these two extremes. In the following two subsections the FIR-submm range is investigated, and the MIR part is addressed in the third subsection.

   
4.3.1 Single modified blackbody

We used the following model:

$$S_{\lambda} = {B_{\lambda}(T)} \cdot (1 - {\rm e}^{- \tau_{\lambda} }) $, with $$$ (1)

$$\tau_{\lambda} = \tau_{ 100~\mu{\rm m}} \cdot (100~\mu{\rm m} / \lambda )^{\beta}.$$ (2)

The SEDs are fitted between 60 and 1300 $\mu$m with a single modified blackbody leaving T, $\beta$ and $\tau_{\rm 100~ \mu m}$ free (minimising $\chi$$^{\rm 2}$ in a grid search).

Figure 2: Distribution of $\tau_{ 100~\mu\rm m}$ versus $\beta$ for those sources with measured submm/mm fluxes. Different symbols correspond to optical spectral types as in Table 4: + Seyfert1, $\times$ Seyfert2, $\blacklozenge$ LINER, $\blacksquare$ HII/SB, and $\bullet$ for not classified; the quasar PG0050+124 denoted by $\ast$ is also included for comparison (from Haas et al. 2000a). Arrows indicate lower limit cases.

Although the emissivity exponent $\beta$ is a free parameter in Eq. (2), it is kept constant over the whole wavelength range. In order to keep the parameter space under control, we have decided not to introduce a $\lambda$ dependence of $\beta$.

The relation of the parameters $\beta$, $\tau_{ 100~\mu\rm m}$ and T with the SED shapes is:

$\bullet$

$\beta$ corresponds to the slope of the Rayleigh-Jeans tail, whether it is flat (small $\beta$) or steep (large $\beta$);

$\bullet$

$\tau_{ 100~\mu\rm m}$ determines, whether the peak plateau of the SED is narrow (small $\tau_{ 100~\mu\rm m}$) or broad (large $\tau_{ 100~\mu\rm m}$);

$\bullet$

Like for normal blackbodies (Wien's displacement law), T is related to the wavelength, at which the SED maximum is located.

Fits were performed only for the "mm-subsample'', i.e. for those 22 sources with submm/mm fluxes available. We also used the 8 cases of upper limits, treating them formally as detections, and after the fitting procedure taking into account that they provide lower limits for $\beta$. In some cases, where the ISO 60 or 90 $\mu$m fluxes show a large error or deviate strongly from the IRAS 60 and 100 $\mu$m fluxes, we also included the IRAS data (e.g. for 16090-0139).

The fitted parameters $\beta$, $\tau_{ 100~\mu\rm m}$ and T are listed in Table 2, together with $\chi^{2}$. A visual impression of the quality of the fits is given in Fig. 1. In summary, the results for the mm-subsample are:

$\bullet$

    1.2 < $\beta$ 2.0 (Fig. 2)

$\bullet$

    0.5 < $\tau_{ 100~\mu\rm m}$ < 5 (Fig. 2)

$\bullet$

    50 K < T < 70 K

with an uncertainty of about 0.2 for $\beta$, 25% for $\tau$ and 5-10 K for T. The quoted uncertainties have been estimated considering "bent'' SEDs with the FIR fluxes reduced and the submm fluxes increased by their measurement uncertainty and vice versa. It should also be noted that for sources with significant MIR flux (an extreme case is Mrk463) additional uncertainties in the determination of T and/or $\tau_{\rm 100~ \mu m}$ are introduced when using the full 60 $\mu$m flux, which is partially due to a dust component not considered here.

The quoted parameter values should be considered with some tolerance and their interdependence borne in mind:

$\bullet$

For two SEDs with maxima located at similar wavelengths, T decreases with $\beta$. Due to the steeper fall-off at long wavelengths the width of the peak plateau can only be maintained, if the maximum is shifted to longer wavelengths. A complementary pair are 19254-7245 ( $\beta = 1.2$) and 20046-0623 ( $\beta = 1.8$).

$\bullet$

For two SEDs with similar $\beta$, but largely different $\tau_{\rm 100~ \mu m}$, T increases with $\tau_{\rm 100~ \mu m}$. Due to the broader width, but similar shape on the long wavelength side, the maximum is shifted towards shorter wavelengths. A complementary pair is Mrk231 and 14348-1447.

On the other hand, there seems to be no correlation between $\tau_{\rm 100~ \mu m}$ and $\beta$ as can be seen from Fig. 2.

The dust parameters were determined via Eq. (1) only for the mm-subsample. For the remaining sources with wavelength coverage limited to 200 $\mu$m (IR-subsample) $\beta$ could not be fitted reliably (as we found from tests with the mm-subsample using only the 60-200 $\mu$m fluxes). For the IR-subsample we kept $\beta$ fixed using the average value $\beta = 1.6$ derived from the mm-subsample. Then $\tau_{ 100~\mu\rm m}$ and T could be determined reasonably well from the 60-200 $\mu$m fluxes alone. (Exceptions are 00262+4251, 15462-0450, 18090+0130 and 19458+0944 which have less complete spectral coverage due to bad quality measurements as flagged in Table 1. In these cases $\tau_{ 100~\mu\rm m}$ was fixed to 6.0). The resulting values lie in the same range as for the mm-subsample (Table 2 and Fig. 1). As a check, we fitted also $\tau_{ 100~\mu\rm m}$ and T of the mm-subsample with a fixed $\beta = 1.6$ using only the 60-200 $\mu$m fluxes. The results are basically consistent with those obtained from the longer wavelength coverage, except for the sources with extremely low or high true $\beta$. Hence, in the discussion below we can mostly use the full sample, and only where $\beta$ plays a role, we confine it to the mm-subsample.

   
4.3.2 Multiple modified blackbodies

As derived in the previous section, for the majority of the "mm-subsample'' sources (11 out of 14, not having lower limits for $\beta$, one of them having $\tau_{100~\mu{\rm m}} = 5$, Fig. 2) it is not possible to fit the FIR-submm SEDs properly with one single modified blackbody with an emissivity law of $\lambda$$^{\rm -2}$, rather the superposition of two or more modified blackbodies is required.

In the low opacity case Eq. (1) can be approximated by

$$S_{\lambda} = {B_{\lambda}(T)} \cdot \tau_{\lambda} \propto {B_{\lambda}(T)} \cdot \lambda ^{- \beta}.$$ (3)

Based on Eq. (3), implicitly assuming $\tau_{100~\mu{\rm m}} \ll 1$, we fitted several modified blackbodies with free T and fixed $\beta = 2$ to all observed SEDs, as illustrated for some examples in Fig. 3. They have temperatures in the range between 30 and 50 K (cool), and between 10 and 30 K (cold). However, the decomposition into the various components is not unique, because of the sparse data coverage in the sub-mm range.

Figure 3: Examples for fits with several optically thin ( $\tau_{100~\mu\rm m} \ll$ 1) modified blackbodies (BBs) with an emissivity law of $\lambda$$^{\rm -2}$. Top row: superposition of two BBs; middle row: possible superposition of three BBs; bottom row: fits can also be done for those SEDs with no submm data: approximation by two BBs.

In the case of multiple blackbodies no direct conclusion about the opacity $\tau_{ 100~\mu\rm m}$ can be drawn (nevertheless, in Sect. 5.1.2 below, $\tau_{ 100~\mu\rm m}$ will be constrained using CO data). The most realistic case might be that of several blackbodies with $\beta \approx 2$, and a range of opacities from low to partly high.

4.3.3 Description of the NIR-MIR emitting dust

The two basic SED shapes in the NIR-MIR outlined in Sect. 4.1 can be formally fitted by a superposition of several warm dust components. For the cases with flat NIR plateau the maximum temperatures are about 100-150 K (e.g. Klaas et al. 1997, 1998a). The power-law-like SEDs can be approximated by a suite of blackbodies up to the dust grain evaporation temperatures of about 1000-1500 K (the hotter the blackbody the less dust mass is involved). Modelling of the continuum is hampered by the presence of strong spectral features like PAH emission and silicate absorption. Using higher spectral resolution, Laurent et al. (2000) investigated this spectral part quantitatively.

   
4.4 Luminosities

Table 2 lists the luminosities derived within various bandpasses in the rest frame of the objects by integrating the spectral energy distribution as outlined by the thick solid and dash-dotted lines shown in Fig. 1 for the indicated wavelength ranges. On the Rayleigh-Jeans tail and around the SED maximum this comprises the single blackbody curve obtained with the Eq. (1) fit, and shortward thereof the lines connecting the data values by linear interpolation.

The total IR-submm luminosity $L_{\rm 10-1000~\mu m}$ is dominated by the FIR in the wavelength range 40-150 $\mu$m, while the 150-1000 $\mu$m submm range plays a minor role ( $L_{\rm 150-1000~\mu m} < 0.1 \cdot L_{\rm 40-150~\mu m}$) as well as the 10-40 $\mu$m MIR range (except for Mrk463 and the z > 0.3 sources which are MIR dominant). The luminosities $L_{\rm 8-1000~\mu m}$ extrapolated from the four IRAS bands (formula cf. Table 1 in Sanders & Mirabel 1996) typically slightly overestimate our IR-submm luminosity values by about 15%; nevertheless this is still a good agreement.

The MIR/FIR luminosity ratio has a median value of about 0.3. Thus, the sample of bright nearby ULIRGs preferentially comprises objects with cool MIR/FIR colours (compared with quasars having $L_{\rm MIR}$/ $L_{\rm FIR} > 1$, cf. Haas et al. 2000a). Though the luminosity range of the ULIRG sample spans about one decade, there is no trend of luminosity with optical spectral type or MIR/FIR colours.

For sources without submm/mm observations available, the submm luminosity $L_{\rm submm} = L_{\rm 150-1000~ \mu m}$ is extrapolated using the average value $\beta = 1.6$ (Sect. 4.3.1), while the actual $L_{\rm submm}$ depends on the actual value of $\beta$. A check on the mm-subsample shows that, for the case of minimum $\beta \approx 1.2$ or maximum $\beta \approx 2$, $L_{\rm submm}$ obtained using the average $\beta = 1.6$ can deviate from the true value by factors of 2 and 0.5, respectively.

   
4.5 Dust masses

In order to derive the dust masses, we used the standard approach based on Hildebrand (1983) and further developed by various authors (e.g. Chini et al. 1986; Krügel et al. 1990):

$$M_{\rm dust} = \frac{ 4 \pi \cdot a^3 \cdot \rho } { 3 \cdot... ...ac{ D ^{2} \cdot S_{850 ~\mu\rm m} } { B_{850 ~\mu\rm m}(T) }$$ (4)

with
$\bullet$ average grain size a = 0.1 $\mu$m;
$\bullet$ grain density $\rho$ = 3 g cm$^{\rm -3}$;
$\bullet$ distance D;
$\bullet$ flux $S_{850~ \mu\rm m}$ (in the restframe of the object);
$\bullet$ Planck function $B_{850~ \mu\rm m}$(T) at restframe 850 $\mu$m; and
$\bullet$ dust grain emission efficiency $Q_{\rm em}$(850 $\mu$m, $\beta$) where

$$Q_{\rm em} (\lambda, \beta ) = Q_0 \cdot a \cdot \{ 250 / \lambda \} ^ \beta$$ (5)

with $Q_{\rm0} = 40$ cm$^{\rm -1}$ for $\lambda = 250~\mu$m.
The first fraction in Eq. (4) is conveniently summarized yielding

$$M_{\rm dust} = \frac { 1 } { \kappa_{850~ \mu\rm m} ( \beta ... ...{ D ^{2} \cdot S_{850~ \mu\rm m} } { {B_{850~ \mu\rm m}(T)} }$$ (6)

with

$$\kappa_{850~ \mu\rm m}(\beta) = \kappa_{850~ \mu\rm m}^{\beta = 2} \cdot \{ 250~ \mu\rm m / 850~ \mu\rm m \}^{\beta - 2}$$ (7)

with

$\bullet$      $\kappa_{850~ \mu\rm m}^{\beta = 2} = 0.865~{\rm cm}^{2}\,{\rm g}^{-1}$

which corresponds to

$\bullet$      $\kappa_{1300~ \mu\rm m}^{\beta = 2} = 0.4~{\rm cm}^{2}\,\rm g^{-1}$.
Our "choice'' of $\kappa$ is consistent with that favoured by Krügel et al. (1990, see their Eq. (10)), and Lisenfeld et al. (2000). Note that we account for $\beta$ in the wavelength dependence of $Q_{\rm em}$ and $\kappa$, respectively. In the case of $\beta < 2$ this leads to a larger value of $\kappa$, hence to dust masses which are smaller than for $\beta = 2$.

 

 

Table 2: Infrared luminosities, dust temperatures, dust masses and IR source sizes. The luminosity distance is determined as D_L = c/$H_{\rm0}$*( $z + z^{\rm 2}$/2), i.e. $q_{\rm0} = 0$, with $H_{\rm0} = 75$ kms^-1/Mpc. The infrared luminosities $L_{\rm MIR}$ (3-40$~\mu$m), $L_{\rm FIR}$ (40-150$~\mu$m) and $L_{\rm submm}$ (150-1000$~\mu$m) are determined from the SED curves shown in Fig. 1, by integrating over the indicated wavelength range shifted to the rest frame of the objects. The dust mass is derived for three models of the IR emission: 1)  $M_{\rm d}^{\rm\tau free}$ for the single modified blackbody with free T, $\tau$ and $\beta$ as described by Eq. (1) (minimising $\chi ^{\rm 2}$) and shown in Fig. 1 ($\beta$ values in brackets $<\,>$ means that a fixed average of $\beta = 1.6$ is used, lower limits are indicated by a > sign). 2)  $M_{\rm d(FIR)}^{\rm\beta = 2}$ for a single modified blackbody with fixed $\beta = 2$ fitted by Eq. (3) to the FIR 60-200 $\mu$m range (for details see Sect. 4.5). 3)  $M_{\rm d(total)}^{\rm\beta = 2}$ for multiple modified blackbodies with fixed $\beta = 2$. The sizes of the dust emitting regions are given for the two cases of a single blackbody as (1) brightness radius $r_{\rm b}$, and (2) as "smallest transparent radius'' $r_{\rm\tau }$, at which $\tau_{\rm 100~\mu m} = 1$ (for details see Sect. 4.6). For comparison M(H$_{\rm 2}$) is listed (using the galactic conversion factor M(H $_{\rm 2}) = 4.6 \cdot L_{\rm CO}$).

Name	z	D	L $_{\rm MIR}$	L $_{\rm FIR}$	L $_{\rm submm}$	T	$\tau$ $_{\rm 100}$	$\beta$	$\chi ^{\rm 2}$	M $_{\rm d}^{\rm\tau free}$	r$_{\rm b}$	r$_{\rm b}$	T	M $_{\rm d(FIR)}^{\rm\beta = 2}$	r $_{\rm\tau}$	r $_{\rm\tau}$	M $_{\rm d(total)}^{\rm\beta = 2}$	M(H$_{\rm 2}$)
			10-40 $\mu$m	40-150	150-1000									single BB			several BBs
		[Mpc]	[10$^{\rm 9}$ $L_{\odot}$]	[10$^{\rm 9}$ $L_{\odot}$]	[10$^{\rm 9}$ $L_{\odot}$]	[K]				[10$^{\rm 6}$ $M_{\odot}$]	[pc]	[ $\hbox {$^{\prime \prime }$ }$]	[K]	[10$^{\rm 6}$ $M_{\odot}$]	[pc]	[ $\hbox {$^{\prime \prime }$ }$]	[10$^{\rm 6}$ $M_{\odot}$]	[10$^{\rm 9}$ $M_{\odot}$]
00199-7426	0.0963	403	307	1139	130	48	2.50	<1.6>	1.07	109	390	0.199	32	160	1097	0.6	549
00262+4251*	0.0971	407	342	675	77	68	6.00	<1.6>	1.24	28	131	0.066	36	48	602	0.3	762	29a
00406-3127	0.3422	1602		2480	101	53	0.50	<1.6>	1.04	51	688	0.088	43	59	666	0.1	157
03068-5346	0.0778	323	126	481	37	60	2.50	<1.6>	1.07	22	172	0.109	36	32	488	0.3	152
03158+4227	0.1343	573	685	1724	71	77	2.00	<1.6>	1.09	26	220	0.079	43	42	563	0.2	115
03538-6432	0.3100	1431		2877	188	68	5.00	<1.6>	1.06	72	263	0.038	39	110	910	0.1	519
04232+1436	0.0799	332	244	540	34	51	1.00	<1.6>	1.04	23	275	0.170	37	33	495	0.3	73	41a
05189-2524	0.0425	173	460	500	33	70	2.50	1.4	1.07	15	128	0.152	38	26	443	0.5	448$^{\rm +}$	23b
06035-7102	0.0794	330	376	785	40	49	0.50	<1.6>	1.09	29	434	0.271	38	35	513	0.3	82	38b
06206-6315	0.0924	386	271	878	54	61	2.00	<1.6>	1.09	30	225	0.120	37	50	611	0.3	130	52b
12112+0305	0.0723	299	264	1052	68	53	1.00	1.5	1.06	42	355	0.244	36	71	731	0.5	856
Mrk 231	0.0417	170	1215	1219	58	54	1.00	1.9	1.02	22	371	0.449	50	28	456	0.5	131$^{\rm +}$	35a
Mrk 273	0.0373	152	248	714	47	62	2.00	1.6	1.14	27	208	0.282	36	47	596	0.8	104$^{\rm +}$	23a
Mrk 463	0.0506	207	347	136	5	52	0.50	2.0	1.06	4	188	0.187	40	5	198	0.2	12
14348-1447	0.0811	337	294	1037	79	67	5.00	2.0	1.12	37	194	0.118	35	81	781	0.5	219$^{\rm +}$	64c
14378-3651	0.0676	279	199	651	49	69	5.00	>1.7	1.10	26	139	0.103	36	44	579	0.4	423	15b
15245+1019	0.0756	314	138	621	39	51	1.00	<1.6>	1.15	27	300	0.197	37	38	534	0.4	97
15250+3609	0.0553	227	304	461	17	59	0.50	1.3	1.10	10	215	0.195	44	11	289	0.3	28
Arp 220	0.0182	73	190	820	86	61	5.00	1.7	1.09	66	214	0.600	40	32	488	1.4	1479$^{\rm +}$	32a
15462-0450*	0.1005	422	397	763	109	59	6.00	<1.6>	1.12	52	179	0.087	34	73	741	0.4	1498
16090-0139	0.1334	569	635	1944	85	49	0.50	>1.9	1.15	43	645	0.233	40	66	708	0.3	182	56a
NGC 6240	0.0245	99	157	347	28	57	2.50	1.5	1.07	21	154	0.320	33	36	520	1.1	580$^{\rm +}$	37a
17208-0014	0.0424	173	235	1226	107	60	3.00	1.7	1.09	64	274	0.326	34	116	936	1.1	602$^{\rm +}$	32a
17463+5806	0.3411	1596		2192	87	59	1.00	<1.6>	1.10	40	434	0.056	43	57	658	0.1	142
18090+0130*	0.0660	273	322	1100	193	46	6.00	<1.6>	1.06	173	319	0.241	28	322	1558	1.2	1746
18470+3234	0.0788	327	270	537	35	74	6.00	<1.6>	1.07	14	92	0.058	38	26	442	0.3	113
19254-7245	0.0615	253	375	473	32	72	3.00	1.2	1.04	15	108	0.088	38	24	429	0.3	1265$^{\rm +}$	35b
19458+0944*	0.0995	418	602	1343	307	46	6.00	<1.6>	1.21	226	373	0.184	28	388	1711	0.8	6405	55a
20046-0623	0.0845	352	344	658	47	60	3.00	>1.8	1.18	26	188	0.110	34	57	654	0.4	123
20087-0308	0.1055	444	337	1370	103	61	4.00	>2.0	1.10	50	254	0.118	35	110	911	0.4	239$^{\rm +}$	74a
20100-4156	0.1295	551	622	1919	135	67	3.50	>1.5	1.07	65	252	0.094	40	86	803	0.3	337
20414-1651	0.0870	363	265	836	38	65	1.50	<1.6>	1.12	19	206	0.117	40	29	471	0.3	88
ESO 286-19	0.0426	174	272	460	16	60	0.50	<1.6>	1.05	8	225	0.266	44	11	289	0.3	28	22b
21130-4446	0.0925	387	130	772	98	52	2.50	>1.2	1.19	71	262	0.140	32	111	916	0.5	932
21504-0628	0.0775	322	214	425	21	75	2.50	>1.6	1.10	8	106	0.068	40	16	343	0.2	39
22491-1808	0.0760	315	268	661	40	73	3.00	1.7	1.10	16	143	0.093	39	32	488	0.3	213$^{\rm +}$	31b
ESO 148-2	0.0446	182	248	502	33	68	3.00	<1.6>	1.05	17	136	0.154	36	34	504	0.6	83	18b
23230-6926	0.1062	447	327	1038	88	66	4.00	1.5	1.12	42	187	0.086	36	66	705	0.3	1565
23365+3604	0.0645	266	250	704	47	68	4.00	2.0	1.12	21	159	0.123	36	47	595	0.5	108$^{\rm +}$	39a
23389-6139	0.0927	388	182	753	56	63	4.00	>1.9	1.12	26	173	0.092	35	59	668	0.4	135
23515-3127	0.3347	1562		2323	105	50	0.50	<1.6>	1.04	62	759	0.100	42	58	662	0.1	184

^* Measurements partly distorted and uncertain, therefore $\tau$ $_{\rm 100~ \mu m}$ has been fixed to 6.0.
⁺ Good determination of total dust mass, data used for estimate of gas-to-dust ratio (see Sect. 5.1.2).
^a From Solomon et al. (1997), Table 2.
^b From Mirabel et al. (1990); recalculated by Gao & Solomon (1999).
^c From Sanders et al. (1991); recalculated by Gao & Solomon (1999).

Table 2 lists the dust masses derived from flux values at 850 $\mu$m, where the emission is certainly transparent (and no mass is "hidden''). For the further discussion, three different kinds of dust mass are determined for each galaxy:

(1) For the assumption that the FIR emission can be modelled by one single blackbody with free $\tau$, $\beta$ and T, as described by Eq. (1) and shown in Fig. 1.

For the case of several components with fixed $\beta = 2$ (Eq. (3)) the temperature decomposition is not unique (Sect. 4.3.2), and the uncertainty in the derived total dust mass becomes large, in particular for those sources without any submm/mm data. Therefore Table 2 lists two estimates:

(2) The total dust mass $M_{\rm d(total)}^{\rm\beta = 2}$ associated with the multiple "optically thin'' blackbodies.

(3) The dust mass associated with the "bulk FIR emission component'' $M_{\rm d(FIR)}^{\rm\beta = 2}$.

While $M_{\rm d(total)}^{\rm\beta = 2}$ may be considered as a maximum dust mass, $M_{\rm d(FIR)}^{\rm\beta = 2}$ is derived under the assumption of one single optically thin $\lambda^{\rm -2}$ modified blackbody of $T \approx 35$-40 K fitted to the FIR 60-200 $\mu$m range. It contributes the bulk of the luminosity and may be considered as a firm lower limit for the dust mass, in particular if other colder dust components exist. $M_{\rm d(total)}^{\rm\beta = 2}$ contains contributions from cold dust components, it is typically much larger than $M_{\rm d(FIR)}^{\rm\beta = 2}$.

For all cases, the dust mass does not show any correlation with the total, mid- or far-infrared luminosity. But the dust mass is quite well correlated with the submm luminosity $L_{\rm 150-1000~ \mu\rm m}$.

   
4.6 Size of dust emitting regions

The smallest possible extent of the FIR emitting region is listed in Table 2. For the case of an opaque blackbody (Eq. (1)), the brightness radius $r_{\rm b}$ is determined via

$$r_{\rm b} = \left\{ \frac {D ^{2} \cdot S_{100~ \mu\rm m} } ... ...- {\rm e}^{- \tau_{100~ \mu\rm m} }) } \right\}^{\rm 1/2}\cdot$$ (8)

For the case of a transparent blackbody with fixed $\beta = 2$ (Eq. (3)), the low opacity condition $\tau_{ 100~\mu\rm m} < 1$ requires that the extent of the dust emission cannot be smaller than a minimum size. Such a "smallest transparent radius'' can be roughly estimated as follows: the dust mass (Eq. (6)) - as determined for the bulk FIR emitting component at $T \approx 30$-40 K - is distributed homogeneously in a "minimum face-on disk'' of radius $r_{\rm\tau }$. Thereby the following standard conversions are used: 0.1 $M_{\odot}$$\cdot$pc $^{\rm -2} \Leftrightarrow \tau_{\rm V} = 0.4$ (Whittet 1992), and $\tau_{\rm 100~\mu m} = 0.006 \cdot \tau_{\rm V}$ (Mathis et al. 1983). Thus

$$r_{\tau} = \left\{ \frac {M_{\rm dust} [M_{\odot}]} { \tau_{... ...~ \mu m} \cdot 41.7 \cdot \pi } \right\}^{\rm 1/2} [{\rm pc}].$$ (9)

Typically $r_{\tau}$, for $\tau_{ 100~ \mu\rm m} = 1$, is a factor 2-5 larger than $r_{\rm b}$ (see Table 2), and we will discuss further below whether a dust emitting region as large as that given by $r_{\tau}$ is still consistent with other data.