3 Results

The results of the model calculations are summarised in Table 2, where we give for each fit also the reduced $\chi^2_{\nu}$, where $\nu$ is the number of free parameters. The best fits obtained for the two mixtures are shown in Fig. 4 together with the fitted spectrum for silicate grains. The emission spectrum is such that measurements at longer wavelengths (e.g. Lundqvist et al. 1999) are too insensitive to probe the emission from this region. The fits can be summarised as follows:

Figure 4: Theoretical emission spectra of dust in the shocked circumstellar medium of SN 1987A for model I ( $n_{\rm H}=300~{\rm cm^{-3}}$) and model II ( $n_{\rm H}=600~{\rm cm^{-3}}$). Shown are spectra from pure silicate (dotted line) and the best fits with a mixture of silicate with graphite (dashed line) and with a mixture of silicate with iron (solid line). The grains are assumed to have a grain size distribution ${\rm d}n\propto a^{-3.5}~{\rm d}a$ with minimum and maximum grain radii of 10 Å and $0.25~\mu{\rm m}$. The ISOCAM fluxes (triangles), shown with the widths of the filters as horizontal lines, have been colour corrected to the spectrum of the silicate-iron mixture. The shown uncertainties are the absolute uncertainties in the integrated flux densities (taken from Table 2 of Paper I) which are $25\%$ ( $6.75~\mu{\rm m}$), $4.3\%$ ( $12~\mu{\rm m}$) and 5.5% ( $14.3~\mu{\rm m}$). These uncertainties are a combination of the systematic calibration uncertainties, which dominates at 12 and 14.3  $\mu {\rm m}$, and the random uncertainties.

 

 

Table 2: Results from the dust modelling.

		results model I ( $n_{\rm H}=300~{\rm cm^{-3}}$)						results model II ( $n_{\rm H}=600~{\rm cm^{-3}}$)
compos.		mixturea	k	$a_{\rm max}[\mu{\rm m}]$	$M_{\rm d} [10^{-6}~{M_{\odot}}]$	$L_{\rm d}$[1028 W]	$\chi^2_{\nu}$	mixture	k	$a_{\rm max}[\mu{\rm m}]$	$M_{\rm d} [10^{-6}~{M_{\odot}}]$	$L_{\rm d}$[1028 W]	$\chi^2_{\nu}$
single variable: $M_{\rm d}$ ($\nu=2$)
silicate		-	3.5	0.25	$0.97\pm 0.03$b	2.94	6.1	-	3.5	0.25	$0.50\pm 0.02$	2.63	5.3
graphite		-	3.5	0.25	$1.04\pm 0.04$	3.35	1.6	-	3.5	0.25	$0.56\pm 0.02$	3.15	3.5
iron		-	3.5	0.25	$1.31\pm 0.05$	3.35	55.	-	3.5	0.25	$0.86\pm 0.03$	3.65	81.
two variables: $M_{\rm d}$ and k ($\nu=1$)
silicate		-	3.99-0.24+0.23	0.25	$0.85\pm 0.03$	2.83	8.9	-	3.39-1.32+0.44	0.25	$0.51\pm 0.02$	2.64	11.
graphite		-	3.56-0.61+0.30	0.25	$1.03\pm 0.04$	3.36	3.1	-	<2.25c	0.25	$0.67\pm0.02$	3.03	1.3
iron		-	<0.53	0.25	$1.51\pm 0.05$	2.54	7.8	-	<0.28	0.25	$1.09\pm 0.04$	2.59	18.
two variables: $M_{\rm d}$ and dust mixture ($\nu=1$)
si.+iron		2.09-0.61+1.09	3.5	0.25	$1.11\pm0.04$	3.18	0.0	2.33-0.73+1.56	3.5	0.25	$0.60\pm 0.02$	2.96	2.4
si.+iron		2.90-0.82+1.57	3.5	0.10	$0.97\pm 0.03$	3.18	0.3	3.09-0.95+2.09	3.5	0.10	$0.51\pm 0.02$	2.95	4.4
si.+iron		3.35-0.95+1.89	3.5	0.06	$0.93\pm0.03$	3.19	0.75	3.39-1.05+2.41	3.5	0.06	$0.48\pm 0.02$	2.96	5.7
si.+gra.		0.25-0.25+0.56	3.5	0.25	$1.03\pm 0.03$	3.27	2.4	0.73-0.38+0.73	3.5	0.25	$0.53\pm 0.02$	2.94	0.2
si.+gra.		0.43-0.33+0.64	3.5	0.10	$0.93\pm0.03$	3.22	1.07	0.84-0.40+0.74	3.5	0.10	$0.47\pm 0.02$	2.92	0.47
si.+gra.		0.52-0.34+0.68	3.5	0.06	$0.89\pm 0.03$	3.20	0.55	0.88-0.40+0.75	3.5	0.06	$0.45\pm 0.02$	2.91	1.45
two variables: $M_{\rm d}$ and $a_{\rm max}$ ($\nu=1$)
silicate		-	3.5	0.03-0.02+0.07	$0.78\pm 0.03$	2.78	9.3	-	3.5	0.51-0.41+1.93	$0.60\pm 0.02$	2.70	10.2
graphite		-	3.5	0.19-0.15+0.30	$1.01\pm 0.03$	3.36	3.1	-	3.5	0.97-0.44+1.60	$0.75\pm0.03$	3.13	3.3
si.+iron		2.09	3.5	0.26-0.15+0.53	$1.12\pm0.04$	3.18	0.0	2.33	3.5	1.55-1.24>10.0	$1.08\pm 0.04$	3.05	0.9
si.+gra.		0.25	3.5	0.05-0.03+0.14	$0.90\pm 0.03$	3.28	1.0	0.73	3.5	0.42-0.33+0.71	$0.59\pm 0.02$	2.95	0.05

^a Relative mass of silicate to iron or silicate to graphite, respectively.
^b The uncertainty of $M_{\rm d}$ was derived with all other parameters fixed.
^c The limit given for k corresponds to $\chi^2(k)-\chi^2(k=0)=1$; here $M_{\rm d}$, $L_{\rm d}$ and $\chi^2_{\nu}$ were calculated for k=0

1.

Fit parameter: $M_{\rm d}$.
We simply compared the measured flux densities in the IR with the theoretical emission spectra of three different grain compositions (pure silicate, graphite or iron) with fixed grain size distribution for $a_{\rm max}=0.25~\mu{\rm m}$ and k=3.5 as proposed for the grains in the ISM (MRN, 1977).
For both model I and model II a better fit is achieved under the assumption of pure graphite grains instead of silicate grains in the circumstellar environment. As seen in Fig. 4 the flux at $6.7~\mu{\rm m}$ is too high to be emitted from silicate grains. Because of the high predicted temperatures a pure iron composition is even more unlikely.

2.

Fit parameter: k, $M_{\rm d}$.
For each of pure silicate, graphite and iron we varied the power k of the size distribution with fixed $a_{\rm max}=0.25~\mu{\rm m}$.
The derived grain size distributions for graphite grains in model I and for silicate grains in both models are consistent with k=3.5. For graphite grains in model II, which reach too high temperatures, a flatter distribution (k<2.25) is required. No grain size distribution could be found which admitted a pure iron solution.

3.

Fit parameter: Dust mixture, $M_{\rm d}$.
For a fixed grain size distribution with power k=3.5 and $a_{\rm max}=0.25~\mu{\rm m}$ we varied the composition of circumstellar grains, for a mixture of silicate with iron grains and a mixture of silicate with graphite grains.
Excellent fits could be achieved for silicate-iron mixture in model I and for the silicate-graphite mixture in model II (see also Fig. 4).

4.

Fit parameter: $a_{\rm max}$, $M_{\rm d}$.
For a given dust composition (silicate, graphite and the two derived mixtures) and a fixed size distribution with power k=3.5 we varied $a_{\rm max}$.
The maximum grain sizes in model I are significantly smaller than in model II. In model I for silicate, graphite or the silicate-graphite mixture a better fit can be achieved with $a_{\rm max}<0.25~\mu{\rm m}$. The estimated maximum grain sizes of all considered compositions in model II are several times larger. We also examined a pure iron composition and found that the maximum grain size would be unrealistically large with $a>10~\mu{\rm m}$.

Due to the large number of free parameters several solutions with different compositions and grain size distributions are possible. The derived values like $a_{\rm max}$, k or the mixture of the composition are interrelated and depend on the plasma density. Despite these uncertainties the following conclusions could be reached largely independent of the assumptions:

1.

Luminosity
For the luminosity of the collisionally heated dust 11 years after outburst we obtained (dependent on composition and the assumed plasma density) values for $L_{\rm d}$ in the range $2.91\times 10^{28}$ to $3.35\times 10^{28}~{\rm W}$. This can be compared with the crude estimate of $L_{\rm d}=2.5\times 10^{28}$ W given in Paper I where we approximated the dust emission spectrum with a simple modified black body spectrum. As already mentioned in Paper I the luminosity in the MIR is larger than in X-rays. Despite the increase of the X-ray luminosity since detection $\sim$1400 days after outburst (Hasinger et al. 1996) the luminosity in the energy range 0.5 to 10 keV even $\sim$13 years after outburst was only $L_X\approx 2\times 10^{28}~{\rm W}$ (Burrows et al. 2000).

2.

Composition
The dust is most likely a mixture of silicate with iron or silicate with graphite. Also pure graphite gives a reasonable fit. Pure silicate on the other hand seems to be unlikely and pure iron can be excluded.

3.

Dust-to-gas ratio
Considering only cases where $a_{\max}$ is not larger than $0.25~\mu{\rm m}$ the dust mass is found to be nearly independent of composition and grain size distribution. The corresponding acceptable fits for model I ( $n_{\rm H}=300~{\rm cm^{-3}}$) yield dust masses in the range $(0.9~{\rm to} ~1.1)\times 10^{-6}~{M_{\odot}}$ and those for model II ( $n_{\rm H}=600~{\rm cm^{-3}}$) yield dust masses in the range $(0.45~{\rm to}~0.67)\times 10^{-6}~ {M_{\odot}}$.

Taking these values the dust-to-gas ratio is only mildly dependent on gas density as one would also expect. The mass of the X-ray emitting gas is given approximately by

$${M_{\rm gas}}\sim \frac{m_{\rm H}EM}{n_{\rm H}},$$ (2)

where EM is the emission measure from X-ray observations and $m_{\rm H}$ the hydrogen mass. We have taken the emission measure from Hasinger et al. (1996) with $(1.4\pm 0.4)\times 10^{57}~{\rm cm^{-3}} \sim 2500$ days after outburst and extrapolated this result with t^2.06 (see Appendix A) to 4000 days. Using Eq. (2) we derive a gas mass of $1.0\times 10^{-2}/n_{\rm H}[300~{\rm cm^{-3}}]~{M_{\odot}}$. Overall, this yields a dust-to-gas ratio in the range $(0.9~{\rm to}~ 1.3)\times 10^{-4}$.

If the grain size distribution is allowed to extend to grains with radii much larger than $0.25~\mu{\rm m}$the acceptable fits imply that the dust-to-gas ratio could be as high as $2.2\times 10^{-4}$. As will be seen in Sect. 4 the pre-supernova dust abundance would however be almost unaffected.