4 Dust heating

In the previous section, we only discussed the dust composition; in this section, we discuss the physical model we have used to account for the dust heating. As pointed out in the paper by Lagage et al. (1996), the dust is probably SN condensates embedded in the FMK's. As discussed by Dwek & Werner (1981), the dominant dust heating mechanism in the FMK's is collisions between the dust and the electrons of the plasma inside the knots. To calculate the heating of the dust, we need to know the physical conditions inside the FMK's. Constraints on these physical conditions can in principle be obtained from optical observations.

4.1 Optical fluxes

There is not yet a model reproducing all the optical observations of a knot. The most developed model is, so far, the model by Sutherland & Dopita (1995) (SD95 hereafter). The basis of the model is the encounter of a FMK with the reverse shock of the SN, which induces the propagation of a shock wave into the knot, preceeded by the propagation of an ionisation front. Oxygen lines arise both at the ionisation front and at the shock, which becomes rapidly radiative. In the following, we adopt the SD95 model with a shock velocity inside the knot of $V_{\rm s}=200$kms^-1. We have chosen this model because it is the one which predicts the right [OIII] 4363 /[OIII] 4959+5007 line flux ratio. Most of the optical [OIII] lines are produced at the ionisation front, which propagates at a high speed of 3300kms^-1. The flux observed in the $\rm [OIII]~4959$+5007Å lines allows us to determine the density in this region using the SD95 model. For the bright optical knot contained in the SWS aperture (see Fig. 1), we measure $F_{\rm Single Knot}([{\rm OIII}])=1.5$10^-11ergcm^-2s^-1from CFHT observations, taking into account the extinction in the same way as Dinerstein et al. (1987). The electronical temperature, derived from SD95 for the part of the ionisation front where [OIII] is present, is 710³K and, given the physical parameters of the lines emission deduced from Osterbrock (1989), we obtain a knot electronic density, $n_{\rm e}$, of 2.410³cm^-3. We assume a spherical geometry and we take Cas A at a distance of 3.4kpc (Reed et al. 1995), and a typical diameter knot size of 1 $^{\prime\prime}$, (size of the brightest optical knot in the SWS aperture, as deduced after deconvolution of the optical image shown in Fig. 1).

4.2 Infrared fluxes and dust mass in the hot component

We consider here the dust heating in the shock cooling region. The electronic temperature of the shocked material decreases rapidly in the cooling region from $T_{\rm e}=10^{6.64}$K to $T_{\rm e}=300$K; the associated electronic density varies from $n_{\rm e}=9.6\,10^{3}$cm^-3 to $n_{\rm e}=3\,10^{7}$cm^-3; (rescaled from SD95). The physical conditions in this region are such that the dust can be collisionally heated at a temperature corresponding to the temperature of the "hot'' dust component discussed in the previous section. Thus we have considered that the hot dust component was associated with this cooling region. In order to derive the "hot'' dust mass from the IR observations, we need to know the size of the dust. We have no observational constraint on these sizes, so that we have taken the sizes calculated by Kozasa et al. (1997) for newly formed SN dust: 70Å for the MgSiO₃, and 10Å for the Al₂O₃ grains; we have also used 70Å for the SiO₂ grains. With these sizes and given the dust heating, the IR luminosities imply 7.810^-9$M_{\odot}$of MgSiO₃, 3.110^-9$M_{\odot}$ of SiO₂ and 1.110^-9$M_{\odot}$ of Al₂O₃. Note that a 30Å change in the size of a 70Å grain leads to a change by a factor 3 of the masses.

Are such masses compatible with dust made from SN material? In DLC99, we have argued that the dust observed from Cas A was silicate dust made from SN material; the layer in the SN where silicate components (Si, Mg and O) are present, was identified in Fig. 2 of DLC99. If we take the relative abundance of elements in the "silicate layer'' (Woosley & Weaver 1995), we can evaluate the mass ratio of the different dust species relatively to the gas. Assuming that the dust formation efficiency is 100%, then almost all the magnesium and part of the silicon is locked in MgSiO₃ dust grains; the remaining silicon is locked in SiO₂ dust grains; all the aluminium is locked in Al₂O₃ dust grains, and the remaining gas is almost entirely made of oxygen. In these conditions, $M_{\rm MgSiO_{3}}/M_{\rm gas}=0.40$, $M_{\rm SiO_{2}}/M_{\rm gas}=0.14$ and $M_{\rm Al_{2}O_{3}}/M_{\rm gas}=0.032$. The size of the cooling region, and the density derived from CFHT observations allow to calculate the mass of gas available in the cooling area. This region is very thin; SD95 obtained a cooling length, $l_{\rm cool}$, of 210¹¹cm for an initial oxygen density of 1.210³cm^-3 (here the oxygen density has been taken equal to $n_{\rm e}$/2 according to the degree of oxygen ionization calculated in SD95 in the ionization area). Then the mass of gas available in the cooling region of a single knot is 1.310^-7$M_{\odot}$; consequently, if we consider a dust formation efficiency of 100%, we obtain $M_{\rm MgSiO_{3}}$ = 5.310^-8$M_{\odot}$, $M_{\rm SiO_{2}}$ = 1.810^-8$M_{\odot}$ and $M_{\rm Al_{2}O_{3}}$ = 4.310^-9$M_{\odot}$. Thus the observed and potentially available dust masses are compatible.

Comparing the observed dust masses and the calculated maximum dust mass produced from SN material in a knot, we can evaluate the dust "formation'' efficiency. When the SWS aperture contains many knots, a map obtained at 9.77$\mu$m with ISOCAM using the smallest pixel field of view of 1.5 $^{\prime\prime}$, shows that the hot component originates mainly (50%) from the bright knot of the SWS aperture of Fig. 1. For this knot, we can derive an estimate of dust formation efficiency of $e_{\rm MgSiO_{3}}$ = 7.4%, $e_{\rm SiO_{2}}$ = 8.6% and $e_{\rm Al_{2}O_{3}}$ = 13%. Given the simplifications in the model and the uncertainties in the parameters of the region (a factor 2 of error on the initial density leads to a factor 2 of error on the efficiency), we can just state that the efficiency is of the order of 10%. Note that the observed $M_{\rm MgSiO_{3}}$/ $M_{\rm SiO_{2}}$ and $M_{\rm MgSiO_{3}}$/ $M_{\rm Al_{2}O_{3}}$ ratios are well predicted by the model.

4.3 Infrared fluxes and dust mass in the cold component

Much more dust is present in the region behind the cooling region. From the SD95 model, we can deduce that the "mean'' physical conditions in that region are $n_{\rm e}=3\,10^{7}$cm^-3 and $T_{\rm e}=300$K. Given these physical conditions and assuming the same dust sizes as for the "hot'' component, we can compute the dust temperature and deduce from the observed fluxes the "cold'' dust masses needed: $M^{\rm cold}_{\rm MgSiO_{3}}$ = 1.110^-4$M_{\odot}$, $M^{\rm cold}_{\rm SiO_{2}}$ = 4.410^-5$M_{\odot}$ and $M^{\rm cold}_{\rm Al_{2}O_{3}}$ = 3.210^-6$M_{\odot}$.

The dust mass in the region behind the cooling area is difficult to determine. An upper limit to the dust mass can be obtained by considering the total dust mass in a knot. From the gas to dust mass ratio (DLC99) and the density previously discussed, we can derive a total dust mass of $M_{\rm MgSiO_{3}}$ = 4.210^-4$M_{\odot}$, $M_{\rm SiO_{2}}$ = 1.510^-4$M_{\odot}$ and $M_{\rm Al_{2}O_{3}}$ = 3.410^-5$M_{\odot}$, in the brightest knot in the SWS aperture. If we assume, as for the hot component, that half of the cold dust is located into this knot, the observed cold component from the knot represents about 14% of the maximum MgSiO₃ and SiO₂ dust masses. We are in the uncomfortable situation that the efficiency deduced for the cold dust is somewhat higher than that deduced from the hot component. The assumption that half of the observed cold dust is present in a bright knot, which cannot be tested because of the poor angular resolution of the SWS instrument, is questionable. Indeed the cold component emission lasts longer than the hot component emission, just because of the rapid cooling of region where the dust hot component is emitted and the rapid propagation of the shock inside the knot. Then it is quite possible that several knots with no or little hot dust emission are present in the SWS beam; this would explain why we observe so much dust in the cold component. Note that the Al₂O₃ cold component mass is only 5% of the total expected dust mass. The origin of such a large depletion compared to MgSiO₃ or SiO₂ may be linked to the small size (10 Å) of the grains, which are more easily destroyed by sputtering effects in the cooling area than the 70 Å sized particles.

One consequence of the model presented here is the temporal correlation between the optical emission at the shock and the hot dust emission. We also predict an increase with time of the IR emission from the cold dust component. Monitoring the knots both in the IR and the optical on the time scale of the knot optical lifetime (about 30 years) should help qualifying the model presented here. High angular resolution at 20$\mu$m are also needed to assess if there are indeed knots emitting mostly at 20$\mu$m, with faint counterparts in the optical or at 10$\mu$m.

Acknowledgements

We thanks A. Jones and J.-P. Chièze for enlightening discussions on shocks and their effects, and D. Péquignot for discussions on the optical emission.