2 Modelling the MIR emission from the shocked CSM

We consider grains that are collisionally heated in the shocked circumstellar plasma downstream of the blast wave.

Figure 1: Cartoon of the basic structure caused by the interaction of the expanding ejecta with the HII-region.

The structure caused by the interaction of the expanding ejecta with its CSM is visualised in Fig. 1. The shocked circumstellar gas is compressed to a thin layer that grows approximately as $0.1v_{\rm S}t$ (Chevalier 1982), where $v_{\rm S}$ is the assumed constant speed of the blast wave and t the time since the shock reached the HII-region. Downstream it is bounded by the contact discontinuity to the shocked outer parts of the expanding ejecta. This ejected material is heated by both the reverse shock and by a shock reflected from the inner boundary of the HII-region (Borkowski et al. 1997). It is thought to be the origin of the strong emission in Ly$\alpha$ and H$\alpha$ at a distance of $\sim$0.6 of the radius of the thick inner ring (Michael et al. 1998), that was measured by the HST close to the time of the ISOCAM observations. We do not expect strong MIR emission from the shocked ejected material for the following reasons:

1.

The material at the reverse shock is mainly hydrogen and has such a low abundance in metals (e.g. Woosley 1988), that, if any, only very few grains may have been formed in this region of the ejecta.

2.

Further, as we show later (see Sect. 4), the grain charge and the strength of the magnetic field in the shocked CSM in the interaction zone makes it unlikely, that an observable amount of circumstellar grains have been overtaken by the outer parts of the ejecta.

3.

The gas density of the ejecta at the reverse shock is, following the numerical results from Borkowski et al. (1997), much less than the density downstream of the blast wave. In consequence, the heating of any ejected grains at the reverse shock should be much weaker than for circumstellar grains downstram of the blast wave.

Figure 2: Temperature distributions of spherical silicate and graphite grains for various grain sizes in the shocked CSM, assuming a density of the shocked gas of $n_{\rm H}=300~{\rm cm^{-3}}$ (model I). The variation in size between two lines is d $\log(a[\mu{\rm m}])=0.1$. The temperature distributions, shown with thicker lines, belong to grains with a radius of $0.001~\mu{\rm m}$ and $0.01~\mu{\rm m}$. The steps in the p(T)-curves are due to the approximation used for ion heating and appear roughly at temperatures where the thermal energy of the grains is equal to the threshold energy of the ions (see also Popescu et al. 2000).

   
2.1 Gas parameters

The collisional heating of a grain in a hot plasma depends mainly on the temperatures of electrons and ions and the number density of the different species of the gas. We fix these properties to be broadly consistent with the observed radio and X-ray emission from the CSM. An analysis of the X-ray observations made with the ROSAT satellite towards SN 1987A up to and including the epoch of the ISOCAM observations is given in Appendix A. For temperatures above $\sim$ $10^7~{\rm K}$, as encountered in the shocked gas, the grain heating of the smaller grains principally contributing to the emission observed by ISOCAM is mainly determined by the plasma density (see e.g. Dwek & Arendt 1992). As the physical conditions are somewhat uncertain we will consider two cases (models I and II) with different temperatures and densities. The parameters of models I and II are summarised in Table 1 and will be justified in the following.

 

 

Table 1: Parameters of the shocked gas.

	model I	model II
shock velocity $v_{\rm S}$	4100 km s-1	2900 km s-1
electron temperature $T_{\rm e}$	$2\times 10^7~{\rm K}$	$2\times 10^7~{\rm K}$
ion temperature $T_{{\rm i}}$	$6\times 10^8~{\rm K}$	$3\times 10^8~{\rm K}$
hydrogen density $n_{{\rm H}}$	$300~{\rm cm^{-3}}$	$600~{\rm cm^{-3}}$
helium density $n_{{\rm He}}$	$2.5~(n_{{\rm He}})_{\odot}$	$2.5~(n_{{\rm He}})_{\odot}$
metallicity Z	$0.3~Z_{\odot}$	$0.3~Z_{\odot}$

Borkowski et al. (1997) have shown that an assumption of constant density, as suggested by Chevalier & Dwarkadas, can explain the soft X-ray emission (Hasinger et al. 1996) until at least 3000 days after outburst. They derived a pre-shock hydrogen density of $75~{\rm cm^{-3}}$ with a corresponding shock speed $v_{\rm S}$ of the blast wave of $4100~{\rm km~s}^{-1}$. In their calculations the HII-region was modelled as a thick torus in which the inner ring seen by the HST is embedded. They also mentioned that for a different structure of the HII-region higher densities would be possible. Indeed, a shock speed of $2900\pm 480~{\rm km~s}^{-1}$ as derived from radio observations (Gaensler et al. 1997) would require a hydrogen density of the HII-region of about $150~{\rm cm^{-3}}$. It might also be possible, that the density increased with time (see Appendix A) although the X-ray measurements are also consistent with a constant external density.

For simplicity we assume that at the time of the ISOCAM observations the blast wave was still expanding into a homogeneous HII-region. We will consider hydrogen densities of $75~{\rm cm^{-3}}$ (model I) and $150~{\rm cm^{-3}}$ (model II) for the pre-shock region. The density of the gas is assumed to be compressed by a factor of four downstream of the blast wave and to stay constant in the whole interaction zone.

In converting the densities given by Borkowski et al. into hydrogen number density and ionic abundances needed for the calculation of dust heating (Sect. 2.2) we adopted the abundances of the thick inner ring given by Lundqvist & Fransson (1996). On this basis we took helium to be a factor of 2.5 more abundant than in the sun, and approximated the metallicity of all heavier elements to be 0.3 solar.

Because of the long equipartition time of the shocked circumstellar plasma the electron temperature should be much lower than the temperature for the ions and should increase with the distance to the outer shock (see e.g. Burrows et al. 2000). Thus, in calculating the grain heating we use two different temperatures for electrons and ions. As for the density, the temperature of the shocked CSM is taken to be independent of position.

The ion temperature $T_{{\rm i}}$ is taken to be (see e.g. Longair 1997)

$$T_{\rm i}=2~\frac{\gamma-1}{\left(\gamma +1\right)^2}~\frac{\mu ~m_{\rm H}}{k_{{\rm B}}}~v_{\rm S}^2$$ (1)

with $\gamma=5/3$ and an atomic weight of $\mu=1.6$ appropriate to the abundances in the CSM.

For the electrons we choose a temperature close to those derived from X-ray observations and predicted by numerical calculations appropriate to SN 1987A. From X-ray observations made with the ROSAT-satellite Hasinger et al. (1996) derived a temperature of approximately $T_{\rm e}\approx 1.2\times 10^7~{\rm K}$. Analysing the X-ray spectrum of SN 1987A, taken later with CHANDRA, using a shock model with a constant $T_{\rm e}$, a higher electron temperature of the order of $\sim$ $3.5\times 10^7~{\rm K}$ was found (Burrows et al. 2000; Park et al. 2002). In our calculation we will adopt $T_{\rm e}=2\times 10^7$ K.

   
2.2 Calculation of the dust emission

The dust in the CSM is taken to be spherical for simplicity as there is no observational evidence for other grain shapes in the CSM of SN 1987A. Spherical grains are also generally assumed in the literature for grains produced in stellar winds (e.g. Gail & Sedlmayr 1999). However, if the grains are not too different from being spherical this assumption should not influence significantly our results about the properties of the grains in the CSM. For example, it has been found, that the temperature of spheroidal grains heated by the interstellar radiation field (ISRF) can only vary by more than 10% if the axial ratio exceeds a value of 2 (Voshchinnikov et al. 1999).

The grains are heated by collisions with electrons and ions of the hot ionized shocked plasma. The gas species is assumed to transfer all (if it sticks) or only a part (if it is not stopped) of the kinetic energy into thermal energy of the grain. The calculation of the energy deposition of non-stopping particles is based on their stopping-distances (ranges) in solids. For the heating by ions we considered in addition to hydrogen and helium also the next most abundant elements oxygen, nitrogen and carbon. The emission from larger grains we derived from their equilibrium temperatures. For smaller grains, where the deposited energy is typically larger than the thermal energy, we took their temperature fluctuations into account. The model for stochastic dust emission from a hot plasma is described in more detail in Popescu et al. (2000). As typical dust species we consider graphite and silicate grains. The temperature distributions of very small silicate and graphite grains in model I are shown in Fig. 2 where it can be seen, that small grains can be heated to very high temperatures. The smallest grains with 10 Å radius will reach temperatures well above their evaporation temperature. The grains contributing to the measured IR emission should therefore not be smaller than this size.

Because iron is expected to form in cool stellar outflows independent of the C/O ratio and is potentially one of the main condensates in the circumstellar environment of oxygen rich stars (see e.g. Gail & Sedlmayr 1999, and references there), we also carried out calculations for pure iron grains. For electrons in iron grains we used the analytical expression for the electron range in graphite derived by Dwek & Smith (1996) on basis of observational data, correcting for the different density. The optical properties of iron spheres we derived using MIE-theory (Bohren & Huffman 1983), whereby we included the dependence of the dielectric function in the IR on size and temperature of the iron grains (Fischera 2000). For the heat capacity of iron grains cooler than 298 K we used values tabulated in the American Institute of Physics Handbook (1972). For iron grains warmer than this we used an analytical expression for the heat capacity given by Chase (1998), assuming that iron is in the $\alpha$-$\delta$-phase.

2.2.1 Grain temperatures in the shocked CSM

The variation of the equilibrium temperature of spherical silicate, graphite, and iron grains in the shocked CSM is shown in Fig. 3. The different behaviour of the three species allows some information about dust composition to be derived from the ISOCAM measurements. Grain temperatures for the case $T_{\rm e}=3.5\times 10^7~{\rm K}$ are also given. As seen in the figure the grain temperature of small silicate and graphite grains is nearly insensitive to $T_{\rm e}$since the electrons are not stopped by the grains. The equilibrium temperatures are consistent with the colour temperatures $T_{\rm c}$ we derived in Paper I for a modified Planck spectrum $F_{\lambda}\propto \lambda^{-\beta}B_{\lambda}(T_{\rm c})$, where $B_{\lambda}(T_{\rm c})$is the Planck function, ranging from $\sim$200 K ($\beta=2$) to $\sim$290 K ($\beta=0$). Because of stronger heating due to the higher grain density and the weaker cooling due to lower emissivities in comparison to silicate and graphite grains pure iron grains attain the highest temperatures. We also note that for the conditions in the shocked CSM, graphite grains are hotter than silicate grains. In less dense plasmas (e.g. $n_{\rm H}=1~{\rm cm^{-3}}$) one would expect a slightly higher temperature for silicate grains (Dwek 1987; Fischera 2000).

Figure 3: Equilibrium temperatures of spherical silicate, graphite and iron grains as a function of grain radius a in the shocked gas downstream of the blast wave. The grain temperatures in models I and II are represented by the solid and the long dashed line, respectively. Also shown is the effect of raising $T_{\rm e}$ from $2\times 10^7$ K to $3.5\times 10^7$ K (short dashed and dotted lines, respectively, for models I and II).

The temperature of small spherical iron grains depends strongly on grain size. This was also noted by Chlewicki & Laureijs (1988) for iron grains heated by the interstellar radiation field.

2.2.2 Grain size distribution in the CSM

In order to construct a SED for comparison with the data a functional form for the grain size distribution must be adopted. The size distribution of the grains in the shocked circumstellar environment of SN 1987A is not known and might be modified in comparison to the initial distribution in the CSM by different processes like evaporation during the UV-flash or sputtering in the shocked gas as will be shown later. However, for simplicity we assume, that the grains in the shocked CSM have a grain size distribution similar to grains in the ISM of our galaxy and can be described by a simple power law ${\rm d}n\propto a^{-k}~{\rm d}a$ with power index kand a minimum and a maximum grain size $a_{\rm min}$ and $a_{\max}$. Following Biermann & Harwit (1980) a power law distribution should be a general description of a grain size distribution resulting from grain-grain collisions and would especially describe the grain size distribution in the atmospheres of red-giants. In contrast to the emission from grains in the ISM, heated by the ISRF, where only stochastically heated small grains can achieve high enough temperatures to emit at shorter wavelengths (Draine & Anderson 1985), in the shocked CSM of SN 1987A larger grains can also contribute to the measured fluxes. The composite SEDs are relatively insensitive to $a_{\rm min}$, which we fixed at 10 Å.

2.2.3 Free dust model parameters

For each of models I and model II for the shocked gas we made calculations taking various combinations of the following parameters as free variables:

• $M_{\rm d}$, the dust mass. This was a scaling parameter in all calculations.
• k, the exponent in the grain size distribution ${\rm d}n(a)\propto a^{-k}~{\rm d}a$.
• $a_{\max}$, the maximum grain size.
• Grain composition. Five compositions were considered: Pure silicate, pure graphite, pure iron, a silicate-iron mixture and a silicate-graphite mixture. The relative composition in the mixtures was considered as a free variable.

The most probable values for the variables were found through a $\chi^2$-fit to the measured flux densities, taken from Paper I, which we colour corrected (see Blommaert et al. 2001) on the basis of the modelled spectrum. In addition we derived for each fit the luminosity $L_{\rm d}$ of the theoretical dust emission spectrum. The one sigma uncertainties of the parameters were calculated from the $\chi^2$-fit by varying $\Delta \chi^2_1=\chi^2_1 - \chi^2_{{\rm min}}$ until $\Delta\chi^2_1=1$ (see e.g. Press et al. 1992). To estimate the uncertainties of the fitted parameters $a_{\rm min}$, k and the dust mixture the dust mass $M_{\rm d}$ was taken to be a free variable. For simplicity the uncertainty of $M_{\rm d}$ itself was derived with all other parameters fixed.