4 Destruction processes due to the supernova

To compare the results for dust abundance and dust composition in the CSM of SN 1987A with expectations for the CSM prior to the supernova event, account must be taken of the grain destruction processes due to the supernova and its remnant.

 Before quantitatively discussing grain evaporation (Sect. 4.1) and grain sputtering (Sect. 4.2) we briefly demonstrate that the grains are dynamically coupled to the gas due to the betatron effect. For example the potential of a graphite grain with $a=0.1~\mu{\rm m}$ at a temperature of $T_{\rm i}=3\times 10^8~{\rm K}$is more than 30 Volt (Draine & Salpeter 1979a). The larmor radius of the grain is given by $R_{\rm L}=m_{\rm gr}v_{\rm gr}/Q_{\rm gr}B$, where $m_{\rm gr}$ is the mass of the grain, $v_{\rm gr}$ the relative velocity of the grain to the gas (initially $\frac{3}{4}v_{\rm S}$), $Q_{\rm gr}$ the grain charge and B the magnetic field. For B we adopt the value for equipartion between magnetic field and relativistic particles (Longair 1997), calculated from an extrapolation of the synchrotron emission after 1200 days (Gaensler et al. 1997). Assuming the radio emission to arise from the same volume as the X-ray emission (see Appendix A), this yields $B\sim 10^{-7}$ T, in agreement with the value found by Ball & Kirk (1992). Taking the shock velocity to be $2900~{\rm km~s}^{-1}$, as derived from radio observations (Gaensler et al. 1997), the larmor radius $R_{\rm L}$is then of the order of $2.1\times 10^{11}~{\rm m}$, which is less then $\sim$10^-4of the diameter of the radio emission region. By comparison, the distance of the outer blast wave to the contact discontinuity increases with $\sim$ $0.1~v_{\rm S}~t$(Chevalier 1982). Thus, the grains comoved with the gas after roughly one month.

To discuss the mass loss due to evaporation and sputtering we assumed, as in our earlier examination of the IR emission, that the grains in the CSM at the time of the supernova explosion had a grain size distribution with k=3.5. Again we choose as minimum grain size $a_{\rm min}=10~$Å.

   
4.1 Evaporation during the UV-flash

The evaporation of grains in the neighbourhood of a supernova in general has already been the subject of earlier examinations (e.g. Draine & Salpeter 1979b; Draine 1981; Pearce & Mayes 1986) and also has been discussed for silicate grains for the SN 1987A (Emmering & Chevalier 1989; Timmermann & Larson 1993). All considered the effect of evaporation for single grain sizes only. Here we are interested in the mass loss of grains with a certain size distribution in the shocked plasma behind the outer shock front. We use more recent theoretical results for the spectrum, duration and luminosity of the UV flash (Ensman & Burrows 1992). Apart from silicate we also consider iron and graphite grains.

As a reasonable assumption, we only consider the evaporation of grains during this UV-flash, when the grains reached their highest temperatures. For the luminosity, temperature, and duration of the UV-flash we take the theoretical results of the model 500full1 from Ensman & Burrows (1992). For our calculation, the spectrum is taken to be a simple black body spectrum with the given colour temperature from Ensman & Burrows normalised to the luminosity.

To simplify the derivation of the mass loss of grains during the UV-flash, we neglect the effect of stochastic heating and assume that all grains are at their equilibrium temperatures. The evaporation of atoms from the surface of a grain with temperature $T_{\rm d}$ leads to a reduction in radius, that can be described by

$$\frac{{\rm d}a[\mu{{\rm m}}]}{{\rm d}t[{\rm s}]}\sim \frac{1}{3}~10^{11} {\rm e}^{-W(a)/k_{{\rm B}}T_{\rm d}(a)},$$ (3)

where $k_{\rm B}$ is the Boltzmann constant and W(a) the energy single atoms need to escape from the surface, which depends on the radius of the grain because of the surface tension of the grain. The coefficient is chosen to be close to the value given by Guhathakurta & Draine (1989) and Voit (1991) for graphite and silicate. The energies W(a) for silicate and graphite are adopted from Guhathakurta & Draine. For iron we adopt

$$W(a) = (50~000-22~000~N(a)^{-1/3})k_{{\rm B}},$$ (4)

where N(a) is the number of atoms in the grain. Here we have chosen a bounding energy of the atoms close to $U=4.29~{\rm eV}$, given in Gerthsen et al. (1989), and a surface tension of $\sigma=1.8~{\rm J/m^2}$, given in Levèfre (1979).

Figure 5: Theoretical grain size distributions (dashed lines) of survived silicate and graphite grains after the UV-flash at various distances D (given in units of 1015 m) to the supernova. The initial size distribution is assumed to have been a power law ${\rm d}n(a)=A a^{-3.5}~{\rm d}a$ with constant A (straight solid line).

The total reduction $a_{\rm ev}(a,t)$ of a grain with an initial radius a during the UV-flash after a time t is found by integrating Eq. (3). If the final radius $\tilde a(a,t)=a-a_{\rm ev}(a,t)$ is smaller than 3 Å, the grain is assumed to have evaporated. The final grain size distribution of non evaporated grains assumed to have an initial size distribution ${\rm d}n(a)=f(a)~{\rm d}a=A~a^{-3.5}~{\rm d}a$ with constant A given by:

$$\tilde{f}(\tilde a)~{\rm d}\tilde a f(a(\tilde a))~\frac{{\rm d}a}{{\rm d}\tilde a}~{\rm d}\tilde a$$ =

$$A~(\tilde a +a_{\rm ev})^{-3.5} \left(1-\frac{{\rm d}a_{{\rm ev}}}{{\rm d}a}\right)^{-1}{\rm d}\tilde a.$$ = (5)

The derived size distributions of silicate and graphite grains for different distances are shown in Fig. 5. As the evaporation is a surface effect, the fractional change in radius of grains at the same temperature is larger for smaller grains. Therefore the number of small grains decreases much more rapidly than the number of bigger grains. In addition, at the same distance to the supernova smaller grains are generally heated to higher temperatures. Due to the exponential dependence of Eq. (3) the evaporation of smaller grains is faster, which causes the maxima in the distribution functions.

How much silicate, iron and graphite dust might have survived the UV-flash in a certain distance to the supernova is shown in Fig. 6 for three different maximum grain sizes (0.25, 0.1 and $0.06~\mu{\rm m}$). It can be seen that silicate grains evaporate out to larger distances from the supernova than graphite grains. This is partly due to the higher bounding energy of graphite grains but mainly caused by the much higher temperatures the silicate grains attain in comparison with graphite grains. This is the opposite of the situation for collisionally heated dust in the CSM.

Figure 6: Survived grain mass of graphite, silicate and iron grains after the UV-flash versus distance r from the supernova. It is assumed, that the grain size distribution before the supernova outburst had k=3.5. $a_{\rm min}$ is chosen to have been 10 Å. The three curves (solid, dashed and dotted line) for each dust composition correspond to different choices for the initial maximum grain size $a_{\rm max}$ (0.25, 0.1 and $0.06~\mu{\rm m}$). Also shown is the approximate range of positions covered by the thick inner ring (dark grey) and the HII-region (light grey) before being caught by the blast wave. The inner radius of the HII-region was taken to be $4.72\times 10^{15}~{\rm m}$. This was calculated from the observed position of the blast wave at epoch 3200 days (Gaensler et al. 1997) extrapolated to the epoch at which the shock first reached the HII-region (1200 days after outburst; Staveley-Smith et al. 1992). The vertical straight line at a radius of $5.43\times 10^{15}~{\rm m}$ indicates the position of the blast wave at the epoch of the ISOCAM observations (4000 days after outburst, also extrapolated from the observed position at 3200 days). The extrapolations were made assuming a velocity of 2900 km s-1 for the blastwave corresponding to model II. If we take the velocity of $4100~{\rm km~s}^{-1}$ (model I) the inner boundary is closer to the supernova and the position of the blast wave further out.

The grains most probably responsible for the measured infrared fluxes originated from between the original inner boundary of the HII-region and the position of the blast wave 4000 days after outburst. The mass loss of iron and silicate grains in this region is comparable and significant even for the largest considered grain size of $0.25~\mu{\rm m}$. This changes at larger distances from the supernova, where evaporation of predominantly small iron grains becomes much stronger. For graphite grains evaporation is unimportant for the whole HII-region.

The radially integrated evaporated dust masses of silicate, graphite and iron grains corresponding to the shown curves is given in Table 3. In the integration it is assumed that the shock surface area is proportional to the square of the distance r of the blast wave to the position of the supernova. The differences in the derived values for the two models are due to the different shock velocities, that give slightly different positions of the inner boundary and the outer shock.

   
4.2 Sputtering in the shocked gas

Behind the blast wave grains undergo sputtering. This is thought to be one of the most important destruction processes in fast moving shocks (see e.g. Dwek et al. 1996). Sputtering time scales appropriate for a plasma with the abundances of the shocked CSM of SN 1987A are given in Appendix B.

The final grain size distribution after a time $\Delta t$ of an initial grain size distribution $f(a)~{\rm d}a$ due to sputtering in a hot plasma is given by:

$$\tilde f(\tilde a, \Delta t)~{\rm d}\tilde a = f(a(\tilde a... ...m d}\tilde a= f(\tilde a+\Delta a(\Delta t))~{\rm d}\tilde a.$$ (6)

Because of the evaporation of grains during the UV-flash, the grain size distribution $f(a)~{\rm d}a$entering the sputtering zone is dependent on the distance to the supernova. The average grain size distribution at the time t after the shock reached the HII-region at time t₀is therefore proportional to:

$$\left<\tilde f(\tilde a,t-t_0)\right>\propto \int_{t_0}^{t}{\rm d}t'~v_{\rm S}~\tilde f(\tilde a,t-t',r(t'))~r(t')^2,$$ (7)

where t₀=1200 days and the surface of the outer shock front is again assumed to increase with the square of the distance r(t) of the blast wave to the supernova position.

The derived mass losses for the considered dust compositions and the three different maximum grain sizes are tabulated in Table 3. For comparison, we derived the sputtered dust mass with and without previous evaporation.

   

Table 3: Mass loss through evaporation and sputtering.

$n_{{\rm H}}$a		$300~{\rm cm^{-3}}$			$600~{\rm cm^{-3}}$
$a_{\rm max}[\mu{\rm m}]$		0.25	0.10	0.06	0.25	0.10	0.06
$\Delta M_{{\rm sputt.}}$b	sil.	27.7%	41.1%	50.3%	44.0%	61.7%	71.9%
$\Delta M_{{\rm evap.}}$	sil.	49.4%	80.4%	98.3%	48.7%	79.2%	98.9%
$\Delta M_{{\rm sputt.}}$	iron	27.2%	40.4%	49.5%	40.9%	58.1%	68.4%
$\Delta M_{{\rm evap.}}$	iron	48.0%	78.6%	99.8%	47.7%	78.1%	100.%
$\Delta M_{{\rm total}}$c	sil.	53.1%	82.4%	98.4%	58.4%	85.1%	99.1%
$\Delta M_{{\rm total}}$	iron	51.8%	81.0%	99.8%	56.4%	83.6%	100%
$\Delta M_{{\rm total}}$	gra.c	11.3%	17.3%	22.0%	19.1%	29.0%	36.3%

^a Density of the shocked gas downstream of the blast wave.
^b Sputtering without previous evaporation.
^c Mass loss after evaporation and following sputtering.
^d For graphite only the total mass loss is given because of an insignificant mass loss due to evaporation.

The mass loss due to sputtering becomes progressively more important for the bigger grains. Whereas sputtering reduces the radius independent of the grain size, evaporation is the dominant destruction process for small grains. Graphite grains which are relatively stable against sputtering (Appendix B) and did not evaporate during the UV-flash (Sect. 4.1) suffer only moderate depletions compared to silicate and iron grains. In model I the initial mass of graphite grains would be less than a factor of 1.3 higher than that infered from the ISOCAM observations for all considered maximum grain sizes.