5 Modelling of the scattered light

   
5.1 The polarising agent

The observation of scattered light in filters containing intrinsically strong lines has the interpretational disadvantage that the data may contain contributions from scattering by both dust grains and gas atoms (on the other hand, they probe both of these circumstellar media). To estimate the amount of total scattered flux due to each scattering agent is therefore not straightforward. In this respect, imaging polarimetry can provide a way to separate the two components.

Rayleigh scattering by small dust particles results in a polarisation degree which increases significantly for scattering angles close to 90 $\hbox{$^\circ$ }$. The polarisation due to line scattering has in principle the same angular behaviour as in the case of Rayleigh scattering by dust. There are particular processes that may decrease the polarisation of line scattering (the presence of a weak magnetic field (Hanle effect), non-coherent scattering due to collisions, and/or interference of atomic sublevels (Nagendra 1988)), but none of them is expected to have any sizeable effect in our case. However, the estimates presented in Paper I point towards (at least partly) optically thick scattering if the observed intensity is due to line scattering. On the contrary, the dust scattering must occur well within the optically thin regime. Thus, optical depth effects may decrease the line polarisation. Dust and line scattering differ in one important aspect. The former has a large forward scattering efficiency which is not present in the latter.

In the following analysis we will assume that the dust grains are the scattering agent responsible for the bulk of the detected polarised light. In Sect. 6 we will put forward more arguments in favour of this interpretation.

5.2 The circumstellar model

5.2.1 The radiative transfer

A modified version of the Monte Carlo scattering code of Ménard (1989) was used to compute model brightness distributions. The code only treats the scattering of stellar photons by dust grains. Thus, information on any possible resonance line scattering by K and Na atoms and on the dust thermal emission is not obtained. The method is in principle simple. A number of photons, emitted by the star, are followed through the dusty CSE, and the new photon paths and Stokes intensities are recalculated after each scattering event. Two-dimensional scattered light images are obtained from the number of photons that escape the CSE, and radial profiles of observed quantities can be derived. In all cases the model results are convolved with a seeing Gaussian of 1 $\hbox{$^{\prime\prime}$ }$.

   
5.2.2 The circumstellar envelope

Based on the results from the CO radio line observations and the analysis of the scattered light we assume that the CSEs are spherically symmetric, and that they are detached from the stars. The detached nature is specified in the models by an inner radius $r_{\rm in}$ at which the dust number density is $n_{\rm in}$. Both quantities are free parameters in the code. The outer radius $r_{\rm out}$ is determined by the fits to the observed total intensity AARPs, which show sharp outer cut-offs (Sect. 4). We have assumed constant mass loss rates during the formation of the shells and uniform expansion, i.e., within the shells the dust density distribution, $\rho_{\rm d}(r)$, follows an r^-2-law. It turned out that the shape of the observed radial profiles of the polarised intensity required a smoother decrease in density inside $r_{\rm in}$ than provided by a step function (see discussion in Sect. 5.5). We parametrise this as a $n_{\rm in}(r/r_{\rm in})^\alpha$ density law for $r < r_{\rm in}$. As an example, the density has decreased by a factor of ten (from its value at $r_{\rm in}$) at 0.75 $r_{\rm in}$, 0.83 $r_{\rm in}$, and 0.87 $r_{\rm in}$ for $\alpha$ equals 8, 12, and 16, respectively. Very likely, the density structure is more complicated than this, but the available constraints are such that a more detailed analysis is not possible.

5.2.3 The dust properties

The dust optical properties are described by the dust absorption and scattering cross sections, which depend on the grain size distribution and the refractive index of the grains. In order to limit the number of free parameters in the models, we have used a single chemical composition of amorphous carbon grains for the dust population. In this context, Bujarrabal & Cernicharo (1994) presented molecular radio line observations towards R Scl. They found, from a comparison of line intensity ratios with those typical in standard AGB-CSEs, that the chemistry of the detached gas shell around this star is C-rich. We note that the presence of carbon grains other than those of amorphous carbon would to some extent modify, through their different polarising characteristics, the results derived here. The optical constants for amorphous carbon at the wavelengths of interest have been obtained from Rouleau & Martin (1991). The corresponding scattering properties of the grains are derived using Mie theory. The grain size distribution is given by a power law with sharp boundaries, i.e., $a^{-\beta}$, where $a_{\rm min} \leq a \leq a_{\rm max}$. We have fixed the values for the minimum and maximum grain sizes to 0.05 and 2 $\mu$m, respectively. The exponent $\beta$ is used as a free parameter.

5.2.4 The fitting procedure

Since for both R Scl and U Ant the scattering is well within the optically thin regime (see below), the dust density at the inner radius, $n_{\rm in}$, is well constrained by the ratio between observed scattered and stellar flux, while the ratio of polarised flux to total scattered flux, i.e., the polarisation degree, is entirely determined by the scattering properties of the dust grains. Once this fact has been established, the rest of the parameters can be determined without influence of optical depth effects, and independently of the absolute calibration.

We start the modelling procedure by fitting the shape of the AARP of the polarised intensity (the fitting range is chosen to lie around the peak flux), since our basic assumption is that all of the scattered polarised flux is due to dust scattering. In this way all parameters, except $n_{\rm in}$, are determined: $r_{\rm in}$, $\rho_{\rm d}(r)$, and $\beta$ (the outer radius is fixed and it is determined by the sharp outer decline of the intensity). The effects of varying these parameters are discussed in Sect. 5.5. Since the total scattered flux and the polarised scattered flux are well calibrated relative to each other, the polarisation degree and the estimate of how much of the total scattered flux can be attributed to dust (assuming that dust scattering is responsible for all the polarised flux) are also relatively accurate. Finally, $n_{\rm in}$ is obtained by fitting the ratios (in the two filters) of scattered flux to stellar flux. Therefore, the uncertainty in the estimate of $n_{\rm in}$ is at least the factor of five which is derived from this ratio.

   
5.3 Results for R Scl

The scattered light images obtained from a model produce total and polarised brightness distributions similar to those observed using as input data $r_{\rm in}=19\hbox {$^{\prime \prime }$ }$, $r_{\rm out}=21\hbox{$^{\prime\prime}$ }$, $n_{\rm in}=3.8~\times~10^{-10}$ cm^-3, $\alpha=7$, and $\beta =-5.5$. The total intensity images show uniform, disk-like brightness distributions very similar to the observed ones. The ring-like structures seen in the images of the polarised intensities are also well reproduced in the model. The central region appears hollow in polarised light since only scattering at $\approx$90 $\hbox{$^\circ$ }$ polarises the light effectively, and there is very little scattering material inside $r_{\rm in}$.

Figure 5: A comparison of the model (solid lines) and observed (dotted lines) AARPs of R Scl in the F77 (left panels) and F59 (right panels) filters. Upper panel: total intensity. Middle panel: polarised intensity. Lower panel: polarisation degree. The model total and polarised intensities are scaled such that the model and observed polarised intensities agree in a region around the peak (see text for details).

For a more detailed comparison between the model results and the observational data, we have calculated the AARPs of the total intensity, the polarised intensity, and the polarisation degree convolved with a seeing Gaussian of 1 $\hbox{$^{\prime\prime}$ }$. The best-fit model profiles in the F77 and F59 filters are shown in Fig. 5. The fits to the AARPs of the polarised intensities are relatively good in both filters. In order to reproduce their shape we allowed for a smoother decrease (rather than instantaneous) in the grain number density inside $r_{\rm in}$. The model total intensity increases inwards in both filters as an effect of forward scattering. Unfortunately, our observations do not probe this inner region, but they seem to indicate a rather uniform total brightness. For a direct comparison of the observed and model total fluxes, the model scattered light has not been considered inside the region which is not probed by the observations (inside $\approx$10 $\hbox{$^{\prime\prime}$ }$). The total fluxes derived from the model are lower than the observed ones by about 40% in the F77 filter and 30% in the F59 filter (note that the inward drop of the AARP in the F77 filter is probably an effect of PSF oversubtraction). Under the assumption that only the circumstellar grains polarise the scattered stellar light, about 60% of the scattering in the F77 filter and 70% in the F59 filter is due to the dust. Thus, there is possibly room for other scattering agents.

The computed scattering optical depths are, in the F77 filter, $3\times10^{-4}$ in the tangential direction and $1.5\times10^{-4}$ in the radial direction. In the F59 filter the corresponding values are $8\times10^{-4}$ and $4\times10^{-4}$. Thus, the dust scattering is optically thin in both filters. The uniform intensity disk appearance is therefore attributed to the large forward scattering efficiency, which also masks the geometrical structure. The model results are summarized in Table 3.

The model results are very sensitive to the grain-size distribution, see Sect. 5.5. A very steep decline in grain size is required to fit the observational data ( $\beta =-5.5$). Such a steep decline has also been found to best fit polarimetric observations of PPNe (Scarrott & Scarrott 1995; Gledhill et al. 2001). However, grains of size <0.1 $\mu$m contribute only marginally to the scattering and extinction at optical wavelengths because their effective cross sections are much smaller than their geometrical ones. As a consequence, the maximum contribution to scattering comes from grains in the size-range 0.1-0.2 $\mu$m.

   
5.4 Results for U Ant

Figure 6: Same as Fig. 5, but for U Ant.

The structure of the circumstellar medium around U Ant appears somewhat complicated (this is further discussed in Sect. 6). The presence of one shell (shell3) is clearly seen in the total intensity images of the scattered light in both filters, but there may be at least two shells inside this. In the polarised intensity images only an external component is evident (shell4), which is just barely visible in the total intensity data. Therefore, modelling the observed brightness distributions is rather tricky. However, the fact that only the light scattered in shell4 appears polarised simplifies matter.

The results of the best-fit model to the observed polarised intensities are shown in Fig. 6. The parameters used are $r_{\rm in}=49\hbox{$^{\prime\prime}$ }$, $r_{\rm out}=54\hbox{$^{\prime\prime}$ }$, $n_{\rm in}=1.8~\times~10^{-10}$ cm^-3, $\alpha=11.5$, and $\beta =-5.5$. The fits are quite good in both filters. However, the model total fluxes (estimated in the region probed by the observations, i.e., outside $\approx$30 $\hbox{$^{\prime\prime}$ }$) are very low compared to the observed ones, only 35% of the total scattered flux in the F77 filter and 25% in the F59 filter. This implies that the bulk of the scattered light (shell1 to shell3) is due to another scattering agent. The derived scattering optical depths in the tangential and radial directions are $3\times10^{-4}$ and $1\times10^{-4}$, respectively, in the F77 filter. At the wavelength of the F59 filter, the corresponding optical depths are $7\times10^{-4}$ and $3\times10^{-4}$. Like in R Scl, the dust scattering in the circumstellar medium of U Ant is optically thin. The model results are summarized in Table 3.

Figure 7: Model AARPs of the total intensity (left panel) and polarised intensity (right panel) in the F59 filter towards U Ant assuming that the bulk of the stellar light scattered in the shell3 component is due to dust scattering (dotted lines give observational results).

In order to check this result, we fitted the total scattered light in shell3 using the scattering code. The result, shown in Fig. 7, is that if the observed light has been scattered by dust grains, it should show clear evidence of polarisation, clearly incompatible with the observations. Dust scattering by grains of different composition, with less effective polarising properties, could explain the fact that the stellar light scattered in shell3 is not polarised. However, resonance line scattering by K and Na atoms seems to be a more plausible interpretation (see Sect. 6).

Figure 8: R Scl model AARPs of the total intensities (left panels) and polarised intensities (right panels) in the F77 filter for different values of the free parameters. Upper panels: results for $r_{\rm in}=2\hbox {$^{\prime \prime }$ }$ (dashed line), $r_{\rm in}=18\hbox {$^{\prime \prime }$ }$ (dash-dot-dot-dot line), $r_{\rm in}=19\hbox {$^{\prime \prime }$ }$ (solid line), and $r_{\rm in}=20\hbox {$^{\prime \prime }$ }$ (dotted line). Middle panels: results for inward density power laws of exponent $\alpha =8$ (dash-dot-dot-dot line), $\alpha =10$ (solid line), $\alpha =16$ (dotted line), and $\alpha =\infty$ (dashed line). Bottom panels: results for $\beta =-4.5$ (dashed line), $\beta =-5.0$ (dash-dot-dot-dot line), $\beta =-5.5$ (solid line), and $\beta =-6.0$ (dotted line) (see text for details).

 

 

Table 3: Model results for R Scl and U Ant (see text for details).

	R	$\Delta R$	$M_{\rm d}$	Filter	Comp.	Total flux	Polarised flux	Scattering by dust1
	[ $\hbox{$^{\prime\prime}$ }$]	[ $\hbox{$^{\prime\prime}$ }$]	[$M_{\odot}$]			[erg s-1 cm-2]	[erg s-1 cm-2]	[%]
R Scl	20	2	$2\times10^{-6}$	F77		$1.3\times10^{-12}$	$4.6\times10^{-13}$	60
				F59		$7.7\times10^{-13}$	$2.6\times10^{-13}$	70
U Ant	52	5	$4\times10^{-6}$	F77	shell4	$1.1\times10^{-12}$	$3.6\times10^{-13}$	35
				F59	shell4	$5.0\times10^{-13}$	$1.2\times10^{-13}$	25

¹ The amount of scattered light attributed to dust based on the polarised emission.

   
5.5 Dependence on parameters

We have computed R Scl models with different values for the free parameters in order to investigate their effects on the results, and also to estimate how well these parameters can be constrained. Figure 8 shows the different radial profiles in the F77 filter obtained when $r_{\rm in}$, $\rho_{\rm d}(r)$, and $\beta$ (from top to bottom) are changed in the optically thin regime. The outer radius is fixed and it is determined by the sharp outer decline of the intensity. The peak of the polarised intensity provides the point to fit.

The peak radius of the polarised intensity depends sensitively on the inner radius. Thus, this parameter can be determined rather accurately in the modelling, and the uncertainty in the $r_{\rm in}$ (and also the $r_{\rm out}$) estimates are dominated by the seeing ($\approx$1 $\hbox{$^{\prime\prime}$ }$). Note here the model results obtained for a CSE which is "attached'' to the star. In this case the total and polarised intensities of the scattered light come mainly from line-of-sights close to the star, and the mismatch with the observed profiles is evident.

The shape of the observed polarised intensity AARP could only be fitted with a rather high value of $n_{\rm in}$ (since $r_{\rm in}$ is fixed by the peak position) for a dust density profile with an instantaneous rise at the shell inner radius (step function profile). Such a high value of $n_{\rm in}$ results in scattered flux to stellar flux ratios that are clearly incompatible with the observed ones. An alternative way to fit the shapes of the polarised intensity profiles with lower optical depths, which are consistent with the observed scattered to stellar flux ratios, is to consider density distributions which decrease more gradually inside $r_{\rm in}$ (see Sect. 5.2.2). Acceptable fits are obtained for high values of $\alpha$, and hence the decrease in density inside $r_{\rm in}$ is rather steep.

A large value of $\beta$ (i.e., less negative) implies an increased importance of larger grains, which makes the scattering process less isotropic. This results in high intensities along line-of-sights close to the star as an effect of increased forward scattering. This parameter also affects the wavelength dependence of the polarisation, and it is therefore rather well constrained by the observations in the two filters.

   
5.6 Thermal dust emission

 

 

Table 4: Observed IRAS fluxes compared to those derived using DUSTY. The model values are separated into the stellar and circumstellar components. The parameters derived from the scattering models are used to describe the circumstellar media around the stars.

		$T_{\rm d}$( $r_{\rm in}$)	$F_{\rm 12}$	$F_{\rm 25}$	$F_{\rm 60}$	$F_{\rm 100}$
			[Jy]	[Jy]	[Jy]	[Jy]
R Scl	star		105	27	5	2
	shell	55	71	16	28	15
	obs		162	82	54	23
U Ant	star		182	48	9	3
	shell	44	81	12	33	29
	obs		168	44	27	21

A possible way of further constraining the modelling is to see whether the dust shells, which we have derived from the scattering modelling, are able to produce the observed IRAS fluxes, which are due to dust thermal emission. We have estimated their fluxes at 12, 25, 60, and 100 $\mu$m using the dust radiative transfer code DUSTY (Ivezic et al. 1999). The parameters derived from the scattering models are used as inputs for the circumstellar medium. For R Scl we used an effective temperature of 2700 K, a luminosity of $5600~L_\odot$ (Hron et al. 1998), and a stellar distance of 360 pc. In the case of U Ant, 2800 K was adopted and $5000~L_\odot$ was derived from the measured $m_{\rm bol}=2.58$ (Bergeat et al. 2001) and the Hipparcos distance of 260 pc.

The model results together with the observed IRAS fluxes are given in Table 4. For both stars the derived fluxes are in good agreement with the observed values. The discrepancies are within a factor of three, which is well within our estimated uncertainty of a factor of five for the $n_{\rm in}$:s. Therefore, we tentatively (considering the uncertainties in the calibration of the scattering data) conclude that the same dust component is responsible for the polarised scattered emission in our images and the thermal emission measured by IRAS.

   
5.7 Dust masses and shell sizes

The dust shell masses are estimated using

$$M_{\rm d}=\frac{16}{3} \pi^2 \rho_{\rm gr} \int_0^{r_{\rm ... ...\rm min}}^{a_{\rm max}} a^3 n(r,a) {\rm d}a~~ r^2 {\rm d}r,$$ (8)

where $n_{\rm in} = \int_{a_{\rm min}}^{a_{\rm max}}n(a){\rm d}a$, and $\rho_{\rm gr}$ is the density of an amorphous carbon grain. We have used $\rho_{\rm gr}=1.85$ g cm^-3 (Bussoletti et al. 1987) and a distance of 360 pc and 260 pc to R Scl and U Ant, respectively. The derived dust shell masses are $2\times10^{-6}~M_{\odot}$ and $4\times10^{-6}~M_{\odot}$ for R Scl and U Ant, respectively, and they are uncertain by at least the factor of five uncertainty in the $n_{\rm in}$ estimate. Izumiura et al. (1997) calculated the masses of the two dust shells seen in high resolution IRAS images of U Ant. For the inner dust shell (which is likely related to shell4 in our observations of stellar scattered light) they obtained a value five times higher than our estimate. This difference is partly due to the fact that they used a higher luminosity and a lower effective temperature of the star.

For U Ant we have tried to derive upper limits to the dust shell masses contained in the inner components shell1, shell2 and shell3, discernible in the images taken with the F59 filter. The shells positions and widths are taken from the observations presented in Paper I due to the higher S/N-ratios of those images. We assume that an upper limit to the thermal emission by the dust grains in each of these components is given by one quarter of the detected circumstellar 60 $\mu$m flux, i.e., $\approx$8 Jy. This is a rather conservative estimate. We derive upper limits of $9\times10^{-7}~M_{\odot}$, $7\times10^{-7}~M_{\odot}$ and $2\times10^{-6}~M_{\odot}$, for shell1, shell2 and shell3, respectively. Therefore, the dust shell masses are estimated to be lower than in shell4 by at least a factor of five in the two innermost components and by at least a factor of two in shell3.

The modelling of the R Scl data results in a dust shell of radius 20 $\hbox{$^{\prime\prime}$ }$ (or $1.1\times10^{17}$ cm) and of width 2 $\hbox{$^{\prime\prime}$ }$ ( $1.1\times10^{16}$ cm). That is, the shell has a small radius/width ratio, $\Delta R/R \approx 0.1$, and in this respect it resembles the CO shells seen towards U Cam (Lindqvist et al. 1999) and TT Cyg (Olofsson et al. 2000). Assuming that the CO gas expansion velocity (15.9 km s^-1, Olofsson et al. 1996) can be used to estimate the age and the time scale of formation of the dust shell, we obtain an age of about 2200 yr and a formation period of about 220 yr (provided that no effects of interacting winds or shell evolution are present). For U Ant we derive from the scattering model a dust shell width of 5 $\hbox{$^{\prime\prime}$ }$ ( $2.0\times10^{16}$ cm) and a radius of 52 $\hbox{$^{\prime\prime}$ }$ ( $2.0\times10^{17}$ cm), i.e., it is also geometrically thin ( $\Delta R/R \approx 0.1$). Using the CO gas expansion velocity (18.1 km s^-1, Olofsson et al. 1996) we estimate age and formation time scales of about 3600 yr and 350 yr, respectively. That is, the two dust shells are characterized by relatively similar time scales.