Astron. Astrophys. 323, 449-460 (1997)

Appendix: brightness distribution calculation of the model dust shell

The brightness distribution of a double dust shell is a simple superposition of those of two single shells in the optically thin limit. We therefore describe here the calculation of the brightness of a single shell for simplicity. The brightness of a model dust shell with spherical symmetry as a function of projected distance b from the central star at frequency , , is calculated under the assumption of the optically thin limit by

where and l denote the grain absorption cross section per unit mass of the grain material at frequency , grain mass density, black body function at , dust temperature, radial distance from the central star, and displacement along the line of sight from the plane which contains the center and is normal to the line of sight, respectively. The variables b and l are related to the radial distance from the star to the relevant point r through

For the definition of , and r see Figure 2. In the figure the plane which contains both the line of sight and the center of the shell is indicated. The integration interval is determined by the shell inner and outer radii, and as

We define here that the grain mass density is a power law function of radial distance from the central star via

We assume that the grain absorption cross section varies as a power law function of , and that the cross section is related to other quantities through

where express the absorption efficiency at frequency , the average grain radius, and the average density of the grain material, respectively. This frequency dependence of the cross section allows us to calculate analytically the temperature distribution throughout the shell for the grains in a steady state (Sopka et al. 1985),

and is given by

where , and c are Planck constant, Boltzman constant, observed stellar flux density at frequency , and the speed of light, respectively. We here normalize the temperature law at the surface of the star. Note that this does not necessarily mean that grains are formed at the surface of the star. In the present paper, we also assume that the spectral energy distribution of the star is approximated by the black body function for the star's effective temperature. Then, we have

where is the effective temperature of the star. Note that the temperature distribution is dependent on only the exponent of the opacity law. The absolute value of is irrelevant. This means that is relevant only to the mass loss rate and mass in the shell. To adjust the calculated brightness distribution to the HIRAS images, we convolved it with a gaussian beam as

where we chose a gaussian pattern with a half-power beam width as the convolution function .

We made the above calculations for the inner and outer dust shells independently, then add the two distributions to obtain

To include the contribution from the central point source, we put a point source which has a total flux density of at frequency smeared by the same gaussian pattern which we used in the above convolution expressed as

Adding the two brightness distributions

we obtain the total brightness which can be compared with the HIRAS image. In order to compare the model predictions with the obtained HIRAS results, we further calculated mean brightness in annular rings for a run of centered at the star using the model brightness as,

where 2 is the width of the rings. The same procedure was applied to the HIRAS images to obtain by replacing with which expresses an direct HIRAS image.

In order to find the best fitted models, we minimise the quantity,

where denotes the internal error estimated with the HIRAS processor, by changing free parameters (, and ). and are flux densities at the frequencies corresponding to the wavelengths of 60µm and 100µm, respectively. Subscripts in and out show that the quantities are of the inner and outer shells, respectively. We fixed =2 in the present study. We do not leave the shell thickness free as well. We examined several cases of shell thickness instead, because the model brightness profile is rather insensitive to the position of the shells outer radii. Afterwards, we calculate the dust mass loss rate and total dust mass in each shell assuming a dust outflow velocity of 21 km s^-1 by

Finally we transformed those quantities to the total mass loss rate and total mass using a dust/gas mass ratio and assuming that the gas outflow velocity is the same as the dust outflow velocity via

where is the dust-to-gas mass ratio.

© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998

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