Two models of a biconical dust nebula with an opaque torus in the symmetry plane are presented: a model containing dust in the whole biconical space ( A ) and an "empty" model with the dust layer on the surface of this empty space (B). Surface brightness distribution and polarization of the emitted light are computed using the Monte Carlo simulation of photon transport in the nebula. The results have features similar to those observed in several compact reflection nebulae.
1.IntroductionDense circumstellar dust clouds illuminated by a central star are of particular interest both for investigating their shape and for studying the solid grains of which they consist. The interpretation of the surface brightness measurements and polarimetric measurements may be strongly affected by the unknown geometry of such clouds. From the morphological point of view, the characteristic features of circumstellar envelopes projected on the celestial sphere may be divided into four classes: (i) circular or ring structure, (ii) flattened disk or torus, (iii) bipolar structure, (iv) combination of the above structures. While circular and elliptical structures are usually modelled by spherical shells and disks (hollow disks), bipolar structure can arise either due to an inclined ring or due to two lobes on opposite sides of the central star. In the present paper we make an attempt to approximate these lobes by a couple of conical segments centered on the star. Thus, only rotational symmetry is introduced into the model, and the inclination of the axis can be varied as one of the model parameters.
2.Radiative transfer treatmentFollowing previous models by VANYSEK and ~L C (1974) and WITT (1977a, b, c, d), we will describe the propagation of light through the nebula as a history of many individual photons moving within the nebula until they reach the surface and escape in the direction to the observer. Each photon trajectory originates in the central star which is represented as a point source with respect to the large dimensions of the envelope. The photon trajectory looks like a 3-D random walk, where the position of the scattering plane cp, the change of photon direction -the scattering angle 9 -, and the free path I between subsequent scattering are random quantities. In the following text the computed random values of these quantities will be denoted as "actual values". The probability densityf(9) of the scattering angle 9 is proportional to the intensity of the light scattered by the grain in a given direction (e.g., it is proportional to the sum of Mie's scattering functions i1(9) + i,(9) in case of spherical grains).The actual value of 9 can be computed fromwhere y is a random number with an uniform distribution in the interval ( 0 , l ) ; y is supplied by a random number generator. Each photon is associated with an orthogonal base of three unit vectors p, I, r. Vector p points to the photon direction.