Monte Carlo radiative transfer methods simulate the interaction of light with matter by treating light as individual "photons" whose small-scale behavior is governed by random sampling from statistical probability functions. One can predict large-scale characteristics of the simulated system by combining results from an ensemble of millions of photons. Since the method is numerical rather than analytic, it does not require symmetry, and is therefore well-suited for problems with complex asymmetric geometries.

With Barb Whitney, I have developed a code, called "DISK," which applies this method to the situation of a binary-disk system: two stars, one surrounded by a disk of scattering material. Wood et al. (1996, AJ, 461, 847) used Monte Carlo modeling to simulate the polarization arising from a disk illuminated by an interior star. We consider two further cases: a self-illuminated disk, and a disk illuminated by an exterior orbiting star. The basic geometry of the systems we model is shown in Figure 1. The disk is the scattering region, and hence the source of the polarization, in the system. In our models, the light scattered in the disk originates from the outer star and from uniform volume emission within the disk itself.

     Figure 1: Schematic representation of the geometry we use for our binary-disk models. The disk is a spherical wedge of constant density with opening angle alpha; the external star is 2 Rdisk away and spherical with a radius 0.65 Rdisk. The inner star is spherical with a radius of 0.1 Rdisk; since Wood et al. (1996) have treated the case of an internally illuminated disk, we here disregard the emission from the inner star.

As a system with this configuration rotates, it produces polarization curves similar to those shown in Figure 2. Our recent work investigates how the shapes of these curves (excluding the eclipses, which depend heavily on the relative sizes of the system components) varies with the opening angle and optical characteristics of the scattering disk.

  
Figure 2: Typical polarization curves of the self- illuminated disk (left) and the externally illuminated disk (right). Key characteristics of these curves are labeled: qDC is the average base level of the disk polarization above zero, while pbump measures the size of the increase in polarization at phases 0.3 and 0.7, produced by 90° scattering of photons from the external star off the disk edge. We use pbump instead of qbump, since Stokes U contributes the externally illuminated polarization at high inclinations.


Here are some of our conclusions; a paper describing these and other results will appear in the ApJ. Figure 3 shows the variation of pbump and qDC with albedo for various inclination angles and high and low optical depths. The thin disk case (left) has alpha=3°; the thick disk (right) has alpha=33°. In most cases, both polarization indexes increase linearly with albedo. However, for the thick disk with high optical depth and high inclination, qDC is negative and decreases with albedo. Here, photons from the equatorial regions of the disk are being absorbed or scattered into the polar regions, where they become negatively polarized. Multiple scattering increases the absolute value of this polarization.

  
Figure 3: Polarization as a function of albedo for various inclination angles and two values of optical depth. In the left plot, the disk is thin with an opening angle of 3°; in the right plot it is thick with an opening angle of 33°. qDC and pbump are defined in Figure 2 above. Dashed lines represent polarization arising from single-scattered photons, while solid lines represent the polarization arising from all scatterings.

Figure 4 shows the variation of pbump and qDC with optical depth for various inclination angles and high and low albedos. qDC is much more sensitive to optical depth than is pbump, and multiple scattering causes larger changes in qDC as well. This is because most of the scattering causing pbump is single scattering off the disk edge. qDC also changes its behavior with optical depth between the thin and thick disks; polar cancellation in the thick disk causes the polarization to decrease rapidly with optical depth, especially for high inclinations.

  
Figure 4: Polarization as a function of optical depth for various inclination angles and two values of albedo. As in Figure 3, thin disk results are on the left and thick disk results on the right. Line types are the same as in Figure 3.

Figure 5 shows the variation of pbump and qDC with inclination (sin2 i) for various optical depths and high and low albedos. According to the formulation of Brown, McLean, & Emslie (1977, A&A, 68, 415), %q is proportional to sin2 i in optically thin binary envelopes symmetric about the orbital plane; our models confirm this result, but show large deviations from the proportionality at higher optical depths. We also see that qDC is larger than pbump for the thin disk, but comparable to or smaller than pbump for the thick disk; again, this phenomenon is due to the fact that, in the thick disk, qDC suffers more polar cancellation at high optical depths than does pbump. Finally, for both polarization quantities, the largest values normally occur not at i=90°, but at intermediate inclinations. This occurs because the polar regions of the disk, where negative polarization arises, become occulted as the disk tilts away from edge-on.

  
Figure 5: Polarization as a function of sin2i for various optical depths and two values of albedo (top: a=0.1; bottom: a=0.9). As in Figure 3, thin disk results are on the left and thick disk results on the right. Line types are the same as in Figure 3.

These results for self- and externally illuminated disks may be combined with models of internally illuminated disks (such as those presented by Wood et al. 1996) disks to model the flux and polarization variations of any binary-disk system in which the disk is the primary source of polarization. With the proper weighting of the three polarization contributions by the relative fluxes of the illuminators, one may constrain the system inclination and disk properties from the observed polarization. We illustrate this method with our models of the eclipsing binary beta Lyr.


Detailed description of the code and analysis of results:
Hoffman et al. 2003, ApJ, 598, 572

Overview of results:     Hoffman 2002, AAS, 201, 2505

Early models using "DISK":     Hoffman et al. 2000, ASP Conf. Ser. 214, 464
For more information, please send email to Jennifer Hoffman.
March 25, 2004