We have generated synthetic eclipse profiles from equation (7) with z given by equation (8). The values of the orbital elements e and were those given in Section D.2. Throughout this section, we assume the disk to be cylindrically symmetric. We model disks with and without central holes and use a variety of midplane opacities and temperatures.
The disk was modeled on a rectangular grid of 101 elements in the z direction and 505 elements in both x and y. The opacity at each grid point was then calculated. The eclipse lightcurve was computed on a grid of 505 time steps separated by approximately 1.3 days, for a range of values of the disk scale height, , the inclination, i, and central hole radius, . The opacity parameter, approximately equal to the ``vertical'' optical depth of the disk, is:
was calculated iteratively to reproduce the maximum eclipse depth of 48% for each combination of h, i and that we modeled. If the disk's mass is 0.1 M and its opacity is similar to that of interstellar matter, then K in the near-IR. A more massive disk, or one made of a more opaque material, could increase K, but realistically only by a few orders of magnitude. No equivalent lower bound on K exists because substantial grain growth, settling of the grains to the midplane and extremely low-mass disks cannot be ruled out. (See Weidenschilling and Cuzzi (1993) for a review of grain growth within protoplanetary disks.) Line of sight optical depth profiles through six of our model disks are shown in Figures and .
Tables and describe the values of the parameters chosen for our simulations and the primary quantitative results. The constant models are numbered consecutively, but the models are given numbers 100 greater than the constant runs with otherwise identical parameters. We present two measures of the dependence of the eclipse depth, D, upon opacity. The ``-color'' is defined as , evaluated at the phase where reached its maximum. The ``-opacity'', , is the factor by which the nominal opacity must be multiplied to give 50%. These parameters allow comparison of model predictions and observations with functions of particle opacity vs. wavelength. A value of = 1 signifies completely gray, since this implies that is the same for both opacities. A value of = also signifies completely gray, since infinite opacity differential means there is no opacity which increases by 2%. Second contact is defined as the first time (in days after first contact) that the eclipse depth D = 46% (vs. a maximum depth of 48%).
The fifth calculated quantity is the fraction of the primary's light which is blocked by the disk at mid-eclipse. Owing to the disk's projected ``bow-tie'' shape, the deepest part of the eclipse usually occurs near second and third contacts, with mid-eclipse brightening in between. The central brightness is a measure of how ``pinched'' the center of the disk appears at the given viewing angle. Representative eclipse profiles at a variety of wavelengths are shown in Figures and ; additional profiles are presented in Figures through .
Our principal conclusion based upon the results presented in Tables and and Figures and is that the particles providing the bulk of the opacity in the Aurigæ secondary are much larger than typical interstellar grains (0.01 -- 0.1 m) and are thus probably the result of a process of solid accretion. Observations of the eclipse imply that the fractional difference in depth from m to m is < 2% (Backman 1985). For an interstellar extinction law, from 1 -- 5 m (Rieke and Lebovsky 1985; this appears to be a robust result, with far less variability than the visible opacity, Whittet 1992). Thus, if the disk's opacity is produced by particles similar to interstellar grains, then the observed colorlessness of the eclipse in the near-IR would imply 1.02; we can only come close to reproducing this value for extremely high opacity disks with small scale heights (compare Figure (c) with Figure (e)). This would imply an extremely massive and cold disk as . If 98% of the opacity at 1.25 m is produced by particles large enough to block radiation with m, then the opacity differential does not provide any constraints on K. However, if the fractional opacity at 1.25 m provided by small particles is S, then mm For plausible particle size distributions, the observed eclipse colorlessness still constrains K from being too small. Thus, unless mm), the principal cause of the decrease in eclipse depth from m to m cannot be increased transparency of the secondary. This reinforces our confidence in temperature estimates of the secondary derived using infrared colors.
If K is small, then the sides of the disk, viewed edge-on, are as much as 10% more effective in blocking light then the center due to a longer path length in the thicker outer portion of the disk. Such a disk produces an eclipse light curve with two minima after 2 and before 3 contact, and a local maximum at mid-eclipse. Substantial mid-eclipse brightening occurs in our synthetic eclipses only if the opacity parameter is small, whereas dense disks appear sharp-edged, producing flat-bottomed light curves. Mid-eclipse brightening has arguably been observed in each of the last three eclipses (e.g., Carroll et al.
1991 and references therein). However, the eclipse depth cannot be precisely measured over periods of less than a few months due to irregular light variations on this time scale, probably caused by pulsations of the primary (Burki 1978, Ferro 1985, Donahue et al. 1985, Carroll et al. 1991). In addition, the apparent mid-eclipse brightening occurs on a time scale more analogous to that of the observed stellar variations than to the gradual change seen in our synthetic profiles. For the cases of low opacity and significant mid-eclipse brightening, color constraints imply that the disk's opacity must be provided almost entirely by particles larger than 5 m.
A 1 disk tilt (i = 89) has little effect on the eclipse profile, although the slight central brightening seen in edge-on disks at moderate to high opacity is eliminated (compare, e.g., models 18 and 21 in Table ). Greater tilt implies that a disk with given physical parameters covers a larger solid angle in the sky plane. Thus, a slightly tilted () thin () disk can produce the observed eclipse depth with a lower opacity parameter than an edge-on disk with the same scale height. However, in the cases where the disk is more significantly tilted and/or thinner, a substantial amount of the disk material does not block the primary, so a larger opacity parameter is needed. We did not even compute eclipse profiles for i = 87.5 and 0.025 because the required values of K are too large. The marginal case of i = 87.5, = 0.03 produces a nearly colorless eclipse because of the sharp cutoff in the optical depth of the projected disk. While a significantly tilted disk with high opacity could produce the observed colorlessness (model #47), it would also produce a very rounded eclipse bottom (Figure (b)) and thus is ruled out by observation. Therefore, we conclude that .
Our results are not sensitive to the radial dependence of the surface density (compare Figures (c) and (f)), unless the opacity parameter is very low (e.g., model 39 vs. model 139). Likewise, central holes, even as large as 0.9, have little influence on eclipse profiles except for low opacity disks, because the optical depth of the outer edge near the midplane remains high enough to block most of the light (Figure (d)). Central holes might have more of an effect on the eclipse light curve if the disk is inclined because this could allow us to see directly through the clear central region (Wilson 1971, Eggleton and Pringle 1985, Kumar 1987, Ferluga (1990). A disk inclined to the system's orbital plane would result in a warp (Kumar 1987) that would make seeing through the central hole difficult if the particles providing the opacity are coupled to the (presumed) gaseous component of the disk.
The models presented above do not depend directly on the size and scale of the Aurigæ system. However, the ratio of the scale height of the disk to its radius as given by eq. (6) depends upon the mass of the secondary as well as the temperature of the outer portion of the disk. Our best matches to observations, using parameters which we consider reasonable, are obtained for (Figure (c) and (c)). Thus, using the temperature of the disk's outer edge measured during the 1982 -- 1984 eclipse by Backman et al. (1984), our results appear to be more consistent with the high-mass model () than with the low-mass model (); however, because of various modeling and observational uncertainties involved, we do not view this result as definitive.