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:
or
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 
m
m
For plausible particle size distributions, the observed eclipse
colorlessness still constrains K
from being too small. Thus, unless 
m
m), 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.
