In this section, we attempt to account for the observed asymmetry in absorption line profiles by using a crude model of the thermal expansion of the outer edge of the disk as it rotates into the radiation field of the primary. The preceeding analysis can be extended to model the temporal variations in the observed absorption line profiles. Ignoring scattering and the emission from the secondary, the fractional depth of an absorption line is equal to
where 
is the continuum optical depth,
is the optical depth of the absorbing species and the integral
is taken over the disk of the primary. For a homogeneous
disk, 
, assuming that the line is not saturated.
Synthetic profiles can be computed from
the above equations for a given constant of proportionality. However,
for an azimuthally symmetric disk
these computed line strengths will be symmetric about the eclipse
center, apart from slight differences resulting from the
eccentricity of the orbit.
Hinkle and Simon (1987) proposed that the asymmetries in
the pre- and post-central eclipse line profiles are a consequence of
the vaporization of grains caused by heating from the primary. Infrared
observations of the
system at quadrature indicate that the primary
heats the side of the disk facing it to an equilibrium temperature of
K (Backman et al.
1996). This value is much hotter than
the 
K
temperature observed on the back side of the disk during eclipse
(Backman et al.
1984, Backman and Gillett 1985). At these high
daytime" temperatures, some grain destruction
should occur. The velocities of the lines imply the disk
rotates in a prograde direction (Lambert and Sawyer 1986), so the
late night/morning"
regions block the primary during ingress, and the afternoon/evening"
regions are in front of the primary during egress
(Figure 
).
Hinkle and Simon's conjecture thus explains the observations in a qualitatively correct manner; however, we believe that it is quantitatively inadequate for the following reasons. First, although the temperature range is large, it does not include the condensation temperatures of the most cosmically abundant grain constituents; ices condense at well below 500 K, whereas most silicates remain in grain form up to at least 1200 K (Lewis 1974). Second, the orbital period at the outer edge of the secondary is given by (cf. eq. 2):
The length of a ``day'' with respect to the primary is slightly longer, because the disk rotates in a prograde direction. Evaporation times of small grains are so short that exposed grains of intermediate volatility are rapidly destroyed in the ``early morning''. Finally, this model does not explain the persistence of absorption profiles well past 4th contact (Lambert and Sawyer 1986).
We agree with Hinkle and Simon (1987) that heating by the
primary probably produced the temporal asymmetry in the absorption lines,
but we
believe that the mechanism is thermal expansion of the disk rather than
grain destruction. Pressure is proportional to the temperature, and
thus rises rapidly in the region of the disk's edge rotating into view
of the primary. This increases the RHS of equation (4) (for z > 0) and
must be balanced by an acceleration of the gas in the vertical direction.
(The second term of the LHS of eq. (4) is quadratic in 
, and as
at ``dawn'', this term is small in the ``early
morning''.)
The disk's edge
thickens until a pressure balance is restored, at which time the scale
height has expanded by a factor of 
.
The time required to restore a balance can be crudely estimated using
dimensional arguments. Eventual thickening is of order unity, thus
Prior to dawn", both terms on the LHS of eq. (4) are
0, assuming
the disk has had time to settle into a static night"
equilibrium. Therefore, the terms of the RHS must be approximately
equal in magnitude just before ``dawn''. During the ``early morning'', the
first term on the RHS of eq. (4) increases by a factor of 
2. Thus,
ignoring the quadratic term on the LHS,
The scalings (12 and 14) can be combined to give
(The second term of the LHS of eq. (4) grows from zero to the order
of the first term during the ``morning'' expansion and therefore has only
an effect of order unity, i.e., it does not alter the above dimensional
arguments.) Disk thickening thus can occur on the required timescale. The
disk does not expand immediately at ``dawn'', but it thickens by a factor
of 
1.4 by ``mid-afternoon''. Contraction at ``night'' occurs on the
same timescale. This follows immediately from a sign change in the above
scalings, or directly from the realization that when pressure support is
removed, gas molecules and dust travel on free-fall orbits, which must
intersect
the disk midplane exactly twice each orbit and thus require 1/4 orbit
from peak to node.
A full treatment of the thermal expansion of the disk would be very complicated, involving 3-D numerical hydrodynamics and radiative transfer. An analysis of disk expansion requires solution of the three components of the momentum equation and the continuity equation, plus computation of temperature using radiative transfer models. These models require input of an optical depth scale consistent with the dynamics. Such a system of coupled nonlinear partial differential equations in three dimensions is prohibitively difficult to solve, even numerically. The uncertainties in many of the required input parameters and the paucity of the observational constraints would make any quantitative results of questionable value. We thus restrict our numerical simulations to the following simple model based on the qualitative arguments presented above: (1) The thickness of the outer edge of the disk is increased as this portion of the disk is exposed to stellar radiation during the ``day''; (2) At ``night'' it cools and contracts. Specifically, we define the outer portion of the disk to be material in the outer 0.1% of the disk radius. We have linearly increased the scale height of the outermost portion of the disk from its nominal value used in Sections D.3 and D.4 at ``6 a.m.'' to 1.4 times that value at ``3 p.m.'' (cf. the discussion prior to eq. 12). Then we linearly decrease the value with the same slope until the quiescent ``night'' value is reached at ``midnight.'' As our hydrodynamic simulation is quite sensitive to variations in a thin outer layer of the disk, we increased the resolution of the grid in the x and y directions by a factor of ten.
In order to estimate the depths of the absorption lines, we must
include in
our models the ratio of line opacity to that in the neighboring continuum.
The persistence of the lines after the end of the continuum eclipse implies
that this ratio is at least of order unity, and possibly much greater. This
contrasts to the situation in the ISM, where the continuum dust opacity is the
larger. The difference is not surprising, as metals are highly depleted in
the gaseous phase in the cold ISM environment, and are thus are probably
substantially more abundant in the much warmer gas of the 
Aurigæ disk. Moreover,
the larger grain sizes in the 
Aurigæ secondary (Section D.4)
imply a smaller continuum opacity than in the ISM.
For illustrative purposes, we consider that lower-abundance
metal lines might be several orders of magnitude weaker than H
which would have a line-center opacity about 
times that in the
neighboring continuum for a Doppler width corresponding to 1000 K
and ISM-like composition.
The larger grain sizes in 
Aurigæ
relative to the ISM would also make the continuum opacity weaker.
We consider line opacities per mass
10
, 10
and 10
times as large as that of the continuum.
Figures , ,
and
illustrate the results of adding thermal expansion to
one of our disk models.
The observed gross temporal asymmetry of the line strength
about the midpoint of the eclipse (Lambert and Sawyer 1986, Hinkle and Simon
1987) is readily reproduced. The persistence of the absorption signatures
after 
contact could be accounted for in a qualitative manner by
allowing the optically thin outer edge of the disk to also expand in
the horizontal direction as a result of heating by the primary;
however, the observation of Ti II for an entire year
after fourth contact (Lambert and Sawyer 1986) exceeds the duration
that would result from an expansion in the
horizontal direction comparable in size to the vertical scale height of
the disk. An optically thin (in the continuum) annulus exterior to
the optically thick disk which expands in the daytime and
contracts at night would help explain both the longevity of the
absorption after fourth contact and the decrease in the absorption
strength observed
just following second contact. An alternative explanation for this
latter feature is gas vertically extended above the disk's outer edge.
