The size, mass and distance to the system are inter-dependent and
uncertain. Constraints derived using the mass function, stellar evolution
models and various other data suggest either a high-mass system
at a distance of
1200 pc or a low-mass system 
at 
600 pc (Webbink 1985).
Note that in the low-mass model, the ``secondary'' is more massive
than the visible ``primary''; however, we shall follow the convention
used in most of the 
Aurigæ literature
and continue to refer to the F star as the primary because it is
clearly the primary source of luminosity of the system. Doppler shifts of
absorption lines seen during eclipse imply an orbital velocity at the
edge of the secondary more compatible with the low-mass model
(Saito et al.
1987, Lambert and Sawyer 1986). A distance
estimate based on astrometry of the primary's orbit of 
pc
(van de Kamp 1978)
also supports the low-mass system; however, this result must be
interpreted with caution due to the difficulty of obtaining such an
estimate from the ground; other evidence suggests a distance of
1000 pc (cf. Carroll et al.
1991). The HIPPARCOS spacecraft
is expected to provide a more accurate estimate of, or lower bound to,
the distance to the 
Aurigæ system.
Many characteristic quantities can be derived for 
Aurigæ that are independent of the system's absolute scale. These
scale-independent
quantities include the ratios between the secondary's length, the primary's
diameter, the component separation, and the system's distance from Earth,
the ratio of the bolometric luminosities of the components, the
radiative flux at the secondary due to the primary, and the color of
the eclipse. Our approach
is to separate scale-dependent from scale-independent quantities, and rely
on the latter as much as possible to construct the model framework.
The scale-dependent quantities (e.g., the thickness of the disk as a
function of temperature, mass of the secondary and the system's dimensions)
then allow us, in principle, to discriminate between the high- and low-mass systems.
A resolution between the two alternative models is
crucial to determining whether the disk is a remnant of
post-main sequence mass transfer (Eggleton and Pringle 1985) or a
protostellar/protoplanetary disk (Carroll et al.
1991). Results in this
paper (Section D.4) lend some support to the high-mass model, but
the scale-independence of our calculations allows them to be
directly applied to either high- or low-mass systems.
The best
known parameters of the 
Aurigæ system are the mass function,
(Morris 1962); the orbital period, P = 9890 days (e.g.,
Carroll et al.
1991); the
eclipse depth, 48% of the primary's light blocked at visual (
0.7 magnitudes extinction; Schmidtke 1985); and the angular size
of the primary, 
str. The mean eclipse
depth varies by 
1% from 1.25 
m -- 4.8 
m (Backman
1985). The upper limit to the color variations is almost as severe
within the visible (Schmidtke 1985), but possible differences between
visible and IR depths are less well constrained because the data were
obtained at different times. The fact that we
can see the eclipse means that the orbital inclination with respect to
the plane of the sky, i, is close to 90
.
Quantitative timing estimates vary from one eclipse to another.
Typical eclipse durations are:
1
-- 4
contact 
670 days, 2
-- 3
contact 
394 days, these numbers are uncertain
and possibly changing (Schmidtke 1985). The time between 1
and
3
(or 2
and 4
) contacts implies the secondary's
length" (i.e., its extent parallel to its orbit) is
where a is the semimajor axis of the primary's orbit around
the center of mass, e the orbital eccentricity
(roughly equal to 0.2, Wright 1970), 
the angle between
periapsis and the line of sight (periapse is estimated to be 14
before we observe mid-eclipse; Wright 1970) and
is the fraction of the system's orbital period which lies
between first and third contacts. Terms of
second order in e or 
have been omitted, and only those
quantities well
known from eclipse observations have been evaluated numerically.
Regarding the geometry of the cross-section of the secondary in view during eclipse, there are several alternatives. A) The secondary may be a sharp-edged thick disk that is parallel to its orbital plane and this plane may also contain our line of sight. This model yields the minimum projected area for the disk. B) The disk could lie parallel to its orbit about the F star and be viewed along a line-of-sight somewhat inclined to that plane, subtending a solid angle larger than the minimum. C) The disk could be warped or inclined to its orbit plane and the line of sight, and be larger in cross-section during eclipse than the minimum size.
The timing between 1
and 2
contact and between 3
and
4
contact gives
the primary's radius, 
= 
138/532 =
0.13 
. The
depth of the eclipse implies a minimum height (extent perpendicular
to its orbit) of the secondary of 
with the equality holding
if and only if Earth is in the orbital plane of the system. Thus, if the
secondary is a thick opaque disk (Huang 1965), the area of the
secondary as viewed from the Earth is 
2.0 times
that of the primary.
If we are
viewing the eclipse from out of the orbital plane (
), then
the most extreme viewing angle consistent with the
observed eclipse depth implies that the secondary blocks the light from
(almost exactly) one hemisphere of the primary and one edge of the
secondary passes nearly across the center of the primary. A thin
secondary in the orbital plane cannot block the center of the primary unless
it extends to the primary's center, which is impossible. A secondary of
thickness H and disk radius 
just eclipses the
primary's center if tan 
.
The area presented to us in such a configuration is
This area is twice that presented in Huang's model if 
and i = 88
.
Comparison of data taken during eclipse with post-eclipse observations
(Backman et al.
1984, Backman and Gillett 1985) seemed to imply that
the
IR fluxes require a larger companion disk than would be
consistent with cases A or B. Alternatively,
these data could be fit by a smaller secondary
which was substantially more transparent at
20 
m than at 4.8 
m, but we show in Section D.4 that such a
possibility is very unlikely given the colorlessness of the eclipse at
shorter wavelengths.
However, recent reanalysis of the thermal-IR data from mid- and post-eclipse
(Backman et al.
1996) indicates that the pre-eclipse 20 
m brightnesses
were too low, and the projected area of the disk companion is possibly
only twice the minimum area allowed by Huang's model.
Models have been proposed of a disk that is substantially warped or
inclined to its orbital plane (e.g. Kumar 1987) due, for example, to
the presence of a secondary binary inside (Lissauer and Backman 1984,
Eggleton and Pringle 1985).
Such a disk might require a hole in its
center (Wilson 1971, Eggleton and Pringle 1985) in order to be
consistent with the flat-bottomed eclipse profile. These sorts of systems are
difficult to model because the available data do not adequately
constrain the conceivable parameter space.
In the remainder of this paper, we will restrict our analysis to models
of disk with midplanes parallel to the orbital plane about the F star,
which either lie within our line of sight (case A), or are inclined
(case B) at angles as high as 3.5
.