Vertical Structure of the Secondary: Hydrostatic Model

In this section, we develop a physical model of the secondary as a disk that is centrifugally supported in the radial direction and hydrostatically supported in the vertical direction. We assume that the disk is vertically isothermal and well mixed. If heating occurs predominantly near the midplane of the disk, e.g. as a result of viscous dissipation, the temperature would be expected to drop with height. However, assuming that the exposed regions are in radiative equilibrium, the material above the disk's photosphere is optically thin in the thermal-IR so it may be expected to be roughly isothermal in z, in analogy with planetary atmospheres (Chamberlain and Hunten 1987). As the visible opacity is likely to be larger than the mid-IR opacity, the assumption of vertical isothermality is likely to hold for the portions of the disk partially transparent to visible and near-IR radiation. Our eclipse profiles are most sensitive to the thermal structure in these regions. If the star(s) at (near) the center of the secondary illuminate(s) and heat(s) the faces of the disk, scattering and absorption at high z in the disk may produce temperature gradients in the optically thin regions of the disk. However, these temperature gradients depend on unknown system parameters; moreover, they are unlikely to be large enough to appreciably affect our analysis.

We use a right-handed Cartesian coordinate system in which the z axis is perpendicular to the plane of the disk and the line of sight from the Earth runs from -y to +y, perpendicular to z.

The vertical component of the momentum equation for a thin" (i.e., centrifugally supported) disk is:

where is the velocity of the gas, t is time, signifies density, p refers to pressure and is the angular frequency of the material relative to the center of the disk. In equilibrium, . If we further assume that the disk is isothermal in the vertical direction and use the perfect gas law, then eq. (4) can be integrated to yield

The Gaussian scale height of the disk is:

where is the mean molecular mass of the gas, is the orbital period at the outer edge of the disk in years, and all of the mass of the secondary has been assumed to be concentrated near its center, so the Keplerian approximation is valid.

The ``dynamical aspect ratio'' of the disk, , where is the scale height at the disk's outer edge, can be computed using eqs. (1), (2) and (5) and the orbital parameters of the system:

where is the mass of a hydrogen atom and = 2.25 is the mean molecular weight of a solar composition gas. The factors contributing to the largest uncertainties to are, in decreasing order, , T, and The mean molecular weight of the gaseous disk is probably similar to because the gas of which the disk is composed is likely to be unprocessed ISM material or mass transferred from the primary when the star overflowed its Roche lobe; both stellar evolution models (Webbink 1985) and the line spectrum of Aurigæ (Castelli 1978) suggest that such gas would be hydrogen-rich. As stated earlier, model estimates of can therefore be useful in estimating and thus in discriminating between the high- and low-mass models of the Aurigæ system.

The radial structure of a centrifugally supported disk is less well defined. Radial profiles for a disk which has settled into a quasi-equilibrium configuration depend on the form of the viscosity law. Temperature profiles are also model-dependent. In an effort to reduce the parameters to a reasonable number, for most of our simulations the surface mass density of the disk, , is assumed to be constant from an inner hole radius (which may be zero) out to the disk radius , beyond which it is zero. We also performed a few runs in which varies as .

The scale height of the disk is assumed to be proportional to the distance from the center (equivalently, and ). The opacity of the gas/dust mixture, (in units of cm/g), is a function of wavelength only. If we also assume that we are viewing the eclipse exactly in the orbital plane, then the optical depth which we observe through the disk at a given point on the projected surface is

or

where the origin of the rectangular coordinate system is the center of the disk, the y axis points toward Earth (see Figure ) and the coordinates x, y and z and hole size have been made nondimensional via division by , , and respectively.

If we are not viewing the disk exactly in the orbital plane, the situation becomes somewhat more complicated. We continue to define the z axis to be perpendicular to the Earth-star line, but the y axis no longer points directly at the Earth. For a given line of sight through the disk, the z coordinate of a ray from the primary to the Earth changes as the ray traverses the disk:

In equation (8), is the z coordinate of the point on the star from which the ray emanates and i is defined relative to the plane" of the sky, i.e., i = 90 when the system is viewed edge-on. Note that we have defined the sign of z such that the center of the primary is at .