2 The model atmospheres

In this section we will discuss the basic assumptions used to calculate the vertical structure of an accretion disc, and the basic equations included in the TLUSDISK code which are used in our calculations.

2.1 The program

In this work we present the fitted observed spectra of the AM CVn systems with theoretical disc spectra computed by the dedicated disc atmosphere model code created by I. Hubeny at Goddard Space Flight Center, NASA. The package includes several programs; the most important are the program for calculating vertical structure of the accretion disc TLUSDISK and the program for calculating an emergent spectrum based on a stellar or disc atmosphere model: SYNSPEC.

The program TLUSDISK is based on the model stellar atmosphere code TLUSTY (Hubeny 1988, 1990a,b). It is designed to calculate the vertical structure of a steady state equilibrium accretion disc and its atmospheres under the assumptions stated in the next section using the CL/ALI (Complete Linearization/Accelerated Lambda Iteration) method (Hubeny & Lanz 1995). The user is given much freedom in choosing parameters for the calculation. This includes the possibility to calculate LTE and NLTE models of different levels of sophistication of atmospheres consisting of elements from hydrogen to zinc. Heavier elements can also be included in the model if that is necessary.

The first run of TLUSDISK produces an LTE model. If further refinement is needed, the output from this run can be used as a starting model for a model with NLTE treatment of continua and lines. The final output from TLUSDISK can be directed to SYNSPEC (Hubeny et al. 1994), which evaluates atmospheric opacity and emissivity and produces a synthetic spectrum. These two programs are accompanied by a set of utility and shell programs, which simplify the process of preparing the initial input and processing the output to give the final disc synthetic spectrum. In addition to that, we developed an utility program RINGRUN to simplify the production of a complete disc model (Semionovas 1998).

All the theoretical models we present in this paper are NLTE models. We will discuss the necessity of the NLTE treatment in more detail in Sect. 5.

2.2 Basic assumptions

The disc is assumed to be in a steady state, geometrically thin and in keplerian rotation. The vertical structure is solved for a set of axially symmetric concentric rings where a plane-parallel 1-D atmosphere calculation is performed. The atmosphere at each disc radius R (specified in the disc mid plane) is in hydrostatic equilibrium, with depth-dependent gravity (g) that arises from the vertical component of the central star's gravitational force on the disc material. Neglecting the self-gravity of the disc and assuming that R is much larger than the distance from the central plane (z):

$$\frac{{\rm d}P}{{\rm d}z} = -g(z) \rho$$ (1)

$$g = \frac{G M_{1} z}{R^{3}}$$ (2)

where P is the pressure, $\rho$ is the mass density, M₁ is the central star mass and G is the gravitational constant. The mass surface density is given by the standard disc model (Shakura & Sunyaev 1973):

$$\Sigma {{\dot M}\over 3\pi{\bar\omega}}\left[1- \left(R_{1}\over R\right)^{1/2}\right],$$ = (3)

$${\bar\omega} {\sqrt{GM_1R_{1}}\over{\it Re}}$$ = (4)

where $\dot M$ is the mass transfer rate, R₁ is the central star radius, and ${\bar\omega}$ is the depth-averaged kinematic viscosity, parameterized in terms of Reynolds number Re of the flow (Lynden-Bell & Pringle 1974). It is useful to calculate the vertical atmosphere structure using the mass-depth variable m(z), expressing the mass of a column of material above a given distance from the central plane z. The viscosity through the disc atmosphere can be expressed by a power law

$$\omega(m)={\bar\omega}(\zeta+1)\left({2m\over\Sigma}\right)^\zeta\cdot$$ (5)

The power law index $\zeta$ is usually set to a positive non-zero value. The disc radiates all the mechanical energy dissipated by the viscous shearing between the keplerian orbits. According to the standard formulae this quantity may be written as follows:

$$D_{\rm mech}(z) = \frac{9 G M_{1} \omega(z)\rho(z)}{4 R_{1}^{3}}$$ (6)

where $\omega(z)$ is the viscosity. The energy balance equation then becomes:

$$\int_{0}^{\infty}(\kappa_{\nu} J_{\nu} - \eta_{\nu}){\rm d}\nu = D_{\rm mech}(z)$$ (7)

where $\kappa_{\nu}$ and $\eta_{\nu}$ are the absorption and emission coefficients, and $J_{\nu}$ is the mean intensity of the radiation, at frequency $\nu$.

As is customary in the stellar atmosphere theory, one expresses the total energy dissipated (and therefore radiated away) from the unit disc face as $D(R) = \int D_{\rm mech}(z){\rm d}z$, through the effective temperature, $T_{\rm eff}$ as $\sigma T_{\rm eff}^{4} = D(r)$, where $\sigma$ is the Stefan-Boltzmann constant. Assuming further a stationary, keplerian disc, the effective temperature is given by:

$$T_{\rm eff} = \bigg(\frac{3 G M_{1}\dot{M}}{8\pi \sigma R^{3}... ...g[1 - \bigg(\frac{R_{1}}{R}\bigg)^{1/2}\bigg]\bigg)^{1/4}\cdot$$ (8)

Here is a summary of the basic assumptions:

$\bullet$ The energy balance is considered as a balance between the net radiation loss (calculated without invoking either optically thin, or optically thick approximations) and the dissipated mechanical energy;

$\bullet$ The dissipated energy is proportional to viscosity, which is given through the Reynolds number;

$\bullet$ The effect of illumination of the disc by the central star may be taken into account using the formalism described in Hubeny (1990b);

$\bullet$ The disc is not considered as a semi-infinite atmosphere but rather as a finite slab of gas. The optical thickness of the disc is a parameter that directly follows from the model;

$\bullet$ The total radiative flux is not constant, but increases upwards (from the central plane of the disc to its surface), its value being determined through the energy balance equation;

$\bullet$ Analogously, the gravity acceleration is not constant and depends on the distance from the central plane;

$\bullet$ The vertical structure of the disc (temperature, density, radiation field etc.) follows from the appropriate model calculations.

The program solves the basic equations (radiative transfer, hydrostatic equilibrium, energy balance, statistical equilibrium, charge and particle conservation) by the use of the hybrid CL/ALI (Complete Linearization/Accelerated Lambda Iteration) method (Hubeny & Lanz 1995).

The thermal structure of the atmosphere, which is determined by the mass accretion rate and the radial distance from the central star, plays a crucial role in the convergence of the models, which consists of a set of concentric rings with variable $T_{\rm eff}$, between the limits shown in Table 1.