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Astron. Astrophys. 344, 639-646 (1999)

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2. The vertical structure

We assume that

  • The disc is geometrically thin [FORMULA] and all gradients in the vertical direction are much greater than the corresponding ones in the radial direction.

  • The radiative transfer can be treated in the grey approximation and only the vertical directions are considered. (`two stream approximation')

  • LTE prevails throughout the entire disc. Deviations from LTE will be discussed in a subsequent paper. We believe that LTE has an effect on the details but not on the general picture discussed here.

The equation for hydrostatic equilibrium in the vertical direction is


where P, T, [FORMULA] and µ are the gas pressure, temperature, density and molecular weight respectively and z the vertical coordinate, [FORMULA] is the gravitational component in the z direction. The extinction coefficient is given by [FORMULA] where [FORMULA] and [FORMULA] are the volume absorption and scattering coefficients respectively. The total radiative flux is F.

The condition of thermal equilibrium is:


where [FORMULA] is the viscous energy dissipation per unit volume.

We assume here that the entire energy dissipated by the viscous forces is carried away by radiation i.e. we ignore at this moment convection and any other non radiative energy transport mechanisms. The radiative transfer equation is:




where B is the wavelength integrated Planck function and J is the wavelength integrated mean intensity. The thermal equilibrium condition can also be written as


where [FORMULA] in the notation of SW and we use the [FORMULA] model approximation for the viscous energy dissipation.

The boundary conditions are:


The above equations, together with the boundary conditions form a closed set, and can be solved for the disc structure for given [FORMULA] and [FORMULA].

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© European Southern Observatory (ESO) 1999

Online publication: March 18, 1999