3. The solution for the disk equations
To obtain the solutions for the disk equations, we shall adopt a different procedure concerning the mass conservation equation, which in steady state under the assumption of azimuthal symmetry and hydrostatic equilibrium in z direction, in cylindrical coordinates, we write
, , R being, respectively, mass density, radial velocity and radial distance. Since we shall be solving this equation with null boundary conditions for the density at z=H (the semi-scale height of the disk), and for the radial velocity at the end of convective region, i.e., z=L , we have
where we'd rather working with volumetric density, instead of column, because we shall be considering the z -structure of the accretion disk and no simplification would result if we do otherwise. Clearly, to the lowest order in z, subject to the null boundary conditions,
where we have used for the accretion rate
z is in units of semi-scale height.
Now, from the angular momentum conservation equation, we have
being the turbulent kinematic viscosity and
is the internal radius of the disk. Null boundary condition for the torque has been used for the torque at . It should be stressed that our dynamical disk is the convective region, extending from the symmetry plane to . Above this, we have the radiative region. We, finally, write the hydrostatic equilibrium equation,
and the energy equation, under the radiative diffusion approximation,
where is the Keplerian angular velocity, is the opacity, T is the temperature, is the hydrogen mass, is the Boltzmann constant, is the Stefan-Boltzmann constant, is the turbulent Prandtl number, is the specific heat at constant pressure. We have used a perfect gas law for the equation of the state. The right hand side of the energy equation is obtained integrating the heat generated from to .
The solution for the disk equations is highly dependent on the kind of process responsible for the opacity which, usually, depends on the values of the density and temperature. Most often, opacity is expressed in powers of density and temperature. However, these powers and constants entering these expressions are dependent on the region of temperature and density we are. So, to avoid these complexities, we shall specialize to regions with , for which the opacity is mainly given by ice, i.e.,
(Lin & Papaloizou 1980).
Making the substitution
where is the central temperature, , in the energy equation, results for the density
As we shall see later on, b is a kind of eigenvalue that determines the behavior of t along z. Inserting these substitutions into the hydrostatic equilibrium equation yields
We now rewrite this equation as
using the behavior of t as and keeping terms only to the sixth order in z, we obtain the following approximate solution
a and c being, respectively, the second and the fourth derivative of t, evaluated at z=0 . and are integration constants to be determined under suitable boundary conditions and
We now make the assumption that convection develops throughout the whole z extent, except in a very narrow region close to the surface of the disk. We,therefore, set z=1 and impose the following boundary conditions: t=1 at z=0 , t=0 at z=1 and at z=1 . The boundary condition on the derivative, with , is the only compatible with the previous boundary conditions on the density, radial velocity and laminar regime at the surface. A detailed analysis of Eq. (34) shows that compatibility with the condition on the derivative is only met if q=1 or . In the following we analyze both cases separately.
q = 1
Physical solutions are only found for . a, b, c, , , t and are given by
It should be mentioned that, since the solutions we have are obtained under the assumption of constant heat generation,i.e.,
which does not depend explicitly on the Prandtl number , the temperature gradient decreases as we increase and so does b.
Physical solutions exist only for . a, b, c, q, , are given by
Since q is not an integer we do not have analytic expressions for both t and its derivative. For the same reason we have already mentioned, b and decrease when increases.
© European Southern Observatory (ESO) 1997