3. Linear analysis
In the framework of the standard linear analysis the pressure, density and velocity are represented as:
In the linear approximation () from Eqs. (1)-(5) the linearized equations become:
As we study only the dynamics of acoustic oscillations, we set . Then the thermal Eq. (6) becomes:
being the entropy of the gas at equilibrium. We eliminate from Eq. (21) with the help of the equation of state . In the linear approximation we obtain:
where , are the specific heats at constant density and pressure.
The short-wave approximation in the radial direction (, k being the radial wave number) allows to write the solution as:
being the complex eigenfrequency.
Taking into account Eqs. (22) and (23), the system Eqs. (17)-(21) is reduced to two ordinary differential equations for the amplitudes of the perturbed pressure and the material displacement from an equilibrium position:
where , , being the epicyclic frequency, , , , , , , , . The prime sign means differentiation with respect to z-coordinate (). is the complex amplitude of the material z-displacement from equilibrium state. We have
Boundary conditions must be defined for solving Eqs. (24) and (25). Given the symmetry, it is natural to consider two types of oscillations: 1) the symmetric oscillations ( or ), and therefore
(such oscillations correspond to the pinch-mode or S-mode); 2) the antisymmetric oscillations ( or ), and therefore
that correspond to the bending oscillations or AS-mode. On the disk unperturbed surface the following condition should be fulfilled:
Solving the system of Eqs. (24), (25) with the boundary conditions Eqs. (28) and (26) (or (27)), we find the eigenvalues of complex frequency for a given distribution of the equilibrium parameters along the vertical coordinate in the disk. A positive imaginary part of the frequency (growth rate) means that the eigen-mode is unstable. This will define the "3D-model", the thin disk model being called the "2D-model".
3.1. Dispersion relation
For a homogeneous z-distribution of the equilibrium quantities, the system of Eqs. (24) and (25) is reduced to the following dispersion relation:
where is the wave number in z-direction. We stress that Eq. (29) at coincides with the dispersion relation obtained earlier for the 2D-model if we replace by the 2D adiabatic index (Wallinder 1991; Khoperskov & Khrapov 1995). Kovalenko and Lukin (1998) have obtained a more accurate formula for the 2D adiabatic index (). This relation takes into account the Keplerian rotation of a thin disk. The adiabatic sound speed in the disk plane is given by the following expression:
Thus, taking into account Eq. (30), the dispersion relation Eq. (29) with describes the dynamics of perturbations within the limits of the flat model (MTD). Obviously, these oscillations correspond to the S-mode.
In the general case where , this dispersion relation gives two unstable acoustic branches of oscillations and two damping modes (viscous and thermal). The damping of the viscous or thermal modes depends on whether or not dissipation and radiative processes are taken into account in the thermal equation. If these factors are taken into account, then the viscous and thermal low-frequency oscillatory branches can be unstable (Shakura & Sunyaev 1976), but this will not change the dispersion properties of sound waves.
© European Southern Observatory (ESO) 1999
Online publication: April 12, 1999