Astron. Astrophys. 327, 57-60 (1997)

2. Basic equations

Let us consider a very flat disk rotating around a central object of mass M, The basic equations are (Shakura & Sunyaev 1973):

In the above expressions, is the accretion rate at infall velocity , H is the density scale heights, is the surface mass density, is the central temperature. is the Keplerian angular velocity, is the mean opacity, is the emergent energy(radiation) flux from one face of the disk, is the release of gravitation potential and viscous dissipation energy, is the component of stress tensor, and is the effective temperature, other symbols , G, b, c have their usual meanings.

where and are the central density and the proton mass respectively. Using the conservation model of the magnetic flux (Sakimoto et al.1981), we have

where is defined as is the viscosity constant.

When we consider the effect of the self-gravitation, g is written as (Paczynski 1978)

where is due to the disk's self-gravitation, is due to the central mass M. we assumed , the large values of a correspond to a strong self-gravity and the small values of a to a weak self-gravity. Here we should point out that the limit value of is marginally acceptable (Paczynski 1978).

For the vertical mechanical stucture, the disk is assumed to be in hydrostatic equilibrium, the equation of hydrostatic equilibrium can then be written as (Paczynski, 1978)

From Eqs. , we get

where , is the mass density of the disk, and P is the total pressure.

If the gas pressure dominate, the Eq. (12) becomes

where is the mean molecular mass. is equal to 0.65(CSD 1990II), k is a constant of Boltzmann.

© European Southern Observatory (ESO) 1997

Online publication: April 8, 1998
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