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

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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):

[EQUATION]

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

[EQUATION]

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

[EQUATION]

where [FORMULA] is defined as [FORMULA] [FORMULA] is the viscosity constant.

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

[EQUATION]

where [FORMULA] is due to the disk's self-gravitation, [FORMULA] is due to the central mass M. we assumed [FORMULA], 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 [FORMULA] 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)

[EQUATION]

From Eqs. [FORMULA], we get

[EQUATION]

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

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

[EQUATION]

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

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© European Southern Observatory (ESO) 1997

Online publication: April 8, 1998
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