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Astron. Astrophys. 356, 363-372 (2000)

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2. Basic non-stationary accretion disk equation

In the approximation of Newtonian potential we assume that the velocity of a free particle orbiting at distance r around a gravitating object is

[EQUATION]

where [FORMULA] is the Kepler angular velocity; M is the mass of the central gravitating object, constant in time; [FORMULA]cm3g-1s-2 is the gravitational constant. This is a good approximation to the law of motion for particles in the standard sub-Eddington disk. In the advection-dominated accretion flow (ADAF) the particles are substantially subjected to the radial gradient of pressure and thus have the velocity different from that given by (1). Following the model by Narayan & Yi (1994), one can assume that the angular velocity in ADAF is [FORMULA].

The height-integrated Euler equation on [FORMULA] and the continuity equation along the height Z are given by:

[EQUATION]

[EQUATION]

where [FORMULA] is the angular velocity in the disk; [FORMULA] - the surface density of the matter, and [FORMULA] is the height-integrated viscous shear stresses between adjacent layers. The time-independent angular velocity is assumed although there can possibly be certain variations of [FORMULA] in the non-Keplerian advective disks when a time-dependent pressure gradient is involved (see, e.g. Ogilvie 1999).

It is convenient to introduce the following variables: [FORMULA], henceforth [FORMULA] means the total moment of viscous forces acting between the adjacent layers, [FORMULA] - the specific angular momentum of the matter in the disk, and [FORMULA]. From Eq. (2) in view of (1) it follows that

[EQUATION]

Substituting (4) in (3) and expressing r in terms of h, we obtain the basic equation of time-dependent accretion:

[EQUATION]

In the case of the Keplerian disk [FORMULA]. The advection-dominated solution by Narayan and Yi (1994) yields [FORMULA], where [FORMULA] is a dimensionless constant.

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© European Southern Observatory (ESO) 2000

Online publication: March 28, 2000
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