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Astron. Astrophys. 328, 247-252 (1997)

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3. Physics of the interacting disk and corona

The evolution of the accretion disk is governed by diffusion. Conservation of mass and angular momentum give the relation between changes of surface density and mass flow in the disk. We consider here a disk which consists of a cool standard dwarf nova accretion disk near the midplane and a hot coronal disk above. The cool disk is geometrically thin, has low viscosity and a low mass flow rate. The self-sustained coronal disk is extended in vertical height, also there friction provides the energy. The mass flow rate is high, so that the situation might rather be described by a coronal flow towards the white dwarf. Close to the white dwarf the cool disk can be completely evaporated and only the hot corona exists.

In the coronal layers additional features appear. The gas pressure that supports the gas has a non-neglegible radial gradient which contributes to the radial support against gravity. This lowers the demand on the centrifugal force and leads to sub-Keplerian rotation speeds, even varying with height above the midplane. The more complex rotation pattern affects the release of energy and the transport of angular momentum in these coronal layers. The deviations from the standard thin disk approximation may be measured by terms of order [FORMULA]. For a standard case (M=1 [FORMULA],r= [FORMULA] cm) this quantity is about 1/4. For a first investigation of the radial structure of an interacting cool disk and a hot corona we here neglect these effects.The viscous relaxation time in the extended corona is much shorter than that in the cool disk. The ratio of the two timescales is of the order of the ratio of the disk thickness to that of the corona squared, [FORMULA] 1/100. Thus the corona will follow the cool disk evolution in a quasi-stationary way.

The changes in the cool and in the hot coronal disk can be described as following from the conservation of mass and angular momentum in these disk layers together. [FORMULA], [FORMULA], [FORMULA] are surface density, viscosity integral , and the mass flow rate of the cool disk. The corresponding values in the corona are [FORMULA], [FORMULA], [FORMULA].

The surface density and viscosity integral are defined as

[EQUATION]

and

[EQUATION]

with density [FORMULA] and vertical height z. The effective viscosity [FORMULA] is parameterized in the standard way as [FORMULA], [FORMULA] viscosity parameter, [FORMULA] sound velocity, [FORMULA] pressure scale height. The integral extends over the cool disk ([FORMULA], [FORMULA]) or the corona ([FORMULA], [FORMULA]), both sides of the midplane together.

3.1. The cool disk

The evolution of the cool disk is governed by conservation of mass and angular momentum,

[EQUATION]

[EQUATION]

where [FORMULA] is the rate of mass evaporating per unit surface from disk to corona.

This gives

[EQUATION]

[EQUATION]

The typical diffusion Eq. (6) now includes the evaporation of the disk to the corona, where the evaporation term [FORMULA] is known from the structure of the corona. If the boundary conditions and the initial distribution of surface density are given, one can evaluate the evolution of the disk with evaporation numerically. Here we change this equation into another form which is used in our numerical computation later:

[EQUATION]

where [FORMULA]

3.2. The corona above the cool disk

Here the corresponding equations are

[EQUATION]

[EQUATION]

where [FORMULA] is the rate of mass taken away by wind and we assume advective wind loss of angular momentum.

We assume that the corona is always at thermal equilibrium and is quasi-stationary. The above two equations lead to

[EQUATION]

[EQUATION]

Meyer & Meyer-Hofmeister (1994) and Liu et al. (1995) have investigated the coronal evaporation in detail and derived a scaling law for [FORMULA] and [FORMULA] from their numerical results. One should note that we assume that the evaporating matter takes all of its local angular momentum from the disk to the corona. From the conservation of angular momentum in both the cool disk and the hot corona it follows that part of the evaporated matter has to condense into the disk at some larger radius. In our computation, we find that the evaporation rate changes from positive to negative with radius although the absolute value of the negative amount is very small. This means that the evaporated matter partially returns to the disk. With these interactions of disk and corona, we have to generalize the results of Liu et al. (1995) derived for a one-zone model. We take their results for the accretion rate onto the white dwarf [FORMULA] at our inner edge of the disk [FORMULA] and deduce from this

[EQUATION]

with [FORMULA] g sec [FORMULA], [FORMULA] radius of the white dwarf.

With this form of [FORMULA] at the boundary, we construct a function of friction for the corona above a cool disk:

[EQUATION]

The new function of friction in the corona is, of course, consistent with the coronal evaporation model (Meyer & Meyer-Hofmeister 1994, Liu et al. 1995)), it has the general form [FORMULA] for [FORMULA] and fulfills [FORMULA] as specified by the one-zone model (Meyer & Meyer-Hofmeister 1994, Liu et al. 1995). Moreover, the angular momentum flow rate derived from the new frictional function [FORMULA] at the boundary is [FORMULA], which is just the quasi-stationary approximation for coronal gas flow.

As to the mass loss by wind, the one-zone model simplifies the innermost disk as a homogeneous area with respect to radius and determines a wind loss fraction [FORMULA], i.e. [FORMULA]. For the generalized formula we require the same scaling with radius which leads us to the following expression:

[EQUATION]

The constant C normalizes the formula such that it yields the same total wind loss as the numerical value in the homogeneous one-zone model, [FORMULA]. This yields [FORMULA].

With the frictional function (11) and wind mass loss expression (12), one obtains the mass evaporation rate from the cool disk to the corona above from Eqs. (8) and (9).

3.3. Boundary conditions

We define the radius [FORMULA] as inner boundary of the disk where the surface density decreases to zero. With the conservation of mass and angular momentum, we deduce that both the friction and mass flow rate in the disk boundary are zero, i.e. [FORMULA], or [FORMULA], [FORMULA]. This means no mass is accreted directly from the disk onto the central white dwarf. The mass in the disk is first evaporated into the corona and then flows towards the central object.

It has been known from Meyer & Meyer-Hofmeister (1994) and Liu et al. (1995) that the evaporation is much more efficient close to the white dwarf than at large distances. This implies that the first inner disk region will be evaporated and a hole is created near the white dwarf. Due to the high efficiency of evaporation near the white dwarf such a hole grows rapidly in the innermost region of the disk. Then the inner boundary of the disk moves outwards more and more gradually. Finally it hardly moves or stalls at some radial distance until mass flowing from secondary and accumulating in the disk has reached the critical surface density and causes the onset of the next outburst.

To follow the motion of the inner boundary by evaporation, we must know the profile of [FORMULA] near the inner boundary. From the diffusion Eq. (7), we find an analytical approximation for the surface density there. Near the inner boundary [FORMULA], [FORMULA], the values of [FORMULA] and b fall to very small values, thus the relation [FORMULA] for the optically thin disk (Ludwig et al. 1994) holds, and we obtain the diffusion equation for b as

[EQUATION]

To solve the above equation we proceed in the following way. The profile moves with a velocity [FORMULA] and only gradually changes its shape. For a small time step we neglect this change of shape. Then the change with time comes from the motion of the profile, [FORMULA], where [FORMULA] is the new coordinate in the co-moving frame. To get an analytical solution for a narrow region near the inner boundary, we neglect the variation of [FORMULA] and [FORMULA] with x and regard these values as constant. Then the differential Eq. (13) becomes

[EQUATION]

where [FORMULA], [FORMULA], [FORMULA].

With the boundary condition [FORMULA] and [FORMULA] at [FORMULA] the solution is

[EQUATION]

The above expression is the analytical "tail" of the evaporating disk near the inner boundary. The given value of b at the grid point closest to [FORMULA] determines [FORMULA], i.e. the velocity [FORMULA] and hence the motion of the inner boundary.

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© European Southern Observatory (ESO) 1997

Online publication: March 24, 1998

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