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Astron. Astrophys. 357, 1123-1132 (2000)

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Appendix A: Thermodynamic considerations for KSG [FORMULA]-disks

For a KSG [FORMULA]-disk, we have from Eq. 26 that

[EQUATION]

If the disk is optically thick and advection is negligible, viscous dissipation leads to local effective temperature of

[EQUATION]

with m the mass of the central star in solar units and [FORMULA] the radius in astronomical units.

An essential thermodynamics requirement is that [FORMULA] or that

[EQUATION]

This condition is satisfied provided that

[EQUATION]

and that the disk is selfgravitating in the vertical direction at [FORMULA]. The latter condition leads to a second requirement on [FORMULA].

For a standard Keplerian disk, the mass flow rate is given (Eqs. 8, 10, 43) by

[EQUATION]

with [FORMULA] the disk's mass. From Eq. 11, the condition that the disk is non-selfgravitating is [FORMULA] and hence, from Eq. A5, that

[EQUATION]

or

[EQUATION]

A selfconsistent and physically acceptable solution can be obtained only if [FORMULA], that is the disk becomes selfgravitating at values of [FORMULA] which are sufficiently high that thermodynamic requirements are not violated. This condition may then be written as

[EQUATION]

Thus thin KSG [FORMULA]-disks appear to be inconsistent with basic thermodynamic requirements if [FORMULA], [FORMULA]. There is no inconsistency if either or both of these quantities are sufficiently large.

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© European Southern Observatory (ESO) 2000

Online publication: June 5, 2000
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