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Astron. Astrophys. 337, 105-112 (1998)

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4. Viscosity in the circumnuclear ring

Before making numerical estimates, we list the basic parameters of the clumpy gas in the circumnuclear ring (CNR), such as the inferred clump size a, the volume filling factor F, and velocity dispersion of the clumps [FORMULA], taken from Jackson et al. (1993) and Güsten at al. (1987):

[EQUATION]

Adopting the average gas density in the clumps [FORMULA] one finds the average clump mass [FORMULA] [FORMULA]. This gives the ratio [FORMULA], which implies that elastic (gravitational) interactions between the clumps are negligible compared to inelastic ones, i.e.  clump-clump collisions. In other words, gravitation plays no role in the interactions between the CNR clumps.

The mean free path of the clumps given by

[EQUATION]

is rather large (even in a marginal conflict with the simplifying assumption (3.1) that [FORMULA]). The clump-clump collision rate in the CNR is given by:

[EQUATION]

i.e.  [FORMULA] implying less than one collision per revolution. The anticipated optical depth is [FORMULA], which, according to Eq. (3.17), results in [FORMULA]. Therefore, the conditions for fully developed turbulence to appear, which are described at the end of Sect. 3, are met to yield the viscosity coefficient

[EQUATION]

It is instructive to compare this result with the upper limit to viscosity derived by von Linden et al. (1993a,b). Their results were obtained by fitting an accretion disk model with arbitrary viscosity to the velocity fields of various molecular clouds, and then inferring the required kinematic viscosity from the fit. Successful fits were made in the radial range from 10 to 100 pc, with an implied kinematic viscosity of [FORMULA] at a distance of 100 pc. Such a high viscosity is just within the limits imposed by the basic assumption of an accretion disk: The thin disk assumption implies, in the context of isotropic turbulence, that the kinematic viscosity has to be clearly less than the circular velocity times the scale height. At [FORMULA] pc the circular velocity is [FORMULA] km/sec, and the scale height is difficult to determine; the z-distribution of clouds gives a lower limit to the scale height, and that is [FORMULA] pc. This means that the kinematic viscosity has to be less than [FORMULA]; the fit by Linden et al. is obviously just at the limit. It follows either, a) that the real scale height is quite a bit larger, with a rather hard limit at roughly 1/3 of the radius, implying a hard limit of the viscosity of [FORMULA], which should not be reached, or b) that the kinematic viscosity cannot be described with an isotropic turbulence.

The viscosity values implied by the fit to the observations of molecular clouds differ for different radii. A fit was made for clouds at 100 pc as well as 10 pc, with the viscosity decreasing for smaller radii. This radial variation may become steeper at smaller radii. The hard upper limit mentioned above would imply that it decreases as approximately [FORMULA], and so would imply that the hard upper limit at 1.5 pc is [FORMULA]. We note that the result (4.4) is a factor of 5 below this upper limit. Assuming the same scaling for the kinematic viscosity derived by von Linden et al. would lead to [FORMULA], which is close to Eq. (4.4); assuming the number derived from a fit at 10 pc we obtain an estimate which is near the limit.

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© European Southern Observatory (ESO) 1998

Online publication: August 6, 1998
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