Astron. Astrophys. 338, 413-434 (1998)

Appendix A: derivation of Eq. (8)

In this Appendix, we derive the expression for the velocity of the radial viscous drag in the gaseous component of a star forming galactic disk undergoing gas infall (Eq. 8).

For a closed system of force-free particles of a compressible gas the momentum balance is given by (e.g. Landau & Lifshitz 1963)

with the stress tensor

and are the coefficients of the viscosity. The sum convention is used. is the gas density; for the sake of clearness we do not use the index g.

For a realistic Galactic disk model we have to take into account the effects of gas infall and interaction between the gaseous and the stellar component. Assuming that the velocity of the infalling gas is constant with time, its contribution to the momentum balance is . Condensation of gas into stars with the rate removes from the gas the momentum . On the other hand, momentum input due to SNe and stellar winds is not described by a separate production term because it is already considered by the viscous stress tensor. (Note that the interpretation of the viscosity is based on the interaction between the stellar and and the gaseous component.) Therewith, we have the momentum balance

the balance for the gas mass

and, from the combination of both, the transport equation

Now, let us consider the -components of Eqs. (A3) and (A5):

where we replaced the dynamic viscosity by the kinematic viscosity . The last equation transforms easily into Eq. (8).

Now we show that Eqs. (A6) and (A7) give the relations used in other studies for the special cases of either vanishing infall or vanishing viscosity. If we consider a closed gaseous disk without star formation and use , Eq. (A6) reads

After multiplication by R, Eq. (A8) gives the Eq. (2.8) in Pringle (1981). Sommer-Larsen & Yoshii (1989) describe the viscous evolution of a closed system with constant rotation curve. In this case our Eq. (A7) is reduced to

Adopting Sommer-Larsen & Yoshii's parametrization of the rotation curve we have

which is easily transformed into Eq. (12) in Sommer-Larsen & Yoshii (1989).

Finally, we consider the case of a disk with vanishing viscosity but undergoing infall. The rotation curve should be constant in time. This corresponds to the models investigated by Pitts & Tayler (1989) and Chamcham & Tayler (1994). From Eq. (A7) we find

Replacing by the specific momentum we have

in agreement with Eq. (4) in Chamcham & Tayler (1994).

© European Southern Observatory (ESO) 1998

Online publication: September 14, 1998
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