## Appendix A: modeling the vertical density profileThe vertical structure of the disk is generally derived by imposing
hydrostatic equilibrium in the where we have taken polar cylindrical coordinates. For simplicity, we assume that the external field associated with is spherical. [In the class of models studied in Sect. 2 and Sect. 4 we have taken the case where .] From the definitions of and of , it is readily shown that, close to the equatorial plane, Eq. (A1) can be rewritten as: The latter expression follows from the exact relation
applicable to a spherical density
distribution. The expression
indicates that the quantity in parentheses should be calculated based
on the radial field given by Eq. (4). For a relatively thin disk the
quantity may be taken to be nearly
independent of where we have introduced the integrated density . Note that the term is just the vertical component of the spherical external field, as might have been anticipated. If is the density on the equatorial plane, we can define two scales, and , and the natural dimensionless parameter , so that Eq. (A5) becomes: to be solved under the conditions
,
; here we have used the
dimensionless variables and
. For each value of The standard non self-gravitating model, where the contribution of
the gas density to the vertical
gravitational field is negligible, corresponds to the limit equation
. Usually the analysis is carried
out with the additional approximation
, but this is not needed; by
retaining the contribution one can
thus keep track of the effects associated with the rest of the disk
mass distribution, which may be significant even where the local disk
density is small. In any case, such
limit equation yields the well known Gaussian profile
, with
, here improved with respect to the
standard expression used in the
so-called Keplerian limit. Note that
is slightly different from the
scaleheight The limit of the homogeneous fully self-gravitating slab
corresponds to the equation , so
that the vertical density profile is given by (Spitzer 1942)
, with
; here the scale The density profile associated with Eq. (A6) is neither Gaussian
nor . For practical purposes it may
be convenient to use a simple interpolation formula for the vertical
scale, which is justified by the following description "biased"
towards the Gaussian limit. If we start from Eq. (A2), naively expand
the vertical field as , and then use
Eq. (A1) to estimate the factor , we
find an equation leading to an unrealistic Gaussian density profile
with scaleheight The quantity contains a
dependence on Note that the exact calculation, from
gives a relation
, with
. The choice of the factor
in Eq. (A8) guarantees the proper
limits for and for
. The accuracy of the interpolation
formula is illustrated in Fig. A1. In turn, since with . This leads to Eq. (17) of the main text, applicable when the external field is produced by a central point mass. This interpolation formula improves on earlier analyses (Sakimoto & Coroniti 1981, Bardou et al. 1998), in several respects, allowing us to better describe the transition between a Keplerian and a self-gravity dominated disk and to extend the treatment to the case where the external field is not just that of a simple point mass at the center.
© European Southern Observatory (ESO) 1999 Online publication: October 4, 1999 |