Appendix A: modeling the vertical density profile
The vertical structure of the disk is generally derived by imposing hydrostatic equilibrium in the z-direction under the assumption of isothermality. The relevant Poisson equation and hydrostatic equilibrium condition for an axisymmetric configuration can be written as (here the symbol denotes volume mass density and should not be confused with the density deviation function of the main text):
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 z, so that the gravitational field obtained by integrating Eq. (A4) includes a term that is approximately linear in z. The result can be inserted in the right hand side of Eq. (A2), which becomes:
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 A one can compute the desired density profile and the associated surface density . One can then introduce a scaleheight h such that , and the value of h can be computed directly from the relation .
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 h defined above, since .
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 h is the same as that defined by the relation . Note that even when no external (spherical) field is present, the solution for an axisymmetric disk obtained from Eq. (A6) is not exactly the one derived in the homogeneous, self-gravitating slab, unless the rotation curve is flat.
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 h given by:
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 A depends on h and on via the quantity , for the purposes of the present paper it may be convenient to reexpress Eq. (A8) as:
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