The Poeckert and Marlborough (1978) (PM) model was one of these ad hoc attempts, in
which the density of the circumstellar disc at the star-disc boundary and the disc’s radial
velocity is specified arbitrarily. By assuming a hydrostatic density distribution perpendicular
to the disc, implying an exponential drop off of density with radius, and using the equation of
continuity radially the equatorial density is calculated. PM assumed a constant temperature
for the circumstellar envelope. The model assumes a steady-state, rotationally and
equatorially symmetric circumstellar envelope. The central star is assumed to be spherically
symmetric. The model was initially used by PM for the Be star 
Cassiopeia, using the
parameters M* =17 M
, R* =10 R and T* =25000K, the circumstellar temperature was
assumed to be 20000K.
Good results were reported from the model; firstly the disc can not be highly ionised as free-bound interactions play a major role in obtaining the correct infrared emission, where in this context highly ionised means less than one in 104 atoms is neutral (Poeckert and Marlborough, 1978). Also the disc does not appear to be, or need to be, rotating at critical velocity, PM were able to reduce the speed from 567kms-1 , the critical velocity, to 427kms-1 and found no significant changes in their results.
Waters (1986b) introduced a simple wedge-shaped and pure hydrogen disc model to describe
the equatorially concentrated wind. The density distribution 
(r) is described as:
 | (1.6) |
(r) is the density at distance r, R* is the radius of the star, 0 is the density at
r = R* and n is a parameter determined from IR observations and typically 2 
n 3.5.
Notably the disc density from equatorial plane to upper edge is a constant, although the
density varies with distance from the star in equi-distant radial arcs. Therefore the density
inside the envelope does not change significantly with increased height above the
equatorial plane. Assuming mass continuity in the out-flowing disc they derive:
 | (1.7) |
- the disc opening angle, which is defined as the
angle between the equatorial plane and either the upper or lower boundary of the disc and
(iii) i - the inclination angle of the rotation axis with respect to the observer. Additionally,
the kinetic temperature of the wind is a free parameter, this determines the free-free and and
free-bound opacities and emissivities. These are needed to interpret the IR excesses used
in the curve of growth methods of Waters et al. (1987), who used these models to
derive density structures and mass-loss rates. Collectively these parameters these
determine the far-IR flux from the circumstellar disc (see Fig 1.7 for the model
geometry).
Waters et al. (1987) fed IRAS data into the disc model and derived typical disc densities of
10-12 
0 
10-11g/cm-3. From curve of growth modelling (see Waters, 1986b) the mass loss
rate of the disc model is:
 | (1.8) |
, 0 in g/cm3, v0 in kms-1 and 
in M yr-1, (see Waters, 1986b).
This produced mass losses of IR emitting material to be in the in region of ~ 10-7 to
10-10 M yr-1. However, the velocity distributions within Be star discs are not well
known and are the greatest uncertainty in estimating mass losses, as an increase
in initial radial velocity leads to an increase in the mass loss rates (Millar and
Marlborough, 1999).
The radial disc velocity and disc opening angle were estimated by Waters (1986b), v0 to be
5kms-1 and to be 15 o . The disc velocity, at r = R*, was estimated from the work on
near-IR line profiles of Cassiopeia by Chalabaev and Maillard (1985). This suggested that
the radial outflow velocities in the winds of Be stars at the photosphere were of the order of
the sound speed or less, i.e., 5 < v0 < 15 kms-1 .
However a number of sources have argued for a much lower outflow velocity in viscous Be
star discs. Marlborough and Cowley (1974) find v0 = 1kms-1 from a study of the H
emission
from B8 shell star 1 Delphinis. Poeckert et al. (1982) have modelled data of o Andromeda in
an attempt to account for observed spectroscopic, photometric and polarimetric changes,
during which the star went through a shell stage, and find v0 ~ 0.3kms-1 . More recently
Marlborough et al. (1997) have modelled the H profile of 
Persei and in order to keep the
profile symmetrical they were forced to use a low value for the initial wind speed of
v0 = 0.34kms-1 . A theoretical estimate for the outflow velocity in viscous Be star
discs has been calculated by Porter (1998), who argues that the velocity could be
very low (v0 
0.01 kms-1 ); this value is arrived at through Be star spin down
considerations due to angular momentum being transfered to the disc from the
star.
Waters et al. (1987) estimated the wind temperature to be Tdisc = 0.8Teff. To check this assumption, and therefore, the appropriateness of the disc model Millar and Marlborough (1999) calculated, using identical parameters to Waters et al., the rates of energy gains and losses throughout the disc. Balancing these rates determined self-consistent temperatures for locations in the disc. Mass loss rates could be determined from the equation of continuity. In addition the disc model was modified to be non-isothermal. In conclusion it is found that the temperature in the equatorial plane is constant with radius to first order approximations. However the assumed temperature of (Tdisc = 0.8Teff; Waters, 1986b) was too high, as in order to obtain equal energy gain and loss rates Millar and Marlborough (1999) found it necessary to lower the wind temperature (Tdisc = 0.5Teff).
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