We take a geometrically thin,
axisymmetric disc as an unperturbed equilibrium disc.
For simplicity, the disc is assumed to be isothermal
with a temperature range of
.
Here, 
is the effective temperature of the star.
We also assume that the unperturbed disc is in hydrostatic equilibrium
in the vertical direction
and extends from the stellar surface (
) to infinity
in the radial direction.
We neglect radial advective motions and viscous effects.
The adopted value of the mean molecular weight is 0.6.
We use a cylindrical coordinate system
to describe the disc;
the origin is at the center of the star and
the z-axis is perpendicular to the disc midplane.
When the distortion of the star is weak,
the potential of the star of mass M with
apsidal motion constant 
can be approximated as
(PSH; SH).
Here, f is the rotation parameter, defined by
the ratio of the rotation velocity of the star
to the Keplerian velocity at the stellar surface.
According to Kogure & Hirata (1982),
for B0-type Be stars
and 
for B5-type Be stars.
The values of 
for rapidly rotating stars are not well known.
Theoretically, 
for non-rotating main-sequence stars has a
maximum value of 
at a mass between 7 and 
(Stothers 1974).
In general, 
is a decreasing function of the degree of
the central condensation of the star:
The value of 
is smaller for a more evolved star
or for a star with larger uniform rotation
(Stothers 1974; Claret & Giménez 1993),
while it can be large for a star of which the inner part
rotates more rapidly than the outer part.
PSH adopted 
and
and SH studied the rotation effect
for 
.
For the radiative force we adopt the parametric form proposed by Chen & Marlborough (1994):
where 
and 
are parameters which characterize
the force due to an ensemble of optically thin lines.
According to Chen & Marlborough (1994),
should be a positive number which is much smaller than unity.
In the above expression,
is the Eddington factor
that accounts for the reduction in the effective gravity
due to electron scattering.
Here, 
is the opacity due to electron scattering
per mass unit and L is the total luminosity of the star.
In the rest of this paper,
we neglect the Eddington factor 
.
This approximation is valid as long as
can be
regarded as a constant throughout the disc
and is much smaller than unity.
Both conditions hold well in Be-star discs.
Since the disc is considered to be optically thin for electron scattering,
the radial dependence of 
can be neglected.
Moreover, using the typical values for Be stars,
is as small as
for a B0V star and 
for a B5V star.
The radiative force is then written as
In the present model, the equation describing the hydrostatic balance in the vertical direction is written as
where 
and 
are the unperturbed density
and pressure, respectively.
Since the disc is isothermal, Eq.(4)
is integrated to give
the density distribution in the vertical direction:
where 
is the unperturbed density in the equatorial plane
and H the vertical scale-height of the disc given by
Here, 
and 
are
the isothermal sound speed and the Keplerian angular velocity,
respectively.
Since the density profiles of Be-star discs are
not well determined,
we adopt a simple power-law form for 
,
where the index 
is a constant.
Observationally, the density-gradient indices for Be stars
have been derived by a curve of growth method,
in which the disc is assumed to have
a constant opening angle and a density distribution
proportional to 
(e.g., Waters 1986).
The obtained value of n is different from star to star,
but is roughly in the range of 
(Waters et al. 1987).
Considering the difference between a disc geometry assumed by
the curve of growth method
and the disc geometry adopted in this paper,
we infer the range of 
to be 
.
Theoretically, 
in the subsonic region
of the isothermal disc,
if the angular momentum distribution of the disc
is due to viscous stress
[see Eq. (11) of Lee et al. (1991)].
The radial distribution of the rotational angular velocity
is derived
from the equation of motion in the radial direction
and is written explicitly in the form
under the approximation 
.
Here, 
is the Mach number of disc rotation
at 
.
For typical Be stars, 
is given by
where we adopted
, 
,
and 
for a B0V star and
, 
,
and 
for a B5V star
(Allen 1973).
In the present model,
the epicyclic frequency 
is given by
Note that the second, third, and fourth terms
between the square brackets in Eqs. (9) and (11)
denote the contributions from
the pressure gradient force in the disc,
the deviation from the point-mass potential due to
the rotational deformation of the star,
and the radiative force due to an ensemble of optically thin lines,
respectively.
The parameters which characterize the problem are
, 
, 
, 
, and 
.
