The Be Star Newsletter, Volume 36 - April 2002
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The Structure and Continuum Emission of Viscous DiscsOne of a series of invited articles currently appearing in the Be Star Newsletter
Astrophysics Research Institute, Liverpool John Moores University,
Twelve Quays House, Egerton Wharf, Birkenhead, CH41 1LD
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ABSTRACT In the last few years a model of the Be star disc maintained by viscous transport of angular momentum has been gaining popularity. In this summing-up of some of the features of this model, a simple analysis of the structure of viscous discs is presented which leads to the generation of radial power-law profiles of the surface density and radial velocity. The infra-red continuum emission from the discs is calculated and compared to typical excess emission observed in Be stars. It is found (in the simple model used) that discs which cool with radius describe observations better than isothermal discs. Some of the positive and negative aspects of the viscous disc model are presented. |
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1. Introduction
Some of the most attractive and successful models for the discs around
Be stars which have been presented in the last decade or so are
viscous discs (e.g. Lee, Saio & Osaki 1991, Okazaki 2001), wind
compressed discs (Bjorkman & Cassinelli 1993), and wind bi-stability
(Lamers & Pauldrach 1991). The wind compressed disc model has been
examined at length, and found to have problems in generating the
required disc densities (Porter 1997), and indeed the mechanism may
not operate at all due to non-radial line forces (Owocki, Cranmer &
Gayley 1996). The bi-stable model is also problematic in that the
disc's velocity field appears to be in conflict with kinematic
observational studies (indeed this also applies to the wind compressed
disc model): the two models predict radial velocities in the disc of
~100 km/s although such high speeds are not observed (for example
see Hanuschik, 2000), also gas in the disc conserves angular momentum
and so the azimuthal velocity is
The remaining model -- the viscous disc model -- seems to be the most attractive: it has azimuthal velocities which are close to Keplerian which is consistent with observation (e.g. Hummel & Vranken, 2000); it allows 1-armed perturbations to grow to generate V/R variations (e.g. Okazaki 2001); it is able to provide enough density to account for IR continuum excesses (e.g. Porter 1999); it doesn't require an excessive amount of angular momentum to sustain it, and hence is not observable via the spin down of Be stars (see Porter 1998, and Steele 1999). However, the viscous disc model has the feature that angular momentum must be continually supplied to it in order that it maintains its structure. The mechanism to achieve this remains obscure, providing the major caveat to the identification of the viscous disc model as the actual Be star disc structure. This contribution outlines some brief arguments pertaining to the viscous disc structure (in some particular cases), along with continuum emission calculations. Much of this analysis has appeared in several papers already, so this is largely a collection of published ideas and results. 2. Disc structure
So, how does the viscous disc model "work"? The gas in the disc is
fully turbulent as it flows around the star. The turbulence
transports angular momentum (which is supplied by the star in an
unknown fashion) outward, and produces a slow drift of gas in the disc
to large radii. As a by-product of this process the disc's azimuthal
velocity
The structure of the disc is determined by the equations of mass,
momentum and angular momentum conservation. Let us consider the
reduced problem where there are no radiative forces, and that the disc
is time independent so
Mass conservation yields the mass-loss rate through the disc (in
cylindrical co-ordinates)
where
where
where C is a constant, or
Numerical solutions of Euler's equation and the conservation of
momentum equations by Okazaki (2001) indicate that the radial
Mach number
The disc scale-height is
The form in which the viscous stresses
2.1 Cooling discs
Why are cooling discs interesting? The time-independent isothermal
discs we have just considered have steep density power laws in radius,
which we might suppose would not account for IR continuum excesses
observed (a correct supposition -- see later). Is there any process
which produces a density profile flatter than
Let us assume that the disc may cool and that the temperature of the
disc varies with radius as a power law
The scale-height for a cooling disc is
What values of m are likely? If the disc is adiabatic, then the
index can be shown to be m = 4/3. Assuming that the
energy balance in the disc lies between the two extremes of isothermal
and adiabatic flows, then we expect
3. Continuum emission Now that expressions for the density and scale height has been obtained, the continuum excess can be calculated. This is done following the prescription of Waters (1986): the density is assumed to decrease exponentially normally to the equatorial plane with the scale-height H.
An example star is used corresponding to typical Be stars and disc
parameters: the underlying photospheric emission is a
Teff = 22,000 K,
log g = 4.0
Kurucz model and the disc's base density
Fig. 1
The four calculations refer to an isothermal model
(m = 0, solid line),
m = 0.5 (dotted line),
m = 1.0 (short dashed line), and an adiabatic model
(m = 4/3 - the long dashed line). As the cooling
index m increases, the surface density variation with radius becomes
less steep which increases the free-free and free-bound optical depth
through the disc
In the right panel of Fig.1 are the excesses of five stars from Waters
et al. (1991) -
4. Summary The viscous disc model seems to be able to provide a density and velocity field which is consistent with observational results. However, cooling must be invoked to to produce the excess IR emission observed - at least for models without radiative forces. After realising the potential of the viscous disc model to account for observations we must remember the limited case considered here: no radiative forces have been included (although Okazaki, 2001, finds very similar density and velocity structures for the line force of Chen & Marlborough, 1994), and the derivation has leant on solutions of expressions for linear and angular momentum conservation and mass conservation in a specific case. Two aspects of the viscous disc model which need attention in the near future are:
Whilst spectral line profiles have not been touched upon in this discussion, current work on disc kinematics seems to favour the viscous disc model's velocity field, which is dominated by rotational support: nearly Keplerian azimuthal speeds and subsonic radial drift to larger radii. The viscous disc model looks promising as an explanation of the dynamics of Be star discs, although there are significant problems to overcome. REFERENCES Bjorkman, J.E. & Cassinelli, J.P., 1993, ApJ, 409, 429 Chen, H. & Marlborough, J.M., 1994, ApJ, 427, 1005 Frank, J., King, A., & Raine, D., 1985, Accretion Power in Astrophysics, Cambridge University Press Gayley, K.G., Ignace, R., & Owocki, S.P., 2001, ApJ, 558, 802 Hanuschik, R.W., 2000, in The Be Phenomenon in Early-Type Stars, IAU Colloquium 175, eds. M.A. Smith, H.F. Henrichs, & J. Fabregat, p. 518 Hummel, W. & Vranken, M., 2000 A&A, 359, 1075 Lamers, H.J.G.L.M. & Pauldrach, A.W.A., 1991, A&A, 244, L5 Lee, U., Saio, H., & Osaki, Y., 1991, MNRAS, 250, 432 Millar, C.E. & Marlborough, J.M., 1998, ApJ, 494, 715 Okazaki, A.T., 2001, PASJ, 53, 119 Owocki, S.P., Cranmer, S.R., & Gayley, K.G., 1996, ApJ, 472, L115 Porter, J.M., 1997, A&A, 324, 597 Porter, J.M., 1998, A&A, 333, L83 Porter, J.M., 1999, A&A, 341, 560 Pringle, J.E., 1981, ARAA, 19, 137 Shakura, N.I & Sunyaev, R.A., 1973, A&A, 24, 337 Steele, I.A., 1999, A&A, 343, 237 Waters, L.B.F.M., 1986, A&A, 162, 121 Waters, L.B.F.M., van der Veen, W.E.C.J., Taylor, A.R., Marlborough, J.M., & Dougherty, S.M., 1991, A&A, 244, 120 |
- again in
conflict with observation (e.g. Hummel & Vrancken 2000), and finally,
there are no obvious sites in the disc where line profile V/R
variations can be generated.
is very close to Keplerian
.
Although turbulent discs have been studied extensively in the context
of accretion onto a central object (where the source of gas and
angular momentum is external to the central object), they have
received less attention when applied to Be star discs (where the
angular momentum and gas source is the star itself). Excellent
reviews of turbulent discs can be found in Pringle (1981), or Frank,
King & Raine (1985).
. Note that
in several previous analyses radiation has been included (Okazaki 2001),
or considered (see Gayley et.al., 2001), and its omission here
should be kept in mind during the following discussion.
,
(
is the radial velocity and
is the surface density of the disc where
is the density, and
is the disc scale height).
Euler's equation in the radial direction produces
is the sound speed,
(for B stars 
)
and conservation of angular momentum gives
is the
component of the viscous stress
(
is the
Shakura-Sunyaev, 1973, viscosity parameter). The expression for the
viscous stress 
was derived in this
form (by Shakura & Sunyaev) assuming that the angular speed
decreases with radius, and so
the replacement 
is used for
all radii. The above (eq.2) can be simply integrated to yield
is small: at the
star-disc boundary
. Assuming that the inner
edge of the disc rotates at its Keplerian speed, then
for B stars.
These lead to
, and hence the
surface density for isothermal discs is
.
, where
is its value at the inner edge.
For parameters appropriate to B stars,
. With this, and the surface
density, the radial dependence of the actual density
can be derived: for
and
, then
.
Using mass conservation, and the surface density, the variation of the
radial velocity 
.
This solution relies on the inequality
the disc becomes transonic
, and the density steepens.
,
which is strictly the correct condition (no shear between two
neighbouring rings in the disc). The simple form of the stresses
produces an angular speed
;
there is nothing mathematically special at
r = R*.
One consequence of this is that no absolute density scaling can
be obtained from mass and momentum conservation (i.e. eq.1 & 2 can be
transformed to be independent of density or mass-loss rate). Future
solutions should include a physical star-disk boundary of some sort.
.
The sound-speed
,
decreases with radius. Inserting this radial variation into the
surface density expression from eq.4 (where
and so the resultant space density of the disc is
.
Finally, the radial velocity of cooling discs is obtained from mass
conservation:
.
.
.
The continuum emission spectrum is calculated for a range of cooling
indices m (from isothermal m = 0 to adiabatic
m = 4/3) and expressed
as the excess flux ratio:
Z - 1 = (flux/photospheric flux) - 1.
The results are shown in the left panel of Fig.1 below.
,
and hence produces more emission (relatively weakly countering this
trend is the less steep increase of scale height of the disc).
However, acting in opposition is that as the gas cools, the total
emission from it decreases (as the flux
).
These competing processes produce the drop off in emission for the
high m calculation.
Cas (open circles),
EW Lac (open squares),
CMi (open triangles),
Per (filled triangles)
and