As mentioned, accretion disks play an extremely important role
in explaining the observational features and properties of
several astrophysical objects, ranging from Cataclysmic
variables and centers of galaxies to protoplanetary disks.
In essence, the gas in the disk accretes towards a central
compact object(s), loosing angular momentum along the way. The
excess angular momentum is carried outwards by an infinitesimal
amount of matter which is believed to be due to the effects of
viscosity. Viscosity plays a central role in the
physical interpretation of the accretion disk
phenomenon, but since its own origin is unclear, is generally
parameterized using the famous a prescription.
We outline the basic equations that govern the evolution of
an Accretion disk, and then specialize on 'steady state' disks.
We assume a thin disk such that in cylindrical coordinates (R,f,z) most of the matter lies close to
the z-plane []. The Keplerian value of the angular velocity of the material comprising the disk is given by:
W = WK = (
G M
R3
)1/2
(1)
We have the mass conservation equation:
R
¶S
¶t
+
¶
¶R
(RSvR) = 0
(2)
and the angular momentum conservation equation:
R
¶
¶t
(SR2 W) +
¶
¶R
(RSvR R2 W) =
1
2p
¶G
¶R
(3)
Here,
S(R,t) is the Surface Density of the disk, vR is the radial `drift' velocity of the disk,
G(R,t) is the viscous torque exerted by the disk on the binary
and is given by...
G(R,t) = 2 pR nSR2 W¢
(4)
with n being the viscosity, which needs to be specified in order to make progress. Once this is done, one can
in principle solve the above equations to get analytical solutions for the disk.
Using (1)2 and (3), we get:
¶S
¶t
= -
1
2 pR
¶
¶R
é ë
1
(R2 W)¢
¶G
¶ R
ù û
(5)
In case of a Keplerian disk, using (1) and (4), we get:
¶S
¶t
=
3
R
¶
¶R
ì í
î
R1/2
¶
¶R
(nSR1/2)
ü ý
þ
(6)
We shall eventually try and obtain solutions for this equation
using self-similar solution techniques. For now though, we
concentrate on steady state solutions from which, we claim, we
can build quasi-steady state solutions which faithfully mimic
the complete SS solutions, except near the boundaries.
To obtain the steady state, we set the time derivative in (6)
to zero. Then by a convenient change of variables, x = (R/R0)1/2,
and setting s = nSR1/2, we get
d2 s
dx2
= 0
(7)
This has the obvious solution:
s(x) = A + Bx
(8)
with A and B being constants to be determined by boundary
conditions. Now, its worth noting that the quantity s(x)
introduced above can be recast as G, the torque, as can be seen
from equations (4) and (5). In other words, equation (7) is
a second order Differential Equation for the torque, and (8) is
a solution for the disk in terms of the variable `x'. To achieve
further insight into the equations, we note that the specific
angular momentum is defined as:
h = (G M R)1/2 = (G M R0)1/2 x
(9)
Thus, essentially h µ x and consequently, one can write
the solution (8) as:
G(h) = G0 -
3 pB
R01/2
h = G0 + C h
(10)
where G0 = -3 p(G M)1/2 A and C = - 3 pB/R01/2 is another
constant.
Now, applying the boundary conditions:
G(xin) = Gin and, G(xout) = Gout, and on
rearranging we obtain:
G(h) =
æ è
h - hin
hin - hout
ö ø
Gout +
æ è
h - hout
hin - hout
ö ø
Gin
(11)
Thus we have the behavior of the torque in the accretion disk as
a function of the specific angular momentum at the location in
the disk. Once we have a prescription for the viscosity, we can
use equation (4) to determine the surface density and using the
energy equation we can obtain the temperature and so on. Should i explicitly write all the equations down here?
We now turn to the time-dependant case, that of an evolving disk
and obtain a general solution for the same using the Self-Similar
solution technique. Then we extend our steady state solution to
an evolving disk, by considering the time-evolution solution to be
successive steady-state solutions.
3 Self-Similar solutions and Quasi Steady-State behavior
We shall treat this case in all generality, relaxing even the
Keplerian nature of the orbit, and adopting a power law
prescription for the viscosity. In general even in the a
prescription, the parameter a is some function of R,
and so it makes sense to circumvent the a-prescription
totally, by simply assuming a power law for the viscosity.
We consequently specialize to specific cases (self-gravity,
a-prescription and so on) and compare the results to
published results.
Using equations (2), (3) and (4) we have a general
evolution equation for the surface density:
¶S
¶t
= -
1
2 pR
¶
¶R
é ë
1
(R2 W)¢
¶G
¶R
ù û
(12)
= -
1
R
¶
¶R
ì í
î
1
(R2W)¢
¶
¶R
(nSR3 W¢)
ü ý
þ
(13)
We now assume the angular velocity to be a power-law in R:
W = W0 (R/R0)a, and change the variable R via
the substitution, x ® (R/R0)-(a+1)1. On substitution and some algebra, one
obtains:
¶S
¶t
= -
(a+1)2 a
(a+2)R02
x[(a+3)/(a+1)]
¶
¶x
ì í
î
x[(2a+3)/(a+1)]
¶
¶x
(nS x-[(a+2)/(a+1)])
ü ý
þ
(14)
Now, we also assume a power law dependance for the viscosity:
I'am working on trying to get a general solution by
appropriate substitutions. I think it is possible. For now, i'll
specialize in the simplest cases to illustrate our simple model.
Lets set a=-3/2 to represent the Keplerian case, and b=1,
c=0 implying the linear viscosity regime. Then Equation
(16) reduces to
¶S
¶t
=
3 n0
4 x3 R02
¶2
¶x2
(x3 S)
(17)
Rearranging and defining the dimensionless time t = (3 n0 t/4 R02), we obtain the familiar diffusion equation
¶s(x,t)
¶t
=
¶2s(x,t)
¶x2
(18)
The solutions to this equation have been dealt with by Pringle
and Lybarskii. We only note that the evolution of the disk can be
thought to occur in 3 stages:
The disk evolves and spreads inwards till it reaches the
inner boundary. This inner edge is
determined by the torque being applied by the central object at that
location.
A quasi-stationary accretion stage near the
inner edge, gradually moving outwards. Here the mass accretion
rate is practically constant or increases slowly.
After a long time," long" being defined as the local
time-scale at the outer edge of the disk, the disk enters a
new stage of equilibrium. The accretion may cease at this
stage and the disk gradually spreads outward (especially in
the case of Circumbinary or black hole disks) with the surface
density falling continuously.
The first stage occurs on relatively short timescales, and is
the transient stage in the entire evolution of the disk. The
second and third stages occur on long time-scales and are
independent of the initial conditions. These stages can be
modelled by Quasi-stationary solutions. In fact if accretion
ceases, i.e. if Macc® 0, as in the third stage, the
solution is a Steady Stage solution mentioned above.
The primary motivation for using quasi-steady state solutions in
mathematical simplicity and physical insight into the disk
evolution. As mentioned, this model should be relevant for stages
2 & 3 where we can appeal to dimensional and time-scale
arguments to achieve mathematical simplicity and physical
insight.
Firstly we briefly examine the various relevant time-scales and
the regime in which we expect this treatment to mimic the
complete time-dependant solution. We then explicitly calculate
various physical quantities of the disk and compare it with the
published complete solutions.
