Angular Momentum Transfer in Accretion Discs

= − 3ν ( ( ))
∂ 1 1 + x2
2x
lnx − + lnI 1/4 .
r 0 ∂x 4 τ
τ
The asymptotic behaviour, I 1/4 (z) ∝ z −1/2 e z at z >> 1 and I 1/4 (z) ∝ z 1/4 at z 0,
τ
at 2x >> τ
v r ∼ − 3ν ( 1
r 0 2x − 2x )
< 0,
τ
at 2x > τ) move outwards, taking away angular momentum
of the inner parts, which move inwards towards the accreting star. Moreover, the radius at which v r
changes its sign moves outwards itself, for if we choose some point at which x initially much greater than
τ, after sufficient time this same point will have x much less than τ. Thus parts of the matter distribution
which are at radii r > r 0 just after the initial release of the ring (t ∼ 0) at first move to larger radii,
but later begin to lose angular momentum to parts of the disc at still larger radii and thus drift inwards.
At very long times (τ >> 1) after the initial release, almost all of the original m has accreted on to the
central star (r ∼ 0), while all of the original angular momentum has been carried to very large radii by
a very small fraction of the mass. We will confirm that this general trend of disc viscous evolution holds
even for radially varying viscosity coefficient.
3.2. Radially varying viscosity
Now consider more general case where ν is given as a power law of radius,
ν = ν 0
( r
r 0
) γ
≡ c 0 r γ .
Such a dependence is expected from the fact that typically in discs sound speed has a power law dependence
itself and, hence scale height also follows this trend (angular velocity of disc rotation is Keplerian
≡ r −3/2 , while the α is approximately constant with radius). For the convenience of further analysis,
following Lynden-bell & Pringle (1974) let us change independent variables: h = r 1/2 and g = νΣr 1/2 .
After this Eq. (3) is rewritten as
∂ 2 g
∂h 2 = 4h2(1−γ) ∂g
3c 0 ∂t .
If we now assume that time variation is given by g ∝ e −st , we have
∂ 2 g
∂h 2 + 4sh2(1−γ)
3c 0
g = 0.
If we now set x = h 2−γ and g 1 = x 1/2(γ−2) g, we again obtain Bessel’s equation
d 2 g 1
dx 2 + 1 (
dg 1 4s
x dx + − 1 )
g 1
3c 0 4x 2 (γ − 2) 2 = 0
The general solution for this equation is given in terms of Bessel functions for each s supplemented with
boundary condition that g vanishes at the origin r = 0 (or more precisely at the edge of the boundary
layer between disc and star surface). On the other hand, time-dependent solution is a linear combination
of Bessel functions with different s. However, from this general solution its hard to extract information on
the key properties of angular momentum transport. In order to clearly understand the latter we restrict
ourselves to the time evolution of a special solution resulting from the following initial distribution for g:
g(h, 0) = Ch · exp(−ah 2(2−γ) ) = Cx 1/(2−γ) exp(−ax 2 ),
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