Accretion disks are an important phenomeneon in astrophysics. They are thought to be involved
in the formation of planetary system, mass transfer in binary system, and also are thought to be associated
with the central engines of AGNs. Though the disk theory is able to explain some of the features observed
at the sites mentioned above, a greater understanding of the disks itself in yet to be achieved. The reason
for this is the uncertainty in the nature and magnitude of the viscosity.
The aim of this exercise to understand the physical and mathematical aspects of the
theory of accretion disks. I plan to work on the formation of the disks, the distribution of mass
and angular momentum in the flow of the gas. Unfortunately, accretion theory has brushed 'under the
carpet' the origin and role of viscocity in the gas-flow. I intend to investigate into these aspects. If
time permits, and i make the required progress, i intend to study the role of self-gravity in disk
dynamics as well.
Lets us concentrate on a binary system since its probably the most understood system in terms of
disk dynamics. The potential is given by the Roche Potential...
$\Phi_R$ = $\frac{G M_1}{\vec r-\vec r1} - \frac{G M_2}{\vec r-\vec r2} -\frac{1}{2} (\omega * \vec r)^2 $
This leads to the typical Roche geometry, and the formation of the Roche lobes. Mass transfer occurs via
Roche lobe overflow. Consequently, the transferring material has a high specific angular momentum and hence
cant accrete directly onto the accreting star. The mass passes through the Lagrangian point L1 as it goes
from the secondary to the primary. So the primary 'sees' the material shoot in through a 'nozzle' situated at L1,
and L1 seems to rotate around it in the binary plane. Unless the binary period is too short, the primary sees
this material enter in almost perpandicularly, along the line joining the centers of the 2 stars. In general, except
in extra-ordinary cases of short-period binaries, $v_perp>>v_parallel$. Since $v_parallel \ sim c_s, v_perp$
is supersonic and in general, the gas shoots into the primary at supersonic speeds. The stream is then further
accelerated by the primary's gravitational force. The stream then follows a balastic trajectory determined by the
Roche potential. If the stream were just a set of non-interacting test particles, each such particle would follow
an elliptical orbit in the plane of the binary. Thus, the particles left to themselves, influenced just by the primary,
would form an elliptical ring around the star. However, the presence of the binary causes this ring to precess slowly.
A continuous string of particles would thus intersect resulting in dissipative loss of energy via shocks. To conserve
angular momentum even as it is losing energy, the gas will be forced into a minimum energy configuration : a circular
orbit. Thus the gas initially sets into a Keplerian orbit at R_circ, where it has the same specific angular momentum as
the gas had whenst it entered in at L1.
In general, we are concerned with compact objects and so almost always we can assume R_circ > R_star.
This, and the presence of dissipative forces due to the presence of the secondary, have interesting implications
on the flow of the gas, and leads to the formation of the disk. Due to the dissipative forces, some of the kinetic
energy of the gas is converted in to heat energy which is eventually radiated away. Thus, the gas looses energy
and in order to maintain orbit, has to sink deeper into the potential well. Now, this it can achieve only if its able
to rid itself of some angular momentum (and reach an angular momentum consistant with its new R_circ). Now,
the timescale for angular momentum redistribution is much longer than the timescales for radiative energy loss
or te orbital time-scale. Consequently, the gas looses as much energy it can for a given angular momentum, which itself
decreases slowly. The gas re-adjusts to the new situation(less energy) by spiralling inwards gradually on the time-scale
dictated by angular momentum loss. Since the flow is continuous, a disk like structure results. The angular momentum
'loss' is essentially transfer of angular momentum outward through the disk. This is achieved via 'internal torques' in the
absence of any 'external torques'.