1. Introduction to Accretion Discs
Andria Rogava
Georgian National Astrophysical Observatory
12/29/2018
School On Astrophysical Fluid Dynamics, Miramare, 2007

2. School On Astrophysical Fluid Dynamics, Miramare, 2007
1. Introductory remarks
The matter accreting onto a compact object will have a significant angular momentum and may form a disc;
The gas elements in the disc lose angular momentum due to friction between adjacent layers of differentially rotating flow;
A part of the released gravitational energy increases the kinetic energy – gas particles spiral inwards;
The other part is converted into thermal energy which is being radiated from the disc surfaces;
Viscosity converts gravitational potential energy into radiation.
12/29/2018
School On Astrophysical Fluid Dynamics, Miramare, 2007

3. Early Studies of disc accretion
Von Weizsaker: ‘The rotation of cosmic gas masses’, Z. Naturforsch, 3a, 524 (1948)
Lust: Z. Naturforsch, 7a, 87 (1952)
Prendergast & Burbridge: ‘On the nature of some galactic X-ray sources’ Ap.J. Lett., 151, L83 (1968).
Shakura: Astron. Zh. 49, 921 (1972).
Pringle & Rees: Astron. Astrophys. 21, 1 (1972).
Shakura & Sunyaev: Astron. Astrophys. 24, 337 (1973).
12/29/2018
School On Astrophysical Fluid Dynamics, Miramare, 2007

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Where do they occur?
Active galaxies
Galactic X-ray sources
Cataclysmic variables
Young stellar objects
12/29/2018
School On Astrophysical Fluid Dynamics, Miramare, 2007

5. Basic build-up of the model
(r, f, z) cylindrical coordinates are used; the z-axis chosen as the axis of rotation: z=0 is the central, equatorial plane of the disc;
the disc is assumed to be axisymmetric:
The disc is supposed to be geometrically thin:
the total pressure is the sum of the gas and radiaton pressures:
the specific internal energy is:
12/29/2018
School On Astrophysical Fluid Dynamics, Miramare, 2007

6. School On Astrophysical Fluid Dynamics, Miramare, 2007
1. Mass conversation
The continutiy equation:
After we define the Surface density as:
leads to:
12/29/2018
School On Astrophysical Fluid Dynamics, Miramare, 2007

7. School On Astrophysical Fluid Dynamics, Miramare, 2007
2. Viscosity tensor
In the standard disc model it is assumed that the main (only?) nonzero component of the viscosity tensor is:
And its vertically averaged value is:
12/29/2018
School On Astrophysical Fluid Dynamics, Miramare, 2007

8. 3. Angular momentum conservation
From the azimuthal component of the equation of motion:
after algebraic manipulations and integration over z:
12/29/2018
School On Astrophysical Fluid Dynamics, Miramare, 2007

9. 4. Radial momentum conversation
Initial equation has the form:
After z-integration it reduces to:
where
12/29/2018
School On Astrophysical Fluid Dynamics, Miramare, 2007

10. School On Astrophysical Fluid Dynamics, Miramare, 2007
5. Energy conversation
Initial equation is:
after integration:
last term on the right hand side is the cooling rate
Note that the Dissipation Function is defined as:
and consequently:
is the energy produced per unit area (‘heating rate’).
12/29/2018
School On Astrophysical Fluid Dynamics, Miramare, 2007

11. 6. Vertical hydrostatic balance
Vertical hydrostatic balance in the disc:
from this equation it follows that rotational motion in the accretion disc is highly supersonic!
In the standard disc model it is assumed that the energy dissipated into heat is totally radiated in the vertical direction:
Which, in other words, means exact balance of cooling and heating rates:
12/29/2018
School On Astrophysical Fluid Dynamics, Miramare, 2007

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7. Keplerian limit
Keplerian limit:
Circular and angular velocities:
Angular momentum conservation reduces to:
In this approximation cooling and heating rates are indeed equal to each other:
12/29/2018
School On Astrophysical Fluid Dynamics, Miramare, 2007

13. School On Astrophysical Fluid Dynamics, Miramare, 2007
8. Steady discs
Continuity:
Angular momentum:
From this equation we find remarkably simple equation for the vertically integrated viscosity coefficient:
12/29/2018
School On Astrophysical Fluid Dynamics, Miramare, 2007

14. 9. Cooling rate and luminosity
Cooling rate is equal to:
While for the luminosity we calculate:
It means that half of the potential energy is radiated away. The other half is in the form of kinetic energy just outside the boundary layer.
12/29/2018
School On Astrophysical Fluid Dynamics, Miramare, 2007

15. School On Astrophysical Fluid Dynamics, Miramare, 2007
10. Standard ‘a-model’
Detailed disc models can be built only if we know the viscosity law!
Accretion disc flow is extremely complex:
Highly supersonic;
Strongly sheared;
Radiative;
Large Reynolds numbers
Usual (molecular) viscosity is not sufficient to generate the observable intense X-ray emission
12/29/2018
School On Astrophysical Fluid Dynamics, Miramare, 2007

16. School On Astrophysical Fluid Dynamics, Miramare, 2007
11. a-prescription
It is believed that in accretion discs angular momentum transport is governed by turbulent viscosity.
Shakura & Sunyaev (1973) introduced a phenomenological parametrization that became the standard model of thin, quasi Keplerian accretion discs;
Alternatively, ‘a-model’ is often written for the only nonzero component of the viscosity tensor:
12/29/2018
School On Astrophysical Fluid Dynamics, Miramare, 2007