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

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

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

School On Astrophysical Fluid Dynamics, Miramare, 2007 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

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

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

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

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

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

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

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

School On Astrophysical Fluid Dynamics, Miramare, 2007 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

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

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

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

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