Abstract
Accretion discs are powerful energy factories in our Universe. They effectively transform the potential energy of gravitational interaction to emission, thereby unraveling the physics of distant objects. This is possible due to the presence of viscosity, driven by turbulent motions in accretion discs. In this chapter, we describe the equations for disc accretion in the framework of the standard model. We outline basic elements of the theory of turbulent viscosity and the emergence of the α-parameter. We further describe the radial and vertical structure of thin stationary accretion discs, and present analytical solutions to the basic equation of the evolution of a viscous accretion disc for both an infinite disc and for a disc in a binary system. Finally, we present a numerical method to solve the equations of disc evolution and vertical structure simultaneously.
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- 1.
With this definition, \(w_{r\varphi }^{\mathrm {t}} \) will be positive in accretion discs. In other literature on the subject, the definition \(t_{r\varphi }=-w_{r\varphi }^{\mathrm {t}}\) is often used instead.
- 2.
They are analogous to the quantities \(v_x^{\prime }\) and \(\xi _z^{\prime }\) discussed in Sect. 1.3.4.
- 3.
- 4.
This property is similar to the one that arises in problems of thermal conductivity, when, due to the non-linearity, the heatwave boundary sharply separates the heated zone from the rest of the region (Zeldovich and Raizer 1967).
- 5.
Note that here F is a factor of 2π larger than in the paper by Lyubarskij and Shakura (1987), and our quantity D is smaller by a factor of (2 π)m.
- 6.
- 7.
- 8.
Fast-rise exponential-decay.
References
Abramowicz MA (2016) Velocity, acceleration and gravity in Einstein’s relativity. ArXiv e-prints 1608.07136
Abramowicz MA, Fragile PC (2013) Foundations of black hole accretion disk theory. Living Rev Relativ 16:1. https://doi.org/10.12942/lrr-2013-1. ArXiv 1104.5499
Anderson DG (1965) Iterative procedures for nonlinear integral equations. J ACM 12(4):547–560. https://doi.org/10.1145/321296.321305. http://doi.acm.org/10.1145/321296.321305
Balbus SA, Hawley JF (1991) A powerful local shear instability in weakly magnetized disks. I - Linear analysis. II - Nonlinear evolution. Astrophys J 376:214–233. https://doi.org/10.1086/170270
Balbus SA, Hawley JF (1998) Instability, turbulence, and enhanced transport in accretion disks. Rev Mod Phys 70:1–53. https://doi.org/10.1103/RevModPhys.70.1
Barenblatt GI (1996) Scaling, Self-similarity, and intermediate asymptotics: dimensional analysis and intermediate asymptotics. Cambridge texts in applied mathematics. Cambridge University Press, Cambridge
Barenblatt G (2003) Scaling. Cambridge texts in applied mathematics. Cambridge University Press, Cambridge. https://books.google.ru/books?id=05zBYET6tR0C
Bisikalo DV, Zhilkin AG, Boyarchuk AA (2013) Gaseous dynamic of close binary stars. FIZMATLIT (in Russian), Moscow
Bisnovatyi-Kogan GS, Blinnikov SI (1976) A hot corona around a black-hole accretion disk as a model for Cygnus X-1. Sov Astron Lett 2:191–193. ArXiv astro-ph/0003275
Bisnovatyi-Kogan GS, Lovelace RVE (2001) Advective accretion disks and related problems including magnetic fields. New Astron Rev 45:663–742. https://doi.org/10.1016/S1387-6473(01)00146-4. ArXiv astro-ph/0207625
Brandenburg A, Nordlund A, Stein RF, Torkelsson U (1996) The disk accretion rate for dynamo-generated turbulence. Astrophys J 458:L45. https://doi.org/10.1086/309913
Cannizzo JK (1992) Accretion disks in active galactic nuclei - vertically explicit models. Astrophys J 385:94–107. https://doi.org/10.1086/170918
Cannizzo JK (1998) The accretion disk limit cycle mechanism in the black hole x-ray binaries: toward an understanding of the systematic effects. Astrophys J 494:366. https://doi.org/10.1086/305210
Cannizzo JK, Lee HM, Goodman J (1990) The disk accretion of a tidally disrupted star onto a massive black hole. Astrophys J 351:38–46. https://doi.org/10.1086/168442
Cantrell AG, Bailyn CD, Orosz JA, McClintock JE, Remillard RA, Froning CS, Neilsen J, Gelino DM, Gou L (2010) The inclination of the soft x-ray transient A0620-00 and the mass of its black hole. Astrophys J 710:1127–1141. https://doi.org/10.1088/0004-637X/710/2/1127. ArXiv 1001.0261
Chandrasekhar S (1960) The stability of non-dissipative Couette flow in hydromagnetics. Proc Natl Acad Sci 46:253–257. https://doi.org/10.1073/pnas.46.2.253
Chen W, Shrader CR, Livio M (1997) The properties of x-ray and optical light curves of x-ray novae. Astrophys J 491:312. https://doi.org/10.1086/304921
Coroniti FV (1981) On the magnetic viscosity in Keplerian accretion disks. Astrophys J 244:587–599. https://doi.org/10.1086/158739
Deardorff JW (1970) A numerical study of three-dimensional turbulent channel flow at large Reynolds numbers. J Fluid Mech 41:453–480. https://doi.org/10.1017/S0022112070000691
Dubrulle B (1993) Differential rotation as a source of angular momentum transfer in the solar nebula. Icarus 106:59. https://doi.org/10.1006/icar.1993.1158
Dubus G, Lasota JP, Hameury JM, Charles P (1999) X-ray irradiation in low-mass binary systems. Mon Not R Astron Soc 303:139–147
Duerbeck HW, Walter K (1976) A periodic brightness variation of the optical counterpart of A 0620-00. NASA Spec Publ 389:343–345
Eardley DM, Lightman AP (1975) Magnetic viscosity in relativistic accretion disks. Astrophys J 200:187–203. https://doi.org/10.1086/153777
Favre AJ (1969) Statistical equations of turbulents gases. 231–267
Felten JE, Rees MJ (1972) Continuum radiative transfer in a hot plasma, with application to Scorpius X-l. Astron Astrophys 17:226
Ferguson JW, Alexander DR, Allard F, Barman T, Bodnarik JG, Hauschildt PH, Heffner-Wong A, Tamanai A (2005) Low-temperature opacities. Astrophys J 623:585–596. https://doi.org/10.1086/428642. ArXiv astro-ph/0502045
Filipov LG (1984) Self-similar problems of the time-dependant discs accretion and the nature of the temporary X-ray sources. Adv Space Res 3:305–313. https://doi.org/10.1016/0273-1177(84)90107-8
Frank J, King A, Raine DJ (2002) Accretion power in astrophysics, 3rd edn. Cambridge University Press, Cambridge
Fridman AM (1989) On the dynamics of a viscous differentially rotating gravitating medium. Sov Astron Lett 15:487
Galeev AA, Rosner R, Vaiana GS (1979) Structured coronae of accretion disks. Astrophys J 229:318–326. https://doi.org/10.1086/156957
Gorbatskii VG (1965) Disk-like envelopes in close binary systems and their effect on stellar spectra. Sov. Astron. 8:680
Gou L, McClintock JE, Steiner JF, Narayan R, Cantrell AG, Bailyn CD, Orosz JA (2010) The spin of the black hole in the soft x-ray transient A0620-00. Astrophys J 718:L122–L126. https://doi.org/10.1088/2041-8205/718/2/L122. ArXiv 1002.2211
Hameury JM, Lasota JP (2005) Tidal torques, disc radius variations, and instabilities in dwarf novae and soft X-ray transients. Astron Astrophys 443:283–289. https://doi.org/10.1051/0004-6361:20053691. ArXiv arXiv:astro-ph/0508509
Hameury JM, Menou K, Dubus G, Lasota JP, Hure JM (1998) Accretion disc outbursts: a new version of an old model. Mon Not R Astron Soc 298:1048–1060
Ichikawa S, Osaki Y (1994) Tidal torques on accretion disks in close binary systems. Publ Astron Soc Jpn 46:621–628
Iglesias CA, Rogers FJ (1996) Updated opal opacities. Astrophys J 464:943. https://doi.org/10.1086/177381
Ivanov PB, Papaloizou JCB, Polnarev AG (1999) The evolution of a supermassive binary caused by an accretion disc. Mon Not R Astron Soc 307:79–90. https://doi.org/10.1046/j.1365-8711.1999.02623.x. ArXiv astro-ph/9812198
Kaluzienski LJ, Holt SS, Boldt EA, Serlemitsos PJ (1977) All-Sky Monitor observations of the decay of A0620-00 /Nova Monocerotis 1975/. Astrophys J 212:203–210. https://doi.org/10.1086/155036
Kato S, Fukue J, Mineshige S (1998) Black-hole accretion disks. Kyoto University Press, Kyoto
Kato S, Fukue J, Mineshige S (2008) Black-hole accretion disks — towards a new paradigm —, 549 pp
Ketsaris NA, Shakura NI (1998) On the calculation of the vertical structure of accretion discs. Astron Astrophys Trans 15:193. https://doi.org/10.1080/10556799808201769
King AR, Ritter H (1998) The light curves of soft X-ray transients. Mon Not R Astron Soc 293:L42–L48
Kotko I, Lasota JP (2012) The viscosity parameter α and the properties of accretion disc outbursts in close binaries. Astron Astrophys 545:A115. https://doi.org/10.1051/0004-6361/201219618. ArXiv 1209.0017
Kurucz RL (1970) Atlas: a computer program for calculating model stellar atmospheres. SAO Special Report. Smithsonian Astrophysical Observatory, Cambridge
Kurucz R (1993) Kurucz CD-ROMs, Smithsonian Astrophysical Observatory, Cambridge
Landau LD, Lifshitz EM (1959) Fluid mechanics, Course of Theoretical Physics, vol 6
Landau LD, Lifshitz EM (1973) The classical theory of fields. Course of theoretical physics, vol 2, Sect. 88. Pergamon, Oxford
Lasota JP (2001) The disc instability model of dwarf novae and low-mass X-ray binary transients. New Astron Rev 45:449–508
Lasota JP (2015) Black hole accretion discs. ArXiv e-prints 1505.02172
Lightman AP, Eardley DM (1974) Black holes in binary systems: instability of disk accretion. Astrophys J 187:L1+
Lin DNC, Bodenheimer P (1982) On the evolution of convective accretion disk models of the primordial solar nebula. Astrophys J 262:768–779. https://doi.org/10.1086/160472
Lin DNC, Pringle JE (1987) A viscosity prescription for a self-gravitating accretion disc. Mon Not R Astron Soc 225:607–613
Lipunova GV (2015) Evolution of finite viscous disks with time-independent viscosity. Astrophys J 804:87. https://doi.org/10.1088/0004-637X/804/2/87. ArXiv 1503.09093
Lipunova GV, Malanchev KL (2017) Determination of the turbulent parameter in accretion discs: effects of self-irradiation in 4U 1543-47 during the 2002 outburst. Mon Not R Astron Soc 468:4735–4747. https://doi.org/10.1093/mnras/stx768. ArXiv 1610.01399
Lipunova GV, Shakura NI (2000) New solution to viscous evolution of accretion disks in binary systems. Astron Astrophys 356:363–372
Lipunova GV, Shakura NI (2002) Non-steady-state accretion disks in x-ray novae: outburst models for Nova Monocerotis 1975 and Nova Muscae 1991. Astron Rep 46:366–379
Lloyd C, Noble R, Penston MV (1977) The light curve of V616 Mon = A0620-00. Mon Not R Astron Soc 179:675–681. https://doi.org/10.1093/mnras/179.4.675
Lüst RZ (1952) Die Entwicklung einer um einen Zentralkörper rotierenden Gasmasse. I. Lösungen der hydrodynamischen Gleichungen mit turbulenter Reibung. Zeitschrift Naturforschung Teil A 7:87
Lynden-Bell D (1969) Galactic nuclei as collapsed old quasars. Nature 223:690
Lynden-Bell D, Pringle JE (1974) The evolution of viscous discs and the origin of the nebular variables. Mon Not R Astron Soc 168:603–637
Lyubarskij YE, Shakura NI (1987) Nonlinear self-similar problems of nonstationary disk accretion. Sov Astron Lett 13:386
MacRobert TM (1932) Fourier integrals. Proc R Soc Edinb 51:116–126
Malanchev KL, Shakura NI (2015) Vertical convection in turbulent accretion disks and light curves of the X-ray nova A0620-00 1975 outburst. Astron Lett 41:797–808. https://doi.org/10.1134/S1063773715120087. ArXiv 1511.02356
Marov M, Kolesnichenko A (2011) Turbulence and self-organization: modeling astrophysical objects. Astrophysics and Space Science Library, Springer, New York. https://books.google.ru/books?id=y1r2sgEACAAJ
Mescheryakov AV, Revnivtsev MG, Filippova EV (2011a) Parameters of irradiated accretion disks from optical and X-ray observations of GS 1826-238. Astron Lett 37:826–844. https://doi.org/10.1134/S1063773711120073
Mescheryakov AV, Shakura NI, Suleimanov VF (2011b) Vertical structure of the outer accretion disk in persistent low-mass X-ray binaries. Astron Lett 37:311–331. https://doi.org/10.1134/S1063773711050045. ArXiv 1108.4222
Meyer F, Meyer-Hofmeister E (1981) On the elusive cause of cataclysmic variable outbursts. Astron Astrophys 104:L10
Meyer F, Meyer-Hofmeister E (1982) Vertical structure of accretion disks. Astron Astrophys 106:34–42
Meyer F, Meyer-Hofmeister E (1984) HZ Her/Her X-1 - an alternative model for the 35d cycle? Astron Astrophys 140:L35–L38
Mihalas D (1978) Stellar atmospheres, 2nd edn. W. H. Freeman and Co., San Francisco
Mihalas D, Mihalas BW (1984) Foundations of radiation hydrodynamics. Courier Corporation, North Chelmsford
Monin A, Yaglom A (1971) Statistical fluid mechanics: mechanics of turbulence. vol 1. Peace Corps. https://books.google.ru/books?id=7BvYQwAACAAJ
Morozov AG, Khoperskov AV (2005) Physics of discs. Volgograd University Press, Volgograd (in Russian)
Nakao Y, Kato S (1995) Vertical dependence of the viscous heating in accretion disks. Publ Astron Soc Jpn 47:451–461
Novikov ID, Thorne KS (1973) Astrophysics of black holes. In: DeWitt C, DeWitt BS (eds) Black holes (Les Astres Occlus), Gordon and Breach, New York, pp 343–450
Ogilvie GI (1999) Time-dependent quasi-spherical accretion. Mon Not R Astron Soc 306:L9–L13
Paczynski B (1977) A model of accretion disks in close binaries. Astrophys J 216:822–826
Paczynski B, Bisnovatyi-Kogan G (1981) A model of a thin accretion disk around a Black Hole. Acta Astron. 31:283
Paczynsky B, Wiita PJ (1980) Thick accretion disks and supercritical luminosities. Astron Astrophys 88:23–31
Papaloizou J, Pringle JE (1977) Tidal torques on accretion discs in close binary systems. Mon Not R Astron Soc 181:441–454
Pletcher R, Tannehill J, Anderson D (1997) Computational fluid mechanics and heat transfer, 2nd edn. Series in computational and physical processes in mechanics and thermal sciences, Taylor & Francis. http://books.google.ru/books?id=ZJPbtHeilCgC
Prandtl L (1925) Bericht uber Untersuchungen zur ausgebildeten Turbulenz. Z Angew Math Mech 5:136–139. http://naca.central.cranfield.ac.uk/reports/1949/naca-tm-1231.pdf
Press WH, Teukolsky SA, Vetterling WT, Flannery BP (2002) Numerical recipes in C: the art of scientific computing, 2nd edn. Cambridge University Press, Cambridge
Pringle JE (1974) PhD thesis, University of Cambridge, 1974
Pringle JE (1991) The properties of external accretion discs. Mon Not R Astron Soc 248:754–759
Pringle JE, Rees MJ (1972) Accretion disc models for compact x-ray sources. Astron Astrophys 21:1
Rafikov RR (2013) Structure and evolution of circumbinary disks around supermassive black hole binaries. Astrophys J 774:144. https://doi.org/10.1088/0004-637X/774/2/144. ArXiv 1205.5017
Rafikov RR (2016) Generalized similarity for accretion/decretion disks. Astrophys J 830:7. https://doi.org/10.3847/0004-637X/830/1/7. ArXiv 1604.07439
Rayleigh L (1917) On the dynamics of revolving fluids. Proc R Soc Lond Ser A 93:148–154. https://doi.org/10.1098/rspa.1917.0010
Richard D, Zahn JP (1999) Turbulence in differentially rotating flows. What can be learned from the Couette-Taylor experiment. Astron Astrophys 347:734–738. ArXiv astro-ph/9903374
Shafee R, Narayan R, McClintock JE (2008) Viscous torque and dissipation in the inner regions of a thin accretion disk: implications for measuring black hole spin. Astrophys J 676:549–561. https://doi.org/10.1086/527346. ArXiv 0705.2244
Shakura NI (1973) Disk model of gas accretion on a relativistic star in a close binary system. Sov Astron 16:756
Shakura NI, Sunyaev RA (1973) Black holes in binary systems. Observational appearance. Astron Astrophys 24:337–355
Shakura NI, Sunyaev RA (1976) A theory of the instability of disk accretion on to black holes and the variability of binary X-ray sources, galactic nuclei and quasars. Mon Not R Astron Soc 175:613–632
Shakura NI, Sunyaev RA, Zilitinkevich SS (1978) On the turbulent energy transport in accretion discs. Astron Astrophys 62:179–187
Shapiro SL, Teukolsky SA (1983) Black holes, white dwarfs, and neutron stars: The physics of compact objects. Research supported by the National Science Foundation. Wiley-Interscience, New York, 663 pp
Shaviv G, Wehrse R (1986) The vertical temperature stratification and corona formation of accretion disc atmospheres. Astron Astrophys 159:L5–L7
Shibazaki N, Hōshi R (1975) Structure and stability of accretion-disk around a black-hole. Prog Theor Phys 54:706–718
Smak J (1984) Accretion in cataclysmic binaries. IV - Accretion disks in dwarf novae. Acta Astron 34:161–189
Sneddon IN (1951) Fourier transforms. International series in pure and applied mathematics. McGraw-Hill. http://books.google.ru/books?id=HRAJAQAAIAAJ
Sobolev VV (1969) Course in theoretical astrophysics, 1, vol. F-531. NASA, nasa technical translation 1
Speith R, Riffert H, Ruder H (1995) The photon transfer function for accretion disks around a Kerr black hole. Comput Phys Commun 88:109–120. https://doi.org/10.1016/0010-4655(95)00067-P
Suleimanov VF (1992) Modeling the accretion disks and spectra of cataclysmic variables - Part One - V603-AQUILAE. Sov Astron Lett 18:104–+
Suleimanov VF, Lipunova GV, Shakura NI (2007) The thickness of accretion α-disks: theory and observations. Astron Rep 51:549–562. https://doi.org/10.1134/S1063772907070049
Suleimanov VF, Lipunova GV, Shakura NI (2008) Modeling of non-stationary accretion disks in X-ray novae A 0620-00 and GRS 1124-68 during outburst. Astron Astrophys 491:267–277. https://doi.org/10.1051/0004-6361:200810155. ArXiv 0805.1001
Syunyaev RA, Shakura NI (1977) Disk reservoirs in binary systems and prospects for observing them. Sov Astron Lett 3:138–141
Tanaka T (2011) Exact time-dependent solutions for the thin accretion disc equation: boundary conditions at finite radius. Mon Not R Astron Soc 410:1007–1017. https://doi.org/10.1111/j.1365-2966.2010.17496.x. ArXiv 1007.4474
Tayler RJ (1980) Vertical energy transport in optically thick steady accretion discs. Mon Not R Astron Soc 191:135–150
Thorne KS, Price RH, MacDonald DA (1986) Black holes: the membrane paradigm. Yale University Press, New Haven
Tout CA, Pringle JE (1992) Accretion disc viscosity - a simple model for a magnetic dynamo. Mon Not R Astron Soc 259:604–612
Velikhov EP (1959) Stability of an ideally conducting liquid flowing between cylinders rotating in a magnetic field. Sov J Exp Theor Phys 9:995–998
Watson G (1944) A treatise on the theory of Bessel functions. Cambridge University Press. http://books.google.ru/books?id=WNoIAQAAIAAJ
Weizsäcker CFV (1948) Die Rotation kosmischer Gasmassen. Zeitschrift Naturforschung Teil A 3:524
Wood KS, Titarchuk L, Ray PS, Wolff MT, Lovellette MN, Bandyopadhyay RM (2001) Disk diffusion propagation model for the outburst of XTE J1118+480. Astrophys J 563:246–254. https://doi.org/10.1086/323768. ArXiv astro-ph/0108189
Zaitsev V, Polyanin A (2012) Handbook of exact solutions for ordinary differential equations. Taylor & Francis. http://books.google.ru/books?id=JjPDfRwOmAIC
Zdziarski AA, Kawabata R, Mineshige S (2009) Viscous propagation of mass flow variability in accretion discs. Mon Not R Astron Soc 399:1633–1640. https://doi.org/10.1111/j.1365-2966.2009.15386.x. ArXiv 0902.4530
Zeldovich YB (1981) On the friction of fluids between rotating cylinders. R Soc Lond Proc Ser 374:299–312. https://doi.org/10.1098/rspa.1981.0024
Zeldovich YB, Kompaneets AS (1950) Sbornik posvyashchennyy 70-leityu A. F. Ioffe (in: Collection of papers celebrating the seventieth birthday of A. F. Ioffe). AN SSSR, Moscow, in Russian
Zeldovich YB, Raizer YP (1967) Physics of shock waves and high-temperature hydrodynamic phenomena. Dover, New York
Zel’dovich YB, Shakura NI (1969) X-ray emission accompanying the accretion of gas by a neutron star. Sov Astron 13:175
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Lipunova, G., Malanchev, K., Shakura, N. (2018). The Standard Model of Disc Accretion. In: Shakura, N. (eds) Accretion Flows in Astrophysics . Astrophysics and Space Science Library, vol 454. Springer, Cham. https://doi.org/10.1007/978-3-319-93009-1_1
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